Categorical Reconstruction Theory

C a te go ri c al Re con st r u ct io n Th eo ry D i s s e r t a t i o n zur Er langung des akademischen Grades Doctor rerum naturalium (Dr . rer . nat.) v orgelegt dem Bereich Mathematik und Naturwissenschaften der T echnischen U niv ersität Dresden v on Tony Z orm an , M . Sc. am 01. April 2025 begutachtet v on Prof. Dr . rer . nat. Ulrich Krähmer Prof. Dr . rer . nat. Catharina Stroppel angef ertigt v on N ov ember 2021 bis September 2025 am Institut für Geometrie I N F O R M A T I O N A B O U T T H E A R X I V V E R S I O N Due to com pilation time constr aints, this is a p df onl y v ersion of the t hesis. The source code is nev ertheless av ailable at the follo wing u rl : codeberg.org/slo tThe/dissertation/src/branch/arXiv iii Für Isabell und meine Eltern I almost wish I hadn’ t gone down that rabbit-hole — and y et — and y et — it’s rather curious, you kno w , this sort of life! Lew is Carol l , Alice in W onder land Ma ybe I’ll start with t he acknow ledgements. . . Mik e Slackene rny ; PhD Comics #149 AC K N O W L E D G E M E N T S It is surely impossible t o write a dissertation without coming into contact with — and indeed becoming indebted to — a larg e number of w onderful people. This non-exhaustiv e list must certainly include Uli, for being the best advisor , teacher , and life coach one could hope for — thank you for tolerating all of m y stupid questions; Isabell, Mum, and Dad, for giving me strength throughout, and for being patient with me when I was t hinking too much about mat hs; Catharina, for ag reeing to be t he second examiner of this t hesis, and for org anising the nev er -ending Bonn –Dresden seminar ; Sebastian, for not only putting up wit h me as an oﬃce mate, but also as m y ﬁrst coaut hor — I’m not sorr y for ha ving inﬂicted Emacs upon you; Matti, for being the best com pan y in t he w or ld, and for talking to me about Hopf monads, ev en though you w ould much rather t hink about socles; Zbiggi, for being a patient teacher of all t hings representation t heory , and for not onl y making our oﬃce 85% more ﬂammable, but also 100% more fun; Benn y , Florian, Julius, and L ukas, for squatting in our oﬃce, asking — and answ ering — lo ts of questions about mat hs and life; Anna, for lear ning ev er ything about duoidal categories in tw o hours ﬂat; Marcel and Sv en, for alwa ys being up for some good old brother ly rivalry ; Florian, for encouraging me to start writing up ear ly , because “ev en just streamlining notation takes more time than you t hink” — y ou w ere right; Iv an and Myriam, for being t he grownups in t he oﬃce in times of need; Philip, for — being kno wledg eable about appro ximately ev er ything — many enlightening con v ersation about mat hematics, philosoph y , music, mo vies, typesetting, and ev en Emacs; and Y ue, for — e v en now — nev er failing to make me smile. I gracefull y acknow ledge t he ﬁnancial support of t he Deutsche Forschungs- gemeinschaft, who supported me from Oct ober 2022 to September 2025 via the g r ant k r 5036/2 –1 “Cocommutativ e Comonoids”. ix C O N T E N T S 1 Int r od ucti on 1 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Pre limi n ar ies 9 2.1 Bicategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Monads and adjunctions . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Comparison functors . . . . . . . . . . . . . . . . . . . . 16 2.2.2 Distributiv e law s . . . . . . . . . . . . . . . . . . . . . . 18 2.3 String diag r ams . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Monoidal and module categories . . . . . . . . . . . . . . . . . 21 2.4.1 Br aidings . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 Closedness . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.3 Rigidity and piv otality . . . . . . . . . . . . . . . . . . . 28 2.4.4 The Drinfeld centre . . . . . . . . . . . . . . . . . . . . . 34 2.5 Linear and abelian categories . . . . . . . . . . . . . . . . . . . 35 2.5.1 Finite and locall y ﬁnite abelian categories . . . . . . . . 36 2.5.2 T ensor and ring categories . . . . . . . . . . . . . . . . . 37 2.6 Algebra and module objects . . . . . . . . . . . . . . . . . . . . 38 2.7 Monoidal bicategories . . . . . . . . . . . . . . . . . . . . . . . 43 2.7.1 String diag r ams in monoidal bicategories . . . . . . . . 46 2.8 Coends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.9 (Co)completions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 Du al ity t heory for m onoid a l ca t egor ies 57 3.1 T ensor representability . . . . . . . . . . . . . . . . . . . . . . . 59 3.1.1 Grothendieck –V erdier duality . . . . . . . . . . . . . . . 61 3.1.2 The free tensor representable category is not rigid . . . 65 3.2 Functor categories . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.1 Cauch y completions . . . . . . . . . . . . . . . . . . . . 76 3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.3.1 Boolean algebr as . . . . . . . . . . . . . . . . . . . . . . 81 3.3.2 Mackey functors . . . . . . . . . . . . . . . . . . . . . . . 85 3.3.3 Crossed modules . . . . . . . . . . . . . . . . . . . . . . 86 xi C on tent s 4 Tw isted cent res 91 4.1 Heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2 Piv otal structures and twisted centres . . . . . . . . . . . . . . 94 4.2.1 T wisted centres and their Picard heaps . . . . . . . . . . 94 4.2.2 Quasi-piv otality . . . . . . . . . . . . . . . . . . . . . . . 98 4.2.3 Piv otality of the Drinfeld centre . . . . . . . . . . . . . . 100 5 Mo n adi c T an n ak a–K rein reconstruct ion 117 5.1 Bimonads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2 Hopf monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3 (Co)module monads . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3.1 Recons truction for comodule monads . . . . . . . . . . 129 6 M on ad ic tw isted cent res 137 6.1 Cross products and distributiv e la ws . . . . . . . . . . . . . . . 138 6.2 Centralisable functors and t he central bimonad . . . . . . . . . 140 6.3 Centralisers and comodule monads . . . . . . . . . . . . . . . . 144 6.4 The Drinfeld and anti-Drinfeld double of a Hopf monad . . . . 149 6.5 P airs in inv olution for Hopf monads . . . . . . . . . . . . . . . 151 7 Duo id al R -matric es 157 7.1 Duoidal categories . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.1.1 Double opmonoidal monads . . . . . . . . . . . . . . . 160 7.2 R -matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.2.1 From R -matrices to duoidal structures and back . . . . 166 7.3 Linear ly distributiv e monads . . . . . . . . . . . . . . . . . . . 171 8 I nfin ite a nd non- rigi d recon s truct ion 177 8.1 Extending module structures . . . . . . . . . . . . . . . . . . . 179 8.1.1 Extendable monads . . . . . . . . . . . . . . . . . . . . . 184 8.1.2 Semisimple monads . . . . . . . . . . . . . . . . . . . . 184 8.1.3 Linton coequalisers via multiactegories . . . . . . . . . 186 8.2 Inter nal projectiv e and injectiv e objects . . . . . . . . . . . . . . 196 8.3 Recons tr uction for lax module endofunctors . . . . . . . . . . . 198 8.3.1 Rigid monoidal and (ﬁnite) tensor categories . . . . . . 201 8.4 An Eilenberg –W atts t heorem for lax module monads . . . . . 202 9 Ho pf t rimod ules 213 9.1 Bicomodules and t heir g r aphical calculus . . . . . . . . . . . . 215 xii Contents 9.2 From Hopf trimodules to lax module functors . . . . . . . . . . 216 9.3 Contramodule reconstruction o v er Hopf trimodule algebras . 226 9.3.1 Morita equivalence for contramodules . . . . . . . . . . 230 9.4 A semisimple example of non-rigid reconstruction . . . . . . . 231 9.5 A Hopf trimodule algebr a constructing t he ﬁbre functor . . . . 233 9.6 The fundamental theorem of Hopf modules . . . . . . . . . . . 237 9.7 Fusion operators for Hopf monads . . . . . . . . . . . . . . . . 241 Bib liog raph y 245 List of F igure s 261 xiii That depends on where y ou want to get to; “W ell it really doesn ’t matter — ”; Then it doesn ’ t really matter which w a y you go. Lew is Carol l , Alice in W onder land I N T R O D U C T I O N 1 Thi s di ssert ation gene rali ses seve ral reconstruct ion r esul ts in classical algebr a to the language of monoidal categories, module categories, and monads. W e s tart with some brief hist orical remar ks, and summarise our main contributions in Section 1.1 belo w . Monoidal and module categories are ubiquitous in man y ﬁelds of math- ematics. Brieﬂy , a monoidal category is a categor y 𝒞 equipped with a tensor product functor ⊗ : 𝒞 × 𝒞 − → 𝒞 and a unit 1 ∈ 𝒞 , satisfying associativity and unitality conditions. A (lef t) module categor y ℳ o v er 𝒞 , in tur n, is endow ed with an associativ e and unital action functor ⊲ : 𝒞 × ℳ − → ℳ . One motiv ating example of these structures comes from linear alg ebra. The categor y V ect of v ector spaces o v er a ﬁeld k is monoidal, and t he right modules ov er a k -algebr a 𝐴 form a lef t V ect -module category . Giv en 𝑉 ∈ V ect and a right 𝐴 -module 𝑀 , the tensor product 𝑉 ⊗ k 𝑀 is ag ain a right 𝐴 -module, with action giv en by ( 𝑣 ⊗ 𝑚 ) . 𝑎 . . = 𝑣 ⊗ 𝑚 . 𝑎 , for all 𝑎 ∈ 𝐴 , 𝑚 ∈ 𝑀 , and 𝑣 ∈ 𝑉 . The areas in which t he languag e and t heory of monoidal and module categories has been applied include quantum ﬁeld t heories [ FRS02 ; Da y07 ], categoriﬁcation in representation t heory [ Str23 ; LMGRSW24 ; SW24 ], algeb- raic geometry [ BZFN10 ; BZBJ18 ; P as24 ], and man y aspects of t he t heory of Hopf algebr as and tensor categories [ Sch00 a ; KK14 ; EGN O15 ; FGJS22 ; Shi23 b ; Str24 b ]. F or a sur v ey on t he role of module categories in applied categor y theor y and computer science, better known as actegories t herein, see [ CG22 ]. Just as in the categor y of v ector spaces, giv en an algebr a object 𝐴 ∈ 𝒞 , the category mod 𝒞 ( 𝐴 ) of right 𝐴 -modules in 𝒞 is naturally a left 𝒞 -module cat- egory . Con v ersely , when considering a given 𝒞 -module category ℳ , it is of ten useful to ﬁnd and study an algebr a object 𝐴 in 𝒞 such t hat t here is an equi- v alence ℳ ≃ mod 𝒞 ( 𝐴 ) of 𝒞 -module categories. This kind of r econstruction process lies at the heart of Chapters 8 and 9 of the present thesis. An ear ly res- ult of t his kind is [ Ost03 , Theorem 1], in which a “nice” module category ov er a ﬁnite tensor categor y 𝒞 — one that is equivalent to t he ﬁnite-dimensional 1 1. Introd uctio n modules ov er a ﬁnite-dimensional algebr a — is expressed as an algebr a object in 𝒞 , see Theorem A of Chapter 8 . Man y g eneralisations and v ariants ha v e ap- peared since; for ins tance [ EGN O15 , Theorem 7.10.1], [ DSPS19 , Theorem 2.24], [ MMMT19 , Theorem 4.7], and [ BZBJ18 , Theorem 4.6]. All of t he reconstruction results cited abov e inv ol v e rigid monoidal cat- egories: those in which ev ery object has a dual. F or example, to ev ery ﬁnite- dimensional vect or space 𝑉 , there exist canonical ev aluation and coev aluation maps ev : 𝑉 ∗ ⊗ k 𝑉 − → k and coev : k − → 𝑉 ⊗ k 𝑉 ∗ , where 𝑉 ∗ . . = Hom k ( 𝑉 , k ) is t he dual space. How ev er , in [ DSPS19 , Example 2.20] it is shown that, in the absence of rigidity , t here exist 𝒞 -modules categories from which it is impossible to reconstruct an algebra object in 𝒞 . This forces us to consider a further generalisation — monads. 07/ Histor icall y , t he stu d y of (co)monads arose out of (co)homology and homo- top y t heory , The ter m “monad” ﬁrst appears in print in Section 5.4 of [ Bén67 ]. Other names include “triple”, “standard construction ”, or “triad”. where one can use comonads to construct simplicial resolutions, see [ God58 ; Hub61 ]. Already noted in [ Hub61 ] is the deep connection of monads with the t heory of adjunctions: ev er y adjunction giv es rise to a monad and, in tur n, ev er y monad arises from an adjunction. In fact, t here exists a category of all adjunctions t hat realise a giv en monad. The initial and ter minal object therein — the Kleisli and Eilenberg –Moore categor y of the monad, [ EM65 ; Kle65 ] — pla y a piv otal role in this w or k. Further applications are manifold and include categor y theory [ Bec69 ; Str72 ; BKP89 ; LS02 ; AM24 ], categorical representation theor y [ Moe02 ; BL V11 ; Str24 b ; BHV24 ], Hopf algebras and quantum groups [ B V07 ; A C12 ; KKS15 ; HL18 ], univ ersal algebr a [ Lin66 ; W al70 ; HP07 ], formal semantics [ Mog89 ; Mog91 ; TP97 ], functional prog r amming [ W ad92 ; W ad93 ; P JW93 ], and algeb- raic eﬀects [ BHM02 ; PP02 ; PP09 ; BP15 ]. F or this t hesis, t he most important aspect of monads lies in monadic re- construction theor y — relating additional structure on a monad 𝑇 on 𝒞 to structure on its Eilenber g –Moore categor y 𝒞 𝑇 . The connection to t he t heory of adjunctions yields a canonical forg etful functor 𝑈 𝑇 : 𝒞 𝑇 − → 𝒞 . This rela- tionship can t hus be seen as a generalisation of T annaka – Kr ein duality [ T an38 ; Kre49 ], in which one reconstructs a com pact g roup from its categor y of rep- resentations. Generalisations of this type of result ha v e been proposed in a v ariety of diﬀerent contexts, see [ SR72 ; W or88 ; Del90 ; L ur04 ; Szl09 ; Sch13 ]. 2 1.1. Summary A cr ucial feature of all of these statements is the in v olv ement of a ﬁbr e funct or : a forg etful strict monoidal functor to a nice underl ying category , like t he category of v ector spaces or bimodules ov er a ring. In t he monadic case, T annaka –Krein-type reconstruction results w ere obtained for bimonads [ Moe02 ; McC02 ], Hopf monads [ B V07 ], as w ell as linear l y distributiv e and ∗ -autonomous monads [ PS09 ; P as12 ]. W e enrich this already broad landscape of results wit h new insights from the mod- ule categor y-theoretic perspectiv e, oﬀering a new reconstruction result for comodule monads ov er bimonads in t he spirit of t he algebr aic v ersion of considering coideal subalgebr as ov er bialgebr as. 1 . 1 s u m m a r y We sha ll now p r o vi de a sh or t out line ; a more detailed description and introduction will be pro vided at the start of each respectiv e chapter . All original contributions of t his thesis are already contained in eit her of t he follo wing articles or preprints. The order in which the articles are listed here is t hat in which they were uploaded to t he arXiv . • [ HZ24 b ]: Sebastian Halbig and T ony Zorman. Piv otality , twisted centres, and the anti-double of a Hopf monad. In: Theory Appl. Categ. 41 (2024), pp. 86 –149. issn: 1201-561 x . • [ HZ24 a ]: Sebastian Halbig and T on y Zorman. Diag r ammatics for Co- module Monads. In: Appl. Categ. S truct. 32 (2024). Id/N o 27, p. 17. issn: 0927-2852. doi: 10.1017/cbo9781139542333. • [ HZ23 ]: Sebastian Halbig and T ony Zorman. Duality in Monoidal Cat- egories; 2023. arXiv: 2301.03545 . • [ SZ24 ]: Mateusz S troiński and T on y Zorman. Reconstruction of module categories in the inﬁnite and non-rigid settings; 2024. arXiv: 2409.00793 . • [ Zor25 ]: T on y Zorman. Duoidal R -Matrices; 2025. arXiv: 2503.03445 . T o keep a common t hematic focal point, t he article [ CSZ25 ] — alt hough created during the aut hor’ s PhD studies — will not be part of this w or k. In the interest of brevity , w e shall refrain from citing eit her of t he art- icles [ HZ23 ; HZ24 a ; HZ24 b ; SZ24 ; Zor25 ] bey ond this introduction. 3 1. Introd uctio n Chapt er 2 : Preliminaries Being an amalgamate of t he preliminaries of all articles, this chap ter ﬁx es notation and recalls basic properties of, for example, 2-categories, monoidal and module categories, t he Drinf eld centre, linear and abelian categories, coends, as w ell as monads. W e furt hermore introduce one of the main tools for computation emplo y ed t hroughout — the g r aphical calculus. Chapt er 3 : Duality theory for monoidal categories Being mainl y based on [ HZ23 ], in t his chapter w e study v arious dualities for monoidal categories and ho w t he y inter act: rigidity , tensor representability , Grothendieck –V erdier duality , and closedness. These notions form a hier - arch y , wit h closedness — t he least restrictiv e type of duality — on one end, and rigidity on t he other . How ev er , it is not ob vious whether all inclusions are strict. T aken together , Propositions 3.7 , 3.12 , and 3.16 ; Examples 3.15 and 3.17 ; and Theorem 3.23 yield Rigid ⊊ T ensor representable ⊊ Grothendieck – V erdier ⊊ Closed , answ ering a question of Heunen. W e t hen more closel y inv estig ate duality structures on ﬁnite-dimensional functor categories endow ed wit h Da y con v olution as its tensor product. Intu- itiv ely , t his generalises the tensor product of modules ov er a commutativ e algebr a, see Example 2.123 . Giv en a nice base category , closedness and Grothendieck –V erdier duality lift to this setting. Proposition 3.27 . Let 𝒞 be a k -linear hom-ﬁnit e Gro thendiec k – V erdier category . U nder the assumptions of Hypothesis 3.25 , the ﬁnite-dimensional functor category [ 𝒞 , vect ] is a Grot hendieck– V er dier category . Being motiv ated b y representation theoretic questions, the Cauchy com- pletion of a category also pla ys an important role for us. For example, the full subcategory of projectiv e modules ov er a ring is the Cauchy completion of the full subcategor y of all free modules. W e ﬁnd t hat t he Cauch y com pletion completel y mirrors t he duality beha viour of its base categor y . Corollar y 3.43 . Let 𝒞 op be a k -linear right closed monoidal category . Then 𝒞 op is right rigid (t ensor repr esentable) if and only if its Cauc hy completion 𝒞 op is right rigid (t ensor r epresent able). F urther , ( 𝒞 op , 𝑑 ) is a right Gro t hendieck– V er dier cat- egory if and only if ( 𝒞 op , よ 𝑑 ) is. 4 1.1. Summary Chapt er 4 : T wist ed centr es In this chapter , mainly based on [ HZ24 b ], w e hoist Theorem 4.1 from t he Hopf-theoretic w orld into that of monoidal categories. It lays the g roundw or k for later monadic gener alisations. Theorem 4.23 . Let 𝒞 be a rigid category . There is a bĳection between (i) eq uivalence classes of quasi-pivo tal structur es on 𝒞 , (ii) t he Picard heap of the anti-centre, and (iii) isomorphism classes of equiv alences of module categories between t he centre and t he anti-centre. W e then more closel y inv estigate the heap structure of t he anti-centre, answ ering a question of Shimizu about t he relationship to quasi-piv otal structures on t he underl ying categor y . Theorem 4.50 . Ther e exis ts a category 𝒞 on which t here exis ts a pivot al structur e t hat is not induced by any element of the Picard heap of the anti-centre of 𝒞 . Chapt er 5 : Monadic T annaka – Kr ein recons truction In t his chapter , which is mainly based on [ HZ24 a ; HZ24 b ; SZ24 ], w e study comodule monads in t he sense of [ A C12 ]. These s tructures can be seen as gener alising comodule algebras ov er a bialgebr a, see Exam ple 5.14 . W e t hen pro v e our main T annaka – Krein reconstruction result for comodule monads in the spirit of [ Moe02 , Theorem 7.1] and [ McC02 , Corollar y 3.13]. Theorem 5.31 . Let 𝐵 be a bimonad on the monoidal cat egory 𝒞 , and 𝑇 a monad on a right 𝒞 -module category ℳ . Coactions of 𝐵 on 𝑇 are in bĳection with right actions of 𝒞 𝐵 on ℳ 𝑇 such t hat 𝑈 𝑇 is a strict comodule functor over 𝑈 𝐵 . In particular , we pro v e statements akin to Kell y’ s doctrinal adjunctions result in t he case of comodule and 𝒞 -module monads — the latter pla y an important role in Chapters 8 and 9 . Theorem 5.28 and P or ism 5.29 . Let 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 be an oplax monoidal adjunc- tion. Lifts of an ordinary adjunction 𝐺 : ℳ ⇄ 𝒩 : 𝑉 t o a comodule adjunction ar e in bĳection wit h lifts of 𝑉 : 𝒩 − → ℳ to a str ong comodule functor . An adjunction 𝐹 : ℳ ⇄ 𝒩 : 𝑈 between 𝒞 -module categories yields a bĳection of oplax 𝒞 -module structur es on 𝐹 and lax 𝒞 -module structur es on 𝑈 . 5 1. Introd uctio n Chapt er 6 : Monadic twist ed centres This chapter , mainly based on [ HZ24 b ], provides a monadic interpretation of Chapter 4 . Based on t he Drinfeld double of a Hopf monad, we dev elop the notion of an anti-double, which realises the anti-centre of ibid as its Eilenberg– Moore category , and present a monadic variant of Theorem 4.23 . Theorem 6.44 . Let 𝒞 be a rigid monoidal category . F or a Hopf monad 𝐻 on 𝒞 t hat admits a double and anti-double, t he following st atements are equiv alent: (i) t he monoidal unit of 𝒞 lif ts to a module over the anti-double, (ii) t he double and anti-double of 𝐻 are isomorphic as comodule monads, and (iii) t he double and anti-double of 𝐻 are isomorphic as monads. If 𝒞 is pivo tal, t he above st atements hold if and only if 𝐻 admits a pair in involution. In v estig ating t he inter pla y betw een the double and anti-double, w e obtain a new criterion for when a rigid monoidal categor y is piv otal. Corollar y 6.45 . Let 𝒞 be a rigid monoidal category . If 𝒞 admits a central Hopf monad D ( 𝒞 ) and an anti-central comodule monad Q ( 𝒞 ) , then it is pivot al if and onl y if D ( 𝒞 )  Q ( 𝒞 ) as monads. Chapt er 7 : Duoidal R -matrices Being mainly based on [ Zor25 ], this chapter introduces R-matrices in duoidal categories, gener alising the w ell-known R-matrices f or bialgebr as, and t hose for bimonads [ B V07 ]. As it tur ns out, R -matrices on a suitable monad corres- pond to duoidal structures its Eilenberg –Moore categor y . Theorem 7.21 . Let 𝒟 be a category with monoidal structur es ◦ and • , and 𝑇 a monad on 𝒟 t hat has a ◦ -oplax monoidal and a • -oplax monoidal structur e. Then quasitriangular structur es on 𝑇 are in bĳection with duoidal structur es on 𝒟 𝑇 . W e t hen explore t he connection betw een normal duoidal categories, and (non-planar) linear l y distributiv e categories [ CS97 ; GLF16 ]. The latter can be seen as an analogue of Grothendieck – V erdier categories wit hout an explicit notion of dual. In Proposition 7.31 , w e relate a cocommutativ e v ersion of duoidal monads to t he linear ly distributiv e monads of [ P as12 ]. 6 1.1. Summary Chapt er 8 : Inﬁnite and non-rigid recons truction This chapter , mainly based on [ SZ24 ], studies reconstruction wit hout a ﬁbre functor , as done for example in [ Ost03 , Theorem 1]. Since ev ery object 𝑚 in a module categor y ℳ o v er 𝒞 giv es rise to a strong 𝒞 -module functor − ⊲ 𝑚 : 𝒞 − → ℳ , and in good cases t his functor has a right adjoint, t he resulting monad is naturall y a lax 𝒞 -module monad. How ev er , it is a priori not clear whether the Eilenberg– Moore categor y of a (right exact) lax 𝒞 -module monad ev en has a canonical 𝒞 -module structure. Using multicategorical techniques, w e establish when such an extension exists. Theorems 8.25 and 8.26 . Let 𝒞 be an abelian monoidal and ℳ an abelian 𝒞 -module category . Suppose t hat t he action functor ⊲ : 𝒞 ⊗ k ℳ − → ℳ is right e xact in bot h variables, and let 𝑇 : ℳ − → ℳ be a right exact lax 𝒞 -module monad. Then ther e exis ts an essentially uniquel y 𝒞 -module structur e on ℳ 𝑇 , suc h that t he canonical inclusion 𝜄 : ℳ 𝑇 − → ℳ 𝑇 is a str ong 𝒞 -module functor . Using t he constructed 𝒞 -module structure, w e t hen establish a reconstruc- tion and classiﬁcation result under mild additional assump tions. Theorems 8.48 to 8.50 . Assume t hat 𝒞 and ℳ hav e enough projectiv es (injectives) and t hat ther e exis ts an object ℓ ∈ ℳ such that: • t here is a right adjoint ⌊ ℓ , −⌋ (left adjoint ⌈ ℓ , −⌉ ) to − ⊲ ℓ ; • for 𝑥 ∈ 𝒞 projective (injective), the object 𝑥 ⊲ ℓ is projectiv e (injective); • any projectiv e (injective) object of ℳ is a direct summand of an object of the form 𝑥 ⊲ ℓ , for 𝑥 projectiv e (injective). Let 𝑇 be the monad ⌊ ℓ , − ⊲ ℓ ⌋ . Then t here is an equiv alence ℳ ≃ 𝒞 𝑇 of 𝒞 -module categories, where t he 𝒞 -module structur e of t he cat egory of 𝑇 -modules is ext ended from t he Kleisli category . Furt hermore, this gives rise to a bĳection { ( ℳ , ℓ ) as above } ⧸ ( ℳ ≃ 𝒩 ) ↔  Right exact lax 𝒞 -module monads on 𝒞  / ( 𝒞 𝑇 ≃ 𝒞 𝑆 ) ( ℳ , ℓ ) ↦− → ⌊ ℓ , − ⊲ ℓ ⌋ ( 𝒞 𝑇 , 𝑇 1 ) ← − [ 𝑇 7 1. Introd uctio n Chapt er 9 : Hopf trimodules The ﬁnal chap ter , mainly based on [ SZ24 ], furt her dev elops t he non-rigid reconstruction results of Chapter 8 in t he case of sev eral examples. F or a Hopf algebr a 𝐵 , t he F undamental Theorem of Hopf Modules sa ys t hat the categor y 𝐵 𝐵 V ect of Hopf modules is naturall y equiv alent to t he category of v ector spaces [ LS69 ]. Likewise, t he category of Hopf trimodules 𝐵 𝐵 V ect 𝐵 is equiv alent t o t hat of right comodules. In comparison to plain Hopf modules, one can gener alise Hopf trimodules to t he quasi-Hopf algebr aic wor ld. If 𝐵 is just a bialgebra — i.e., does not ha v e an antipode — then w e can still extract interesting structure from such modules. Theorem 9.2 . F or a bialg ebra 𝐵 and 𝒱 . . = 𝐵 V ect , ther e is a monoidal equiv alence 𝐵 𝐵 V ect 𝐵 ≃ Le xfLax 𝒱 Mod ( 𝒱 , 𝒱 ) between the category of Hopf trimodules, and t he cat egor y of left exact ﬁnitary lax 𝒱 -module endofunctors on 𝒱 . This allo ws us to giv e a categorical inter pretation of t he aforementioned Fundamental Theorem of Hopf Modules. Proposition 9.38 and Corollar y 9.45 . The functor 𝐵 V ect − → 𝐵 𝐵 V ect 𝐵 corr esponds to t he inclusion of s trong 𝐵 V ect -module endofunctors int o the category of lax 𝐵 V ect - module endofunctors. It is an equivalence if and only if 𝐵 vect is left rigid, which is t he case if and only if 𝐵 has a twist ed antipode. W e also giv e applications in t he setting of semigroup algebr as, see Sec- tions 9.4 and 9.5 . This can be seen as an exam ple of a non-rigid categor y where the reconstruction procedure described in [ EGN O15 , Chapter 7] fails. Lastl y , t he fusion operators of a bimonad ma y be expressed in ter ms of its canonical module structure. Lemma 9.47 and Proposition 9.49 . Let 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 be an oplax monoidal adjunction. The str ong monoidal s tructure of 𝑈 turns 𝒞 into a 𝒟 -module category , by deﬁning − ⊲ = . . = 𝑈 (−) ⊗ = . W it h respect to t his 𝒟 -module structur e, t he bimonad 𝑇 . . = 𝑈 𝐹 on 𝒞 becomes an oplax 𝒟 -module monad. The right fusion oper ator is given by evaluating the 𝒟 -module coherence mor phism at a fr ee 𝑇 -module, and furt hermore 𝑇 is right Hopf if and only if it is a str ong 𝒞 𝑇 -module monad. 8 Das Rad gehört nicht zur Maschine, das man drehen kann, ohne daß etw as anderes sich mitbew egt. Lud wig Wi ttgenste in ; Philosophische U ntersuchungen P R E L I M I N A R I E S 2 We as sume t he rea der’s f a mili arit y wit h basic category theoretical concep ts, as w ell as t he main features of the t heories of monoidal and 2-categories, as discussed for exam ple in [ ML98 ; EGN O15 ; Rie17 ; JY21 ]. N ev ert heless, w e start wit h a brief review of t he most important ter minology and notation. 2 . 1 b i c at e g o r i e s The the or y o f b ica t egor ies i s e ssen tial to t he graphical calculus w e will use throughout, see Section 2.3 , as w ell as to some formal arguments w e shall emplo y . Follo wing [ JY21 ], we recall the most important deﬁnitions. Deﬁnition 2.1. A bicategory B has as data: • A collection of objects Ob B , where we often write 𝑥 ∈ B for 𝑥 ∈ Ob B ; • for all 𝑥 , 𝑦 ∈ B a hom-cat egor y B ( 𝑥 , 𝑦 ) , whose objects are called 1-cells (or 1-morphisms) and whose mor phisms are called 2-cells (or 2-morphisms); • for all 𝑥 , 𝑦 , 𝑧 ∈ B a horizontal composition functor ⊗ 𝑦 : B ( 𝑦 , 𝑧 ) × B ( 𝑥 , 𝑦 ) − → B ( 𝑥 , 𝑧 ) ; • for all 𝑥 ∈ B an identity 1 𝑥 ∈ B ( 𝑥 , 𝑥 ) ; • a natural isomor phism 𝛼 𝑀 ,𝑁 ,𝑃 : ( 𝑀 ⊗ 𝑦 𝑁 ) ⊗ 𝑥 𝑃 − → 𝑀 ⊗ 𝑦 ( 𝑁 ⊗ 𝑥 𝑃 ) ; • natural isomor phisms 𝜆 𝑀 : 1 𝑦 ⊗ 𝑦 𝑀 − → 𝑀 and 𝜌 𝑀 : 𝑀 − → 𝑀 ⊗ 𝑥 1 𝑥 . 9 2. Prel iminari es This data is subject to t he commutativity of the follo wing diag r ams, for all suitable 1-cells 𝑀 , 𝑁 , 𝑃 , 𝑅 : ( 𝑀 ⊗ 𝑧 ( 𝑁 ⊗ 𝑦 𝑃 )) ⊗ 𝑥 𝑅 𝑀 ⊗ 𝑧 (( 𝑁 ⊗ 𝑦 𝑃 ) ⊗ 𝑥 𝑅 ) (( 𝑀 ⊗ 𝑧 𝑁 ) ⊗ 𝑦 𝑃 ) ⊗ 𝑥 𝑅 𝑀 ⊗ 𝑧 ( 𝑁 ⊗ 𝑦 ( 𝑃 ⊗ 𝑥 𝑅 )) ( 𝑀 ⊗ 𝑧 𝑁 ) ⊗ 𝑦 ( 𝑃 ⊗ 𝑥 𝑅 ) ( 𝑀 ⊗ 𝑥 1 𝑥 ) ⊗ 𝑥 𝑁 𝑀 ⊗ 𝑥 ( 1 𝑥 ⊗ 𝑥 𝑁 ) 𝑀 ⊗ 𝑥 𝑁 𝑀 ⊗ 𝑥 𝑁 𝛼 𝑀 , 𝑁 ⊗ 𝑦 𝑃 ,𝑅 𝑀 ⊗ 𝛼 𝑁 𝑃 𝑅 𝛼 𝑀 𝑁 𝑃 ⊗ 𝑅 𝛼 𝑀 ⊗ 𝑧 𝑁 , 𝑃 ,𝑅 𝛼 𝑀 , 𝑁 ,𝑃 ⊗ 𝑥 𝑅 𝛼 𝑀 , 1 𝑥 , 𝑁 𝑀 ⊗ 𝜆 𝑁 𝜌 𝑀 ⊗ 𝑁 If 𝛼 , 𝜆 , and 𝜌 are all identities, w e call B a 2-category . The composition of morphisms inside of t he category B ( 𝑥 , 𝑦 ) will be called the vertical composition . W e denote t he terminal bicategor y consisting of a single object, a single 1-, and a single 2-morphism by ♥ . Exam ple 2.2. The category C at of (small) categories, functors, and natur al transf or mations is a 2-categor y . Horizontal composition is given b y horizontal composition of functors and natural transf ormations. Examples of bicategories are plentiful t hroughout t his w or k. W e refrain from spelling t hem out here before ha ving introduced more of t he necessar y notation, but see Examples 2.103 , 2.111 , and 2.113 and Remar k s 4.8 and 5.2 Remark 2.3. The use of ⊗ for t he horizontal com position is non-standard, and inspired by [ Gar22 ]. W e use it here to highlight the parallels to t he deﬁnition of monoidal categories, see Deﬁnition 2.31 . Notation 2.4. T o syntacticall y diﬀerentiate bicategories from ordinary cat- egories, w e will gener ally start t he name of a bicategor y wit h a blackboard bold letter . For example, when talking about the 1-categor y of all (small) categories and functors, w e write Cat instead of C at . Much like monoidal categories, see Theorem 2.40 , bicategories admit a coher ence and strictiﬁcation result, in t hat ev er y bicategor y is biequivalent to a 2-category , see for example [ JY21 , Theorem 8.4.1 and Corollary 8.4.2]. 10 2.1. Bicategories Deﬁnition 2.5. A lax functor 𝐹 : B − → C of bicategories B , C has as data: • an object assignment 𝐹 : Ob B − → Ob C ; • for all 𝑥 , 𝑦 ∈ B , a functor 𝐹 𝑥 , 𝑦 : B ( 𝑥 , 𝑦 ) − → C ( 𝐹 𝑥 , 𝐹 𝑦 ) ; • a famil y of natural transf ormations 𝐹 2 : 𝐹 ⊗ 𝐹 = ⇒ 𝐹 ◦ ⊗ , with components 𝐹 2 , 𝑀 , 𝑁 : 𝐹 𝑦 𝑧 𝑀 ⊗ 𝐹 𝑦 𝐹 𝑥 𝑦 𝑁 − → 𝐹 𝑥 𝑧 ( 𝑀 ⊗ 𝑦 𝑁 ) , for all 𝑥 , 𝑦 , 𝑧 ∈ B , 𝑀 ∈ B ( 𝑦 , 𝑧 ) , and 𝑁 ∈ B ( 𝑥 , 𝑦 ) ; and • for all 𝑥 ∈ B , arro ws 𝐹 0 ,𝑥 : 1 𝐹 𝑥 − → 𝐹 1 𝑥 . This data is subject to t he follo wing axioms in C ( 𝐹 𝑤 , 𝐹 𝑧 ) and C ( 𝐹 𝑥 , 𝐹 𝑦 ) , re- spectiv ely , for all admissible 𝑀 , 𝑁 , 𝑃 : ( 𝐹 𝑀 ⊗ 𝐹 𝑦 𝐹 𝑁 ) ⊗ 𝐹 𝑥 𝐹 𝑃 𝐹 𝑀 ⊗ 𝐹 𝑦 ( 𝐹 𝑁 ⊗ 𝐹 𝑥 𝐹 𝑃 ) 𝐹 ( 𝑀 ⊗ 𝑦 𝑁 ) ⊗ 𝐹 𝑥 𝐹 𝑃 𝐹 𝑀 ⊗ 𝐹 𝑦 𝐹 ( 𝑁 ⊗ 𝑥 𝑃 ) 𝐹 (( 𝑀 ⊗ 𝑦 𝑁 ) ⊗ 𝑥 𝑃 ) 𝐹 ( 𝑀 ⊗ 𝑦 ( 𝑁 ⊗ 𝑥 𝑃 )) 𝛼 𝐹 2; 𝑀 ,𝑁 ⊗ id id ⊗ 𝐹 2; 𝑁 ,𝑃 𝐹 2; 𝑀 ⊗ 𝑁 ,𝑃 𝐹 2; 𝑀 ,𝑁 ⊗ 𝑃 𝐹 𝛼 1 𝐹 𝑦 ⊗ 𝐹 𝑦 𝐹 𝑀 𝐹 𝑀 𝐹 𝑀 ⊗ 𝐹 𝑥 1 𝐹 𝑥 𝐹 1 𝑦 ⊗ 𝐹 𝑦 𝐹 𝑀 𝐹 𝑀 ⊗ 𝐹 𝑥 𝐹 1 𝑥 𝐹 ( 1 𝑦 ⊗ 𝑦 𝑀 ) 𝐹 𝑀 𝐹 ( 𝑀 ⊗ 𝑥 1 𝑥 ) 𝜆 𝐹 0; 𝑦 ⊗ id 𝜌 id ⊗ 𝐹 0; 𝑥 𝐹 2;1 𝑦 , 𝑀 𝐹 2; 𝑀 , 1 𝑥 𝐹 𝜆 𝐹 𝜌 A lax functor is called a pseudofunctor if all 𝐹 2 ’ s and 𝐹 0 ’ s are in v ertible, and a 2-functor if they are all identities. Analogousl y , one deﬁnes oplax functor s betw een 2-categories. Deﬁnition 2.6. An oplax transf ormation 𝜔 : 𝐹 = ⇒ 𝐺 betw een lax functors ( 𝐹 , 𝐹 2 , 𝐹 0 ) , ( 𝐺 , 𝐺 2 , 𝐺 0 ) : B − → C consists of a 1-cell 𝜔 𝑥 : 𝐹 𝑥 − → 𝐺 𝑥 in C for ev ery 𝑥 ∈ B , and for all 𝑥 , 𝑦 ∈ B a natural transf ormation 𝜔 : 𝜔 𝑦 ⊗ 𝐹 (−) = ⇒ 𝐺 (−) ⊗ 𝜔 𝑥 : B ( 𝑥 , 𝑦 ) − → C ( 𝐹 𝑥 , 𝐺 𝑦 ) 11 2. Prel iminari es with component 2-cells 𝜔 𝑓 : 𝜔 𝑦 ⊗ 𝐹 𝑦 𝐹 𝑓 = ⇒ 𝐺 𝑓 ⊗ 𝐺 𝑥 𝜔 𝑥 for all 𝑓 : 𝑥 − → 𝑦 in B . Graphicall y , w e dra w these 2-cells like t he follo wing: 𝐹 𝑥 𝐹 𝑦 𝐺 𝑥 𝐺 𝑦 𝐹 𝑓 𝜔 𝑥 𝜔 𝑓 𝜔 𝑦 𝐺 𝑓 An oplax transf ormation is called a pseudonatur al transf ormation 1 if all 𝜔 𝑓 ’ s 1 Pseudonatural transf ormations are called strong transf ormations in [ JY21 ]. are isomorphisms, and a 2-natural transf ormation if they are all identities. Analogousl y , one deﬁnes lax transformations betw een lax functors, as w ell as lax and oplax transformations betw een oplax functors. Deﬁnition 2.7 ([ Lac10 ]) . Let 𝐹 , 𝐺 : B − → C be lax functors betw een bicategor - ies. An icon betw een 𝐹 and 𝐺 consists of an assertion t hat 𝐹 𝑥 = 𝐺 𝑥 , for all 𝑥 ∈ B ; and an oplax transf ormation 𝜔 : 𝐹 = ⇒ 𝐺 , such t hat for all 𝑥 ∈ B , t he 1-cell 𝜔 𝑥 : 𝐹 𝑥 − → 𝐺 𝑥 is the identity . Deﬁnition 2.7 might seem a bit contrived at ﬁrs t, but it is an important concept: t here is no 2-category with (small) bicategories as 0-cells, lax (ev en pseudo-) functors as 1-cells, and 2-natutral transf ormations as 2-cells. In- stead, icon s yield t he “right” kind of 2-cells for t his construction, see [ JY21 , Section 4.6] for a more extensiv e account. Deﬁnition 2.8. Let B and C be bicategories, 𝐹 , 𝐺 : B − → C lax functors, and suppose that 𝛼 , 𝛽 : 𝐹 = ⇒ 𝐺 are oplax transf ormations. A modiﬁcation Υ : 𝛼 ⇛ 𝛽 betw een 𝛼 and 𝛽 consists of a 2-cell Υ 𝑥 : 𝛼 𝑥 = ⇒ 𝛽 𝑥 for ev er y 𝑥 ∈ B , such that t he follo wing diagram commutes for all 1-cells 𝑓 ∈ B ( 𝑥 , 𝑦 ) : 𝐹 𝑥 𝐹 𝑦 𝐹 𝑥 𝐹 𝑦 = 𝐺 𝑥 𝐺 𝑦 𝐺 𝑥 𝐺 𝑦 𝐹 𝑓 𝛼 𝑥 𝛽 𝑥 𝛼 𝑓 𝛼 𝑦 𝐹 𝑓 𝛽 𝑥 𝛽 𝑓 𝛽 𝑦 𝛼 𝑦 𝐺 𝑓 𝐺 𝑓 Υ 𝑥 Υ 𝑦 In fact, bicategories, lax functors, lax natural tr ansformations, and modiﬁc- ations betw een them form a tricategory . W e shall not giv e the formal deﬁnition here, but see [ GPS95 ; Gur06 ] and [ JY21 , Chapter 11]. 12 2.2. Monads and adjunctions 2 . 2 m o na d s a n d a dj u nc t i o n s Deﬁnition 2.9. A monad on a categor y 𝒞 consists of an endofunctor 𝑇 on 𝒞 and tw o natural tr ansformations 𝜇 : 𝑇 2 = ⇒ 𝑇 and 𝜂 : Id 𝒞 = ⇒ 𝑇 , called t he multiplication and unit of 𝑇 , satisfying associativity and unitality axioms: 𝑇 3 𝑇 2 𝑇 2 𝑇 𝑇 𝑇 2 𝑇 𝑇 𝑇 𝜇 𝜇 𝜇 𝑇 𝜇 𝜂 𝑇 𝜇 𝑇 𝜂 𝜇 Deﬁnition 2.10. An adjunction consists of a pair of funct ors 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 , tog ether wit h tw o natural transformations, t he unit 𝜂 : Id 𝒞 = ⇒ 𝑈 𝐹 and t he counit 𝜀 : 𝐹 𝑈 = ⇒ Id 𝒟 , satisfying t he snake or triang le identities; see for ex- ample [ Rie17 , Deﬁnition 4.2.5]. Alternativ ely , one could deﬁne an adjunction betw een 𝐹 : 𝒞 − → 𝒟 and 𝐺 : 𝒟 − → 𝒞 to require t he existence of a natural isomorphism 𝒟 ( 𝐹 (−) , = )  𝒞 (− , 𝑈 ( = )) , from which one recov ers the unit and counit, see [ Rie17 , Proposition 4.2.6]. N ote t hat, a priori, diﬀerent natural isomor phisms lead to diﬀerent adjunc- tions betw een the same functors. Ho we v er , ﬁxing for example 𝐺 , then for tw o lef t adjoints 𝐹 and 𝐹 ′ there exists a unique isomorphism Θ : 𝐹 = ⇒ 𝐹 ′ that commutes with t he respectiv e units and counits; see [ Rie17 , Proposition 4.4.1]. Exam ple 2.11. If 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 is an adjunction with unit 𝜂 and counit 𝜀 , t hen 𝑈 𝐹 : 𝒞 − → 𝒞 is a monad; t he multiplication 𝜇 𝑇 : 𝑈 𝐹 𝑈 𝐹 = ⇒ 𝑈 𝐹 is giv en b y 𝑈 𝜀 𝐹 , and t he unit 𝜂 𝑇 : Id 𝒞 = ⇒ 𝑈 𝐹 is equal to 𝜂 . Deﬁnition 2.12. Giv en a monad ( 𝑇 , 𝜇 , 𝜂 ) on a categor y 𝒞 , a 𝑇 -alg ebr a consists of an object 𝑥 ∈ 𝒞 and a mor phism 𝛼 : 𝑇 𝑥 − → 𝑥 , satisfying 𝛼 ◦ 𝑇 𝛼 = 𝛼 ◦ 𝜇 𝑥 and 𝛼 ◦ 𝜂 𝑥 = id 𝑥 . Giv en tw o 𝑇 -algebr as ( 𝑥 , 𝛼 ) and ( 𝑦 , 𝛽 ) , a morphism between t hem consists of a morphism 𝑓 : 𝑥 − → 𝑦 in 𝒞 , such t hat 𝛽 ◦ 𝑇 𝑓 = 𝑓 ◦ 𝛼 . 13 2. Prel iminari es Remark 2.13. For any monad 𝑇 , the 𝑇 -algebr as and t heir mor phisms form a category: the Eilenberg– Moor e category of 𝑇 . Our notation for the Eilenberg– Moore category already ascribes a certain “terminal” quality to it; see Remark 2.23 . W e shall denote it by 𝒞 𝑇 . The Eilenberg– Moore categor y of 𝑇 is also often called the category of 𝑇 -alg ebr as or , follo wing for exam ple [ B V07 ], t he category of modules ov er 𝑇 . W e use all three ter minologies interchangeabl y . A monad is intimatel y connected to its Eilenberg –Moore categor y . Exam ple 2.14. There is a 2 -category M on ( C at ) of monads in C at , [ Str72 , § 1]. The inclusion 2-functor maps a categor y to its identity monad: C at − → M on ( C at ) , 𝒞 ↦− → ( Id 𝒞 , id Id 𝒞 , id Id 𝒞 ) . By assump tion, C at admits t he construction of algebr as : there exists a right adjoint to t he abov e functor: M on ( C at ) − → C at , ( 𝑇 : 𝒞 − → 𝒞 , 𝜇 , 𝜂 ) ↦− → 𝒞 𝑇 , where 𝒞 𝑇 ∈ C at is t he Eilenberg –Moore categor y of 𝑇 . Using the previous 2-adjunction, one pro v es that to ev ery monad ( 𝑇 , 𝜇 , 𝜂 ) on 𝒞 there exist an Eilenber g –Moore adjunction 𝐹 𝑇 : 𝒞 − → 𝒞 𝑇 and 𝑈 𝑇 : 𝒞 𝑇 − → 𝒞 , such t hat 𝑇 = 𝑈 𝑇 𝐹 𝑇 , 𝜇 = 𝐹 𝑇 𝜀 𝑈 𝑇 , 𝜂 = 𝜂 , where 𝜂 : 1 𝒞 = ⇒ 𝑈 𝑇 𝐹 𝑇 and 𝜀 : 𝐹 𝑇 𝑈 𝑇 = ⇒ 1 𝒞 𝑇 are t he unit and counit of t he Eilenberg– Moore adjunction. W e shall call 𝐹 𝑇 and 𝑈 𝑇 the free and for getful functor associated to 𝑇 , respectivel y . F or the following deﬁnition, w e follo w [ B V07 ; TV17 ]. Deﬁnition 2.15. Suppose t hat 𝑇 and 𝑆 are tw o monads on t he categor y 𝒞 . A morphism of monads betw een 𝑇 and 𝑆 is a natural transf ormation 𝑓 : 𝑇 = ⇒ 𝑆 , such that t he follo wing diagrams commute Id 𝑇 𝑇 𝑇 𝑇 𝑆 𝑆 𝑆 𝑆 𝑇 𝑆 𝜂 𝑇 𝜂 𝑆 𝑓 𝑇 𝑓 𝜇 𝑇 𝑓 𝑆 𝜇 𝑆 𝑓 (2.2.1) Remark 2.16. The terminology of Deﬁnition 2.15 is slightl y non-s tandard. What w e call a morphism of monads is often called a oplax (or colax) monad morphism, see for example [ Str72 , § 1]. 14 2.2. Monads and adjunctions Remark 2.17. One can deﬁne a monad in an y bicategory B b y considering ( 𝐶 , 𝑡 , 𝜂 , 𝜇 ) , where 𝐶 ∈ B is an object, 𝑡 : 𝐶 − → 𝐶 is a 1-cell, and 𝜂 : Id 𝐶 = ⇒ 𝑡 and 𝜇 : 𝑡 𝑡 = ⇒ 𝑡 are 2-cells, satisfying relations analogous to Deﬁnition 2.9 . An oplax mor phism of monads from ( 𝐶 , 𝑡 , 𝜂 𝑡 , 𝜇 𝑡 ) to ( 𝐷 , 𝑠 , 𝜂 𝑠 , 𝜇 𝑠 ) then consists of a 1-cell 𝑢 : 𝐶 − → 𝐷 and a 2-cell 𝜙 : 𝑢 𝑡 = ⇒ 𝑠 𝑢 , subject to identities reminiscent of Diag r am ( 2.2.1 ). A lax morphism of monads in v olv es a 1-cell 𝑢 : 𝐶 − → 𝐷 and a 2-cell 𝜙 : 𝑠 𝑢 = ⇒ 𝑢 𝑡 , satisfying similar properties. The follo wing example sheds some additional light on t his terminology . Exam ple 2.18. Monads can alternativ ely be deﬁned as lax functors — in t he sense of Deﬁnition 2.5 — from t he ter minal 2-category ♥ to C at . U nra v elling this deﬁnition, a lax functor T : ♥ − → C at consists of: • an object assignment T : Ob ♥ − → Ob C at , sending the unique object ∗ to a categor y 𝒞 ; • a functor T (∗ , ∗) : ♥(∗ , ∗) − → C at ( 𝒞 , 𝒞 ) from t he ter minal 1-category ♥(∗ , ∗) to t he categor y of endofunctors on 𝒞 , sending t he unique 1- morphism id ∗ : ∗ − → ∗ to 𝑇 : 𝒞 − → 𝒞 and the unique 2-mor phism 1 id ∗ : id ∗ = ⇒ id ∗ to t he identity natural transformation 𝑇 = ⇒ 𝑇 ; • a 2-cell Recall that ⊗ is the horizontal composition in B . T 2 : T id ∗ ⊗ T id ∗ = ⇒ T id ∗ , which w e write as 𝜇 : 𝑇 𝑇 = ⇒ 𝑇 ; and • a 2-cell T 0 : 1 T (∗) = ⇒ T id ∗ that w e write as 𝜂 : Id 𝒞 = ⇒ 𝑇 . The properties of Deﬁnition 2.5 f or T 2 and T 0 translate to the associativity and unitality properties of 𝜇 and 𝜂 . In t his setting, a mor phism of monads becomes an oplax transf ormation in t he sense of Deﬁnition 2.6 . Exam ple 2.19. A monad 𝑇 on 𝒞 has another canonical categor y associated to it: its Kleisli category 𝒞 𝑇 . On objects, it is giv en b y Ob ( 𝒞 𝑇 ) . . = Ob ( 𝒞 ) , and for 𝑥 , 𝑦 ∈ 𝒞 𝑇 w e ha v e 𝒞 𝑇 ( 𝑥 , 𝑦 ) . . = 𝒞 ( 𝑥 , 𝑇 𝑦 ) . Composition is deﬁned by ◦ : 𝒞 𝑇 ( 𝑦 , 𝑧 ) × 𝒞 𝑇 ( 𝑥 , 𝑦 ) − → 𝒞 𝑇 ( 𝑥 , 𝑧 ) ( 𝑔 , 𝑓 ) ↦− →  𝑥 𝑓 − − − → 𝑇 𝑦 𝑇 𝑔 − − − → 𝑇 2 𝑧 𝜇 𝑧 − − − − → 𝑇 𝑧  . Proposition 2.20. Let 𝑇 be a monad on a category 𝒞 . Ther e exis ts an adjunction 𝒞 𝒞 𝑇 𝐹 𝑇 𝑈 𝑇 ⊣ wher e 𝐹 𝑇 is identity on objects and sends 𝑓 ∈ 𝒞 ( 𝑥 , 𝑦 ) to 𝜂 𝑦 ◦ 𝑓 , and 𝑈 𝑇 sends 𝑥 t o 𝑇 𝑥 and 𝑓 ∈ 𝒞 ( 𝑥 , 𝑦 ) to 𝜇 𝑦 ◦ 𝑇 𝑓 . 15 2. Prel iminari es 2.2.1 Com parison functor s Giv en a m onad 𝑇 coming f r om the adjunction 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 , w e might ask how much the functors 𝐹 and 𝑈 “diﬀer” from t he free and for getful functors 𝐹 𝑇 : 𝒞 − → 𝒞 𝑇 and 𝑈 𝑇 : 𝒞 𝑇 − → 𝒞 of 𝑇 . R oughly summarised w e are interested in t he follo wing: 𝒟 𝒞 𝑇 𝒞 𝑈 “compare” 𝑈 𝑇 𝐹 𝐹 𝑇 Proposition 2.21 ([ Fre69 , Theorem 2]) . Let 𝑇 and 𝑆 be monads on a category 𝒞 . Then ther e exists a bĳection between monad morphisms from 𝑇 to 𝑆 and functors 𝐹 : 𝒞 𝑆 − → 𝒞 𝑇 between their categories of algebr as, such that 𝑈 𝑇 𝐹 = 𝑈 𝑆 . N ote that, if w e w ere to deﬁne morphisms of monads as lax transforma- tions, Proposition 2.21 w ould yield a cov ariant assignment. Lemma 2.22 ([ Str72 , Theorem 3]) . Let 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 be an adjunction and set 𝑇 . . = 𝑈 𝐹 . Ther e exis ts a unique functor 𝐾 𝑇 : 𝒟 − → 𝒞 𝑇 satisfying 𝐾 𝑇 𝐹 = 𝐹 𝑇 and 𝑈 𝑇 𝐾 𝑇 = 𝑈 . On objects it is given by 𝐾 𝑇 𝑑 = ( 𝑈 𝑑 , 𝑈 𝜀 𝑑 ) , for all 𝑑 ∈ 𝒟 . W e call t he unique functor from Lemma 2.22 t he comparison functor . An adjunction is called monadic if its comparison functor is an equivalence. There exists an analogous v ersion of Lemma 2.22 for t he Kleisli categor y of a monad 𝑇 . The comparison functor in this situation is giv en b y 𝐾 𝑇 : 𝒞 𝑇 − → 𝒟 , 𝑥 ↦− → 𝐹 𝑥 , 𝑓 ∈ 𝒟 ( 𝑥 , 𝑇 𝑦 ) ↦− → 𝜀 𝐹 𝑦 ◦ 𝐹 𝑓 . Remark 2.23. In fact, the Eilenberg –Moore and t he Kleisli categor y are the terminal and initial objects in the suitably deﬁned categor y of adjunctions 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 producing the monad 𝑇 . Thus, the diag r am 𝒞 𝑇 𝒟 𝒞 𝑇 𝒞 𝐾 𝑇 𝑈 𝑇 𝐾 𝑇 𝑈 𝑈 𝑇 𝐹 𝑇 𝐹 𝐹 𝑇 ⊣ ⊣ ⊣ 16 2.2. Monads and adjunctions commutes, and its commutativity characterises 𝐾 𝑇 and 𝐾 𝑇 completel y . The com posite 𝐾 𝑇 ◦ 𝐾 𝑇 is full y faithful, its full image consisting of fr ee modules . W e will denote the canonical inclusion of 𝒞 𝑇 into 𝒞 𝑇 b y 𝜄 : 𝒞 𝑇 ↩ − → 𝒞 𝑇 , 𝑥 ↦− → ( 𝑇 𝑥 , 𝜇 𝑥 ) , 𝑓 ∈ 𝒞 ( 𝑥 , 𝑇 𝑦 ) ↦− → 𝜇 𝑦 ◦ 𝑇 𝑓 . (2.2.2) Exam ple 2.24. A comonad on a category 𝒞 consists of an endofunctor 𝑆 on 𝒞 tog ether wit h natural transf ormations Δ : 𝑆 = ⇒ 𝑆 2 and 𝜀 : 𝑆 = ⇒ Id 𝒞 , satisfying axioms dual to t hose of Deﬁnition 2.9 . F or a comonad 𝑆 on 𝒞 an 𝑆 -comodule or 𝑆 -coalg ebra is an object 𝑥 ∈ 𝒞 tog ether wit h a morphism 𝑥 − → 𝑆 𝑥 satisfying axioms analogous to Deﬁni- tion 2.12 . W e will denote the categor y of 𝑆 -comodules b y 𝒞 𝑆 and the Kleisli category of 𝑆 by 𝒞 𝑆 . They ha v e analogously deﬁned adjunctions 𝒞 𝑆 𝒞 and 𝒞 𝑆 𝒞 . 𝐹 𝑆 𝑈 𝑆 𝐹 𝑆 𝑈 𝑆 ⊣ ⊣ The Eilenberg– Moore category and t he Kleisli categor y for the comonad 𝑆 . . = 𝐹𝑈 can be characterised as a ter minal and initial object, respectiv ely . This yields functors 𝐾 𝑆 and 𝐾 𝑆 such that t he follo wing diagram commutes 𝒟 𝒞 𝑆 𝒞 𝒞 𝑆 𝑈 𝑆 𝑈 𝑈 𝑆 𝐹 𝑆 𝐾 𝑆 𝐹 𝐾 𝑆 𝐹 𝑆 ⊣ ⊣ ⊣ Similar ly to Example 2.11 , giv en an adjunction 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 , w e obtain a comonad 𝐹 𝑈 on 𝒟 . It ma y be that a monad 𝑇 on 𝒞 has a right adjoint 𝐺 — in t his case, one can automaticall y equip 𝐺 with a canonical comonad structure. Proposition 2.25 ([ MLM92 , Theorem V .8.2]) . If 𝑇 is a monad on a category 𝒞 and 𝐺 is a right adjoint to 𝑇 , t hen 𝐺 is a comonad and ther e is a canonical isomorphism between the Eilenberg– Moor e categories: 𝒞 𝑇  𝒞 𝐺 . Proposition 2.26 ([ Kle90 , Theorem 3]) . If 𝑇 is a monad on 𝒞 and 𝑆 is a left adjoint to 𝑇 , then 𝑆 is a comonad and ther e is a canonical isomor phism 𝒞 𝑇  𝒞 𝑆 . In gener al, it is not true t hat 𝒞 𝑇  𝒞 𝑆 in the setting of Proposition 2.26 . 17 2. Prel iminari es 2.2.2 Dis tributive law s Bec k’s t heory of distr ibut ive l a ws concer ns itself wit h t he question when, giv en a monad 𝑇 on a category 𝒞 , a functor 𝑆 : 𝒞 − → 𝒞 lifts to a functor on t he Eilenberg– Moore category of 𝑇 , [ Bec69 ]. That is, t here is some ˜ 𝑆 : 𝒞 𝑇 − → 𝒞 𝑇 , that satisﬁes t he lif ting condition 𝒞 𝑇 𝒞 𝑇 𝒞 𝑈 𝑇 ˜ 𝑆 𝑆 ◦ 𝑈 𝑇 If 𝑆 is a monad t his in turn is equiv alent to 𝑆𝑇 being a monad itself. Dis- tributiv e la ws thus yield a witness for t he composability of tw o monads. Street further dev eloped t he theor y of distributiv e la ws intrinsic to certain w ell-beha v ed bicategories, see [ Str72 ]. Deﬁnition 2.27. Let 𝑇 and 𝑆 be tw o monad on a categor y 𝒞 . A distributive law of 𝑇 o v er 𝑆 consists of a natural transformation Ω : 𝑇 𝑆 = ⇒ 𝑆𝑇 such that the follo wing diagrams commute: 𝑇 𝑆 𝑆 𝑆𝑇 𝑆 𝑆 𝑆𝑇 𝑇 𝑇 𝑆 𝑇 𝑆 𝑆𝑇 𝑆𝑇 𝑇 𝜇 𝑆 Ω Ω 𝑆 𝑆 Ω 𝜇 𝑆 𝑇 𝑇 𝜂 𝑆 Ω 𝜂 𝑆 𝑇 𝑇 𝑇 𝑆 𝑇 𝑆 𝑇 𝑆 𝑇 𝑇 𝑆 𝑇 𝑆 𝑇 𝑆 𝑆𝑇 𝑆𝑇 𝑇 Ω Ω 𝑇 𝑆 𝜇 𝑇 𝜇 𝑇 𝑆 Ω 𝑆 𝜂 𝑇 𝜂 𝑇 𝑆 Ω One analogousl y deﬁnes a distributiv e law betw een tw o comonads, and mixed distributiv e laws betw een a monad and a comonad. Theorem 2.28 ([ Bec69 ]) . Let 𝑆 and 𝑇 be monads on a cat egory 𝒞 . Then the f ol- lowing ar e equiv alent: (i) dis tributive law s of 𝑇 over 𝑆 ; (ii) monad s tructures 𝑆 ◦ Ω 𝑇 . . = ( 𝑆𝑇 , 𝜇 , 𝜂 ) on 𝑆 𝑇 , such t hat 𝑆 𝜂 𝑇 and 𝜂 𝑆 𝑇 ar e morphisms of monads and 1 𝑆𝑇 = 𝜇 ◦ 𝑆 𝜂 𝑇 𝜂 𝑇 𝑇 ; and 18 2.3. String diagrams (iii) lifts of 𝑆 to the Eilenber g – Moor e category of 𝑇 , such t hat 𝑆 is a monad on 𝒞 𝑇 . Sketch of proof. W e outline t he main constructions. Giv en a dis tributiv e law Ω : 𝑇 𝑆 = ⇒ 𝑆𝑇 , for tw o monads on 𝒞 , t he monad structure on 𝑆𝑇 is giv en by 𝜇 : 𝑆 𝑇 𝑆 𝑇 𝑆 Ω 𝑇 − − − → 𝑆 𝑆𝑇 𝑇 𝜇 𝑆 𝜇 𝑇 − − − → 𝑆𝑇 and 𝜂 : Id 𝒞 𝜂 𝑇 − → 𝑇 𝜂 𝑆 𝑇 − − → 𝑆𝑇 . The lif t ˜ 𝑆 of 𝑆 to 𝒞 𝑇 is deﬁned by ˜ 𝑆 ( 𝑥 , ∇ 𝑥 ) . . = ( 𝑆 𝑥 , 𝑇 𝑆 𝑥 Ω 𝑥 − − → 𝑆𝑇 𝑥 𝑆 ∇ 𝑥 − − − → 𝑆 𝑥 ) . Its multiplication and unit is giv en by those of 𝑆 . Giv en a monad structure ( 𝑆 𝑇 , 𝜇 , 𝜂 ) , one deﬁnes a distributiv e la w b y 𝑇 𝑆 𝜂 𝑆 𝑇 𝑆 𝜂 𝑇 − − − − − → 𝑆𝑇 𝑆𝑇 𝜇 − − → 𝑆 𝑇 . Finall y , if ˜ 𝑆 is a lif t of 𝑆 , w e obtain a distributiv e la w via 𝑇 𝑆 𝑇 𝑆 𝜂 𝑇 − − − → 𝑇 𝑆𝑈 𝑇 𝐹 𝑇 = 𝑈 𝑇 𝐹 𝑇 𝑈 𝑇 ˜ 𝑆 𝐹 𝑇 𝑈 𝑇 𝜀 𝑇 ˜ 𝑆 𝐹 𝑇 − − − − − − − → 𝑈 𝑇 ˜ 𝑆 𝐹 𝑇 = 𝑆𝑇 . □ 2 . 3 s t r i n g d i ag r a m s Reca ll from Exam ple 2 .2 that C at is t he 2-categor y of all (small) categories, functors, and natural transf ormations. W e use juxtaposition for t he hori- zontal com position of C at , or , in case w e want to emphasise t he direction of composition, − ⊙ = : C at ( ℬ , 𝒞 ) × C at ( 𝒜 , ℬ ) − → C at ( 𝒜 , 𝒞 ) , 𝒜 , ℬ , 𝒞 ∈ C at . String diagrams will ser v e as an important tool for doing computations. More generall y , our applications of the graphical calculus can be formulated in any bicategor y that admits t he construction of algebr as in the sense of Example 2.14 . For ease of presentation w e shall stick wit h C at . In t he case of C at , a string diagram consists of regions labelled with categories, strings labelled wit h functors, and v ertices betw een the strings labelled with natural transf ormations. If tw o string diagrams can be transf ormed into each other — t hat is, if they are iso topic — t hen the natur al transformations t he y represent are equal. A more detailed description is giv en in [ JS91 ; Sel11 ]. Our conv ention is to read diag r ams from bott om to top and right to left. Horizontal and v ertical com position are giv en b y horizontal and v ertical gluing of diagrams, respectiv ely . Identity natural transformations are giv en b y unlabelled v ertices. The edg e for t he identity functor of a category will not be dra wn. If the inv olv ed categories are clear from the context, w e omit writing them explicitly . Figure 2.1 details our con v entions. 19 2. Prel iminari es ⊙ 𝛼 = 𝛼 𝛽 ◦ 𝛼 = 𝛼 𝛽 𝐹 𝐹 𝐻 𝐺 𝐹 𝐺 𝐹 𝐻 Horizontal composition of natural trans- formations id 𝐹 : 𝐹 = ⇒ 𝐹 and 𝛼 : 𝐺 = ⇒ 𝐻 where 𝐹 : 𝒞 − → 𝒟 and 𝐺 , 𝐻 : 𝒟 − → ℰ . V ertical composition of tw o natural transformations 𝛼 : 𝐹 = ⇒ 𝐺 and 𝛽 : 𝐺 = ⇒ 𝐻 , where 𝐹 , 𝐺 , 𝐻 : 𝒞 − → 𝒟 . 𝐺 𝐻 𝐹 𝐺 𝐹 𝐻 𝒞 𝒟 𝒟 ℰ 𝒞 𝒟 𝒞 𝒟 𝒞 𝒟 𝛾 = 𝛾 A natural transformation 𝛾 : Id 𝒞 = ⇒ 𝐹 for Id 𝒞 , 𝐹 : 𝒞 − → 𝒞 . Id 𝒞 𝐹 𝐹 𝒞 𝒞 𝒞 𝒞 𝒟 ℰ Figure 2.1: Basic string diagrammatic con v entions. Exam ple 2.29. Let 𝑇 be a monad on 𝒞 . W e represent t he multiplication and unit of 𝑇 in terms of string diagrams: 𝑇 𝑇 𝑇 𝑇 and Their associativity and unitality then equate to = = = 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 and 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 W e depict t he unit and counit of t he Eilenberg– Moore adjunction as def = 𝑈 𝑇 𝐹 𝑇 𝐹 𝑇 𝜂 𝑈 𝑇 Id 𝒞 def = 𝑈 𝑇 𝐹 𝑇 𝐹 𝑇 𝜀 𝑈 𝑇 Id 𝒞 𝑇 𝒞 𝒞 𝒞 𝑇 𝒞 𝒞 𝑇 𝒞 𝑇 and The deﬁning equations of adjunctions translate to t he snake equations 𝑈 𝑇 𝑈 𝑇 = 𝑈 𝑇 𝑈 𝑇 𝐹 𝑇 𝐹 𝑇 = 𝐹 𝑇 𝐹 𝑇 and 20 2.4. Monoidal and module categories Remark 2.30. Giv en a monad 𝑇 , its categor y of algebr as can be incor por ated into t he g r aphical calculus, see [ Wil08 ]. Deﬁne t he natural transf ormation ∇ : 𝑇 𝑈 𝑇 = 𝑈 𝑇 𝐹 𝑇 𝑈 𝑇 𝑈 𝑇 𝜀 − − − − → 𝑈 𝑇 . (2.3.1) N o w , one ma y write ∇ def = 𝑇 𝑈 𝑇 𝒞 𝒞 𝒞 𝑇 𝑈 𝑇 𝑇 W e t hink of ∇ : 𝑇 𝑈 𝑇 − → 𝑈 𝑇 as an action of 𝑇 on 𝑈 𝑇 due to t he identities = = 𝑇 𝑇 𝑇 𝑇 and By abuse of notation, w e shall often simpl y speak of an object 𝑥 ∈ 𝒞 𝑇 , lea ving t he action ∇ 𝑥 implicit. 2 . 4 m o n o i da l a n d m o d u l e c at e g o r i e s For a mo re e xten sive acc ou nt concerning the notions of monoidal and module categories, w e refer t he reader to [ EGNO15 , Chapter 2]. Deﬁnition 2.31. A monoidal category consists of a categor y 𝒞 tog ether wit h a tensor product functor ⊗ : 𝒞 × 𝒞 − → 𝒞 , a unit 1 ∈ 𝒞 , and for all 𝑥 , 𝑦 , 𝑧 ∈ 𝒞 , natural isomor phisms 𝛼 𝑥 , 𝑦 , 𝑧 : ( 𝑥 ⊗ 𝑦 ) ⊗ 𝑧 ∼ = ⇒ 𝑥 ⊗ ( 𝑦 ⊗ 𝑧 ) , 𝜆 𝑥 : 1 ⊗ 𝑥 ∼ = ⇒ 𝑥 , 𝜌 𝑥 : 𝑥 ⊗ 1 ∼ = ⇒ 𝑥 , satisfying coher ence axioms; see [ EGNO15 , Deﬁnition 2.1.1]. A monoidal category is called strict if 𝛼 , 𝜆 , and 𝜌 are identities. Exam ple 2.32. Giv en a monoidal categor y ( 𝒞 , ⊗ , 1 ) , w e denote b y 𝒞 rev the monoidal category ( 𝒞 , ⊗ op , 1 ) , where for all 𝑥 , 𝑦 ∈ 𝒞 one sets 𝑥 ⊗ op 𝑦 . . = 𝑦 ⊗ 𝑥 . Recall Sweedler not ation : if 𝐵 is a bialgebra in V ect with comultiplication Δ : 𝐵 − → 𝐵 ⊗ k 𝐵 , t hen f or 𝑏 ∈ 𝐵 w e write 𝑏 ( 1 ) ⊗ 𝑏 ( 2 ) . . = Δ ( 𝑏 ) ∈ 𝐵 ⊗ k 𝐵 . F or example, in this notation coassociativity translates to 𝑏 ( 1 ) ⊗ ( 𝑏 ( 2 ) ) ( 1 ) ⊗ ( 𝑏 ( 2 ) ) ( 2 ) = 𝑏 ( 1 ) ⊗ 𝑏 ( 2 ) ⊗ 𝑏 ( 3 ) = ( 𝑏 ( 1 ) ) ( 1 ) ⊗ ( 𝑏 ( 1 ) ) ( 2 ) ⊗ 𝑏 ( 2 ) . This naturall y extends to left and right coactions on 𝐵 , which w e respectiv ely denote b y 𝑥 ↦− → 𝑥 (− 1 ) ⊗ 𝑥 ( 0 ) and 𝑥 ↦− → 𝑥 ( 0 ) ⊗ 𝑥 ( 1 ) . 21 2. Prel iminari es Exam ple 2.33 ([ Y et90 ]) . Let 𝐻 ∈ V ect be a bialg ebra The categor y 𝐻 𝐻 𝒴𝒟 of (left-left) Y etter – Drinf eld modules is deﬁned as the categor y of t hose 𝑌 ∈ V ect that are lef t 𝐻 -modules with action ⊲ and also left 𝐻 -comodules with coaction 𝑦 ↦− → 𝑦 (− 1 ) ⊗ 𝑦 ( 0 ) , such t hat t he follo wing compatibility condition holds for all ℎ ∈ 𝐻 and 𝑦 ∈ 𝑌 : ℎ ( 1 ) 𝑦 (− 1 ) ⊗ ( ℎ ( 2 ) ⊲ 𝑦 ( 0 ) ) = ( ℎ ( 1 ) ⊲ 𝑦 ) (− 1 ) ℎ ( 2 ) ⊗ ( ℎ ( 1 ) ⊲ 𝑦 ) ( 0 ) (2.4.1) Then 𝐻 𝐻 𝒴𝒟 is monoidal: giv en Y etter– Drinf eld modules 𝑌 and 𝑍 , their tensor product 𝑌 ⊗ k 𝑍 can be equipped wit h t he structure of a Y etter – Drinf eld module. F or all ℎ ∈ 𝐻 , 𝑦 ∈ 𝑌 , and 𝑧 ∈ 𝑍 , deﬁne ℎ ⊲ ( 𝑦 ⊗ 𝑧 ) . . = ( ℎ ( 1 ) ⊲ 𝑦 ) ⊗ ( ℎ ( 2 ) ⊲ 𝑧 ) and 𝑦 ⊗ 𝑧 ↦− → 𝑦 (− 1 ) 𝑧 (− 1 ) ⊗ 𝑦 ( 0 ) ⊗ 𝑧 ( 0 ) . In fact, if 𝐻 is a Hopf alg ebra wit h in vertible antipode 𝑆 — for example, when 𝐻 is ﬁnite-dimensional — then Equation ( 2.4.1 ) is equivalent to ( ℎ ⊲ 𝑦 ) (− 1 ) ⊗ ( ℎ ⊲ 𝑦 ) ( 0 ) = ℎ ( 1 ) 𝑦 (− 1 ) 𝑆 ( ℎ ( 3 ) ) ⊗ ( ℎ ( 2 ) ⊲ 𝑦 ( 0 ) ) , (2.4.2) see [ Mon93 , Proposition 10.6.16]. Deﬁnition 2.34. A lax monoidal functor betw een monoidal categories 𝒞 and 𝒟 consists of a funct or 𝐹 : 𝒞 − → 𝒟 tog et her with a morphism 𝐹 0 : 1 − → 𝐹 1 and a natural transf or mation 𝐹 2 : 𝐹 (−) ⊗ 𝐹 ( = ) = ⇒ 𝐹 (− ⊗ = ) , satisfying associativity and unitality conditions, see [ EGNO15 , Deﬁnition 2.4.1]. Analogousl y , an oplax monoidal functor is a functor 𝐺 : 𝒞 − → 𝒟 tog ether with mor phisms 𝐺 0 : 𝐺 1 − → 1 and 𝐺 2 : 𝐺 (− ⊗ = ) = ⇒ 𝐺 (−) ⊗ 𝐺 ( = ) , making 𝐺 op : 𝒞 op − → 𝒟 op lax monoidal. W e call 𝐹 str ong monoidal if 𝐹 0 and 𝐹 2 and in v ertible, and strict monoidal if t he y are identities. Observe t hat an oplax monoidal functor with in v ertible coherence mor ph- isms canonicall y deﬁnes a strong monoidal functor . Remark 2.35. Let 𝑇 be an oplax monoidal endofunctor on a monoidal categor y 𝒞 . By coassociativity of 𝑇 2 , t here exists a w ell-deﬁned natural transf ormation 𝑇 3 : 𝑇 (− ⊗ = ⊗ ≡) = ⇒ 𝑇 (−) ⊗ 𝑇 ( = ) ⊗ 𝑇 (≡) , which, for all 𝑥 , 𝑦 , 𝑧 ∈ 𝒞 , has components 𝑇 3; 𝑥 , 𝑦 ,𝑧 . . = ( 𝑇 2; 𝑥 , 𝑦 ⊗ 𝑇 𝑧 ) ◦ 𝑇 2; 𝑥 ⊗ 𝑦 , 𝑧 = ( 𝑇 𝑥 ⊗ 𝑇 2; 𝑦 , 𝑧 ) ◦ 𝑇 2; 𝑥 , 𝑦 ⊗ 𝑧 . 22 2.4. Monoidal and module categories Deﬁnition 2.36. Let 𝒞 and 𝒟 be tw o monoidal categories and suppose t hat 𝐹 , 𝐺 : 𝒞 − → 𝒟 are two lax monoidal functors. A (lax) monoidal natural trans- formation is a natural transf ormation 𝜑 : 𝐹 = ⇒ 𝐺 such that t he following tw o diagrams commute for all 𝑥 , 𝑦 ∈ 𝒞 : 𝐹 𝑥 ⊗ 𝐹 𝑦 𝐹 ( 𝑥 ⊗ 𝑦 ) 1 𝐹 1 𝐺 𝑥 ⊗ 𝐺 𝑦 𝐺 ( 𝑥 ⊗ 𝑦 ) 𝐺 1 𝐹 2; 𝑥 , 𝑦 𝜑 𝑥 ⊗ 𝑦 𝜑 𝑥 ⊗ 𝜑 𝑦 𝐺 2; 𝑥 , 𝑦 𝐺 0 𝐹 0 𝜑 1 Duall y , giv en tw o oplax monoidal functors 𝐹 , 𝐺 : 𝒞 − → 𝒟 , one deﬁnes an oplax monoidal natural transf ormation as an arrow 𝜑 : 𝐹 = ⇒ 𝐺 making t he follo wing diag r ams commute for all 𝑥 , 𝑦 ∈ 𝒞 : 𝐹 ( 𝑥 ⊗ 𝑦 ) 𝐹 𝑥 ⊗ 𝐹 𝑦 𝐹 1 1 𝐺 ( 𝑥 ⊗ 𝑦 ) 𝐺 𝑥 ⊗ 𝐺 𝑦 𝐺 1 𝐹 2; 𝑥 , 𝑦 𝜑 𝑥 ⊗ 𝑦 𝜑 𝑥 ⊗ 𝜑 𝑦 𝐺 2; 𝑥 , 𝑦 𝐺 0 𝐹 0 𝜑 1 Deﬁnition 2.37. An adjunction 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 is called lax monoidal if 𝐹 and 𝑈 are lax monoidal functors, and t he unit and counit of the adjunction are monoidal natural transf ormations. Analogousl y to Deﬁnition 2.37 one deﬁnes oplax monoidal adjunctions . Deﬁnition 2.38. Let 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 be an ordinary adjunction between mono- idal categories. A lif t of 𝐹 ⊣ 𝑈 to a lax monoidal adjunction consists of choices of lax monoidal structures on 𝐹 and 𝑈 , tog ether with assertions t hat t he unit and counit of the adjunction are lax monoidal natur al transformations. In other w ords, it is t he structure necessar y for a lax monoidal adjunction whose left adjoint is 𝐹 and whose right adjoint is 𝑈 . Similar l y , one can deﬁne a lift of 𝐹 ⊣ 𝑈 to an oplax monoidal adjunction. Remark 2.39. A strong monoidal functor is called a monoidal equiv alence if it additionall y is an equivalence of categories. The quasi-in v erse 𝐹 − 1 : 𝒟 − → 𝒞 of a monoidal equiv alence 𝐹 : 𝒞 − → 𝒟 is again monoidal, and t here are monoidal natural isomor phisms 𝐹 ◦ 𝐹 − 1 ∼ − → Id 𝒟 and 𝐹 − 1 ◦ 𝐹 ∼ − → Id 𝒞 . 23 2. Prel iminari es U nless other wise speciﬁed, all monoidal categories in t he rest of t his t hesis are assumed to be strict. This is jus tiﬁed b y t he follo wing strictiﬁcation result of Mac Lane, [ ML63 ], see also [ EGN O15 , Theorem 2.8.5]. Theorem 2.40. Every monoidal cat egory is monoidally equivalent t o a strict mo- noidal category . Deﬁnition 2.41. A (left) 𝒞 -module category of a monoidal category 𝒞 consists of a category ℳ tog ether wit h a functor ⊲ : 𝒞 × ℳ − → ℳ and isomorphisms ℳ a ; 𝑥 , 𝑦 ,𝑚 : ( 𝑥 ⊗ 𝑦 ) ⊲ 𝑚 ∼ = ⇒ 𝑥 ⊲ ( 𝑦 ⊲ 𝑚 ) , natural in 𝑚 ∈ ℳ and 𝑥 , 𝑦 ∈ 𝒞 , satisfying coherence axioms similar to those of a monoidal category; see [ EGNO15 , Deﬁnition 7.1.1]. Analogousl y , one can deﬁne right module categories o v er 𝒞 , which in v olv e a functor ⊳ : ℳ × 𝒞 − → ℳ and isomorphisms 𝑚 ⊳ ( 𝑥 ⊗ 𝑦 ) ∼ = ⇒ ( 𝑥 ⊳ 𝑦 ) ⊳ 𝑚 . For brevity , w e ma y sometimes speak of just “ 𝒞 -module categories ”; t his will alw a ys mean “lef t 𝒞 -module categories”. Exam ple 2.42 ([ HKRS04 ]) . Let 𝐻 be a Hopf algebra wit h in v ertible antipode 𝑆 . Recall from Example 2.33 t hat the categor y 𝐻 𝐻 𝒴𝒟 of Y etter – Drinf eld modules is a monoidal categor y . A left-left anti-Y etter– Drinf eld module is some 𝑀 ∈ V ect equipped with a lef t 𝐻 -module structure ⊲ and a left 𝐻 -comodule structure 𝑚 ↦− → 𝑚 (− 1 ) ⊗ 𝑚 ( 0 ) such that t he follo wing holds for all ℎ ∈ 𝐻 and 𝑚 ∈ 𝑀 : N otice t hat the only diﬀerence to Equation ( 2.4.2 ) is that w e use 𝑆 − 1 instead of 𝑆 . ( ℎ ⊲ 𝑚 ) (− 1 ) ⊗ ( ℎ ⊲ 𝑚 ) ( 0 ) = ℎ ( 1 ) 𝑚 (− 1 ) 𝑆 − 1 ( ℎ ( 3 ) ) ⊗ ( ℎ ( 2 ) ⊲ 𝑚 ( 0 ) ) The category of anti-Y etter –Drinfeld modules do no t f orm a monoidal category , but rather a 𝒞 -module category ov er the Y etter –Drinfeld modules. Giv en 𝑀 ∈ 𝐻 𝐻 a 𝒴𝒟 and 𝑌 ∈ 𝐻 𝐻 𝒴𝒟 , deﬁne 𝑌 ⊲ 𝑀 ∈ 𝐻 𝐻 a 𝒴𝒟 as the v ector space 𝑌 ⊗ k 𝑀 , equipped wit h t he follo wing action and coaction: ℎ ⊲ ( 𝑦 ⊗ 𝑚 ) . . = ( ℎ ( 1 ) ⊲ 𝑦 ) ⊗ ( ℎ ( 2 ) ⊲ 𝑚 ) , 𝑦 ⊗ 𝑚 ↦− → 𝑦 (− 1 ) 𝑚 (− 1 ) ⊗ 𝑦 ( 0 ) ⊗ 𝑚 ( 0 ) . Deﬁnition 2.43. Let 𝒞 and 𝒟 be monoidal categories. A ( 𝒞 , 𝒟 ) -bimodule category consists of a lef t 𝒞 -module categor y ℳ that is simultaneously a right 𝒟 -module category , such t hat t here exists a natural isomorphism ( 𝑥 ⊲ 𝑚 ) ⊳ 𝑦 ∼ = ⇒ 𝑥 ⊲ ( 𝑚 ⊳ 𝑦 ) , for all 𝑥 ∈ 𝒞 , 𝑦 ∈ 𝒟 , and 𝑚 ∈ ℳ , called t he middle inter chang e , which satisﬁes appropriate associativity constr aints; see [ EGN O15 , Deﬁnition 7.1.7]. 24 2.4. Monoidal and module categories Exam ple 2.44. Ev ery monoidal categor y 𝒞 is a ( 𝒞 , 𝒞 ) -bimodule categor y , whose actions are both giv en by t he tensor product of 𝒞 . W e call t his module category t he r egular bimodule , and denote it by 𝒞 𝒞 𝒞 . Analogously , one deﬁnes the lef t and right regular 𝒞 -module categories 𝒞 𝒞 and 𝒞 𝒞 . Exam ple 2.45. Let 𝒞 be a monoidal categor y . The regular action is not the onl y w a y in which w e can consider 𝒞 as a bimodule ov er itself. Suppose t hat 𝑅 : 𝒞 − → 𝒞 is a strong monoidal functor . The action ⊳ : 𝒞 × 𝒞 𝒞 × 𝑅 − − − → 𝒞 × 𝒞 ⊗ − − → 𝒞 endo ws 𝒞 with the structure of a right module categor y ov er itself, which w e shall denote b y 𝒞 𝑅 . In other w ords, we ha ve 𝑥 ⊲ 𝑦 . . = 𝑅 𝑥 ⊗ 𝑦 , 𝑓 ⊲ 𝑔 . . = 𝑅 𝑓 ⊗ 𝑔 , for 𝑣 , 𝑤 , 𝑥 , 𝑦 ∈ 𝒞 and 𝑓 : 𝑣 − → 𝑤 , 𝑔 : 𝑥 − → 𝑦 . W e can also twist the action from t he lef t with ano t her monoidal functor 𝐿 . The twist ed bimodule obtained in this manner is denoted b y 𝐿 𝒞 𝑅 . Remark 2.46. One can easil y imagine a more in v olv ed setting than Ex- ample 2.45 b y twisting wit h an oplax monoidal functor 𝐿 : 𝒞 − → 𝒞 from the left and a lax monoidal functor 𝑅 from t he right. In t his setting, 𝐿 𝒞 𝑅 is a lax left and oplax right 𝒞 -bimodule categor y , see for example [ Szl12 , Section 2]. Since in most of this thesis w e onl y consider t he strong case, w e refr ain from more formall y treating this case; how ev er , see Remar k 6.13 . Deﬁnition 2.47. Let ℳ and 𝒩 be left module categories o v er a monoidal category 𝒞 . A lax 𝒞 -module functor from ℳ to 𝒩 is a functor 𝐹 : ℳ − → 𝒩 , tog ether wit h a collection of mor phisms 𝐹 a ; 𝑥 , 𝑚 : 𝑥 ⊲ 𝐹 𝑚 − → 𝐹 ( 𝑥 ⊲ 𝑚 ) , natur al for all 𝑥 ∈ 𝒞 and 𝑚 ∈ ℳ , satisfying associativity and unit ality conditions, see [ EGN O15 , Deﬁnition 7.2.1]. Oplax and s trong 𝒞 -module functors are deﬁned similarl y ; in the former case, one considers arro ws 𝐹 a ; 𝑥 , 𝑚 : 𝐹 ( 𝑥 ⊲ 𝑚 ) − → 𝑥 ⊲ 𝐹 𝑚 , while for t he latter 𝐹 a should be in v ertible. Deﬁnition 2.48. Let 𝐹 , 𝐺 : ℳ − → 𝒩 be 𝒞 -module functors betw een t he left 𝒞 -module categories ℳ and 𝒩 . A natural transf ormation 𝜙 : 𝐹 = ⇒ 𝐺 is called a 𝒞 -module tr ansformation if 𝐺 a ; 𝑥 , 𝑚 ◦ ( 𝜙 𝑥 ⊲ 𝑚 ) = 𝜙 𝑥 ⊲ 𝑚 ◦ 𝐹 a ; 𝑥 , 𝑚 , for all 𝑥 ∈ 𝒞 and 𝑚 ∈ ℳ . 25 2. Prel iminari es Notation 2.49. F or 𝒞 -module categories ℳ and 𝒩 , we obtain t he follo wing categories of 𝒞 -module functors: • 𝒞 Mod ( ℳ , 𝒩 ) . . = Str 𝒞 Mod ( ℳ , 𝒩 ) : strong 𝒞 -module functors from ℳ to 𝒩 , • Lax 𝒞 Mod ( ℳ , 𝒩 ) : lax 𝒞 -module functors from ℳ to 𝒩 , • Oplax 𝒞 Mod ( ℳ , 𝒩 ) : oplax 𝒞 -module functors from ℳ to 𝒩 . Exam ple 2.50. F or an object 𝑥 is a (non-strict) monoidal category 𝒞 , the functor 𝐹 . . = − ⊗ 𝑥 is a strong lef t 𝒞 -module functor . The transf ormation 𝐹 a is giv en by the associator of 𝒞 . Further , a mor phism 𝑓 : 𝑥 − → 𝑦 in 𝒞 yields a 𝒞 -module transf ormation − ⊗ 𝑓 : − ⊗ 𝑥 = ⇒ − ⊗ 𝑦 . In fact, Example 2.50 generalises and completel y describes module func- tors from the regular module. W e emphasise t hat t his result is a consequence of the bicategorical Y oneda lemma, see [ JY21 , Lemma 8.3.16]. Proposition 2.51. Let ℳ be a lef t module cat egor y over the monoidal category 𝒞 . Then t here is an equiv alence of module categories Str 𝒞 Mod ( 𝒞 , ℳ ) ∼ − → ℳ , 𝐹 ↦− → 𝐹 1 , − ⊲ 𝑚 ← − [ 𝑚 . In particular , t his yields a monoidal equiv alence Str 𝒞 Mod ≃ 𝒞 rev . 2.4.1 Br aidings Bra idings are n a tur al t rans f or ma ti ons relating t he tensor product to its opposite. They where introduced b y Joy al and Street in [ JS85 ; JS86 ; JS93 ], and build on the notion of symmetries studied in [ ML63 ; EK66 ]. Deﬁnition 2.52. A braiding on a monoidal category 𝒞 is a natural isomorphism 𝜎 − , = : − ⊗ = = ⇒ = ⊗ − , satisfying 𝜎 𝑥 , 1 = id 𝑥 and the hexagon axioms 2 . 2 The name “hexagon axioms ” is due to the fact that in t he non-strict setting, t he deﬁning equations can be organised as a hexagon-shaped diagram, see [ JS93 ]. The pair ( 𝒞 , 𝜎 ) will be called a braided monoidal category . Remark 2.53. For a braiding 𝜎 , t he condition t hat 𝜎 𝑥 , 1 = id 𝑥 is actuall y implied b y the hexagon identities: The calculation 𝜎 𝑥 , 1 = 𝜎 𝑥 , 1 ⊗ 1 = ( id 1 ⊗ 𝜎 𝑥 , 1 ) ◦ ( 𝜎 𝑥 , 1 ⊗ id 1 ) = 𝜎 𝑥 , 1 ◦ 𝜎 𝑥 , 1 sho ws t hat 𝜎 𝑥 , 1 is an in v ertible idempo tent, hence the claim follo ws. 26 2.4. Monoidal and module categories Exam ple 2.54 ([ Y et90 , Theorem 7.2]) . The categor y of left-left Y etter – Drinf eld modules of Example 2.33 is braided monoidal when equipped wit h 𝜎 . . = { 𝜎 𝑌 , 𝑍 : 𝑌 ⊗ 𝑍 − → 𝑍 ⊗ 𝑌 , 𝑦 ⊗ 𝑧 ↦− → ( 𝑦 (− 1 ) ⊲ 𝑧 ) ⊗ 𝑦 ( 0 ) } 𝑌 , 𝑍 ∈ 𝐻 𝐻 𝒴𝒟 , called the Y ett er – Drinf eld braiding . No te that 𝜎 is in v ertible because 𝐻 has an in v ertible antipode. Its in v erse is giv en for all 𝑌 , 𝑍 ∈ 𝐻 𝐻 𝒴𝒟 b y 𝜎 − 1 . . = { 𝜎 − 1 𝑌 , 𝑍 : 𝑍 ⊗ 𝑌 − → 𝑌 ⊗ 𝑌 , 𝑧 ⊗ 𝑦 ↦− → 𝑦 ( 0 ) ⊗ ( 𝑆 − 1 ( 𝑦 (− 1 ) ) ⊲ 𝑧 ) } 𝑌 , 𝑍 ∈ 𝐻 𝐻 𝒴𝒟 . Deﬁnition 2.55. A braiding 𝜎 on a monoidal categor y 𝒞 is called symmetric if 𝜎 − 1 𝑥 , 𝑦 = 𝜎 𝑦 , 𝑥 for all 𝑥 , 𝑦 ∈ 𝒞 . A monoidal categor y equipped wit h a symmetric braiding will be referred to as a symmetric monoidal category . Braidings are depicted in t he g r aphical calculus by crossings of s trings subject to Reidemeis ter-esque identities, see [ Sel11 ]. The follo wing ﬁgure sho ws a braiding, its in verse, the hexagon identity , and t he naturality of t he braiding in its ﬁrst argument: 𝜎 𝑥 , 𝑦 𝑥 𝑦 𝑦 𝑥 𝜎 − 1 𝑥 , 𝑦 𝑥 𝑦 𝑦 𝑥 ( id 𝑦 ⊗ 𝜎 𝑥 , 𝑧 ) ◦ ( 𝜎 𝑥 , 𝑦 ⊗ id 𝑧 ) = 𝜎 𝑥 , 𝑦 ⊗ 𝑧 𝑥 𝑦 𝑧 𝑧 𝑥 𝑦 = 𝑥 𝑦 𝑧 𝑧 𝑥 𝑦 𝑥 𝑦 𝑦 𝑥 𝑓 𝑥 𝑦 𝑦 𝑥 𝑓 = 𝜎 𝑤 , 𝑦 ◦ ( 𝑓 ⊗ id 𝑦 ) = ( 𝑦 ⊗ 𝑓 ) ◦ 𝜎 𝑥 , 𝑦 2.4.2 Closedness Deﬁnition 2.56. Let 𝒞 be a monoidal categor y . An object 𝑚 in a 𝒞 -module category ℳ is called closed if t he functor − ⊲ 𝑚 : 𝒞 − → ℳ has a right adjoint ⌊ 𝑚 , −⌋ : ℳ − → 𝒞 , called the internal hom from 𝑚 . W e refer to − ⊲ 𝑚 : 𝒞 ⇄ ℳ : ⌊ 𝑚 , −⌋ as the internal tensor – hom adjunction . Similar ly to Deﬁnition 2.56 , an object 𝑚 ∈ ℳ is called coclosed if the functor − ⊲ 𝑚 : 𝒞 − → ℳ has a left adjoint ; w e denote t his internal cohom from 𝑚 b y ⌈ 𝑚 , −⌉ : ℳ − → 𝒞 , and call ⌈ 𝑚 , −⌉ : ℳ ⇄ 𝒞 : − ⊲ 𝑚 as the internal cohom – t ensor adjunction . If ev er y object of ℳ is closed, w e call ℳ a closed 𝒞 -module category . 27 2. Prel iminari es Deﬁnition 2.57. A monoidal categor y 𝒞 is called left closed if t he lef t regular 𝒞 -module categor y 𝒞 𝒞 is closed, right closed if the right regular 𝒞 -module category 𝒞 𝒞 is closed, and closed or biclosed if it is both left and right closed. In these cases, w e write [ 𝑥 , −] ℓ and [ 𝑥 , −] 𝑟 for ⌊ 𝑥 , −⌋ , respectiv ely . Exam ple 2.58. A symmetric monoidal categor y is lef t closed if and only if it is right closed if and onl y if it is closed. The object-wise tensor– hom adjunctions of a closed 𝒞 -module categor y specify a unique functor ⌊ − , = ⌋ : ℳ op × ℳ − → 𝒞 such t hat w e ha v e ℳ ( 𝑐 ⊲ 𝑚 , 𝑛 )  𝒞 ( 𝑐 , ⌊ 𝑚 , 𝑛 ⌋) , natural in all t hree v ariables, see [ ML98 , Section iv .7] for the monoidal case. Exam ple 2.59. For an y object 𝑥 ∈ 𝒞 of a right closed monoidal category , a combination of the unit and counit 𝜂 ( 𝑥 ) 𝑦 : 𝑦 − → [ 𝑥 , 𝑥 ⊗ 𝑦 ] 𝑟 , 𝜀 ( 𝑥 ) 𝑦 : 𝑥 ⊗ [ 𝑥 , 𝑦 ] 𝑟 − → 𝑦 , for all 𝑦 ∈ 𝒞 giv es rise to the mor phism 𝜙 ( 𝑥 ) 𝑦 . . = [ 𝑥 , 1 ] 𝑟 ⊗ 𝑦 [ 𝑥 , 𝑥 ⊗ [ 𝑥 , 1 ] 𝑟 ⊗ 𝑦 ] 𝑟 [ 𝑥 , 𝑦 ] 𝑟 . 𝜂 ( 𝑥 ) [ 𝑥 , 1 ] 𝑟 ⊗ 𝑦 [ 𝑥 , 𝜀 ( 𝑥 ) 1 ⊗ 𝑦 ] (2.4.3) 2.4.3 Rigidity and pivot ality Rig idit y Rigid monoidal categories are also called autonomous , or , in the symmetric case, compact closed . in the conte xt of mono id al ca te gori es refers to a concept of duality similar to t hat of ﬁnite-dimensional v ector spaces. Importantl y , notions like dual bases and ev aluations ha v e t heir analogues in t his setting. Piv otal categories — also known as balanced or sover eign categories — are those where t here exists an identiﬁcation betw een objects and their double duals that is compatible wit h t he tensor product. Recall from Equation ( 2.4.3 ) there is a canonical mor phism Φ ( 𝑥 ) 𝑦 : [ 𝑥 , 1 ] 𝑟 ⊗ 𝑦 − → [ 𝑥 , 𝑦 ] 𝑟 for all objects 𝑥 , 𝑦 in t he right closed monoidal categor y 𝒞 . The next results links t he in v ertibility of t his mor phism to conditions t hat 𝒞 is rigid monoidal; see for example [ Kel72 ], and [ NW17 , Proposition 2.1] for a concise proof. 28 2.4. Monoidal and module categories Proposition 2.60. Let 𝒞 be a monoidal category . For any object 𝑥 ∈ 𝒞 , the following st atements are equivalent: (i) t he right internal hom of 𝑥 exists, and the canonical arr ows 𝜙 ( 𝑥 ) 𝑦 ar e inv ertible for all 𝑦 ∈ 𝒞 ; (ii) t he right internal hom of 𝑥 exists and 𝜙 ( 𝑥 ) 𝑥 : [ 𝑥 , 1 ] 𝑟 ⊗ 𝑥 − → [ 𝑥 , 𝑥 ] 𝑟 is an isomorphism; and (iii) t here exis ts an object 𝑥 ∨ ∈ 𝒞 tog ether with morphisms ev 𝑟 𝑥 : 𝑥 ⊗ 𝑥 ∨ − → 1 and coev 𝑟 𝑥 : 1 − → 𝑥 ∨ ⊗ 𝑥 , satisfying t he snake identities id 𝑥 = ( ev 𝑟 𝑥 ⊗ id 𝑥 ) ◦ ( id 𝑥 ⊗ coev 𝑟 𝑥 ) , id 𝑥 ∨ = ( id 𝑥 ∨ ⊗ ev 𝑟 𝑥 ) ◦ ( coev 𝑟 𝑥 ⊗ id 𝑥 ∨ ) . (2.4.4) Deﬁnition 2.61. W e call 𝑥 right (rigidly) dualisable if an y of the equiv alent conditions of Proposition 2.60 are met. In t his case, w e hav e [ 𝑥 , 𝑦 ] 𝑟  𝑥 ∨ ⊗ 𝑦 for all 𝑥 , 𝑦 ∈ 𝒞 and in particular 𝑥 ∨  [ 𝑥 , 1 ] 𝑟 . The lef t (rigid) dualisability of an object 𝑥 ∈ 𝒞 can be deﬁned similarl y by eit her t he in v ertibility of a canonical morphism 𝜓 ( 𝑥 ) 𝑦 : 𝑦 ⊗ [ 𝑥 , 1 ] ℓ − → [ 𝑥 , 𝑦 ] ℓ or b y t he existence of an object ∨ 𝑥 ∈ 𝒞 endo w ed with morphisms ev ℓ 𝑥 : ∨ 𝑥 ⊗ 𝑥 − → 1 and coev ℓ 𝑥 : 1 − → 𝑥 ⊗ ∨ 𝑥 , subject to suitable variants of the snake identities. When the context leav es t he choice unambiguous, w e — in the interest of brevity — omit t he left and right superscripts for t he (co)ev aluation morphisms. The pair ( 𝑦 , 𝑥 , ev , coev ) , where 𝑦 is a left dual of 𝑥 and, t hus, 𝑥 is a right dual of 𝑦 is called a dual pair . Deﬁnition 2.62. A monoidal category 𝒞 is called lef t (right) rigid if ev ery object is left (right) rigidly dualisable, and rigid if it is left and right rigid. W e denote t he lef t and right dualising functor s of a rigid categor y 𝒞 b y ∨ (−)  [− , 1 ] ℓ : 𝒞 op , rev − → 𝒞 and (−) ∨  [− , 1 ] 𝑟 : 𝒞 op , rev − → 𝒞 . Exam ple 2.63. Let 𝒞 be a lef t rigid monoidal category . Giv en a morphism 𝑓 : 𝑥 − → 𝑦 in 𝒞 , t he induced mor phism ∨ 𝑓 : ∨ 𝑦 − → ∨ 𝑥 is giv en b y ( ev ℓ 𝑦 ⊗ id ∨ 𝑥 ) ◦ ( id ∨ 𝑦 ⊗ 𝑓 ⊗ id ∨ 𝑥 ) ◦ ( id ∨ 𝑦 ⊗ coev ℓ 𝑥 ) . 29 2. Prel iminari es There are categories wit h onl y one-sided closedness or rigidity , see [ Lor21 , Theorem 6.3.3] and [ TV17 , Example 1.6.2]. T aking the opposite of a rigid category rev erses t he roles of ev aluation and coev aluation. Thus the opposite 𝒞 op of the lef t rigid categor y 𝒞 is right rigid. Lemma 2.64 ([ EGN O15 , Exercise 2.10.6]) . Let 𝐹 : 𝒞 − → 𝒟 be a str ong monoidal functor . The image 𝐹 𝑥 of any (rigidly) dualisable object 𝑥 ∈ 𝒞 is dualisable. Sketch of proof. If 𝑥 ∈ 𝒞 is an object wit h right dual 𝑥 ∨ , deﬁne ( 𝐹 𝑥 ) ∨ . . = 𝐹 ( 𝑥 ∨ ) ; the ev aluation map is giv en by 𝐹 𝑥 ⊗ 𝐹 ( 𝑥 ∨ ) 𝐹 2 − − → 𝐹 ( 𝑥 ⊗ 𝑥 ∨ ) 𝐹 ev ( 𝑟 ) 𝑥 − − − − → 𝐹 1 𝐹 − 1 0 − − → 1 , and t he coev aluation is deﬁned analogously . A str aightforwar d calculation sho ws t he snake identities to be satisﬁed. □ Graphicall y , ev aluations and coev aluations will be represented by semi- circles, possibly decorated wit h arro ws to emphasise whet her w e consider their lef t or right v ersion. 𝑥 ∨ 𝑥 ev ℓ 𝑥 : ∨ 𝑥 ⊗ 𝑥 − → 1 𝑥 ∨ 𝑥 coev ℓ 𝑥 : 1 − → 𝑥 ⊗ ∨ 𝑥 𝑥 𝑥 ∨ ev 𝑟 𝑥 : 𝑥 ⊗ 𝑥 ∨ − → 1 𝑥 𝑥 ∨ coev 𝑟 𝑥 : 1 − → 𝑥 ∨ ⊗ 𝑥 Deﬁnition 2.65. An object 𝑥 ∈ 𝒞 in a rigid monoidal category 𝒞 is called in vertible if its (lef t) ev aluation and coev aluation are isomorphisms. It follo ws t hat the right ev aluations and coev aluations of an in v ertible objects are isomorphisms as w ell. Further , tensor products and duals of inv ert- ible objects are in v ertible, such that t he full subcategor y Inv ( 𝒞 ) of in v ertible object of 𝒞 is rigid monoidal. Deﬁnition 2.66 ([ Cas05 ]) . The Picard group Pic 𝒞 of a rigid monoidal category 𝒞 is t he group of isomorphism classes of in v ertible objects in 𝒞 . Its multiplic- ation is induced by the tensor product; i.e., [ 𝛼 ] · [ 𝛽 ] . . = [ 𝛼 ⊗ 𝛽 ] for 𝛼 , 𝛽 ∈ Inv ( 𝒞 ) . The unit of Pic 𝒞 is [ 1 ] and for an y 𝛼 ∈ Inv ( 𝒞 ) we ha ve that [ 𝛼 ] − 1 = [ ∨ 𝛼 ] . Proposition 2.67. F or every object 𝑥 ∈ 𝒞 in a rigid category 𝒞 we obtain two chains of adjoint endofunctor s of 𝒞 : . . . ⊣ − ⊗ 𝑥 ∨ ⊣ − ⊗ 𝑥 ⊣ − ⊗ ∨ 𝑥 ⊣ . . . . . . ⊣ ∨ 𝑥 ⊗ − ⊣ 𝑥 ⊗ − ⊣ 𝑥 ∨ ⊗ − ⊣ . . . F urther , − ⊗ 𝑥 and 𝑥 ⊗ − are equivalences of categories if and only if 𝑥 is in vertible. 30 2.4. Monoidal and module categories Proof. The existence of t he stated chains of adjunctions follo ws from [ EGN O15 , Proposition 2.10.8]. A str aightforw ard calculation sho ws that tensoring (from the lef t or right) wit h an in v ertible object yields an equiv alence of categories. Con v ersely , suppose that 𝑥 ∈ 𝒞 is such t hat 𝐹 . . = − ⊗ 𝑥 is an equivalence of categories. The functor 𝐹 and its quasi-in v erse 𝑈 are part of an adjunc- tion with in v ertible unit 𝜂 : Id 𝒞 = ⇒ 𝑈 𝐹 and counit 𝜀 : 𝐹 𝑈 = ⇒ Id 𝒟 , see f or example [ Rie17 , Proposition 4.4.5]. By [ Rie17 , Proposition 4.4.1], t here exists a natural isomorphism 𝜃 : 𝑈 = ⇒ − ⊗ ∨ 𝑥 that commutes wit h the respectiv e counits and units. Applied to t he monoidal unit 1 ∈ 𝒞 , w e obtain coev ℓ 𝑥 = 𝜃 𝑥 ◦ 𝜂 1 and ev ℓ 𝑥 ◦ ( 𝜃 1 ⊗ id 𝑥 ) = 𝜀 1 . It follo ws t hat 𝑥 is in v ertible. An analogous argument shows t hat 𝑥 ⊗ − being an equiv alence of categories also entails 𝑥 being in vertible. □ Remark 2.68. Giv en a monoidal category 𝒞 , t he mere presence of adjunctions − ⊗ 𝑥 : 𝒞 ⇄ 𝒞 : − ⊗ 𝐿 𝑥 and 𝑥 ⊗ − : 𝒞 ⇄ 𝒞 : 𝑅 𝑥 ⊗ − for objects 𝐿 𝑥 , 𝑅 𝑥 ∈ 𝒞 does not lead to rigidity , but to t he w eaker notion of tensor r epresent ability , see Deﬁnitions 3.1 and 3.2 . T o elucidate t he underl ying problem, let us assume for a moment t hat w e are giv en objects 𝑥 , 𝑅 𝑥 ∈ 𝒞 , such t hat 𝑥 ⊗ − : 𝒞 ⇄ 𝒞 : 𝑅 𝑥 ⊗ − . One can show that an y right rigid dual of 𝑥 — if it exists — has to be isomorphic to 𝑅 𝑥 ; i.e., one has 𝑅 𝑥  𝑥 ∨ on objects. Hence, t he unit 𝜂 ( 𝑥 ) 𝑧 : 𝑧 − → 𝑅 𝑥 ⊗ 𝑥 ⊗ 𝑧 and counit 𝜀 ( 𝑥 ) 𝑧 : 𝑥 ⊗ 𝑅 𝑥 ⊗ 𝑧 − → 𝑧 of the adjunction provide us wit h natural candidates for t he coev aluation and ev aluation morphisms: coev 𝑥 . . = 𝜂 ( 𝑥 ) 1 : 1 − → 𝑅 𝑥 ⊗ 𝑥 and e v 𝑥 . . = 𝜀 ( 𝑥 ) 1 : 𝑥 ⊗ 𝑅 𝑥 − → 1 . Ev aluating t he triangle identities of the adjunction at the monoidal unit yields id 𝑥 = 𝜀 ( 𝑥 ) 𝑥 ◦ ( 𝑥 ⊗ 𝜂 ( 𝑥 ) 1 ) and id 𝑅 𝑥 = ( 𝑅 𝑥 ⊗ 𝜀 ( 𝑥 ) 1 ) ◦ 𝜂 ( 𝑥 ) 𝑅 𝑥 . Ho w ev er , if 𝑅 𝑥 is to be a dual of x in t he rigid sense, the snake identities ( 2.4.4 ) must hold. For t his, w e ought to require the stronger condition t hat 𝜀 ( 𝑥 ) 𝑥 = 𝜀 ( 𝑥 ) 1 ⊗ id 𝑥 and 𝜂 ( 𝑥 ) 𝑅 𝑥 = 𝜂 ( 𝑥 ) 1 ⊗ 𝑅 𝑥 . 31 2. Prel iminari es Remark 2.69. A left closed monoidal such t hat for ev er y 𝑥 ∈ 𝒞 , the functor − ⊗ 𝑥 admits a right dual [ 𝑥 , −] ℓ as a 𝒞 -module functor — see Example 5.26 — is already rigid. Explicitl y , t his in vol v es the follo wing map natural in 𝑦 , 𝑧 ∈ 𝒞 : [ 𝑥 , 𝑧 ] ℓ ⊗ 𝑦 ∼ − → [ 𝑥 , 𝑧 ⊗ 𝑦 ] ℓ , the resulting compatibility conditions of which impl y the snake identities. W e will study t hese questions in greater details in Chapter 3 ; see also Remar k 9.39 for a characterisation of rigidity in terms of 𝒞 -module functors. Deﬁnition 2.70. A rigid monoidal categor y 𝒞 is called strict rigid if t he dual functors ∨ (−) , (−) ∨ : 𝒞 op , rev − → 𝒞 are strict and ∨  (−) ∨  = Id 𝒞 =  ∨ (−)  ∨ . Notation 2.71. Let 𝒞 be a rigid monoidal categor y ; for 𝑥 ∈ 𝒞 and 𝑛 ∈ Z , write ( 𝑥 ) 𝑛 . . =          The 𝑛 -f old left dual of 𝑥 , if 𝑛 > 0; 𝑥 , if 𝑛 = 0; The 𝑛 -f old right dual of 𝑥 , if 𝑛 < 0 . Our next result w as conjectured in [ Sch01 , Section 5], and is a slight v ari- ation of [ NS07 , Theorem 2.2]. It show s that ev er y rigid category admits a rigid strictiﬁcation ; i.e., a monoidally equiv alent strict rigid categor y . The compat- ibility betw een t he respectiv e lef t and right duality functors is an immediate consequence of t he fact that for an y strong monoidal functor 𝐹 : 𝒞 − → 𝒟 betw een rigid categories t here are natural monoidal isomor phisms 𝜙 𝑥 : 𝐹 ( ∨ 𝑥 ) ∼ − → ∨ 𝐹 𝑥 , and 𝜓 𝑥 : 𝐹 ( 𝑥 ∨ ) ∼ − → 𝐹 𝑥 ∨ , for all 𝑥 ∈ 𝒞 . Theorem 2.72. Every rigid category admits a rigid strictiﬁcation. Proof. Suppose that 𝒞 is a rigid strict monoidal category 𝒞 . Build a monoid- all y equiv alent strict rigid categor y 𝒟 as follo ws: t he objects of 𝒟 are (possibly emp ty) ﬁnite sequences ( 𝑥 𝑛 1 1 , . . . , 𝑥 𝑛 𝑖 𝑖 ) of objects 𝑥 1 , . . . , 𝑥 𝑖 ∈ 𝒞 , adorned with integers 𝑛 1 , . . . , 𝑛 𝑖 ∈ Z . T o deﬁne t he morphisms of 𝒟 , recall Notation 2.71 and set 𝒟 (( 𝑥 𝑛 1 1 , . . . , 𝑥 𝑛 𝑖 𝑖 ) , ( 𝑦 𝑚 1 1 , . . . , 𝑦 𝑚 𝑗 𝑗 )) . . = 𝒞 (( 𝑥 1 ) 𝑛 1 ⊗ · · · ⊗ ( 𝑥 𝑖 ) 𝑛 𝑖 , ( 𝑦 1 ) 𝑚 1 ⊗ · · · ⊗ ( 𝑦 𝑗 ) 𝑚 𝑗 ) . 32 2.4. Monoidal and module categories The category 𝒟 is strict monoidal when equipped wit h t he concatenation of sequences as tensor product and t he emp ty sequence as unit. By construction, there exis ts a strict monoidal equivalence of categories 𝐹 : 𝒟 − → 𝒞 , which maps an y object ( 𝑥 𝑛 1 1 , . . . , 𝑥 𝑛 𝑖 𝑖 ) ∈ 𝒟 to ( 𝑥 1 ) 𝑛 1 ⊗ · · · ⊗ ( 𝑥 𝑖 ) 𝑛 𝑖 ∈ 𝒞 , as w ell as ev ery morphism to itself; the unit of 𝒞 is t he emp ty tensor product. Deﬁne the lef t dual of an object 𝑥 . . = ( 𝑥 𝑛 1 1 , . . . , 𝑥 𝑛 𝑖 𝑖 ) ∈ 𝒟 as ∨ 𝑥 . . = ( 𝑥 𝑛 𝑖 + 1 𝑖 , . . . , 𝑥 𝑛 1 + 1 1 ) , with ev aluation and coev aluation mor phisms giv en b y 𝜙 1 𝜙 𝑛 ( 𝑥 1 ) 𝑛 1 + 1 ( 𝑥 1 ) 𝑛 1 ( 𝑥 𝑖 ) 𝑛 𝑖 + 1 ( 𝑥 𝑖 ) 𝑛 𝑖 . . . . . . . . . 𝜓 1 𝜓 𝑛 ( 𝑥 𝑖 ) 𝑛 𝑖 ( 𝑥 𝑖 ) 𝑛 𝑖 + 1 ( 𝑥 1 ) 𝑛 1 ( 𝑥 1 ) 𝑛 1 + 1 . . . . . . . . . F or all 1 ≤ 𝑘 ≤ 𝑖 , w e set 𝜙 𝑘 . . =  ev ℓ ( 𝑥 𝑘 ) 𝑛 𝑘 , if 𝑛 𝑘 ≥ 0; ev 𝑟 ( 𝑥 𝑘 ) 𝑛 𝑘 + 1 , if 𝑛 𝑘 < 0; and 𝜓 𝑘 . . =  coev ℓ ( 𝑥 𝑘 ) 𝑛 𝑘 , if 𝑛 𝑘 ≥ 0; coev 𝑟 ( 𝑥 𝑘 ) 𝑛 𝑘 + 1 , if 𝑛 𝑘 < 0 . The right dual of 𝑥 if deﬁned similarl y as 𝑥 ∨ . . = ( 𝑥 𝑛 𝑖 − 1 𝑖 , . . . , 𝑥 𝑛 1 − 1 1 ) . Hence, 𝒟 is strict rigid, which completes t he proof. □ Module functors ov er rigid categories beha v e in particular ly nice w a ys. Proposition 2.73 ([ Ost03 , Remar k 4], [ DSPS19 , Lemma 2.10]) . If 𝒞 is a left rigid monoidal, t hen every lax 𝒞 -module functor is str ong. Dually , if 𝒞 is a right rigid monoidal category , then every oplax 𝒞 -module functor is str ong. Man y applications require t hat the objects of a rigid monoidal categor y are isomor phic to their double duals in a wa y which is compatible wit h t he monoidal structure. In Corollar y 6.45 w e prov e a result for detecting this. Deﬁnition 2.74. A pivot al category is a rigid monoidal categor y 𝒞 equipped with a monoidal natural isomor phism 𝜌 : Id 𝒞 ∼ = ⇒ ∨∨ − , a pivo tal structur e of 𝒞 . Rigid monoidal categories do not ha v e to admit a piv otal structure and, if they do, it need not be unique. Examples coming from Hopf algebr a theor y are given in [ KR93 ; HK19 ; Hal21 ]. Ho w ev er , ev ery rigid monoidal categor y admits a univ ersal piv otal categor y , called its pivot al cover , see [ Shi15 ]. 33 2. Prel iminari es 2.4.4 The Drinfeld centr e Cla ssica ll y , th e c entre constructi on is used to build a braided mono- idal categor y from a monoidal one, see f or exam ple [ EGN O15 , Chap ter 7]. Throughout especiall y Chapters 4 and 5 , w e w ork in a slightly more general setting, see for example [ GNN09 ; B V12 ; HKS19 ; FH23 ; K ow24 ]. Deﬁnition 2.75. Let 𝒞 be a monoidal category , ℳ a 𝒞 -bimodule category , and 𝑚 ∈ ℳ . Analogously to Remar k 2.53 , one can show t hat 𝜎 𝑀 , 1 = id 𝑀 is implied by the hexagon axiom. A half-br aiding on 𝑚 is a natural isomorphism 𝜎 − , = : − ⊳ = = ⇒ − ⊲ = , such that for all 𝑥 , 𝑦 ∈ 𝒞 w e ha v e 𝜎 𝑀 , 1 = id 𝑀 and 𝜎 𝑚 , 𝑥 ⊗ 𝑦 = ( id 𝑥 ⊲ 𝜎 𝑚 , 𝑦 ) ◦ ( 𝜎 𝑚 , 𝑥 ⊳ id 𝑦 ) . Deﬁnition 2.76. The centr e of a 𝒞 -bimodule categor y ℳ is t he category Z ( ℳ ) deﬁned as follo ws: • Objects are pairs ( 𝑚 , 𝜎 𝑚 , − ) of an object 𝑚 ∈ ℳ and a half-braiding 𝜎 𝑚 , − . • A morphism 𝑓 : ( 𝑚 , 𝜎 𝑚 , − ) − → ( 𝑛 , 𝜎 𝑛 , − ) consists of an 𝑓 ∈ ℳ ( 𝑚 , 𝑛 ) that commutes with t he half-braidings: ( id 𝑥 ⊲ 𝑓 ) ◦ 𝜎 𝑚 , 𝑥 = 𝜎 𝑛 , 𝑥 ◦ ( 𝑓 ⊳ id 𝑥 ) , for all 𝑥 ∈ 𝒞 . There is a canonical forg etful functor 𝑈 ( 𝑀 ) : Z ( ℳ ) − → ℳ . U nlike classical representation theor y where the centre of a bimodule is a subset of t he bimodule, 𝑈 ( 𝑀 ) need not be injectiv e on objects in general. Exam ple 2.77. The centre Z ( 𝒞 ) of t he regular bimodule of a monoidal categor y 𝒞 , see Example 2.44 , is t he Drinfeld centre of 𝒞 . The tensor product is deﬁned b y ( 𝑥 , 𝜎 𝑥 , − ) ⊗ ( 𝑦 , 𝜎 𝑦 , − ) . . = ( 𝑥 ⊗ 𝑦 , 𝜎 𝑥 ⊗ 𝑦 , − ) , with 𝜎 𝑥 ⊗ 𝑦 , 𝑧 . . = ( 𝜎 𝑥 , 𝑧 ⊗ id 𝑦 ) ◦ ( id 𝑥 ⊗ 𝜎 𝑦 , 𝑧 ) , for all 𝑧 ∈ 𝒞 . The centre is braided monoidal, with braiding given b y gluing tog ether t he respectiv e half-braidings. The hexagon axioms follo w from t he deﬁnition of the half-braiding and the tensor product of Z ( 𝒞 ) . Exam ple 2.78. Let 𝐻 ∈ V ect be a ﬁnite-dimensional Hopf algebr a. Then t he category of left-lef t Y etter –Drinfeld modules of Examples 2.33 and 2.54 is, as a braided monoidal categor y , equivalent to the Drinfeld centre of the categor y 𝐻 V ect of left 𝐻 -modules, [ Dri87 ; Y et90 ]. Further , by [ Dri87 ] t here exists another Hopf algebr a 𝐷 ( 𝐻 ) , t he Drin- feld double of 𝐻 , t hat has Z ( 𝐻 V ect ) ≃ 𝐻 𝐻 𝒴𝒟 as its category of modules. W e refer t o [ Kas98 , Deﬁnition i x .4.1] f or a complete deﬁnition, and to [ Kas98 , Theorem x iii .5.1] for a proof of t he abov e equiv alence. 34 2.5. Linear and abelian categories Remark 2.79. The category of anti-Y etter – Drinf eld modules of Example 2.42 is also the category of modules ov er a certain Hopf algebra: t he anti-Drinfeld double [ CMZ97 ; Sch99 ]. In analogy to how t he anti-Y etter – Drinf eld modules are a module categor y o v er t he Y etter– Drinfeld modules, the anti-Drinfeld double is a comodule algebr a ov er the Drinfeld double. F or the next result, recall Notation 2.71 for iterated duals. Proposition 2.80 ([ JS91 , Lemma 7]) . The centr e of a (s trict) rigid category 𝒞 is (s trict) rigid: for all ( 𝑥 , 𝜎 𝑥 , − ) ∈ Z ( 𝒞 ) , we ha ve t hat 𝑈 ( 𝑍 )  ∨ ( 𝑥 , 𝜎 𝑥 , − )  = ∨ 𝑥 and 𝑈 ( 𝑍 )  ( 𝑥 , 𝜎 𝑥 , − ) ∨  = 𝑥 ∨ . Moreov er , for every 𝑛 ∈ Z and 𝑥 ∈ Z ( 𝒞 ) , the equality 𝜎 ( 𝑥 ) 𝑛 , ( 𝑦 ) 𝑛 = ( 𝜎 𝑥 , 𝑦 ) 𝑛 holds for all 𝑦 ∈ Z ( 𝒞 ) . 2 . 5 l i n e a r a n d a b e l i a n c at e g o r i e s We s hall p a y sp ecia l attent ion to k -linear categories, for some ﬁeld k . Indeed, k -linear functor categories encompass man y phenomena in repres- entation theor y ; see for example Section 3.2 and Chapter 9 . W e shall denote the categor y k -v ector spaces b y V ect , and write vect for t he full subcategory of ﬁnite-dimensional k -v ector spaces. 3 3 This will be a gener al theme throughout the thesis: whenev er there exists a full subcategory of “ﬁnite-dimensional objects” inside of a lager category of all objects, t he ﬁnite-dimensional subcategory will start with a lo w er case letter , and w e use a capital for the larg er category . Deﬁnition 2.81. A k -linear category is a categor y enriched in V ect Exam ple 2.82. Let 𝒞 be an ordinary category . Its linearisation k 𝒞 has t he same objects, and t he space of mor phisms betw een tw o objects 𝑎 , 𝑏 ∈ k 𝒞 is k 𝒞 ( 𝑎 , 𝑏 ) . . = span k 𝒞 ( 𝑎 , 𝑏 ) . T o deﬁne t he composition in k 𝒞 , note that for any 𝑎 , 𝑏 , 𝑐 ∈ k 𝒞 there is a unique linear map − ◦ = : k 𝒞 ( 𝑏 , 𝑐 ) ⊗ k k 𝒞 ( 𝑎 , 𝑏 ) − → k 𝒞 ( 𝑎 , 𝑐 ) , such that 𝑔 ◦ 𝑓 = 𝑔 𝑓 for all 𝑔 ∈ 𝒞 ( 𝑏 , 𝑐 ) and 𝑓 ∈ 𝒞 ( 𝑎 , 𝑏 ) . Let 𝜄 : 𝒞 − → k 𝒞 be t he functor that is t he identity on objects and maps an y mor phism to t he corresponding basis v ector . For an y functor 𝐹 : 𝒞 − → 𝒟 whose codomain is a k -linear categor y , w e obtain a commuting triangle: 𝒞 𝒟 k 𝒞 𝐹 𝜄 ∃ ! 𝐺 ( linear ) As a consequence, if w e endow t he ordinary functor categor y [ 𝒞 , V ect ] with the pointwise k -linear structure, w e obtain an isomor phism of linear categor- ies betw een it and t he categor y of k -linear functors from k 𝒞 to V ect . 35 2. Prel iminari es 2.5.1 F inite and locally ﬁnite abelian categories In th is s ecti on , we assume all categories and functors to be k -linear , for some ﬁeld k . W e call a functor 𝐹 : 𝒜 − → ℬ betw een abelian categories 𝒜 and ℬ left exact if it preser v es ﬁnite limits, right exact if it preser v es ﬁnite colimits, and exact if it is right and left exact. The categor y of right exact functors from 𝒜 to ℬ will be denoted b y Re x ( 𝒜 , ℬ ) , and w e write Le x ( 𝒜 , ℬ ) for the categor y of left exact functors. Further , let 𝒜 -proj denote the full subcategory of 𝒜 consisting of its projectiv e objects, and similarl y for 𝒜 -inj. W e recall basic properties of projective and injectiv e objects in this setting. Proposition 2.83 ([ GKKP22 , Lemma 3.3]) . Let 𝒜 and ℬ be abelian cat egories, and assume t hat ℬ has enough projectiv es. Let 𝜄 : 𝒜 -proj ↩ − → 𝒜 be t he inclusion functor . Then the r estriction funct or Re x ( 𝒜 , ℬ ) −◦ 𝜄 − − → [ 𝒜 -proj , ℬ ] is an equivalence. Proposition 2.84 ([ GKKP22 , Lemma 3.3]) . Let 𝒜 and ℬ be abelian categories, and assume t hat ℬ has enough injectives. Let 𝜄 : 𝒜 -inj ↩ − → 𝒜 be the inclusion functor . Then Lex ( 𝒜 , ℬ ) −◦ 𝜄 − − → [ 𝒜 -inj , ℬ ] is an equiv alence. W e refer to [ EGN O15 , Chapter 1] for t he follo wing deﬁnitions and results. An object of a k -linear category is said to be simple if an y non-zero endo- morphism t hereof is an isomor phism. A category 𝒞 is said to be semisimple if an y of its objects is a direct sum of simple objects. Proposition 2.85 ([ TV17 , Theorem C.6]) . A semisimple category is abelian. Any epimorphism or monomor phism in a semisim ple category splits. In particular , any functor from a semisimple category to an abelian category is exact. An object 𝑉 in an abelian categor y 𝒜 has ﬁnite length if t here is a ﬁltration 0 = 𝑉 0 ⊆ 𝑉 1 ⊆ · · · ⊆ 𝑉 𝑛 = 𝑉 , such that 𝑉 𝑖 / 𝑉 𝑖 − 1 is simple, for all 𝑖 ∈ { 1 , . . . , 𝑛 } . Deﬁnition 2.86. A categor y 𝒞 is said to be ﬁnite abelian if it is abelian, hom- ﬁnite 4 , has enough projectiv es, onl y ﬁnitel y many isomorphism classes of 4 A ( k -linear) category 𝒞 is called hom-ﬁnite if the hom space 𝒞 ( 𝑥 , 𝑦 ) is ﬁnite-dimensional for all 𝑥 , 𝑦 ∈ 𝒞 . simple objects, and all of its objects are of ﬁnite length. Lemma 2.87. A category 𝒜 is ﬁnite abelian if and only if ther e is a ﬁnite-dimensional k -alg ebra 𝐴 suc h that 𝒜 ≃ 𝐴 -mod . 36 2.5. Linear and abelian categories Deﬁnition 2.88. A categor y is said to be locally ﬁnite abelian if it is abelian, hom-ﬁnite, and all of its objects are of ﬁnite length. Lemma 2.89. A cat egor y 𝒞 is locall y ﬁnite abelian if and only if ther e exists a k -coalg ebra 𝐶 such that 𝒞 ≃ 𝐶 vect . Locall y ﬁnite abelian categories satisfy a Krull – Schmidt-type theorem. Proposition 2.90 ([ EGN O15 , Theorem 1.5.7]) . Every object of ﬁnite length admits a unique decomposition into a direct sum of indecomposable objects. Deﬁnition 2.91. An abelian categor y 𝒞 that satisﬁes Proposition 2.90 is called a Krull –Schmidt category . General abelian monoidal categories admit a v er y useful special case of Beck’ s monadicity theor em , see for example [ BZBJ18 , Section 4.1]. Theorem 2.92. Let 𝐹 : 𝒜 ⇄ ℬ : 𝑈 be an adjunction of abelian categories. If 𝑈 is right e xact and reﬂects zero objects 5 , t he comparison functor 𝐾 : ℬ − → 𝒜 𝑈 𝐹 is an 5 A functor 𝐹 reﬂects zero objects if 𝐹 𝑥  0 = ⇒ 𝑥  0 . equiv alence. Likewise, if 𝐹 is lef t exact and reﬂects zero objects then the comparison functor 𝐾 : 𝒜 − → ℬ 𝐹𝑈 is an equiv alence. 2.5.2 T ensor and ring categories W e refer t he reader to [ EGN O15 , Chapter 4] for a comprehensiv e account of the t heory of (multi)tensor and ring categories. All categories and functors in this section are again assumed to be k -linear . Deﬁnition 2.93. • A tensor category is a locall y ﬁnite abelian rigid monoidal category 𝒞 such that End 𝒞 ( 1 )  k . • A ﬁnite tensor category is a ﬁnite abelian rigid monoidal categor y 𝒞 such that End 𝒞 ( 1 )  k . • A ring category is a locally ﬁnite abelian separately exact monoidal category 𝒞 such t hat End 𝒞 ( 1 )  k . • A ﬁnite ring category is a ﬁnite abelian separatel y exact monoidal category 𝒞 such that End 𝒞 ( 1 )  k . 37 2. Prel iminari es Despite apparent similarity in terminology , t he notion of a ring categor y is not related to t hat of a rig categor y , such as t hose considered in [ JY21 ]. Deﬁnition 2.94. Let 𝒞 be a ring category . A ﬁbre functor for 𝒞 is a f ait hful and exact monoidal functor 𝑈 : 𝒞 − → v ect . The next result is a v ariant of T annaka –Krein duality for ring categories. Theorem 2.95 ([ EGN O15 , Theorem 5.4.1]) . Let 𝒞 be a ring category , and assume it admits a ﬁbre functor 𝑈 : 𝒞 − → vect . Then ther e exis ts a bialg ebra 𝐵 such that t here is a monoidal equivalence 𝐾 𝐵 : 𝒞 − → 𝐵 -comod and a monoidal natural iso- morphism 𝑈 𝐵 ◦ 𝐾 𝐵  𝑈 , where 𝑈 𝐵 : 𝐵 -comod − → v ect is the for g etful functor . F urther , 𝐵 is uniq ue up to isomorphism, Hopf if and onl y if 𝒞 is a tensor category , and ﬁnite-dimensional if and onl y if 𝒞 is a ﬁnite ring category . 2 . 6 a lg e b r a a n d m o d u l e o b j e c t s Let 𝒞 be a monoi d a l category . An algebr a object in 𝒞 consists of an object 𝐴 ∈ 𝒞 tog ether with a multiplication 𝜇 : 𝐴 ⊗ 𝐴 − → 𝐴 and a unit 𝜂 : 1 − → 𝐴 , satisfying associativity and unitality axioms analogous to those for a k -algebr a. In fact, a k -algebr a is the same as an algebra object in V ect . A coalg ebra object in 𝒞 is an object 𝐶 ∈ 𝒞 equipped with a comultiplication Δ : 𝐶 − → 𝐶 ⊗ 𝐶 and a counit morphism 𝜀 : 𝐶 − → 1 , satisfying coassociativity and counitality axioms, such that 𝐶 becomes an algebra object in 𝒞 op . Exam ple 2.96. An object 𝐴 in a monoidal categor y 𝒞 is an algebra object in 𝒞 if and onl y if − ⊗ 𝐴 : 𝒞 − → 𝒞 is a monad. Exam ple 2.97. Let 𝒞 be a monoidal categor y , and suppose t hat 𝑥 ∈ 𝒞 has a left dual ∨ 𝑥 ∈ 𝒞 . Then this induces an algebra object ( 𝑥 ⊗ ∨ 𝑥 , 𝑥 ⊗ ev ℓ 𝑥 ⊗ ∨ 𝑥 : 𝑥 ⊗ ∨ 𝑥 ⊗ 𝑥 ⊗ ∨ 𝑥 − → 𝑥 ⊗ ∨ 𝑥 , coev ℓ 𝑥 : 1 − → 𝑥 ⊗ ∨ 𝑥 ) and, duall y , a coalgebr a object ( ∨ 𝑥 ⊗ 𝑥 , ∨ 𝑥 ⊗ coev ℓ 𝑥 ⊗ 𝑋 , ev ℓ 𝑥 ) in 𝒞 . A special case of Example 2.97 is when w e consider t he monoidal category [ 𝒞 , 𝒞 ] of endofunctors. Then a left dual of some 𝑈 ∈ [ 𝒞 , 𝒞 ] is a lef t adjoint, with ev ℓ 𝑈 and coev ℓ 𝑈 giving the coev aluation and ev aluation, respectiv ely . the resulting algebra and coalgebr a structures are t he monad and comonad corresponding to t he adjunction. 38 2.6. Algebr a and module objects Deﬁnition 2.98. Let ( ℳ , ⊲ ) be a left 𝒞 -module categor y and ( 𝐴 , 𝜇 , 𝜂 ) an algebr a object in 𝒞 . A left 𝐴 -module in ℳ is an object 𝑀 ∈ ℳ tog ether wit h an action morphism 𝛼 : 𝐴 ⊲ 𝑀 − → 𝑀 such that t he follo wing diagrams commute ( 𝐴 ⊗ 𝐴 ) ⊲ 𝑀 𝐴 ⊲ ( 𝐴 ⊲ 𝑀 ) 𝐴 ⊲ 𝑀 1 ⊲ 𝑀 𝑀 𝐴 ⊲ 𝑀 𝑀 𝐴 ⊲ 𝑀 ℳ a 𝜇 ⊲ 𝑚 𝐴 ⊲ 𝛼 𝛼 ℳ u 𝜂 ⊲ 𝑀 𝛼 𝛼 A morphism of modules is an arrow 𝑓 : 𝑀 − → 𝑁 in ℳ commuting wit h t he respectiv e actions in t he sense t hat 𝑓 ◦ 𝛼 𝑚 = 𝛼 𝑛 ◦ ( 𝐴 ⊲ 𝑓 ) . Exam ple 2.99. Clearl y , left 𝐴 -modules in ℳ and their mor phisms form a category , which w e shall denote by 𝐴 - Mod ℳ . The obvious forg etful functor 𝐴 -For get ℳ : 𝐴 - Mod ℳ − → ℳ admits a lef t adjoint 𝐴 -Free ℳ : ℳ − → 𝐴 - Mod ℳ 𝑀 ↦− →  𝐴 ⊲ 𝑀 , 𝐴 ⊲ ( 𝐴 ⊲ 𝑀 ) ℳ − 1 a − − − → ( 𝐴 ⊗ 𝐴 ) ⊲ 𝑀 𝜇 ⊲ 𝑀 − − − − → 𝐴 ⊲ 𝑀  The unit of this adjunction is giv en by 𝑀 ℳ u − − − → 1 ⊲ 𝑀 𝜂 ⊲ 𝑀 − − − → 𝐴 ⊲ 𝑀 , and t he counit is 𝛼 𝑀 : 𝐴 ⊲ 𝑀 − → 𝑀 . Giv en a coalgebra object 𝐶 in 𝒞 , a left 𝐶 -comodule in ℳ is an object 𝑁 ∈ 𝒞 satisfying axioms dual to t hose for a module in ℳ , so that a lef t 𝐶 -comodule in ℳ is the same as a lef t 𝐶 -module in the 𝒞 op -module category ℳ op . Comodule morphisms are deﬁned similarl y . W e denote the category of 𝐶 -comodules in ℳ b y 𝐶 - Comod ℳ . There is an adjunction analogous to t hat for modules: 𝐶 -Coforg et ℳ : ℳ ⇄ 𝐶 - Comod ℳ : 𝐶 -Cofree ℳ . Similar ly , one can deﬁne right modules and comodules in a right module category , and bimodules and bicomodules in a bimodule categor y . Deﬁnition 2.100. Let 𝐴 be an algebra in 𝒞 . A (lef t) 𝐴 -comodule algebr a is an algebr a 𝐶 , together with a left 𝐴 -comodule structure 𝜌 : 𝐶 − → 𝐴 ⊗ 𝐶 , such that 𝜌 is a mor phism of 𝐴 -algebr as. Analogousl y to Deﬁnition 2.100 one deﬁnes right comodule algebr as, as w ell as comodule coalgebr as and module algebr as. 39 2. Prel iminari es Notation 2.101. If the underl ying categor y is t he categor y of v ector spaces, then w e use the following notation, to diﬀerentiate it from t he gener al case: 𝐶 V ect . . = 𝐶 - Comod V ect . for t he category of lef t 𝐶 -comodules of a coalgebr a 𝐶 . Analogousl y , w e for example deﬁne 𝐴 V ect , 𝐶 V ect 𝐴 ; or 𝐶 vect , 𝐴 vect , and 𝐵 𝐵 vect 𝐵 𝐵 for t he ﬁnite-dimensional case. Remark 2.102. Let 𝐴 be an algebr a object in 𝒞 and let ℳ and 𝒩 be left 𝒞 - module categories. A lax 𝒞 -module functor 𝐹 : 𝒩 − → ℳ can be lif ted to 𝐴 - Mod 𝐹 : 𝐴 - Mod 𝒩 − → 𝐴 - Mod ℳ on t he respectiv e categories of 𝐴 -modules by sending 𝑁 ∈ 𝐴 - Mod 𝒩 to the lef t 𝐴 -module 𝐹 𝑁 , whose action is giv en by 𝛼 𝐹 𝑁 : 𝐴 ⊲ ℳ 𝐹 𝑁 𝐹 a; 𝐴 ,𝑁 − − − − → 𝐹 ( 𝐴 ⊲ 𝒩 𝑁 ) 𝐹 𝛼 𝑁 − − − → 𝐹 𝑁 . This functor is a lif t of 𝐹 — it satisﬁes the follo wing relation: 𝐴 - Mod 𝒩 𝐴 - Mod ℳ 𝒩 ℳ 𝐴 - Mod 𝐹 𝐴 -Forg et 𝒩 𝐴 -Forg et ℳ 𝐹 Exam ple 2.103. Let 𝒞 be a cocomplete symmetric monoidal categor y , such that its is right exact in both v ariables. Then one ma y deﬁne t he bicategory B imod ( 𝒞 ) of bimodules in 𝒞 as follo ws: • Objects are algebr a objects in 𝒞 . • F or 𝐴 and 𝐵 in 𝒞 , t he hom-categor y B imod ( 𝒞 )( 𝐴 , 𝐵 ) is giv en b y 𝐵 - 𝐴 - bimodules in 𝒞 and their homomor phisms. Horizontal com position is giv en by the balanced tensor product of bimodules. Analogousl y to Example 2.103 , there is a bicategory B icomod ( 𝒞 ) of 𝒞 - bicomodules, for 𝒞 a complete symmetric monoidal categor y , wit h t he tensor product being left exact in both variables. Deﬁnition 2.104. A categor y ℐ is ﬁlter ed if ev er y ﬁnite diagram has a cocone. 40 2.6. Algebr a and module objects Equiv alentl y , Deﬁnition 2.104 could be stated in t he follo wing w a y: a category ℐ is ﬁltered if t he follo wing tw o conditions hold: • for all 𝑥 , 𝑦 ∈ ℐ there exist arrow s 𝑘 𝑥 : 𝑥 − → 𝑘 and 𝑘 𝑦 : 𝑦 − → 𝑘 in ℐ ; and • for all parallel arrow s 𝑓 , 𝑔 : 𝑥 − → 𝑦 in ℐ there exists a 𝑘 𝑓 𝑔 : 𝑦 − → 𝑘 , such that 𝑘 𝑓 𝑔 ◦ 𝑔 = 𝑘 𝑓 𝑔 ◦ 𝑓 . Deﬁnition 2.105. A diag r am 𝐹 : ℐ − → 𝒞 is called ﬁlter ed if t he domain ℐ is a ﬁltered categor y . A ﬁlter ed colimit is a colimit of a ﬁltered diag r am, and a functor is called ﬁnitary if it preser v es ﬁltered colimits. Ordinaril y , limits and colimits do not commute. For example, for sets 𝑋 , 𝑌 , 𝑊 , and 𝑉 , t he canonical morphism ( 𝑋 × 𝑌 )  ( 𝑊 × 𝑉 ) − → ( 𝑋  𝑉 ) × ( 𝑌  𝑉 ) is not isomor phism in gener al. How ev er , it tur ns out that ﬁltered colimits do commute with ﬁnite limits, see for example [ ML98 , Theorem i x .2.1]. Proposition 2.106. F ilter ed colimits commute with ﬁnite limits in Set : given a functor 𝐹 : 𝒥 × ℐ − → Set of two diagram categories, where 𝒥 is ﬁnit e and ℐ is small and ﬁlter ed, the following canonical mor phism is an isomorphism: colim 𝑖 lim 𝑗 𝐹 ( 𝑗 , 𝑖 ) ∼ − → lim 𝑗 colim 𝑖 𝐹 ( 𝑗 , 𝑖 ) . Lemma 2.107. Let 𝒜 and ℬ be additive categories. F inite bipr oducts endow 𝒜 and ℬ with the structur e of vect -module categories. Any functor 𝐹 : 𝒜 − → ℬ is a vect -module functor with respect to these vect -actions. If ℬ admits ﬁlter ed colimits, 𝐹 ext ends essentially uniq uel y to a ﬁnitary V ect - module functor Ind ( 𝒜 ) − → ℬ , and any ﬁnitary V ect -module functor is of this form. Proof. Let add ( 𝒜 ) denote the cocompletion of 𝒜 under ﬁnite biproducts — the additive closure of 𝒜 . As ﬁnite biproducts are absolute colimits, the left ad- joint add ( 𝒜 ) − → 𝒜 of the canonical inclusion 𝒜 ↩ − → add ( 𝒜 ) , coming from the univ ersal property of Equation ( 2.9.1 ), is an equivalence of vect -module categories. Hence, so is t he inclusion 𝒜 ↩ − → add ( 𝒜 ) . W e ﬁnd the equiv alences Cat k ( 𝒜 , −) ≃ v ect -Mod ( add ( 𝒜 ) , −) ≃ vect -Mod ( 𝒜 , −) , establishing t he ﬁrst claim. The second claim follo ws in a similar wa y , using that V ect ≃ Ind ( vect ) and vect -Mod ( 𝒜 , ℬ ) ≃ V ect -Mod ﬁlt ( Ind ( 𝒜 ) , ℬ ) . □ 41 2. Prel iminari es Exam ple 2.108. Let 𝐶 be a coalgebr a in V ect . Giv en tw o 𝐶 -comodules ( 𝑋 , 𝜌 ) and ( 𝑌 , 𝜆 ) , one ma y form their cot ensor product — the v ector space 𝑋 □ 𝑌 giv en b y the equaliser 𝑋 □ 𝑌 𝑋 ⊗ k 𝑌 𝑋 ⊗ k 𝐶 ⊗ k 𝑌 . 𝜌 ⊗ k 𝑌 𝑋 ⊗ k 𝜆 Sometimes, t he formalism of module categories — ev en t hose ov er t he seemingl y trivial monoidal categories vect and V ect — provides additional clarity to classical statements about algebraic and coalgebr aic k -linear struc- tures. W e giv e a brief proof of the result below as an example of t his. Proposition 2.109 ([ T ak77 , Proposition 2.1]) . Let 𝒟 be an abelian category ad- mitting ﬁlter ed colimits, and let 𝐶 be a coalgebr a over k . There is an equivalence Le xf ( 𝐶 V ect , 𝒟 ) ∼ − → Comod 𝒟 𝐶 𝐹 ↦− → 𝐹 𝐶 𝑁 □ − ← − [ 𝑁 , wher e t he left-hand side is the category of lef t exact ﬁnitary functor s fr om 𝐶 V ect , and t he right-hand side — f ollowing t he not ation of Example 2.99 — deno tes the category of right 𝐶 -comodules in 𝒟 . Her e, 𝒟 is endowed with the V ect -module structur e described in Proposition 2.131 . In particular , for a coalg ebra 𝐷 over k , we have Le xf ( 𝐶 V ect , 𝐷 V ect ) ≃ 𝐷 V ect 𝐶 . Proof. Since 𝐶 V ect ≃ Ind ( 𝐶 vect ) , the functor 𝐹 can be seen as induced to ﬁltered colimits from its res triction to 𝐶 vect , and as such is a V ect -module functor , follo wing Lemma 2.107 . Obser v e that 𝐶 , being a bicomodule o v er itself, is a right 𝐶 -comodule in 𝐶 V ect . Thus, 𝐹 𝐶 is a right 𝐶 -comodule in 𝒟 . T o see t hat 𝐹  𝐹 𝐶 □ − , let 𝑁 ∈ 𝐶 V ect and recall that t he bar construction pro vides a functorial injectiv e resolution 𝑁  eq ( 𝐶 ⊗ 𝑁 𝐶 ⊗ 𝐶 ⊗ 𝑁 Δ 𝐶 ⊗ 𝑁 𝐶 ⊗ Δ 𝑁 ) . Thus, 𝐹 𝑁  𝐹  eq ( 𝐶 ⊗ 𝑁 𝐶 ⊗ 𝐶 ⊗ 𝑁 Δ 𝐶 ⊗ 𝑁 𝐶 ⊗ Δ 𝑁 )   eq  𝐹 ( 𝐶 ⊗ 𝑁 ) 𝐹 ( 𝐶 ⊗ 𝐶 ⊗ 𝑁 ) 𝐹 ( Δ 𝐶 ⊗ 𝑁 ) 𝐹 ( 𝐶 ⊗ Δ 𝑁 )   eq  𝐹 𝐶 ⊗ 𝑁 𝐹 𝐶 ⊗ 𝐶 ⊗ 𝑁 𝐹 ( Δ 𝐶 )⊗ 𝑁 𝐹 𝐶 ⊗ Δ 𝑁  = 𝐹 𝐶 □ 𝑁 , 42 2.7. Monoidal bicategories where t he ﬁrst isomor phism is using the bar construction, the second follow s from left exactness of 𝐹 , the t hir d uses the fact t hat 𝐹 is a right V ect -module functor , and t he fourth follo ws by obser ving t hat 𝐹 ( Δ 𝐶 ) = Δ 𝐹 ( 𝐶 ) . □ 2 . 7 m o n o i da l b i c at e g o r i e s We brie fl y recal l t he noti on of a m onoid al b ica t egory , which will pla y an im portant role in t he string diag r ams used t hroughout. More detailed accounts are giv en [ JY21 , Chatper 12], see also [ GPS95 ; Gur06 ; GS16 ]. Deﬁnition 2.110. A monoidal bicat egor y consists of a bicategor y B tog ether with t he follo wing data: • A pseudofunctor ⊠ : B × B − → B . • A pseudofunctor 1 B : ♥ − → B . The image of t he unique object ♥ in B will be referred to as the identity object of B , and, abusing notation, also denoted b y 1 B . The pseudofunctoriality of 1 B yields additional structure: a 1-mor phism I B : 1 B − → 1 B and in v ertible 2-morphisms ∇ I : I B ◦ I B ∼ − → I B and 𝜂 I : Id 1 B ∼ − → I B , such t hat ( 1 B , ∇ I , 𝜂 I ) form an idempo tent monad in B . • An adjoint pseudonatural equiv alence ( A , A ♦ , 𝜂 A , 𝜀 A ) : B × B × B B × B B × B B ⊠ × B B × ⊠ A ⊠ ⊠ which will be referred to as t he associator for B . Its components are 1-morphisms of t he form A 𝒜 , ℬ , 𝒞 : ( 𝒜 ⊠ ℬ ) ⊠ 𝒞 ∼ − → 𝒜 ⊠ ( ℬ ⊠ 𝒞 ) . • Pseudonatural adjoint equiv alences ( L , L ♦ , 𝜂 L , 𝜀 L ) and ( R , R ♦ , 𝜂 R , 𝜀 R ) , referred to as lef t and right unitors , respectiv ely : B B × B B ⊠ id B × 1 B 1 B × id B L R Its components are 1-mor phisms of B of the f orm R : 1 B ⊠ 𝒜 ∼ − → 𝒜 and L : 𝒜 ⊠ 1 B ∼ − → 𝒜 , for 𝒜 ∈ Ob B . 43 2. Prel iminari es • An inv ertible modiﬁcation p , as indicated by the components of p dis- pla y ed below , where 𝒜 , ℬ , 𝒞 , 𝒟 ∈ Ob B : ( 𝒟 ⊠ ( 𝒞 ⊠ ℬ )) ⊠ 𝒜 𝒟 ⊠ (( 𝒞 ⊠ ℬ ) ⊠ 𝒜 ) (( 𝒟 ⊠ 𝒞 ) ⊠ ℬ ) ⊠ 𝒜 𝒟 ⊠ ( 𝒞 ⊠ ( ℬ ⊠ 𝒜 )) ( 𝒟 ⊠ 𝒞 ) ⊠ ( ℬ ⊠ 𝒜 ) p 𝒟 , 𝒞 , ℬ , 𝒜 (2.7.1) The arrow s in t he diag r am are formed using t he associator A as indicated. W e refer to p as t he pentagonat or . • In v ertible modiﬁcations l , m , and r ; t he lef t, middle, and right 2-unitor s : ( ℬ ⊠ 1 B ) ⊠ 𝒜 ℬ ⊠ ( 1 B ⊠ 𝒜 ) ( ℬ ⊠ 𝒜 ) ⊠ 1 B ℬ ⊠ ( 𝒜 ⊠ 1 B ) ℬ ⊠ 𝒜 ℬ ⊠ 𝒜 ℬ ⊠ 𝒜 ( 1 B ⊠ ℬ ) ⊠ 𝒜 1 B ⊠ ( ℬ ⊠ 𝒜 ) ℬ ⊠ 𝒜 A ℬ , 1 B , 𝒜 ℬ ⊠ L 𝒜 A ℬ , 𝒜 , 1 B R ♦ ⊠ 𝒜 R ♦ ℬ ⊠ 𝒜 ℬ ⊠ R ♦ 𝒜 A 1 B , ℬ , 𝒜 L ℬ ⊠ 𝒜 L ℬ ⊠ 𝒜 m ℬ , 𝒜 r ℬ , 𝒜 l ℬ , 𝒜 This data is subject to coherence axioms, see [ JY21 , Explanation 12.1.3]. Exam ple 2.111. The 2-categor y C at k of k -linear categories, k -linear functors, and natural transf ormations betw een t hem is a monoidal bicategory when endo w ed with t he tensor product of k -linear categories: giv en 𝒜 and ℬ , deﬁne the k -linear categor y 𝒜 ⊗ k ℬ b y • Ob ( 𝒜 ⊗ k ℬ ) . . = Ob 𝒜 × Ob ℬ , • ( 𝒜 ⊗ k ℬ )(( 𝑎 , 𝑏 ) , ( 𝑎 ′ , 𝑏 ′ )) . . = 𝒜 ( 𝑎 , 𝑎 ′ ) ⊗ k ℬ ( 𝑏 , 𝑏 ′ ) . Giv en k -linear functors 𝐹 : 𝒜 → ℬ and 𝐺 : 𝒞 → 𝒟 , t he tensor product 𝐹 ⊗ k 𝐺 is deﬁned b y Ob ( 𝐹 ⊗ k 𝐺 ) = Ob ( 𝐹 ) × Ob ( 𝐺 ) and ( 𝐹 ⊗ k 𝐺 ) ( 𝑎 , 𝑏 ) , ( 𝑐 , 𝑑 ) : 𝒜 ( 𝑎 , 𝑐 ) ⊗ k ℬ ( 𝑏 , 𝑑 ) 𝐹 𝑎 , 𝑐 ⊗ k 𝐺 𝑏 , 𝑑 − − − − − − − → 𝒞 ( 𝐹 𝑎 , 𝐹 𝑐 ) ⊗ k 𝒟 ( 𝐺 𝑏 , 𝐺 𝑑 ) . 44 2.7. Monoidal bicategories The tensor product of natural transf ormations extends similarl y . The associators and unitors are lifted from Set and V ect , and t he pentagon- ators and 2 -unitors are trivial. F or example, giv en k -linear categories 𝒜 , ℬ , and 𝒞 , the associator A 𝒜 , ℬ , 𝒞 is the equivalence ( 𝒜 ⊗ k ℬ ) ⊗ k 𝒞 ∼ − → 𝒜 ⊗ k ( ℬ ⊗ k 𝒞 ) (( 𝑎 , 𝑏 ) , 𝑐 ) ↦− → ( 𝑎 , ( 𝑏 , 𝑐 )) ( 𝑓 ⊗ k 𝑔 ) ⊗ k ℎ ↦− → 𝑓 ⊗ k ( 𝑔 ⊗ k ℎ ) . Diagram ( 2.7.1 ) commutes in this case, and so t he pentagonator ma y be chosen to be t he identity . Generalising the abov e exam ple, f or a symmetric closed monoidal categor y ( 𝒱 , ⊗) , t he bicategory 𝒱 - C at is monoidal, with t he monoidal structure 𝒜 ⊗ ℬ deﬁned analogousl y to 𝒜 ⊗ k ℬ in the case of 𝒱 . . = V ect . Exam ple 2.112. The bicategor y B imod ( 𝒞 ) of Example 2.103 is a monoidal bicategory wit h respect to the under lying tensor product of 𝒞 . T ensoring bimodules 𝐵 𝑀 𝐴 and 𝐷 𝑁 𝐶 is giv en b y t he ( 𝐵 ⊗ 𝐷 ) - ( 𝐴 ⊗ 𝐶 ) -bimodule 𝐵 𝑀 𝐴 ⊗ 𝐷 𝑁 𝐶 , and this extends to bimodule homomorphisms in t he obvious w a y . F or details on the follo wing example, together with its more general v ariant where 𝒱 is only required to be braided, see [ DS97 , Section 7]. Exam ple 2.113. The bicategor y P rof k has as objects k -linear categories, and the hom-categories are giv en b y P rof k ( 𝒞 , 𝒟 ) . . = C at k ( 𝒟 op ⊗ k 𝒞 , V ect ) . W e call 𝐹 ∈ P rof k ( 𝒞 , 𝒟 ) a profunct or , and write 𝐹 : 𝒞 − ↦ → 𝒟 . Composition of profunctors 𝐹 : 𝒞 − ↦ → 𝒟 and 𝐺 : 𝒟 − ↦ → ℰ is giv en by the coend ( 𝐺 ⋄ 𝐹 )( 𝑒 , 𝑐 ) . . =  𝑑 ∈ 𝒟 𝐺 ( 𝑒 , 𝑑 ) ⊗ k 𝐹 ( 𝑑 , 𝑐 ) . Further , P rof k is monoidal when endow ed t he monoidal structure ⊗ k . On objects, it coincides wit h t hat in Example 2.111 : giv en categories 𝒜 and ℬ , their tensor product is 𝒜 ⊗ k ℬ . Giv en profunctors Φ : 𝒜 − ↦ → 𝒜 ′ and Ψ : ℬ − ↦ → ℬ ′ , the profunctor Φ ⊗ k Ψ : 𝒜 ⊗ k 𝒜 ′ − ↦ → ℬ ⊗ k ℬ ′ is deﬁned via ( Φ ⊗ k Ψ )(( 𝑏 , 𝑏 ′ ) , ( 𝑎 , 𝑎 ′ )) = Φ ( 𝑎 ′ , 𝑎 ) ⊗ k Ψ ( 𝑏 ′ , 𝑏 ) . This extends similar l y to natural transf ormations betw een profunctors. 45 2. Prel iminari es 2.7.1 S tring diagrams in monoidal bicategories Thi s s ecti on ex tend s t he gr aphic al l angu a ge of Section 2.3 b y incor por - ating t he tensor product of C at , which turns it into a monoidal 2-category . In our presentation, w e closely f ollow W illerton [ W il08 ], see [ DS03 ; Str03 ; BMS24 ; DS25 ] for other examples of “sheet” or “surface” diagrams. As before, w e consider strings and v ertices betw een them. These are labelled with funct ors and natural transf ormations, respectiv ely . The strings and v ertices are embedded into bounded rectangles, which w e will call sheets. Each (connected) region of a sheet is decorated with a categor y . The same mechanics as for ordinary 2-dimensional string diag r ams appl y: horizontal and v ertical gluing represents com position of functors and natural transf ormations. On top of t hese operations, w e add stacking sheets behind each other to depict t he monoidal product of C at . Our con vention is to read diagrams from front to back, right to left, and bottom to top. Giv en a monoidal categor y 𝒞 , tw o of t he most vital building blocks in this new g r aphical languag e are its tensor product ⊗ : 𝒞 × 𝒞 − → 𝒞 and unit 1 ∈ 𝒞 ; they are shown in F igure 2.2 . On t he left, t here are tw o sheets equating ⊗ def = 𝒞 𝒞 1 1 𝒞 𝒞 𝒞 𝒞 𝒞 𝒞 𝒞 1 Figure 2.2: Graphical representation of the tensor product and unit of a monoidal categor y ( 𝒞 , ⊗ , 1 ) . to tw o copies of 𝒞 joined b y a line: t he tensor product of 𝒞 . On t he right, w e ha v e the unit of 𝒞 considered as a functor from t he terminal categor y to 𝒞 . W e represent 1 by the empty sheet, and the unit of 𝒞 by a dashed line. Exam ple 2.114. Consider an oplax monoidal functor ( 𝐹 , 𝐹 2 , 𝐹 0 ) : 𝒞 − → 𝒟 . Figure 2.3 depicts t he comultiplication 𝐹 2 as a “time ev olution”, where w e start wit h 𝐹 (− ⊗ = ) on t he bottom, and end up with 𝐹 (−) ⊗ 𝐹 ( = ) at t he top. The coassociativity and counitality conditions are depicted in Figure 2.4 . 46 2.7. Monoidal bicategories 𝐹 𝒟 𝒞 𝒞 𝐹 𝑡 𝑦 𝑥 𝒞 𝒞 𝐹 𝐹 𝒞 𝒞 𝒟 𝒟 𝐹 2 𝐹 𝒟 ⊗ ⊗ 𝐹 𝒟 𝒞 Figure 2.3: The comultiplication of an oplax monoidal functor 𝐹 : 𝒞 − → 𝒟 . 𝐹 𝐹 𝐹 𝐹 = = 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 𝐹 = 𝐹 𝐹 Figure 2.4: Coassociativity and counitality conditions of an oplax monoidal functor . Remark 2.115. Giv en tw o oplax monoidal functors 𝐹 , 𝐺 : 𝒞 − → 𝒟 , the string diagrammatic conditions for a natural transf ormation 𝜂 : 𝐹 = ⇒ 𝐺 to be one of oplax monoidal functors look like 𝐹 𝐺 𝐺 = 𝐹 𝐺 𝐺 𝐹 𝐹 = , Exam ple 2.116. Suppose t hat 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 is an oplax monoidal adjunction. In string diagrams, the conditions t hat t he unit 𝜂 and the counit 𝜀 are oplax monoidal natural transf ormation is giv en in Figure 2.5 . 47 2. Prel iminari es 𝐶 𝐹 = 𝐹 𝐹 𝑈 𝑈 𝑈 𝑈 𝐹 = 𝐹 𝑈 𝑈 Figure 2.5: Conditions for 𝜂 and 𝜀 to be oplax transf ormations. 2 . 8 c o e n d s In this sect ion w e g ive t he de fini tion of an enriched coend in t he special case of k -linear categories. More gener al accounts can be found for example in [ Gra80 , Section 2.3], [ Kel05 , Chapter 3], and [ Lor21 , Chapter 4]. Deﬁnition 2.117. Let 𝒞 be an essentially small k -linear categor y . The coend of a k -linear functor 𝑃 : 𝒞 op ⊗ k 𝒞 − → V ect is follo wing t he coequaliser in V ect :  𝑎 , 𝑏 ∈ 𝒞 𝒞 ( 𝑎 , 𝑏 ) ⊗ k 𝑃 ( 𝑏 , 𝑎 )  𝑎 ∈ 𝒞 𝑃 ( 𝑎 , 𝑎 )  𝑎 ∈ 𝒞 𝑃 ( 𝑎 , 𝑎 ) . F or 𝑎 , 𝑏 ∈ 𝒞 the tw o parallel mor phisms are giv en b y 𝒞 ( 𝑎 , 𝑏 ) ⊗ k 𝑃 ( 𝑏 , 𝑎 ) − → 𝑃 ( 𝑎 , 𝑎 ) , 𝑓 ⊗ 𝑥 ↦− → 𝑃 ( 𝑓 , 𝑎 ) 𝑥 , 𝒞 ( 𝑎 , 𝑏 ) ⊗ k 𝑃 ( 𝑏 , 𝑎 ) − → 𝑃 ( 𝑏 , 𝑏 ) , 𝑔 ⊗ 𝑦 ↦− → 𝑃 ( 𝑏 , 𝑔 ) 𝑦 . The end  𝑎 ∈ 𝒞 𝑃 ( 𝑎 , 𝑎 ) of 𝑃 is deﬁned analogously as an equaliser in V ect . Remark 2.118. A consequence of 𝒞 being essentiall y small is that w e can rewrite t he (co)equalisers deﬁning t he respectiv e (co)ends in Deﬁnition 2.117 to index ov er a set of objects, instead of a proper class. Since V ect is com plete and cocomplete, the end and coend of any 𝑃 : 𝒞 op ⊗ k 𝒞 − → V ect exists. In case t he indexing categor y 𝒞 can be inferred unambiguously from the context, w e will omit writing it explicitly . 48 2.8. Coends Remark 2.119. Ends and coends hav e a number of useful properties, tw o of which w e are going to frequentl y use in t his thesis. Firs t, t he y are functorial: An y natural transformation 𝛼 : 𝑃 = ⇒ 𝑃 ′ betw een functors 𝑃 , 𝑃 ′ : 𝒞 op × 𝒞 − → V ect induces a unique mor phism  𝜇 :  𝑃 − →  𝑃 ′ b y the univ ersal property of (co-)equalisers. Second, for appropriate functors 𝑃 , t he F ubini –T onelli int er chang e law holds: if an y of t he follo wing coends exist:  𝑎 ∈ 𝒞  𝑏 ∈ 𝒞 𝑃 ( 𝑎 , 𝑎 , 𝑏 , 𝑏 ) ,  ( 𝑎 , 𝑏 )∈ 𝒞 × 𝒞 𝑃 ( 𝑎 , 𝑎 , 𝑏 , 𝑏 ) ,  𝑏 ∈ 𝒞  𝑎 ∈ 𝒞 𝑃 ( 𝑎 , 𝑎 , 𝑏 , 𝑏 ) , then t he y all do and are isomorphic; w e often simpl y write  𝑎 , 𝑏 𝑃 ( 𝑎 , 𝑎 , 𝑏 , 𝑏 ) . The follo wing lemma is also known as t he enriched Y oneda lemma . Lemma 2.120. Let 𝒞 be a category , and let 𝐹 : 𝒞 − → V ect and 𝐺 : 𝒞 op − → V ect be functor s. Then t here ar e natural isomorphisms  𝑐 𝒞 ( 𝑐 , −) ⊗ k 𝐹 𝑐  𝐹   𝑐 V ect ( 𝒞 (− , 𝑐 ) , 𝐹 𝑐 ) , (2.8.1) and  𝑐 𝒞 (− , 𝑐 ) ⊗ k 𝐺 𝑐  𝐺   𝑐 V ect ( 𝒞 ( 𝑐 , −) , 𝐺 𝑐 ) . (2.8.2) As t he name sugges ts, t he abo v e lemma is an analogue of the Y oneda lemma in t he enriched case; as such, w e will only refer to it as “t he Y oneda lemma ” in subsequent considerations. F or a proof, see [ Kel05 , Section 2.4]. Deﬁnition 2.121. Let 𝒞 be a k -linear monoidal category . Giv en tw o k -linear functors 𝐹 , 𝐺 : 𝒞 − → V ect , t he Day convolution of 𝐹 and 𝐺 is the k -linear functor 𝐹 ∗ 𝐺 : 𝒞 − → V ect that is given b y the coend 𝐹 ∗ 𝐺 . . =  𝑎 , 𝑏 𝒞 ( 𝑎 ⊗ 𝑏 , −) ⊗ k 𝐹 𝑎 ⊗ k 𝐺 𝑏 . 49 2. Prel iminari es Remark 2.122. If 𝒞 is left or right closed monoidal, w e ma y simplify the Da y con v olution product using t he Y oneda lemma for coends: 𝐹 ∗ 𝐺 =  𝑎 , 𝑏 𝒞 ( 𝑎 ⊗ 𝑏 , −) ⊗ k 𝐹 𝑎 ⊗ k 𝐺 𝑏   𝑎 , 𝑏 𝒞 ( 𝑎 , [ 𝑏 , −] ℓ ) ⊗ k 𝐹 𝑎 ⊗ k 𝐺 𝑏 ( 2.8.1 )   𝑏 𝐹 ([ 𝑏 , −] ℓ ) ⊗ k 𝐺 𝑏 , (2.8.3) 𝐹 ∗ 𝐺 =  𝑎 , 𝑏 𝒞 ( 𝑎 ⊗ 𝑏 , −) ⊗ k 𝐹 𝑎 ⊗ k 𝐺 𝑏   𝑎 , 𝑏 𝒞 ( 𝑏 , [ 𝑎 , −] 𝑟 ) ⊗ k 𝐹 𝑎 ⊗ k 𝐺 𝑏 ( 2.8.1 )   𝑎 𝐹 𝑎 ⊗ 𝐺 ([ 𝑎 , −] 𝑟 ) . (2.8.4) Exam ple 2.123. Consider a k -linear monoidal categor y 𝒳 with a single object 𝑥 . Then 𝐴 . . = End 𝒳 ( 𝑥 ) is a commutativ e algebra and [ 𝒳 , V ect ]  Mod- 𝐴 . Let 𝐹 , 𝐺 : 𝒳 − → V ect be two functors. W riting 𝑀 . . = 𝐹 𝑥 and 𝑁 . . = 𝐺 𝑥 for the corresponding modules ov er 𝐴 , and 𝑎 𝑛 . . = 𝑛 𝑎 for all 𝑎 ∈ 𝐴 and 𝑛 ∈ 𝑁 , w e obtain the follo wing using Equation ( 2.8.3 ) and the deﬁnition of coends: ( 𝐹 ∗ 𝐺 ) 𝑥  𝑀 ⊗ k 𝑁 ⧸ ⟨ 𝑚 𝑎 ⊗ k 𝑛 − 𝑚 ⊗ k 𝑎 𝑛 | 𝑚 ∈ 𝑀 , 𝑛 ∈ 𝑁 , 𝑎 ∈ 𝐴 ⟩ = 𝑀 ⊗ 𝐴 𝑁 . Thus, one recov ers t he tensor product of modules ov er commutativ e algebras. Theorem 2.124 ([ Da y71 ]) . F or any monoidal category 𝒞 , the category [ 𝒞 op , V ect ] is closed monoidal with Day conv olution as its tensor pr oduct, 𝒞 ( 1 , −) as its unit, and t he internal homs given for all 𝐹 , 𝐺 : 𝒞 op − → V ect by [ 𝐹 , 𝐺 ] ℓ . . =  𝑎 , 𝑏 V ect ( 𝒞 (− ⊗ 𝑎 , 𝑏 ) , V ect ( 𝐹 𝑎 , 𝐺 𝑏 )) , (2.8.5) [ 𝐹 , 𝐺 ] 𝑟 . . =  𝑎 , 𝑏 V ect ( 𝒞 ( 𝑎 ⊗ − , 𝑏 ) , V ect ( 𝐹 𝑎 , 𝐺 𝑏 )) . (2.8.6) Remark 2.125. If t he monoidal categor y 𝒞 is closed, then the formulas for the internal homs of [ 𝒞 , V ect ] ma y be simpliﬁed b y means of t he Y oneda lemma: [ 𝐹 , 𝐺 ] 𝑟 =  𝑎 , 𝑏 V ect ( 𝒞 ( 𝑎 ⊗ − , 𝑏 ) , V ect ( 𝐹 𝑎 , 𝐺 𝑏 ))   𝑎 , 𝑏 V ect ( 𝒞 ( 𝑎 ⊗ − , 𝑏 ) ⊗ k 𝐹 𝑎 , 𝐺 𝑏 )   𝑏 V ect   𝑎 𝒞 ( 𝑎 ⊗ − , 𝑏 ) ⊗ k 𝐹 𝑎 , 𝐺 𝑏    𝑏 V ect   𝑎 𝒞 ( 𝑎 , [− , 𝑏 ] ℓ ) ⊗ k 𝐹 𝑎 , 𝐺 𝑏  ( 2.8.1 )   𝑏 V ect ( 𝐹 [− , 𝑏 ] ℓ , 𝐺 𝑏 ) , (2.8.7) 50 2.9. (Co)completions [ 𝐹 , 𝐺 ] ℓ   𝑏 V ect   𝑎 𝒞 ( 𝑎 , [− , 𝑏 ] 𝑟 ) ⊗ k 𝐹 𝑎 , 𝐺 𝑏  ( 2.8.1 )   𝑎 V ect ( 𝐹 [− , 𝑏 ] 𝑟 , 𝐺 𝑏 ) . (2.8.8) Remark 2.126. Whenev er w e treat  𝒞 op . . = [ 𝒞 , V ect ] as a (closed) monoidal category , w e implicitl y equip with t he con v olution tensor product. Analog- ousl y , one ma y deﬁne a closed monoidal structure on  𝒞 . . = [ 𝒞 op , V ect ] . N ote, ho w ev er , that in t his case we cannot simplify the inter nal hom in the same w a y as in Remar k 2.125 ; t his w ould require the categor y 𝒞 op , as opposed to 𝒞 itself, to be closed monoidal. The con v olution structure is particularl y w ell-beha v ed on representables: 𝒞 (− , 𝑥 ) ∗ 𝒞 (− , 𝑦 ) =  𝑎 , 𝑏 ∈ 𝒞 𝒞 (− , 𝑎 ⊗ 𝑏 ) ⊗ k 𝒞 ( 𝑎 , 𝑥 ) ⊗ k 𝒞 ( 𝑏 , 𝑦 ) = 𝒞 (− , 𝑥 ⊗ 𝑦 ) This connection extends to t he entire functor category , see [ Da y71 ; IK86 ]. Proposition 2.127. F or a monoidal category 𝒞 , the Y oneda embedding よ : 𝒞 − →  𝒞 = [ 𝒞 op , V ect ] , 𝑥 ↦− → 𝒞 (− , 𝑥 ) (2.8.9) is a str ong monoidal functor . 2 . 9 ( co ) co m p l e t i o n s In th is su bsec tion we gi ve a br ief — infor mal — a c c ou nt of the results regar ding t he monoidal pseudofunctoriality of cocompletions and the res- ulting (co)completion operations for monoidal and module categories. W e refer to [ Kel05 ; KS06 ] for gener alities on (co)limits and (co)completions. W e implicitl y assume all categories and functors to be k -linear . Let Φ be a class of diagrams. W e sa y t hat a categor y is Φ -cocomplet e if it admits colimits of functors wit h domain in Φ , and w e sa y that a functor is Φ -cocontinuous if it preserves such colimits. Deﬁnition 2.128. A monoidal categor y 𝒞 is called separat el y Φ -cocontinuous if 𝒞 is Φ -cocomplete and its tensor product is separatel y Φ -cocontinuous. Similar ly , f or a Φ -cocomplete monoidal categor y 𝒟 , a 𝒟 -module categor y ℳ is said to be separatel y Φ -cocontinuous if ℳ is Φ -cocomplete and t he action − ⊲ ℳ = is separatel y Φ -cocontinuous. 51 2. Prel iminari es Let C at Φ - Cocts be the 2 -category of Φ -cocomplete categories, Φ -cocontinu- ous functors, and transformations betw een them. Let 𝒞 be a small categor y . By [ Kel82 , Section 5.7], t here is a categor y Φ ( 𝒞 ) , the Φ -cocompletion of 𝒞 , and an embedding 𝜄 : 𝒞 ↩ − → Φ ( 𝒞 ) , such t hat for an y Φ -cocomplete category 𝒟 the restriction functor C at Φ - Cocts ( 𝒞 , 𝒟 ) −◦ 𝜄 − − → C at ( 𝒞 , 𝒟 ) is an equiv alence. By t he main results of [ Zöb76 ; K oc95 ], t he inclusion 2 -functor of C at Φ - Cocts into C at is 2 -monadic, wit h a lef t adjoint giv en by the pseudofunctor of Φ - cocompletion, Φ (−) : C at − → C at Φ - Cocts . This 2 -monad is a lax-idem potent 2 -monad in the sense of [ KL97 ]. By [ KL00 ], t here is a similar 2 -monad Φ 𝑐 on C at , whose algebr as are categories wit h a pr escribed choice of Φ -colimits. Strict mor phisms are functors strictl y preserving t he chosen Φ -colimits, and pseudomor phisms are functors preserving Φ -colimits in t he ordinary sense. By [ LF11 , Theorem 6.2], t his yields a pseudoclosed s tructure the 2 -category C at Φ 𝑐 - Cocts of categories wit h a choice of Φ -colimits, Φ -cocontinuous functors, and transformations between them. The category underl ying the internal hom from 𝒜 to ℬ is giv en b y C at Φ - Cocts ( 𝒜 , ℬ ) , and t he functors comprising the 2 -monad are closed. Further , in some cases, for instance if Φ is the class of ﬁnite colimits, the pseudoclosed structure becomes pseudoclosed monoidal. Since monoidal categories and module categories can be formulated as pseudomonoids internally to a pseudoclosed monoidal structure, see [ HP02 ], w e ﬁnd t he follo wing result. Proposition 2.129. Let 𝒞 be a monoidal category . Then Φ ( 𝒞 ) is separat ely Φ -cocon- tinuous monoidal, the inclusion 𝜄 : 𝒞 ↩ − → Φ ( 𝒞 ) is str ong monoidal, and induces t he following equivalence for any separ atel y Φ -cocontinuous monoidal category 𝒟 : StrMon Φ - Cocts ( Φ ( 𝒞 ) , 𝒟 ) −◦ 𝜄 − − → StrMon ( 𝒞 , 𝒟 ) . Similar equiv alences ar e induced for lax and oplax monoidal functors. Likewise, given a 𝒞 -module category ℳ , the Φ -cocompletion Φ ( ℳ ) is a separ - atel y Φ -cocontinuous Φ ( 𝒞 ) hypmodule category , t he inclusion 𝜄 : ℳ ↩ − → Φ ( ℳ ) is a str ong 𝒞 -module functor , and the functor Str Φ ( 𝒞 ) Mod Φ - Cocts ( Φ ( ℳ ) , 𝒩 ) −◦ 𝜄 − − → Str 𝒞 Mod ( ℳ , 𝒩 ) , (2.9.1) is an equiv alence; similar equiv alences exis t for lax and oplax module functor s. 52 2.9. (Co)completions W e remark t hat explicit constructions of t he monoidal and module struc- tures obtained in Proposition 2.129 and direct proofs of t heir univ ersal prop- erties are giv en in a variety of cases and w a ys in t he literature, see for in- stance [ MMMT19 , Section 3] and [ CG22 , Example 3.2.9]. Proposition 2.130. Let 𝒜 be an additive category . The copower of 𝒜 over vect — see Example 5.22 — is the essentiall y unique vect -module structur e on 𝒜 . Proof. Let add denote t he collection of ﬁnite discrete categories. An add - cocomplete category is sim ply an additiv e category , and an add -cocontinuous functor is an additiv e functor . Since any k -linear functor preser v es direct sums, an additiv e vect -module category is necessarily separately additiv e in the sense of Deﬁnition 2.128 . The result then follo ws by obser ving t hat vect ≃ add ({ ⊛ }) , where { ⊛ } is the terminal monoidal categor y , so StrMon add ( vect , C at add ( 𝒜 , 𝒜 )) ≃ StrMon ({∗} , C at add ( 𝒜 , 𝒜 )) ≃ {∗} , where { ∗} is the ter minal category and StrMon add denotes t he category of strong monoidal additiv e functors. □ Proposition 2.131. Let 𝒜 be an additive category admitting ﬁlter ed colimits. The action of v ect on 𝒜 ext ends uniquel y to a separ atel y ﬁnit ar y V ect -module cat egory structur e on 𝒜 . Proof. Let { add , ﬁlt } denote t he collection of diagrams which are ﬁltered or ﬁnite discrete. Then w e ha v e V ect ≃ { add , ﬁlt }({ ⊛ }) , and StrMon ﬁlt ( V ect , C at add , ﬁlt ( 𝒜 , 𝒜 )) = S trMon add , ﬁlt ( V ect , C at add , ﬁlt ( 𝒜 , 𝒜 )) ≃ S trMon ({ ⊛ } , C at add , ﬁlt ( 𝒜 , 𝒜 )) = S trMon ({ ⊛ } , C at ﬁlt ( 𝒜 , 𝒜 )) ≃ {∗} . □ In view of Lemma 2.89 , ﬁltered colimits and cocompletions under them, which w e shall call ind-completions , are particularl y important in t he study of locally ﬁnite abelian categories. F or example, t he ind-completion Ind ( 𝒞 ) of a locall y ﬁnite abelian category 𝒞 alw a ys has enough injectiv es. Another reason for our study of t hese completions is that categories of comodules are locall y ﬁnitely presentable, whence w e can make use of t he characterisation of adjoint functors, see Propositions 2.133 and 2.134 below . Deﬁnition 2.132. An object 𝑥 in a category 𝒞 is called compact or ﬁnitel y pr esented if 𝒞 ( 𝑥 , −) is ﬁnitary; i.e., preser v es ﬁltered colimits. W e denote t he full subcategory of compact objects by 𝒞 c . A categor y 𝒞 is called locall y ﬁnit ely pr esentable if ev er y object in 𝒞 is a ﬁltered colimit of compact objects. 53 2. Prel iminari es Examples of ind-completions arise from the fact t hat Ind ( 𝐶 vect ) ≃ 𝐶 V ect . Proposition 2.133 (Special adjoint functor theorem for right adjoints) . If 𝒞 and 𝒟 ar e locally ﬁnite abelian categories, then ther e are equiv alences Re x ( 𝒞 , 𝒟 ) Ind − − → Cocont ( Ind ( 𝒞 ) , Ind ( 𝒟 )) ∼ − → Map ( Ind ( 𝒞 ) , Ind ( 𝒟 )) , wher e Map ( 𝒜 , ℬ ) denot es the category of functor s admitting a right adjoint. In particular , a functor from Ind ( 𝒞 ) t o Ind ( 𝒟 ) admits a right adjoint if and only if it is cocontinuous, and the ext ension of a functor 𝐹 : 𝒞 − → 𝒟 t o Ind -cocompletions is cocontinuous if and onl y if 𝐹 is right exact. Proposition 2.134 (Special adjoint functor t heorem for left adjoints) . Let 𝒞 and 𝒟 be locally ﬁnite abelian cat egories. If a functor 𝐺 : Ind ( 𝒞 ) − → Ind ( 𝒟 ) is continuous and pr eserves ﬁlter ed colimits, then it admits a lef t adjoint. If a functor 𝐺 : Ind ( 𝒞 ) − → Ind ( 𝒟 ) admits a lef t adjoint 𝐹 , then 𝐺 preserv es ﬁlter ed colimits if and only if 𝐹 pr eserves compact objects in 𝒞 . The latter part of Proposition 2.133 follo ws from Ind ( 𝐹 ) preserving both ﬁnite and ﬁltered colimits, and thus preser ving all colimits. Since ﬁltered colimits in a locally ﬁnitel y presentable category are exact, the extension Ind ( 𝐺 ) of a left exact functor 𝐺 is lef t exact. How ev er , it is not clear that Ind ( 𝐺 ) preserves arbitrary products. Deﬁnition 2.135. An object 𝑋 ∈ Ind ( 𝒞 ) for a locally ﬁnite abelian categor y 𝒞 is ﬁnitel y cogener ated if t he functor Ind ( 𝒞 )(− , 𝑋 ) preserves arbitrary products. One can verify that for a coalgebra 𝐷 such that Ind ( 𝒞 ) ≃ Comod- 𝐷 , a comodule 𝑀 is ﬁnitely cogener ated if and only if 𝑀 embeds into a comodule of the form 𝐷 ⊕ 𝑚 for some 𝑚 ∈ N . Observe t hat an object 𝑀 ∈ Ind ( 𝒟 ) is compact if and onl y if it lies in the essential image of the embedding 𝒟 ↩ − → Ind ( 𝒟 ) . Deﬁnition 2.136. Let 𝒞 and 𝒟 be locall y ﬁnite abelian categories. A func- tor 𝐹 : Ind ( 𝒞 ) − → Ind ( 𝒟 ) is called quasi-ﬁnit e if for an y ﬁnitel y cog enerated injectiv e object 𝐼 in Ind ( 𝒞 ) and compact object 𝑀 of 𝒟 , the k -v ector space Ind ( 𝒟 )( 𝑀 , 𝐹 𝐼 ) is ﬁnite-dimensional. Using the equivalence 𝒞 ≃ 𝐶 vect of Lemma 2.89 , t he follo wing result is a direct consequence of [ T ak77 , Proposition 1.3]. Proposition 2.137. Let 𝐺 : 𝒞 − → 𝒟 be a left exact functor of locall y ﬁnite abelian categories. The functor Ind ( 𝐺 ) has a lef t adjoint if and only if it is quasi-ﬁnit e. 54 2.9. (Co)completions A strong er result holds for ﬁnite abelian categories, as a direct consequence of t he Eilenberg –W atts t heorem for categories of bimodules, see for ex- ample [ FSS20 , Lemma 2.1]. Proposition 2.138. A functor 𝐹 : 𝒜 − → ℬ of ﬁnit e abelian categories is right exact if and onl y if it has a right adjoint, and left exact if and only if it has a left adjoint. Recall that an object 𝐴 in a categor y 𝒜 that admits Φ -colimits is said to be Φ -small if the functor 𝒜 ( 𝐴 , −) preserves them. Lemma 2.139. Let 𝐺 : 𝒞 − → 𝒟 be a Φ -cocontinuous functor , and assume 𝐺 has a left adjoint 𝐹 . Then 𝐹 sends Φ -small objects to Φ -small objects. Proof. Let 𝑥 ∈ 𝒟 be a Φ -small object. Then 𝒞 ( 𝐹 𝑥 , −)  𝒟 ( 𝑥 , 𝐺 (−)) = 𝒟 ( 𝑥 , −) ◦ 𝐺 . The functor 𝒞 ( 𝐹 𝑥 , −) is t hus naturall y isomorphic to a com position of tw o Φ -cocontinuous functors, and hence it itself is a Φ -cocontinuous functor . □ 55 Ma y it be a light to y ou in dark places, when all other lights go out. J. R. R Tolki en ; The F ellow ship of the Ring D UA L I T Y T H E O R Y F O R M O N O I DA L C AT E G O R I E S 3 The a im of thi s ch apte r i s to comp a re sev eral notions of “duality” in monoidal categories. On t he one hand, w e ha v e closedness, exem plifying the tensor – hom adjunction in t he categor y of sets; on the other lies rigidity , gener alising the duals of ﬁnite-dimensional v ector spaces. Grothendieck – V erdier duality [ BD13 ], also called ∗ -autonom y [ Bar79 ], describes a duality theor y betw een the strict conﬁnements of rigidity and the v er y g eneral notion of monoidal closedness. Roughl y speaking, Grothendieck –V erdier categories consist of a closed monoidal categor y and a chosen dualising object, such t hat the inter nal hom into this object induces an anti-equivalence of categories. While closed and rigid categories ha v e left and right variants, Grothendieck – V erdier duality is an inherentl y ambidextrous concept. Our ﬁrst goal is to more closel y in v estigate t he con v erse of Proposition 2.67 , see also Remar k 2.68 . That is, w e study whet her one can decide if a closed monoidal category is rigid solel y by v erifying t hat t he inter nal hom is giv en b y tensoring wit h an object. W e t hank Chris Heunen for sugges ting t his ma y not hold in gener al at the bcqt 2022 summer school. Theorem 3.23 . Ther e exis ts a non-rigid tensor repr esentable category . For the second inclusion we assume t hat the natural transf ormations “comparing” the left and right-handed v ersion of tensor representability are in v ertible, see Propositions 3.7 and 3.16 . For rigid categories t his is alw a ys the case. In particular , taking Theorem 3.23 , Proposition 3.16 and Exam ple 3.17 , as w ell as Proposition 3.12 and Example 3.15 together yields a strict hierarch y: Rigid ⊊ T ensor representable ⊊ Grothendieck – V erdier ⊊ Closed . W e t hen inv estig ate closed and Grothendieck – V erdier structures on func- tor categories endow ed with Day con volution as its tensor product. Giv en a suﬃcientl y nice base category , t he functor category inherits t hese dualities. Proposition 3.27 . Let 𝒞 be a k -linear hom-ﬁnite Gro t hendieck– V er dier cat egor y . If Hypot hesis 3.25 holds — all r elevant coends exist and ar e ﬁnit e-dimensional vect or spaces — then [ 𝒞 , v ect ] is a Gro t hendieck– V er dier category . 57 3. Du al ity t heory for m onoid a l ca t egor ies W e further more in v estig ate t he duality properties of Cauchy comple- tions. Since w e are w or king wit h t he functor categor y [ 𝒞 , vect ] instead of the preshea v es [ 𝒞 op , vect ] , w e w or k with t he Cauch y completion 𝒞 op of 𝒞 op , it being a subcategor y of the free cocompletion  𝒞 op = [ 𝒞 , vect ] of 𝒞 op . Recall the deﬁnition of t he Y oneda embedding from Equation ( 2.8.9 ). Corollar y 3.43 . Let 𝒞 op be a k -linear right closed monoidal category . (i) 𝒞 op is right rigid if and onl y if its Cauchy completion 𝒞 op is. (ii) 𝒞 op is right tensor r epresent able if and only if 𝒞 op is. (iii) ( 𝒞 op , 𝑑 ) is a right Grot hendieck – V erdier category if and only if ( 𝒞 op , よ 𝑑 ) is. The chapter concludes wit h an in v estigation of t hree examples of abelian closed monoidal functor categories. The ﬁrst is ﬁnite Boolean algebras, which hist orically arose in the study of order t heory and univ ersal algebr a; t he connection to Grothendieck – V erdier duality w as already noted in [ Bar79 ]. Proposition 3.54 . The pat h algebr a of a ﬁnite Boolean algebr a is qf -2. See Deﬁnition 3.53 for when an algebra is qf -2. W e subsequently study t he categor y of ﬁnite-dimensional Macke y func- tors, objects abstracting and gener alising the induction, restriction, and con- jugation functors in classical representation t heory [ Lin76 ; TW95 ; W eb00 ]. Proposition 3.57 . Let 𝐺 be a ﬁnit e group and suppose k is a ﬁeld. The cat egor y mky k ( 𝐺 ) is a Gr othendiec k–V er dier category , and its rigid objects are precisel y t he ﬁnitel y-gener ated projectiv e Mackey functors. F urthermor e, mky k ( 𝐺 ) is rigid if and onl y if t he Mackey alg ebra M 𝐺 is semisimple, which is equivalent to char k not dividing t he or der of 𝐺 . Finall y , w e concer n ourselv es wit h strict 2-g roups and their equivalent formulation as crossed modules [ W ag21 ]. These are related to 𝑟 -categories , a special case of Grothendieck – V erdier duality in which the dualising object is isomorphic to the monoidal unit. Proposition 3.65 . Let 𝐺 and 𝐻 be ﬁnite groups, and suppose Suppose t hat  𝐺 , 𝐻 , 𝑡 : 𝐻 − → 𝐺 , 𝛼 : 𝐺 − → A ut ( 𝐻 )  is a crossed module. Let 𝒢 be its associat ed strict 2 -group. The category rep 𝒢 ( ker 𝑡 ) of ﬁnite-dimensional r epresent ations of k er 𝑡 is a right 𝑟 -category . 58 3.1. T ensor representability 3 . 1 t e n s o r r e p r e s e n ta b i l i t y Deﬁnition 3.1. W e call an endofunctor 𝐹 : 𝒞 − → 𝒞 of a monoidal categor y right tensor r epresent able if t here exists an 𝑥 ∈ 𝒞 such t hat 𝐹  𝑥 ⊗ − . Proposition 2.67 states that t he inter nal hom of a rigid monoidal categor y is tensor -representable at ev ery object. From no w on, for t he sake of brevity , w e will sim ply speak of “(monoidal) tensor representable categories” instead of “closed monoidal categories wit h tensor representable inter nal homs”. As sho wn in Lemma 3.5 below , t his is equivalent to the follo wing deﬁnition, which does not rely on closedness. Deﬁnition 3.2. A monoidal categor y 𝒞 is called right tensor repr esentable if for all 𝑥 ∈ 𝒞 there exists an object 𝑅 𝑥 ∈ 𝒞 , called t he right tensor dual of 𝑥 , such that 𝑥 ⊗ − ⊣ 𝑅 𝑥 ⊗ − . A monoidal categor y 𝒞 is called lef t tensor repr esentable if 𝒞 rev is right tensor representable. A lef t and right tensor representable category will be referred to as tensor r epresent able . T ensor representable categories encompass all rigid monoidal categories. As such, some results in t he t heory of t he latter carr y ov er to the f ormer . For example, t he follo wing result is w ell-known in the case of rigid monoidal abelian categories, see for example [ EGNO15 , Corollar y 4.2.13]. Lemma 3.3. All objects of an abelian monoidal right t ensor repr esentable category ar e projectiv e if and onl y if its unit is projectiv e. Proof. As t he ﬁrst claim implies the second, w e only ha v e to sho w its con v erse. Fix an object 𝑥 in an abelian right tensor representable categor y 𝒜 . There exists a chain of adjunctions 𝑥 ⊗ − ⊣ 𝑅 𝑥 ⊗ − ⊣ 𝑅 2 𝑥 ⊗ − ⊣ . . . . Hence, t he functor 𝑅 𝑥 ⊗ − : 𝒜 − → 𝒜 has left and right adjoints and is therefore exact. Consequently , 𝒜 ( 𝑥 , −)  𝒜 ( 1 , 𝑅 𝑥 ⊗ −) is — as a composite of exact functors — exact itself. □ While Lemma 2.64 holds in t he rigid case, it need not be true for tensor representable categories: giv en a strong monoidal functor 𝐹 , the image 𝐹 𝑥 of an object 𝑥 admitting a (right) tensor dual ma y not ha v e a (right) tensor dual itself. Ho w ev er , the property is preser v ed b y monoidal equiv alences. 59 3. Du al ity t heory for m onoid a l ca t egor ies Lemma 3.4. Let 𝒞 be a right tensor r epresent able cat egory , and suppose that 𝒟 is monoidal. Any str ong monoidal adjoint equiv alence 𝐹 : 𝒞 ⇄ 𝒟 : 𝐹 − 1 equips 𝒟 wit h a right tensor r epresent able structur e. Proof. F or an y 𝑥 ∈ 𝒟 , deﬁne 𝐷 𝑥 . . = 𝐹 𝑅 𝐹 − 1 𝑥 ∈ 𝒟 , where 𝑅 𝐹 − 1 𝑥 is a right tensor dual of 𝐹 − 1 𝑥 ∈ 𝒞 . Then 𝐷 𝑥 ⊗ − is right adjoint to 𝑥 ⊗ − : 𝒟 ( 𝑥 ⊗ 𝑦 , 𝑧 )  𝒞 ( 𝐹 − 1 ( 𝑥 ⊗ 𝑦 ) , 𝐹 − 1 𝑧 )  𝒞 ( 𝐹 − 1 𝑥 ⊗ 𝐹 − 1 𝑦 , 𝐹 − 1 𝑧 )  𝒞 ( 𝐹 − 1 𝑦 , 𝑅 𝐹 − 1 𝑥 ⊗ 𝐹 − 1 𝑧 )  𝒟 ( 𝐹 𝐹 − 1 𝑦 , 𝐹 ( 𝑅 𝐹 − 1 𝑥 ⊗ 𝐹 − 1 𝑧 ))  𝒟 ( 𝑦 , 𝐹 𝑅 𝐹 − 1 𝑥 ⊗ 𝑧 ) = 𝒟 ( 𝑦 , 𝐷 𝑥 ⊗ 𝑧 ) . Thus, 𝒟 is right tensor representable. □ Just like in t he rigid case, t he relationship between closedness and tensor representability is go v er ned by a famil y of natural isomor phisms. Lemma 3.5. A monoidal category 𝒞 is right tensor repr esentable if and only if it is right closed monoidal and for all 𝑥 ∈ 𝒞 t here ar e isomorphisms 𝜑 ( 𝑥 ) 𝑦 : [ 𝑥 , 𝑦 ] 𝑟 − → [ 𝑥 , 1 ] 𝑟 ⊗ 𝑦 natur al in 𝑦 ∈ 𝒞 . (3.1.1) Proof. F or ev er y 𝑥 ∈ 𝒞 in a right tensor representable category 𝒞 , b y deﬁnition there exists an object 𝑅 𝑥 ∈ 𝒞 such t hat 𝑅 𝑥 ⊗ − is right adjoint to 𝑥 ⊗ − . Hence, the categor y 𝒞 is right closed monoidal, and t he inter nal hom from 𝑥 ∈ 𝒞 is giv en by 𝑅 𝑥 ⊗ − . The claim follo ws by noticing t hat Equation ( 3.1.1 ) is now ev en the functorial equality 𝑅 𝑥 ⊗ − = 𝑅 𝑥 ⊗ 1 ⊗ − . Con v ersely , for all 𝑥 ∈ 𝒞 t here is an isomorphism 𝒞 ( 𝑦 , [ 𝑥 , 𝑧 ] 𝑟 )  𝒞 ( 𝑥 ⊗ 𝑦 , 𝑧 )  𝒞 ( 𝑦 , [ 𝑥 , 1 ] 𝑟 ⊗ 𝑧 ) natural in 𝑦 , 𝑧 ∈ 𝒞 ; t he result follo ws b y the Y oneda lemma. □ Deﬁnition 3.6. Assume 𝒞 to be a right tensor representable category . By Lemma 3.5 , it is right closed monoidal. Then 𝑅 . . = [− , 1 ] 𝑟 : 𝒞 op − → 𝒞 is called the right (tensor -)dualising functor of 𝒞 . Analogousl y , 𝐿 . . = [− , 1 ] ℓ : 𝒟 op − → 𝒟 is the lef t (tensor -)dualising functor of a lef t tensor representable categor y 𝒟 . A right rigid monoidal category is simultaneously left rigid if and only if its right dualising functor is an equivalence of categories. This is more complicated in t he tensor representable case. Assume 𝒞 to be of t his type. 60 3.1. T ensor representability W rite 𝑅 : 𝒞 op − → 𝒞 and 𝐿 : 𝒞 op − → 𝒞 for t he right and left tensor dualising functors, respectiv el y . Set 𝜂 ( 𝑥 ) 𝑦 : 𝑦 − → 𝑅 𝑥 ⊗ 𝑥 ⊗ 𝑦 , 𝜀 ( 𝑥 ) 𝑦 : 𝑥 ⊗ 𝑅 𝑥 ⊗ 𝑦 − → 𝑦 , 𝑢 ( 𝑥 ) 𝑦 : 𝑦 − → 𝑦 ⊗ 𝑥 ⊗ 𝐿 𝑥 , 𝑐 ( 𝑥 ) 𝑦 : 𝑦 ⊗ 𝐿 𝑥 ⊗ 𝑥 − → 𝑦 for t he units and counits of t he correspondig adjunctions. The left and right tensor dualising functors are related to each other b y 𝑥 𝑢 ( 𝑅 𝑥 ) 𝑥 − − − → 𝑥 ⊗ 𝑅 𝑥 ⊗ 𝐿 𝑅 𝑥 𝜀 ( 𝑥 ) 1 ⊗ 𝐿𝑅 𝑥 − − − − − − → 𝐿 𝑅 𝑥 , (3.1.2) 𝑥 𝜂 ( 𝐿 𝑥 ) 𝑥 − − − → 𝑅 𝐿 𝑥 ⊗ 𝐿 𝑥 ⊗ 𝑥 𝑅 𝐿 𝑥 ⊗ 𝑐 ( 𝑥 ) 1 − − − − − − → 𝑅 𝐿 𝑥 . (3.1.3) Proposition 3.7. The right dualising funct or 𝑅 : 𝒞 op − → 𝒞 of a tensor r epresent- able category 𝒞 is right adjoint t o 𝐿 op : 𝒞 − → 𝒞 op . F urther , 𝑅 is an equivalence if and onl y if the canonical morphisms of Equations ( 3.1.2 ) and ( 3.1.3 ) are inv ertible. In t his case 𝐿 op is a quasi-in verse of 𝑅 . Proof. For an y tw o objects 𝑥 , 𝑦 ∈ 𝒞 , w e ha v e 𝒞 op ( 𝐿 𝑥 , 𝑦 ) = 𝒞 ( 𝑦 , 𝐿 𝑥 )  𝒞 ( 𝑦 ⊗ 𝑥 , 1 )  𝒞 ( 𝑥 , 𝑅 𝑦 ) . A direct computation show s t hat t he unit and counit of t his adjunction are giv en by the natural transf ormations displa y ed in Equations ( 3.1.2 ) and ( 3.1.3 ). If these mor phisms are in v ertible, 𝑅 and 𝐿 op are quasi-in v erse to each other . Con v ersely , assume 𝑅 to be an equivalence of categories. Since any equi- v alence can be bettered to an adjoint equivalence and adjoints are unique up to unique natural isomorphism, 𝐿 op must be quasi-in v erse to 𝑅 and the unit and counit morphisms of Equations ( 3.1.2 ) and ( 3.1.3 ) are inv ertible. □ 3.1.1 Gr ot hendieck –V erdier duality We are g oing to loo sel y fo llo w the notation and terminology of [ BD13 ]. Deﬁnition 3.8. A (right) Grot hendieck – V er dier category consists of a pair ( 𝒞 , 𝑑 ) of a monoidal category 𝒞 and an object 𝑑 ∈ 𝒞 , such t hat for all 𝑥 ∈ 𝒞 there exists a representing object 𝐷 𝑟 𝑥 ∈ 𝒞 for 𝒞 ( 𝑥 ⊗ − , 𝑑 ) ; i.e., 𝒞 ( 𝑥 ⊗ − , 𝑑 )  𝒞 (− , 𝐷 𝑟 𝑥 ) , (3.1.4) and the induced right dualising functor 𝐷 𝑟 : 𝒞 op − → 𝒞 is an equiv alence of categories. If 𝑑 = 1 is the monoidal unit, one speak s of an 𝑟 -category . 61 3. Du al ity t heory for m onoid a l ca t egor ies Because 𝐷 𝑟 is an equiv alence, one also obtains 𝒞 ( 𝑥 ⊗ 𝑦 , 𝑑 )  𝒞 ( 𝑦 , 𝐷 𝑟 𝑥 )  𝒞 op ( 𝐷 − 1 ℓ 𝑦 , 𝑥 ) = 𝒞 ( 𝑥 , 𝐷 − 1 𝑟 𝑦 ) , for all 𝑥 , 𝑦 ∈ 𝒞 . (3.1.5) Remark 3.9. In [ BD13 ], t he aut hors instead consider a pair ( 𝒞 , 𝑑 ) subject to the natural isomorphism 𝒞 (− ⊗ 𝑦 , 𝑑 )  𝒞 (− , 𝐷 ℓ 𝑦 ) , such t hat t he induced functor 𝐷 ℓ : 𝒞 op − → 𝒞 is an equiv alence. In our fr amew or k, w e shall call this a lef t Grothendieck – V erdier category structure on 𝒞 , wit h lef t dualising functor 𝐷 ℓ . Equations ( 3.1.4 ) and ( 3.1.5 ) become 𝒞 ( 𝑥 ⊗ 𝑦 , 𝑑 )  𝒞 ( 𝑥 , 𝐷 ℓ 𝑦 )  𝒞 ( 𝑦 , 𝐷 − 1 ℓ 𝑥 ) . (3.1.6) Remark 3.10. No tice t hat t he nomenclature of Remar k 3.9 might be mislead- ing at ﬁrst sight. Because the dualising functor of a Grothendieck –V erdier category is assumed to be an equivalence, Grothendieck –V erdier structures are alwa ys tw o-sided. In view of Equation ( 3.1.5 ), a right Grothendieck – V erdier category ( 𝒞 , 𝑑 ) is also a left Grothendieck – V erdier categor y wit h t he same dualising object 𝑑 , as w ell as 𝐷 − 1 𝑟 = 𝐷 ℓ . Like wise, all lef t Grothendieck – V erdier categories are also right-sided. F or the sake of clarity and to denote t he directional bias, Grothendieck – V erdier structures will of ten nev ert heless explicitl y carr y a preﬁx. In particular , in Section 3.2 w e consider the induced Grothendieck – V erdier structure of the categor y of copresheav es  𝒞 op o v er a Grothendieck–V erdier base categor y 𝒞 . In this case, t he lef t dualising functor of 𝒞 naturall y giv es rise to a right dualising functor for  𝒞 op ; see Proposition 3.27 . The symmetric nature of Grothendieck –V erdier categories is in star k con- tras t to t he one-sidedness of rigidity and tensor representability . Exam ple 3.11. Consider a Hopf alg ebra 𝐻 whose antipode is not in vert- ible, see [ Sch00 b ]. Its ﬁnite-dimensional right comodules are a left-rigid category comod- 𝐻 which, as discussed in [ Ulb90 , Remar k 2], is not right rigid. If comod- 𝐻 w as a Grothendieck –V erdier categor y , there w ould as a consequence of Remar k 3.13 exist a comodule 𝑀 ∈ comod- 𝐻 , such t hat ∨ (−) ⊗ 𝑀 : ( comod- 𝐻 ) op − → comod- 𝐻 is an equivalence of categories. This w ould impl y t hat 𝑀 is one-dimensional, and t heref ore the left dualising func- tor ∨ (−) : comod- 𝐻 op − → comod- 𝐻 w ould be an equivalence — contradiction. The follo wing proposition, also discussed in [ BD13 , Section 2], is a direct consequence of Equations ( 3.1.4 ) and ( 3.1.5 ). 62 3.1. T ensor representability Proposition 3.12. Every (right) Gro thendiec k–V erdier category ( 𝒞 , 𝑑 ) is closed monoidal; for all 𝑥 , 𝑦 ∈ 𝒞 , the left and right internal homs are given by [ 𝑥 , 𝑦 ] ℓ . . = 𝐷 − 1 𝑟 ( 𝑥 ⊗ 𝐷 𝑟 𝑦 ) = 𝐷 ℓ ( 𝑥 ⊗ 𝐷 − 1 ℓ 𝑦 ) , [ 𝑥 , 𝑦 ] 𝑟 . . = 𝐷 𝑟 ( 𝐷 − 1 𝑟 𝑦 ⊗ 𝑥 ) = 𝐷 − 1 ℓ ( 𝐷 ℓ 𝑦 ⊗ 𝑥 ) . (3.1.7) Remark 3.13. If ( 𝒞 , 𝑑 ) is a Grothendieck – V erdier categor y , one recov ers its dualising object b y applying t he dualising functor to t he monoidal unit of 𝒞 . In fact, as shown in [ BD13 , Remar k 2.1.(4)], this relationship is bidirectional: 𝐷 𝑟 1  𝑑 , 𝐷 𝑟 𝑑  1 , 𝐷 ℓ 1  𝑑 , 𝐷 ℓ 𝑑  1 , 𝐷 2 𝑟 1  1 , 𝐷 2 ℓ 𝑑  𝑑 . (3.1.8) In combination wit h Proposition 3.12 , one obtains an explicit description of the dualising functors of 𝒞 : 𝐷 𝑟 𝑥  [ 𝑥 , 𝑑 ] 𝑟 and 𝐷 ℓ 𝑥 = 𝐷 − 1 𝑟 𝑥  [ 𝑥 , 𝑑 ] ℓ , for all 𝑥 ∈ 𝒞 . (3.1.9) While Grothendieck –V erdier duality is an extra bit of structure on a mono- idal category — a priori making it quite distinct from the other dualities — in practise it still beha v es v er y similar ly to how a property w ould, as t he inv ert- ible objects go v er n t he possible choices for dualising objects. Lemma 3.14 ([ BD13 , Proposition 2.3]) . Given a right Gro t hendieck– V er dier cat- egory ( 𝒞 , 𝑑 ) , ther e is an anti-equiv alence between its full subcategories of in vertible and dualising objects given by In v ( 𝒞 ) − → Dual ( 𝒞 ) , 𝛼 ↦− → 𝛼 − 1 ⊗ 𝑑 . Exam ple 3.15. The categor y ( Set , × , 1 ) of sets wit h its usual Cartesian mono- idal structure is closed monoidal. If some set 𝑋 is in v ertible, t hen it must be isomorphic to t he terminal set 1 . Thus, to deduce t hat t here is no Grothen- dieck –V erdier structure on Set , it is suﬃcient to see t hat t he functor Set (− , 1 ) is not an anti-equiv alence, which follo ws b y 1 being terminal. Let us now characterise t he relationship betw een Grothendieck – V erdier and tensor representable categories. Proposition 3.16. Let 𝒞 be a t ensor repr esentable category. Then the following st atements are equivalent: (i) t he right dualising functor admits a quasi-in verse, 63 3. Du al ity t heory for m onoid a l ca t egor ies (ii) t he category 𝒞 is an 𝑟 -category , and (iii) t he category 𝒞 is a Gro thendiec k – V erdier category . Proof. By deﬁnition w e hav e t hat (ii) = ⇒ (iii) , and (i) = ⇒ (ii) follo ws by Deﬁnition 3.6 and Remar k 3.13 . T o prov e (iii) = ⇒ (ii) , assume that 𝒞 is Grothendieck –V erdier with dual- ising object 𝑑 ∈ 𝒞 . W rite 𝐿 , 𝑅 : 𝒞 op − → 𝒞 for its left and right tensor dualising functors and 𝐷 𝑟 : 𝒞 op − → 𝒞 for t he dualising functor induced b y 𝑑 ∈ 𝒞 . Then 𝑅 𝑑 ⊗ 𝑑 ( 3.1.1 )  [ 𝑑 , 𝑑 ] 𝑟 ( 3.1.7 )  𝐷 𝑟 ( 𝐷 − 1 𝑟 𝑑 ⊗ 𝑑 ) ( 3.1.8 )  𝐷 𝑟 𝑑 ( 3.1.8 )  1 , 𝑑 ⊗ 𝐿 𝑑 ( 3.1.1 )  [ 𝑑 , 𝑑 ] ℓ ( 3.1.7 )  𝐷 − 1 𝑟 ( 𝑑 ⊗ 𝐷 𝑟 𝑑 ) ( 3.1.8 )  𝐷 − 1 𝑟 𝑑 ( 3.1.8 )  1 , and t heref ore 𝑑 ⊗ 𝑅 𝑑  𝑑 ⊗ 𝑅 𝑑 ⊗ 𝑑 ⊗ 𝐿 𝑑  1 . In other w ords, 𝑑 is inv ertible and 1  𝑅 𝑑 ⊗ 𝑑 is another dualising object due to Lemma 3.14 . T o complete the proof notice t hat the right dualising functor of 𝒞 con- sidered as an 𝑟 -category coincides with its right tensor dualising functor . □ Exam ple 3.17. The con v erse of Proposition 3.7 is not true. Let 𝐴 be a commut- ativ e Frobenius algebr a o v er a ﬁeld k . W e write 𝐴 -mod f . g . for t he categor y of ﬁnitel y-gener ated left 𝐴 -modules. One can show , see for example [ Lam99 , Sections 15 and 16], that for all 𝑀 ∈ 𝐴 -mod f . g . the canonical mor phism 𝜙 𝑀 : 𝑀 − → 𝑀 ∨∨ = Hom 𝐴 ( Hom 𝐴 ( 𝑀 , 𝐴 ) , 𝐴 ) , 𝜙 𝑀 ( 𝑥 ) 𝛼 = 𝛼 ( 𝑥 ) is an isomorphism; t he module 𝑀 is called reﬂe xive . Put diﬀerentl y , 𝐴 -mod f . g . is an 𝑟 -category wit h Hom 𝐴 (− , 𝐴 ) : ( 𝐴 -mod f . g . ) op − → 𝐴 -mod f . g . as dualising functor . It is w ell-known t hat for any 𝐴 -module 𝑀 there exists an 𝐴 -module 𝑁 such t hat 𝑀 ⊗ − ⊣ 𝑁 ⊗ − if and only if 𝑀 is ﬁnitely-g enerated projectiv e; see for example [ NW17 , Proposition 2.1]. Thus, 𝐴 -mod f . g . is tensor representable if and onl y if all (ﬁnitely-g enerated) 𝐴 -modules are projectiv e, which is equiv alent to 𝐴 being semisimple. Ho w ev er , not all commutativ e F robenius alg ebras are semisimple. For example, t he g roup algebr a k [ 𝐺 ] of a ﬁnite abelian g roup 𝐺 is Frobenius. By Maschke’ s theorem, it is semisimple if and only if t he characteristic of k does not divide t he order of 𝐺 . 64 3.1. T ensor representability Remark 3.18. Let 𝒞 be a right tensor representable categor y whose dualising functor 𝑅 : 𝒞 op − → 𝒞 is an equiv alence. F ollowing [ BD13 , Section 4], for all 𝑥 , 𝑦 ∈ 𝒞 t here exists a canonical isomor phism 𝜏 𝑥 , 𝑦 : 𝒞 ( 𝑦 ⊗ 𝑥 ⊗ 𝑅 𝑥 ⊗ 𝑅 𝑦 , 1 ) ∼ − → 𝒞 ( 𝑅 𝑥 ⊗ 𝑅 𝑦 , 𝑅 ( 𝑦 ⊗ 𝑥 )) ∼ − → 𝒞 op ( 𝑦 ⊗ 𝑥 , 𝑅 − 1 ( 𝑅 𝑥 ⊗ 𝑅 𝑦 )) . As sho wn in [ BD13 , Lemma 4.1 and Remar k 4.2], t he morphisms 𝑓 . . = 𝜏 𝑥 , 𝑦  𝜀 ( 𝑦 ) 1 ◦ ( 𝑦 ⊗ 𝜀 ( 𝑥 ) 1 ⊗ 𝑅 𝑦 )  : 𝑅 − 1 ( 𝑅 𝑥 ⊗ 𝑅 𝑦 ) − → 𝑦 ⊗ 𝑥 𝜇 𝑥 , 𝑦 . . = 𝑅 𝑓 : 𝑅 𝑥 ⊗ 𝑅 𝑦 − → 𝑅 ( 𝑦 ⊗ 𝑥 ) = 𝑅 ( 𝑥 ⊗ op 𝑦 ) endo w 𝑅 : 𝒞 op , rev − → 𝒞 wit h t he structure of a lax monoidal functor : 𝜇 𝑥 , 𝑦 ⊗ op 𝑧 ◦ ( 𝑅 𝑥 ⊗ 𝜇 𝑦 , 𝑧 ) = 𝜇 𝑥 ⊗ op 𝑦 , 𝑧 ◦ ( 𝜇 𝑥 , 𝑦 ⊗ 𝑅 𝑧 ) , 𝜇 1 ,𝑥 ◦ ( 𝑅 1 ⊗ 𝑅 𝑥 ) = id 𝑅 𝑥 = 𝜇 𝑥 , 1 ◦ ( 𝑅 𝑥 ⊗ 𝑅 1 ) , for all 𝑥 , 𝑦 , 𝑧 ∈ 𝒞 . In the situation of t he abov e remar k and as a consequence of [ BD13 , Pro- position 4.4], w e obtain t he follo wing characterisation of rigidity . Proposition 3.19. Let t he dualising functor 𝑅 : 𝒞 op , rev − → 𝒞 of a right tensor r epresent able category be an equivalence. Then 𝒞 is rigid if and onl y if t he morphism 𝜇 𝑥 , 𝑦 : 𝑅 (−) ⊗ 𝑅 ( = ) = ⇒ 𝑅 (− ⊗ op = ) is an isomor phism, making 𝑅 str ong monoidal. 3.1.2 The free tensor repr esentable category is not rigid The aim o f t his s ecti on is t o concretise Remar k 2.68 . That is, w e explicitly construct a “free” tensor representable categor y 𝒲 and show that it is not rigid; see Theorem 3.23 . Due to the reliance of t he proof on some technical arguments, w e will ﬁrst provide an ov erview of the underl ying strategy . As an inter mediate step in t he construction of 𝒲 , w e deﬁne a strict mono- idal category 𝒱 , which has the same generat ors as 𝒲 but less relations. In other wor ds, 𝒲 itself will be a quotient of 𝒱 . The categor y 𝒱 will hav e t he property t hat all presentations of a giv en mor phism ha ve the same lengt h. In order to show t hat 𝒲 is not rigid, w e will ﬁx an object 𝑡 ∈ 𝒲 and assume it admits a rigid dual. In particular , 𝑡 w ould hav e to admit ev aluation and coe- v aluation morphisms ev 𝑡 : 𝑡 ⊗ 𝑡 ∨ − → 1 and coev 𝑡 : 1 − → 𝑡 ∨ ⊗ 𝑡 . By sho wing that any mor phism in 𝒱 representing the class ( ev 𝑡 ⊗ 𝑡 ) ◦ ( 𝑡 ⊗ coev 𝑡 ) ∈ 𝒲 ( 𝑡 , 𝑡 ) has to ha v e lengt h at least 2, w e conclude that the snake identities cannot hold. Hence, 𝒲 is not rigid. 65 3. Du al ity t heory for m onoid a l ca t egor ies W e deﬁne t he strict monoidal categor y 𝒱 in terms of g enerators and rela- tions. For details of t his type of construction w e refer to [ Kas98 , Chapter xi i ]. The monoid of objects of t he categor y 𝒱 is freel y gener ated b y the alphabet { . . . , 𝑅 − 1 𝑡 , 𝑡 , 𝑅 𝑡 , . . . } . That is, its elements are w ords of ﬁnite lengt h whose letters are 𝑅 𝑛 𝑡 for some integer 𝑛 ∈ Z . W rite 1 ∈ Ob ( 𝒱 ) for t he monoidal unit of 𝒱 , giv en b y t he emp ty w ord. For an object 𝑤 . . = ( 𝑅 𝑘 1 𝑡 , . . . , 𝑅 𝑘 𝑛 𝑡 ) ∈ Ob ( 𝒱 ) , set 𝑅 𝑤 . . = ( 𝑅 𝑘 𝑛 + 1 𝑡 , . . . , 𝑅 𝑘 1 + 1 𝑡 ) ∈ Ob ( 𝒱 ) . This assignment extends to a g roup action of t he free abelian group ⟨ 𝑅 ⟩  ( Z , +) on Ob ( 𝒱 ) . For any 𝑖 ∈ Z , w e write 𝑅 𝑖 𝑤 for t he action of 𝑅 𝑖 on 𝑤 ∈ Ob ( 𝒱 ) . The mor phisms of 𝒱 are tensor products and compositions of identities and the gener ating mor phisms . These are giv en for all 𝑥 ≠ 1 , 𝑦 ∈ Ob ( 𝒱 ) b y 𝜂 ( 𝑥 ) 𝑦 : 𝑦 − → 𝑅 𝑥 ⊗ 𝑥 ⊗ 𝑦 , 𝜀 ( 𝑥 ) 𝑦 : 𝑥 ⊗ 𝑅 𝑥 ⊗ 𝑦 − → 𝑦 , 𝑢 ( 𝑥 ) 𝑦 : 𝑦 − → 𝑦 ⊗ 𝑥 ⊗ 𝑅 − 1 𝑥 , 𝑐 ( 𝑥 ) 𝑦 : 𝑦 ⊗ 𝑅 − 1 𝑥 ⊗ 𝑥 − → 𝑦 . These relations are tailored to implement natural transf ormations 𝜂 ( 𝑥 ) : Id 𝒱 = ⇒ 𝑅 𝑥 ⊗ 𝑥 ⊗ − , 𝜀 ( 𝑥 ) : 𝑥 ⊗ 𝑅 𝑥 ⊗ − = ⇒ Id 𝒱 , 𝑢 ( 𝑥 ) : Id 𝒱 = ⇒ − ⊗ 𝑥 ⊗ 𝑅 − 1 𝑥 , 𝑐 ( 𝑥 ) : − ⊗ 𝑅 − 1 𝑥 ⊗ 𝑥 = ⇒ Id 𝒱 . Explicitl y , for an y gener ating mor phism 𝑔 : 𝑐 − → 𝑑 , an y 𝑦 = 𝑎 ⊗ 𝑐 ⊗ 𝑏 , 𝑧 = 𝑎 ⊗ 𝑑 ⊗ 𝑏 , an y arrow 𝑓 = ( id 𝑎 ⊗ 𝑔 ⊗ id 𝑏 ) : 𝑦 − → 𝑧 , and all 𝑥 ∈ Ob ( 𝒱 ) \ { 1 } , w e require t he follo wing conditions to hold: 𝜂 ( 𝑥 ) 𝑧 ◦ 𝑓 = ( 𝑅 𝑥 ⊗ 𝑥 ⊗ 𝑓 ) ◦ 𝜂 ( 𝑥 ) 𝑦 , 𝜀 ( 𝑥 ) 𝑧 ◦ ( 𝑥 ⊗ 𝑅 𝑥 ⊗ 𝑓 ) = 𝑓 ◦ 𝜀 ( 𝑥 ) 𝑦 , 𝑢 ( 𝑥 ) 𝑧 ◦ 𝑓 = ( 𝑓 ⊗ 𝑥 ⊗ 𝑅 − 1 𝑥 ) ◦ 𝑢 ( 𝑥 ) 𝑦 , 𝑐 ( 𝑥 ) 𝑧 ◦ ( 𝑓 ⊗ 𝑅 − 1 𝑥 ⊗ 𝑥 ) = 𝑓 ◦ 𝑐 ( 𝑥 ) 𝑦 . W e obtain t he tensor representable categor y 𝒲 b y quotienting out t he triangle identities. 6 Explained in more detail, the strict monoidal categor y 𝒲 6 The categor y is obtained by monoidal localisation , see [ Day73 ]. has the same objects and g enerating mor phisms as 𝒱 , and the same identities hold. In addition, for an y 𝑥 , 𝑦 ∈ Ob ( 𝒲 ) with 𝑥 ≠ 1 , w e require 𝜀 ( 𝑥 ) 𝑥 ⊗ 𝑦 ◦ ( 𝑥 ⊗ 𝜂 ( 𝑥 ) 𝑦 ) = id 𝑥 , ( 𝑅 𝑥 ⊗ 𝜀 ( 𝑥 ) 𝑦 ) ◦ 𝜂 ( 𝑥 ) 𝑅 𝑥 ⊗ 𝑦 = id 𝑅 𝑥 , 𝑐 ( 𝑥 ) 𝑦 ⊗ 𝑥 ◦ ( 𝑢 ( 𝑥 ) 𝑦 ⊗ 𝑥 ) = id 𝑥 , ( 𝑐 ( 𝑥 ) 𝑦 ⊗ 𝑅 − 1 𝑥 ) ◦ 𝑢 ( 𝑥 ) 𝑦 ⊗ 𝑅 − 1 𝑥 = id 𝑅 − 1 𝑥 , ( 𝜀 ( 𝑥 ) 1 ⊗ 𝑥 ) ◦ 𝑢 ( 𝑅 𝑥 ) 𝑥 = id 𝑥 , ( 𝑥 ⊗ 𝑐 ( 𝑥 ) 1 ) ◦ 𝜂 ( 𝑅 − 1 𝑥 ) 𝑥 = id 𝑥 . (3.1.10) The next result succinctl y summarises the obser v ations made so far con- cerning t he inter nal hom of 𝒲 . 66 3.1. T ensor representability Lemma 3.20. The category 𝒲 is tensor repr esentable. F urthermor e, its right dual- ising functor 𝑅 : 𝒲 op − → 𝒲 is an equiv alence of categories. Proof. By construction, w e ha v e 𝑥 ⊗ − ⊣ 𝑅 𝑥 ⊗ − and − ⊗ 𝑥 ⊣ − ⊗ 𝑅 − 1 𝑥 for all 𝑥 ∈ 𝒲 . Thus, 𝒲 is tensor representable. Equation ( 3.1.10 ) and Proposition 3.7 impl y t hat 𝑅 : 𝒲 op − → 𝒲 and 𝑅 − 1 : 𝒲 op − → 𝒲 are quasi-in v erses. □ T o analyse t he mor phisms in 𝒲 and show t hat it is no t rigid monoidal, w e will rely on tw o tools. The ﬁrst is t he length of an arrow 𝑓 ∈ 𝒲 ( 𝑥 , 𝑦 ) . It is deﬁned as t he minimal number of generating mor phisms needed to present 𝑓 . The second tool will be giv en by in v ariants for mor phisms in 𝒲 arising from functors into the rigid monoidal categor y vect of ﬁnite-dimensional v ector spaces ov er a ﬁeld k . W e will write (−) ∗ : v ect op − → v ect for the right rigid dualising functor of vect . The 𝑛 -fold dual of a ﬁnite-dimensional v ector space 𝑉 will be denoted b y 𝑉 ( 𝑛 ) . Lemma 3.21. F or all 𝑉 ∈ vect t here is a str ong monoidal functor 𝐹 𝑉 : 𝒲 − → vect satisfying 𝐹 𝑉 ( 𝑅 𝑛 𝑡 )  𝑉 ( 𝑛 ) for all 𝑛 ∈ Z and 𝐹 𝑉 ( 𝜂 ( 𝑥 ) 𝑦 ) = coev 𝑟 𝐹 𝑉 𝑥 ⊗ 𝐹 𝑉 𝑦 , 𝐹 𝑉 ( 𝜀 ( 𝑥 ) 𝑦 ) = ev 𝑟 𝐹 𝑉 𝑥 ⊗ 𝐹 𝑉 𝑦 , 𝐹 𝑉 ( 𝑢 ( 𝑥 ) 𝑦 ) = 𝐹 𝑉 𝑦 ⊗ coev ℓ 𝐹 𝑉 𝑥 , 𝐹 𝑉 ( 𝑐 ( 𝑥 ) 𝑦 ) = 𝐹 𝑉 𝑦 ⊗ ev ℓ 𝐹 𝑉 𝑥 , for suitabl y chosen evaluation and coevaluation mor phisms. Proof. Theorem 2.72 yields a monoidal equiv alence 𝐺 : vect ⇄ 𝒜 : 𝐻 betw een the categor y of ﬁnite-dimensional vect or spaces and a rigid monoidal categor y 𝒜 whose dualising functors 𝐷 ℓ , 𝐷 𝑟 : 𝒜 op − → 𝒜 satisfy 𝐷 ℓ 𝐷 op 𝑟 = Id 𝒜 = 𝐷 op 𝑟 𝐷 ℓ . Fix a quasi-in v erse 𝐻 : 𝒜 − → v ect and choose an object 𝑣 ∈ 𝒜 with 𝐻 ( 𝑣 )  𝑉 , for some 𝑉 ∈ vect . A direct computation using the univ ersal property of 𝒲 , see for example [ Kas98 , Proposition x ii .1.4], sho ws t hat for any object 𝑣 ∈ 𝒜 there is a unique s trict monoidal funct or 𝐹 ′ 𝑣 : 𝒲 − → 𝒜 such that 𝐹 ′ 𝑣 ( 𝑡 ) = 𝑣 and for all 𝑥 , 𝑦 ∈ 𝒲 , 𝑥 ≠ 1 , w e ha v e 𝐹 ′ 𝑣 𝜂 ( 𝑥 ) 𝑦 = coev 𝑟 𝐹 ′ 𝑣 𝑥 ⊗ 𝐹 ′ 𝑣 𝑦 , 𝐹 ′ 𝑣 𝜀 ( 𝑥 ) 𝑦 = ev 𝑟 𝐹 ′ 𝑣 𝑥 ⊗ 𝐹 ′ 𝑣 𝑦 , 𝐹 ′ 𝑣 𝑢 ( 𝑥 ) 𝑦 = 𝐹 ′ 𝑣 𝑦 ⊗ coev ℓ 𝐹 ′ 𝑣 𝑥 , 𝐹 ′ 𝑣 𝑐 ( 𝑥 ) 𝑦 = 𝐹 ′ 𝑣 𝑦 ⊗ ev ℓ 𝐹 ′ 𝑣 𝑥 . The claim no w follo ws b y setting 𝐹 = 𝐻 𝐹 ′ 𝐺 ( 𝑉 ) . □ 67 3. Du al ity t heory for m onoid a l ca t egor ies Corollar y 3.22. The following arrow s cannot be isomor phisms, for any 1 ≠ 𝑥 ∈ 𝒲 and any morphism 𝑔 ∈ 𝒲 ( 𝑎 1 ⊗ 𝑦 ⊗ 𝑏 1 , 𝑎 2 ⊗ 𝑧 ⊗ 𝑏 2 ) : ( id 𝑎 2 ⊗ 𝑢 ( 𝑥 ) 𝑧 ⊗ id 𝑏 2 ) ◦ 𝑔 , 𝑔 ◦ ( id 𝑎 1 ⊗ 𝜀 ( 𝑥 ) 𝑦 ⊗ id 𝑏 1 ) . Proof. Suppose t hat 𝑔 ∈ 𝒲 ( 𝑎 1 ⊗ 𝑦 ⊗ 𝑏 1 , 𝑎 2 ⊗ 𝑧 ⊗ 𝑏 2 ) and consider 𝑓 . . = 𝑔 ◦ ( id 𝑎 1 ⊗ 𝜀 ( 𝑥 ) 𝑦 ⊗ id 𝑏 1 ) . Let 𝑉 ∈ vect be a v ector space of dimension at least tw o. Applying 𝐹 𝑉 to 𝑓 , w e get 𝐹 𝑉 𝑓 = 𝐹 𝑉 𝑔 ◦ 𝐹 𝑉 ( id 𝑎 1 ⊗ 𝜀 ( 𝑥 ) 𝑦 ⊗ id 𝑏 1 ) . Ho w ev er , due to the diﬀerence in the dimensions of its source and targ et, 𝐹 𝑉 ( id 𝑎 1 ⊗ 𝜀 ( 𝑥 ) 𝑦 ⊗ id 𝑏 1 ) must ha v e a non-trivial kernel and thus 𝑓 cannot be an isomorphism. A similar argument in vol ving t he cokernel prov es t hat t he composition ( id 𝑎 2 ⊗ 𝑢 ( 𝑥 ) 𝑧 ⊗ id 𝑏 2 ) ◦ 𝑔 is not inv ertible. □ Theorem 3.23. The category 𝒲 is not rigid. Proof. Assume t hat 𝑡 ∈ 𝒲 admits a right rigid dual 𝑥 ∈ 𝒲 . By t he uniqueness of adjoints, t here exist isomor phisms 𝜗 : 𝑅 𝑡 ⊗ 𝑡 − → 𝑥 ⊗ 𝑡 and 𝜃 : 𝑥 − → 𝑅 𝑡 such that t he ev aluation and coev aluation mor phisms are giv en b y coev . . = 𝜗 ◦ 𝜂 ( 𝑡 ) 1 : 1 − → 𝑥 ⊗ 𝑡 , e v . . = 𝜀 ( 𝑡 ) 1 ◦ ( 𝑡 ⊗ 𝜃 ) : 𝑡 ⊗ 𝑥 − → 1 . Let 𝜋 : 𝒱 − ↠ 𝒲 be the projection functor . W e consider t he set 𝑆 . . =  ( 𝜀 ( 𝑡 ) 1 ⊗ 𝑡 ) 𝜙 ( 𝑡 ⊗ 𝜂 ( 𝑡 ) 1 ) ∈ 𝒱 ( 𝑡 , 𝑡 )     𝜙 ∈ 𝒱 ( 𝑡 ⊗ 𝑅 𝑡 ⊗ 𝑡 , 𝑡 ⊗ 𝑅 𝑡 ⊗ 𝑡 ) such that 𝜋 ( 𝜙 ) is in v ertible  . N otice that 𝑆 ⊆ Mor ( 𝒱 ) . By construction, there exists a morphism 𝑠 ∈ 𝑆 such that t he com posite arro w ( ev ⊗ 𝑡 ) ◦ ( 𝑡 ⊗ coev ) = 𝜋 ( 𝑠 ) corresponds to one of the snake identities. F urther more, ev ery element of 𝑆 has lengt h at least tw o. 7 Thus, b y proving t hat 𝑆 is closed under t he relations arising from 7 N ote t hat the relations of 𝒱 lea v e the number of gener ating morphisms in any presentation of a giv en arrow in variant. Equation ( 3.1.10 ), it follo ws t hat 𝜋 ( 𝑠 ) ≠ id 𝑡 , which concludes the proof. Let us consider an arbitrary element 𝑓 . . = ( 𝜀 ( 𝑡 ) 1 ⊗ 𝑡 ) ◦ 𝜙 ◦ ( 𝑡 ⊗ 𝜂 ( 𝑡 ) 1 ) ∈ 𝑆 . There are tw o types of “mo v es” w e ha v e to study . Firs t, suppose w e expand an identity into one of t he mor phisms displa yed in Equation ( 3.1.10 ). This equates to eit her pre or postcomposing 𝜙 with an arrow 𝜓 ∈ 𝒱 ( 𝑡 ⊗ 𝑅 𝑡 ⊗ 𝑡 , 𝑡 ⊗ 𝑅 𝑡 ⊗ 𝑡 ) that projects onto an isomorphism in 𝒲 , leading to another element in 𝑆 . 68 3.2. Funct or categories Second, an y of t he mor phisms of Equation ( 3.1.10 ) might be contr acted to an identity . A priori, t here are t hree wa ys in which t his might occur: 𝑓 = ( 𝜀 ( 𝑡 ) 1 ⊗ 𝑡 ) ◦ 𝜙 ′ ◦ 𝜀 ( 𝑡 ) 𝑡 ◦ ( 𝑡 ⊗ 𝜂 ( 𝑡 ) 1 ) , with 𝜙 = 𝜙 ′ ◦ 𝜀 ( 𝑡 ) 𝑡 , or (3.1.11) 𝑓 = ( 𝜀 ( 𝑡 ) 1 ⊗ 𝑡 ) ◦ 𝑢 ( 𝑅 𝑡 ) 𝑡 ◦ 𝜙 ′ ′ ◦ ( 𝑡 ⊗ 𝜂 ( 𝑡 ) 1 ) , wit h 𝜙 = 𝑢 ( 𝑅 𝑡 ) 𝑡 ◦ 𝜙 ′ ′ , or (3.1.12) 𝑓 = ( 𝜀 ( 𝑡 ) 1 ⊗ 𝑡 ) ◦ 𝜙 2 ◦ 𝑔 ◦ 𝜙 1 ◦ ( 𝑡 ⊗ 𝜂 ( 𝑡 ) 1 ) , with 𝜙 = 𝜙 2 ◦ 𝑔 ◦ 𝜙 1 and 𝜋 ( 𝑔 ) = id . (3.1.13) Due to Corollary 3.22 , neither 𝜋 ( 𝜙 ′ ) ◦ 𝜋 ( 𝜀 ( 𝑡 ) 𝑡 ) nor 𝜋 ( 𝑢 ( 𝑅 𝑡 ) 𝑡 ) ◦ 𝜋 ( 𝜙 ′ ′ ) are isomor ph- isms, contradicting Cases ( 3.1.11 ) and ( 3.1.12 ). N o w , assume 𝑓 = ( 𝜀 ( 𝑡 ) 1 ⊗ 𝑡 ) ◦ 𝜙 2 ◦ 𝑔 ◦ 𝜙 1 ◦ ( 𝑡 ⊗ 𝜂 ( 𝑡 ) 1 ) and 𝜙 = 𝜙 2 ◦ 𝑔 ◦ 𝜙 1 . Using the functoriality of 𝜋 : 𝒱 − → 𝒲 , w e get 𝜋 ( 𝜙 ) = 𝜋 ( 𝜙 2 ◦ 𝑔 ◦ 𝜙 1 ) = 𝜋 ( 𝜙 2 ) ◦ 𝜋 ( 𝑔 ) ◦ 𝜋 ( 𝜙 1 ) = 𝜋 ( 𝜙 2 ) ◦ 𝜋 ( 𝜙 1 ) = 𝜋 ( 𝜙 2 ◦ 𝜙 1 ) . Thus, 𝜋 ( 𝜙 2 ◦ 𝜙 1 ) is an isomor phism and ( 𝜀 ( 𝑡 ) 1 ⊗ 𝑡 ) ◦ 𝜙 2 ◦ 𝜙 1 ◦ ( 𝑡 ⊗ 𝜂 ( 𝑡 ) 1 ) is an element of 𝑆 . □ 3 . 2 f u nc t o r c at e g o r i e s We will no w inv estiga te the previousl y discussed types of dualities in t he context of functor categories. The starting point is t he fact that Da y con v o- lution is closed monoidal, see Theorem 2.124 . Examples of such categories of functors are abound, and include modules o v er path algebras, Macke y functors, and g roup-graded representations arising from crossed modules; w e shall more closely in v estigate t hese examples in Section 3.3 . Hypothesis 3.24. F ix a ﬁeld k , and write (−) ∗ : v ect op − → v ect for t he k -linear dualising functor . U nless explicitly stated, for the remainder of t his section all (monoidal) categories and functors will be enriched ov er V ect , and 𝒞 will denote an essentially small monoidal k -linear categor y . For t he sake of brevity , the preﬁx es “enriched”, “ k -linear”, and “(essentially) small” will often be omitted. The category of functors from 𝒞 to V ect will be denoted by  𝒞 op . Instead of s tudying functors 𝐹 ∈  𝒞 op = [ 𝒞 , V ect ] from a hom-ﬁnite cat- egory 𝒞 to all v ector spaces, w e are interested in t he full subcategor y  𝒞 op ﬁn of (point-wise) ﬁnite-dimensional functors. That is, 𝐹 ∈  𝒞 op such that 𝐹 𝑥 is ﬁnite-dimensional for all 𝑥 ∈ 𝒞 . 69 3. Du al ity t heory for m onoid a l ca t egor ies Hypothesis 3.25. Within this section, w e assume that all relev ant ends and coends are ﬁnite-dimensional v ector spaces. In particular , Hypothesis 3.25 im plies that  𝒞 op ﬁn is closed monoidal; t hat is, t he tensor product and t he inter nal hom are ﬁnite-dimensional at ev er y point. These assump tions impose a certain ﬁniteness condition on 𝒞 itself, or — as t he next example will show — on a dense 8 subcategory of it. 8 A subcategor y 𝒪 of 𝒞 is dense if the restricted Y oneda embedding from 𝒞 to [ 𝒪 op , V ect ] is full y faithful. Exam ple 3.26. Let 𝒞 be a hom-ﬁnite categor y , and assume t hat t here is a full subcategory 𝒮 such t hat ev ery object of 𝒞 ma y be written as a ﬁnite direct sum of objects in 𝒮 . In t his setting, one can reformulate Lemma 2.120 to index o v er t his ﬁnite set: for exam ple, giv en 𝐹 : 𝒞 − → v ect , w e hav e 𝐹 𝑥  𝐹  𝑛  𝑖 = 1 𝑠 𝑖   𝑛  𝑖 = 1 𝐹 𝑠 𝑖  𝑛  𝑖 = 1  𝑠 ∈ 𝒮 𝒮 ( 𝑠 , 𝑠 𝑖 ) ⊗ 𝐹 𝑠  𝑛  𝑖 = 1  𝑠 ∈ 𝒮 𝒞 ( 𝑠 , 𝑠 𝑖 ) ⊗ 𝐹 𝑠   𝑠 ∈ 𝒮 𝒞  𝑠 , 𝑛  𝑖 = 1 𝑠 𝑖  ⊗ 𝐹 𝑠   𝑠 ∈ 𝒮 𝒞 ( 𝑠 , 𝑥 ) ⊗ 𝐹 𝑠 . P anchadcharam and Street used t his to show t hat ﬁnite-dimensional Mack ey functors are a Grothendieck – V erdier categor y , see [ PS07 , Section 9]. Hypothesis 3.25 does not only impl y a closed structure on  𝒞 op ﬁn , but yields a canonical notion of a dual. Proposition 3.27. Let ( 𝒞 , 𝑑 ) be a hom-ﬁnite lef t Grot hendieck – V erdier category wit h dualising functor 𝐷 ℓ . Then  𝒞 op ﬁn is a right Gro thendiec k–V er dier category wit h dualising object 𝒞 (− , 𝑑 ) ∗ ∈  𝒞 op ﬁn and dualising functor D 𝑟 :  𝒞 op op ﬁn − →  𝒞 op ﬁn , 𝐹 ↦− → 𝐹 ( 𝐷 ℓ −) ∗ . Proof. Using Equation ( 3.1.9 ), one can recov er the right dualising functor of a Grothendieck – V erdier categor y from its right inter nal hom b y ev aluating it at the dualising object; i.e., D 𝑟  [− , 𝒞 (− , 𝑑 ) ∗ ] 𝑟 . In our case, w e ha v e [ 𝐹 , 𝒞 (− , 𝑑 ) ∗ ] 𝑟 𝑥 ( 2.8.7 )   𝑏 vect  𝐹 [ 𝑥 , 𝑏 ] ℓ , vect ( 𝒞 ( 𝑏 , 𝑑 ) , k )    𝑏 vect  𝒞 ( 𝑏 , 𝑑 ) ⊗ 𝐹 [ 𝑥 , 𝑏 ] ℓ , k   v ect   𝑏 𝒞 ( 𝑏 , 𝑑 ) ⊗ 𝐹 [ 𝑥 , 𝑏 ] ℓ , k  ( 2.8.1 )  vect  𝐹 [ 𝑥 , 𝑑 ] ℓ , k  ( 3.1.9 )  𝐹 ( 𝐷 ℓ 𝑥 ) ∗ . 70 3.2. Funct or categories T o conclude t he proof, notice t hat D 𝑟 :  𝒞 op op ﬁn − →  𝒞 op ﬁn has a quasi-in v erse D − 1 𝑟 :  𝒞 op ﬁn − →  𝒞 op op ﬁn , 𝐹 ↦− → 𝐹 ( 𝐷 − 1 ℓ −) ∗ . □ While w e will dev elop eﬃcient means to detect tensor representability and rigidity based on properties which exist in the abelian case, it is w orthwhile to explore ho w such structures arise as an interpla y betw een t he duality type of the base categor y and t hat of vect . Corollar y 3.28. N ot all assump tions are strictly needed, see Remark 3.34 . Let ( 𝒞 , 𝑑 ) be a hom-ﬁnite lef t Grot hendieck – V erdier category with dualising functor 𝐷 ℓ . Then t he category  𝒞 op ﬁn is right tensor r epresent able if ther e ar e isomorphisms 𝐷 2 ℓ 𝑎  𝑎 , 𝐷 ℓ ( 𝑎 ⊗ 𝑥 )  𝐷 ℓ 𝑥 ⊗ 𝐷 ℓ 𝑎 , natural in 𝑎 , 𝑥 ∈ 𝒞 , (3.2.1) and for all objects 𝐹 , 𝐺 ∈  𝒞 op ﬁn we hav e  𝑎 vect ( 𝐹 [− , 𝑎 ] ℓ , 𝐺 𝑎 )   𝑎 vect ( 𝐹 [− , 𝑎 ] ℓ , 𝐺 𝑎 ) . (3.2.2) Proof. Let D 𝑟 be the dualising functor of Proposition 3.27 , and ﬁx functors 𝐹 , 𝐺 ∈  𝒞 op ﬁn . Then, b y t he follo wing calculation, w e obtain an isomor phism ( D 𝑟 𝐹 ∗ 𝐺 ) 𝑥  [ 𝐹 , 𝐺 ] 𝑟 𝑥 , and hence t he claim follo ws: ( D 𝑟 𝐹 ∗ 𝐺 ) 𝑥 ( 2.8.3 )   𝑎 D 𝑟 𝐹 ([ 𝑎 , 𝑥 ] ℓ ) ⊗ 𝐺 𝑎 ( 3.1.7 )   𝑎 D 𝑟 𝐹 ( 𝐷 ℓ ( 𝑎 ⊗ 𝐷 ℓ 𝑥 )) ⊗ 𝐺 𝑎 3 . 27   𝑎 𝐹 ( 𝐷 2 ℓ ( 𝑎 ⊗ 𝐷 ℓ 𝑥 )) ∗ ⊗ 𝐺 𝑎 ( 3.2.1 )   𝑎 vect  𝐹 ( 𝑎 ⊗ 𝐷 ℓ 𝑥 ) , 𝐺 𝑎  ( 3.2.1 )   𝑎 vect  𝐹 ( 𝐷 2 ℓ 𝑎 ⊗ 𝐷 ℓ 𝑥 ) , 𝐺 𝑎  ( 3.2.1 )   𝑎 vect  𝐹 ( 𝐷 ℓ ( 𝑥 ⊗ 𝐷 ℓ 𝑎 )) , 𝐺 𝑎  ( 3.1.7 ) + ( 3.2.1 )   𝑎 vect  𝐹 ([ 𝑥 , 𝑎 ] ℓ ) , 𝐺 𝑎  ( 3.2.2 )   𝑎 vect  𝐹 ([ 𝑥 , 𝑎 ] ℓ ) , 𝐺 𝑎  ( 2.8.8 ) = [ 𝐹 , 𝐺 ] 𝑟 𝑥 . □ Remark 3.29. Before w e discuss conditions for the interchangeability of ends and coends, let us brieﬂy mention some cases where the dualising functor of a left Grothendieck – V erdier categor y ( 𝒞 , 𝑑 ) admits natural isomor phisms as stated in Equation ( 3.2.1 ). 71 3. Du al ity t heory for m onoid a l ca t egor ies The requirement 𝐷 ℓ ( 𝑥 ⊗ 𝑦 )  𝐷 ℓ 𝑦 ⊗ 𝐷 ℓ 𝑥 implies that 𝒞 must be lef t tensor representable, since then for all 𝑥 , 𝑦 , 𝑧 ∈ 𝒞 w e ha ve 𝒞 ( 𝑥 ⊗ 𝑦 , 𝑧 ) ( 3.1.6 )  𝒞 ( 𝑥 ⊗ 𝑦 ⊗ 𝐷 − 1 ℓ 𝑧 , 𝑑 ) ( 3.1.6 )  𝒞 ( 𝑥 , 𝐷 ℓ ( 𝑦 ⊗ 𝐷 − 1 ℓ 𝑧 )) ( 3.2.1 )  𝒞 ( 𝑥 , 𝑧 ⊗ 𝐷 ℓ 𝑦 ) . In the absence of furt her coherence assump tions, t his does not impl y t hat 𝒞 must be rigid. The condition 𝐷 2 ℓ  Id 𝒞 is met for exam ple if 𝒞 is braided. For all 𝑥 , 𝑦 ∈ 𝒞 w e ha v e 𝒞 ( 𝑥 , 𝑦 ) ( 3.1.6 )  𝒞 ( 𝐷 ℓ 𝑦 ⊗ 𝑥 , 𝑑 )  𝒞 ( 𝑥 ⊗ 𝐷 ℓ 𝑦 , 𝑑 ) ( 3.1.6 )  𝒞 ( 𝑥 , 𝐷 2 ℓ 𝑦 ) , and the Y oneda lemma implies that Id 𝒞  𝐷 2 ℓ . The follo wing constructions are an adaptation of [ Da y06 ]. F or simplicity , w e replaced the promonoidal structure in ibid with t he one induced from a monoidal structure. That is, 𝐽 . . = 𝒞 ( 1 , −) , 𝑃 ( 𝑥 , 𝑦 , 𝑧 ) . . = 𝒞 ( 𝑥 ⊗ 𝑦 , 𝑧 ) , for 𝐽 and 𝑃 the promonoidal unit and multiplication. Remark 3.30. Let 𝒞 be a categor y . Giv en any functor 𝑇 : 𝒞 op ⊗ k 𝒞 − → V ect , there exists a canonical map  𝑎 ∈ 𝒞  𝑏 ∈ 𝒞 𝒞 ( 𝑏 , 𝑎 ) ⊗ 𝑇 ( 𝑎 , 𝑏 ) − →  𝑏 ∈ 𝒞  𝑎 ∈ 𝒞 𝒞 ( 𝑏 , 𝑎 ) ⊗ 𝑇 ( 𝑎 , 𝑏 ) . (3.2.3) This follo ws from t he fact t hat ends and coends are functorial; in particular , for an y arro w 𝑓 : 𝑥 − → 𝑦 , t he follo wing diag r am commutes:  𝑏 𝒞 ( 𝑏 , 𝑥 ) ⊗ 𝑇 ( 𝑦 , 𝑏 )  𝑏 𝒞 ( 𝑏 , 𝑦 ) ⊗ 𝑇 ( 𝑦 , 𝑏 )  𝑏 𝒞 ( 𝑏 , 𝑥 ) ⊗ 𝑇 ( 𝑥 , 𝑏 )  𝑎  𝑏 𝒞 ( 𝑏 , 𝑎 ) ⊗ 𝑇 ( 𝑎 , 𝑏 )  𝑏  𝑎 𝒞 ( 𝑏 , 𝑎 ) ⊗ 𝑇 ( 𝑎 , 𝑏 )  𝑏 𝒞 ( 𝑏 , 𝑓 )⊗ 𝑇 ( 𝑦 ,𝑏 )  𝑏 𝒞 ( 𝑏 , 𝑥 )⊗ 𝑇 ( 𝑓 , 𝑏 ) can can ∃ !  𝑏 can  𝑏 can Lemma 3.31 ([ Da y06 , p. 1]) . Let 𝒞 be a hom-ﬁnite category , 𝑃 : 𝒞 op ⊗ k 𝒞 − → v ect a funct or , and suppose t hat t he map given in Equation ( 3.2.3 ) is inv ertible. If for all 𝑎 , 𝑏 ∈ 𝒞 ther e is a natural isomor phism 𝜑 : 𝒞 ( 𝑏 , 𝑎 ) ∼ − → 𝒞 ( 𝑎 , 𝑏 ) ∗ , t hen  𝑎 𝑃 ( 𝑎 , 𝑎 ) ∼ − →  𝑏 𝑃 ( 𝑏 , 𝑏 ) . 72 3.2. Funct or categories Proof. W e calculate  𝑎 𝑃 ( 𝑎 , 𝑎 )   𝑎  𝑏 vect ( 𝒞 ( 𝑎 , 𝑏 ) , 𝑃 ( 𝑎 , 𝑏 ))   𝑎  𝑏 𝒞 ( 𝑎 , 𝑏 ) ∗ ⊗ 𝑃 ( 𝑎 , 𝑏 ) 𝜑 − 1   𝑎  𝑏 𝒞 ( 𝑏 , 𝑎 ) ⊗ 𝑃 ( 𝑎 , 𝑏 ) ( 3.2.3 )   𝑏  𝑎 𝒞 ( 𝑏 , 𝑎 ) ⊗ 𝑃 ( 𝑎 , 𝑏 )   𝑏 𝑃 ( 𝑏 , 𝑏 ) . The ﬁrst and last isomorphisms follo w from Lemma 2.120 and the second one is a consequence of 𝒞 ( 𝑎 , 𝑏 ) being ﬁnite-dimensional and vect being rigid. □ In gener al, it seems diﬃcult to decide whet her a natural isomorphism as that in Lemma 3.31 exists. How ev er , for certain classes of exam ples, trace maps pro vide us with viable candidates. Remark 3.32. Let 𝒞 be a hom-ﬁnite and piv otal categor y wit h piv otal structure 𝜓 : (−) ∨ − → ∨ (−) , and suppose that 𝒞 ( 1 , 1 )  k . F or all 𝑎 , 𝑏 ∈ 𝒞 , consider 𝜑 𝑎 , 𝑏 : 𝒞 ( 𝑏 , 𝑎 ) − → 𝒞 ( 𝑎 , 𝑏 ) ∗ 𝑓 ↦− →  𝑔 ↦− → tr ( 𝑓 ◦ 𝑔 ) . . = ev ℓ 𝑎 ◦ ( 𝜓 𝑎 ⊗ ( 𝑓 ◦ 𝑔 )) ◦ coev 𝑟 𝑎  . (3.2.4) The dual of 𝜑 is giv en by ( 𝜑 𝑎 , 𝑏 ) ∗ : 𝒞 ( 𝑎 , 𝑏 ) ∗∗  𝒞 ( 𝑎 , 𝑏 ) − → 𝒞 ( 𝑏 , 𝑎 ) ∗ 𝑔 ↦− →  𝑓 ↦− → tr ( 𝑓 ◦ 𝑔 ) = tr ( 𝑔 ◦ 𝑓 ) = 𝜑 𝑏 , 𝑎 ( 𝑔 )( 𝑓 )  . Thus, 𝜑 𝑎 , 𝑏 is injectiv e if and only if 𝜑 𝑏 , 𝑎 is sur jectiv e. Suppose k has characteris tic zero and 𝒞 . . = 𝐻 -mod is t he category of ﬁnite- dimensional modules of a semisim ple Hopf algebr a 𝐻 . As a consequence of [ LR88 , Theorem 4], lef t and right duals coincide and w e can chose t he “quantum tr ace” of Equation ( 3.2.4 ) to ag ree with the usual tr ace of endo- morphisms betw een ﬁnite-dimensional v ector spaces. Let 𝑓 : 𝑀 − → 𝑁 be a morphism in 𝒞 . By semisimplicity , t he follo wing short exact sequence splits: 0 ker 𝑓 𝑀 im 𝑓 0 𝑓 𝜄 The arro w 𝑔 = 𝜄 ⊕ 0 : 𝑁  im 𝑓 ⊕ 𝑁 / im 𝑓 − → 𝑀 satisﬁes tr ( 𝑓 𝑔 ) = tr ( 𝑓 𝜄 ) = tr ( id im 𝑓 ) = dim im 𝑓 , impl ying t hat 𝜑 𝑀 ,𝑁 is injectiv e. 73 3. Du al ity t heory for m onoid a l ca t egor ies Exam ple 3.33. Consider a ﬁeld k of characteris tic 𝑝 , as w ell as 𝑞 ∈ N a multiple of 𝑝 . The group algebra 𝐻 . . = k [ GL 𝑞 ( 𝑝 )] of the g roup of in v ertible 𝑞 × 𝑞 -matrices ov er k pro vides us with an example of a piv otal hom-ﬁnite category where the mor phism of Equation ( 3.2.4 ) is not in v ertible. Matrix- v ector multiplication turns k 𝑞 into a simple 𝐻 -module whose endomorphism algebr a is one-dimensional. This implies that for all 𝑓 , 𝑔 ∈ End GL 𝑞 ( 𝑝 ) ( k 𝑝 ) there exists some 𝜆 ∈ k such t hat tr ( 𝑓 𝑔 ) = 𝜆 tr ( id k 𝑞 ) = 𝜆 · 0 = 0 . Remark 3.34. One can use a combination of Corollary 3.28 and Lemma 3.31 to deduce that a giv en category of ﬁnite-dimensional functors is right tensor representable. T o reduce t he number of axioms to be checked, note t hat the square of t he left dualising functor being isomorphic t o t he identity is implied b y t he other assump tions. Let ( 𝒞 , 𝑑 ) be a hom-ﬁnite left Grothendieck – V erdier categor y wit h dualising functor 𝐷 ℓ , such t hat there are isomorphisms 𝒞 ( 𝑎 , 𝑏 )  𝒞 ( 𝑏 , 𝑎 ) ∗ and 𝐷 ℓ ( 𝑎 ⊗ 𝑏 )  𝐷 ℓ 𝑏 ⊗ 𝐷 ℓ 𝑎 , natural in 𝑎 , 𝑏 ∈ 𝒞 . Using t hat 𝒞 ( 𝑑 , 𝑎 ⊗ 𝐷 ℓ 𝑏 ) ( 3.1.6 )  𝒞 ( 𝑑 ⊗ 𝐷 − 1 ℓ ( 𝑎 ⊗ 𝐷 ℓ 𝑏 ) , 𝑑 )  𝒞 ( 𝑑 ⊗ 𝑏 ⊗ 𝐷 − 1 ℓ 𝑎 , 𝑑 ) ( 3.1.6 )  𝒞 ( 𝐷 − 1 ℓ 𝑎 , 𝐷 − 1 ℓ ( 𝑑 ⊗ 𝑏 )) ( 3.1.8 )  𝒞 ( 𝐷 − 1 ℓ 𝑎 , 𝐷 − 1 ℓ 𝑏 )  𝒞 ( 𝑏 , 𝑎 ) , one obtains 𝒞 ( 𝑎 , 𝐷 2 ℓ 𝑏 ) ( 3.1.6 )  𝒞 ( 𝑎 ⊗ 𝐷 ℓ 𝑏 , 𝑑 )  𝒞 ( 𝑑 , 𝑎 ⊗ 𝐷 ℓ 𝑏 ) ∗  𝒞 ( 𝑏 , 𝑎 ) ∗  𝒞 ( 𝑎 , 𝑏 ) ∗∗  𝒞 ( 𝑎 , 𝑏 ) . The claim follo ws from t he Y oneda lemma. Lemma 3.35. Assume k to be a perfect ﬁeld. If the hom-ﬁnite category 𝒞 has ﬁnit el y many objects and  𝒞 op ﬁn is semisimple, then the canonical map  𝑎  𝑏 𝑇 ( 𝑏 , 𝑏 , 𝑎 , 𝑎 )   𝑏  𝑎 𝑇 ( 𝑎 , 𝑎 , 𝑏 , 𝑏 ) is in vertible for all functors 𝑇 : 𝒞 op ⊗ k 𝒞 ⊗ k 𝒞 op ⊗ k 𝒞 − → v ect . Proof. W e endow t he v ector space 𝐴 . . =  𝑎 , 𝑏 ∈ 𝒞 𝒞 ( 𝑎 , 𝑏 ) with t he structure of an associativ e unital algebra via t he multiplication follo wing multiplication, for 𝑓 ∈ 𝒞 ( 𝑎 , 𝑏 ) , 𝑔 ∈ 𝒞 ( 𝑐 , 𝑑 ) : 𝑓 · 𝑔 . . =  𝑔 ◦ 𝑓 , if b = c, 0 , otherwise. 74 3.2. Funct or categories The unit  𝑎 ∈ 𝒞 id 𝑎 of 𝐴 is a sum of ort hogonal idempo tents. Thus, an y right module 𝑀 of 𝐴 decomposes as a v ector space into a direct sum 𝑀   𝑎 ∈ 𝒞 𝑀 𝑎 , where 𝑀 𝑎 = 𝑀 · id 𝑎 , and the action of any 𝑓 ∈ 𝒞 ( 𝑎 , 𝑏 ) deﬁnes a linear map 𝑀 𝑎 − → 𝑀 𝑏 . A ccordingl y , a mor phism of right 𝐴 -modules corresponds to a collection of homomorphisms { 𝜙 𝑎 : 𝑀 𝑎 − → 𝑁 𝑎 } 𝑎 ∈ 𝒞 such that 𝜙 𝑏 ( 𝑚 · 𝑓 ) = 𝜙 𝑎 ( 𝑚 ) · 𝑓 for all 𝑚 ∈ 𝑀 𝑎 and 𝑓 ∈ 𝒞 ( 𝑎 , 𝑏 ) . This deﬁnes a functor Θ : Mod- 𝐴 − →  𝒞 op . Its quasi-in v erse Ω maps an y 𝐹 ∈  𝒞 op to t he module  𝑎 ∈ 𝒞 𝐹 𝑎 , and an y arro w { 𝜓 𝑎 : 𝐹 𝑎 − → 𝐺 𝑎 } 𝑎 ∈ 𝒞 to t he module homomorphism  𝑎 ∈ 𝒞 𝜓 𝑎 :  𝑎 ∈ 𝒞 𝐹 𝑎 − →  𝑎 ∈ 𝒞 𝐺 𝑎 . Thus,  𝒞 op ﬁn corresponds to t he categor y mod- 𝐴 of ﬁnite-dimensional right 𝐴 -modules. Since  𝒞 op ﬁn is semisimple, so is 𝐴 and 𝐴 op . Furthermore, k being perfect implies t hat the tensor product of an y tw o ﬁnite-dimensional semisim ple algebr as is semisimple, see [ FD93 , Section 3]. In particular , w e ha v e t hat the algebr a 𝐵 . . = 𝐴 op ⊗ k 𝐴 ⊗ k 𝐴 op ⊗ k 𝐴 is semisimple, and [ 𝒞 op ⊗ k 𝒞 ⊗ k 𝒞 op ⊗ k 𝒞 , vect ]  mod- 𝐵 . W rite 𝒟 . . = [ 𝒞 op ⊗ k 𝒞 , [ 𝒞 op ⊗ k 𝒞 , vect ]] ; t hen t he functor [ 𝒞 op ⊗ k 𝒞 ⊗ k 𝒞 op ⊗ k 𝒞 , vect ] − → 𝒟 , 𝐹 ↦− →  𝐹 , where  𝐹 ( 𝑥 , 𝑦 )[ 𝑢 , 𝑣 ] = 𝐹 ( 𝑢 , 𝑣 , 𝑥 , 𝑦 ) , is a k -linear equiv alence of categories. Further , since limits commute with limits, t he functor end : 𝒟 − → [ 𝒞 op ⊗ k 𝒞 , vect ] , 𝐹 ↦− → ( 𝑢 , 𝑣 ↦− →   𝑏 𝐹 ( 𝑏 , 𝑏 )  ( 𝑢 , 𝑣 )) is lef t exact. By semisimplicity , an y short exact sequence in 𝒟 splits and split epimorphisms are preser v ed b y all functors. Thus, end : 𝒟 − → [ 𝒞 op ⊗ k 𝒞 , vect ] is exact and must preserve colimits. Giv en 𝐹 ∈ [ 𝒞 op ⊗ k 𝒞 ⊗ k 𝒞 op ⊗ k 𝒞 , vect ] , w e now compute  𝑎  𝑏 𝐹 ( 𝑎 , 𝑎 )( 𝑏 , 𝑏 )   𝑎  𝑏  𝐹 ( 𝑏 , 𝑏 )[ 𝑎 , 𝑎 ]   𝑏  𝑎  𝐹 ( 𝑏 , 𝑏 )( 𝑎 , 𝑎 )   𝑏  𝑎 𝐹 ( 𝑎 , 𝑎 , 𝑏 , 𝑏 ) . □ 75 3. Du al ity t heory for m onoid a l ca t egor ies 3.2.1 Cauc hy completions Ha vi n g a ssem bled all of th e n eces s ary tool s , w e can no w state explicit criteria for t he Cauch y completion of (the opposite of) a k -linear category to carry certain duality s tructures. As wit h modules ov er (commutativ e) rings, these notions are closely connected with objects being ﬁnitely-g enerated projectiv e. Ag ain, w e implicitl y assume all categories and functors to be k -linear , for a ﬁeld k . How ev er , w e do not require ﬁniteness of hom-spaces and w ork solely under the assump tions made in Hypothesis 3.24 . Deﬁnition 3.36. Suppose t hat 𝒞 is a k -linear categor y . The Cauchy completion of 𝒞 is the full subcategory 𝒞 of  𝒞 that consists of all preshea v es 𝐹 , such t hat  𝒞 ( 𝐹 , −) commutes wit h all small limits. Remark 3.37. Deﬁnition 3.36 diﬀers from t he usual deﬁnition of t he Cauch y completion of a k -linear categor y 𝒞 as t he additiv e and idempotent completion of 𝒞 . How ev er , b y [ BD86 , Proposition 2] and [ L T22 , Corollary 4.22], these tw o notions are equiv alent. W e choose t he former because it is more conv enient to w or k with for the pur poses of t his t hesis. Exam ple 3.38. Let 𝜄 : 𝒞 ↩ − → 𝒞 denote the inclusion of a category 𝒞 into its Cauch y completion, sending 𝑥 ∈ 𝒞 to 𝒞 (− , 𝑥 ) . There is an adjoint equivalence  𝒞 (− , 𝜄 = ) , 𝒞 ( 𝜄 − , = ) , 𝜂 , 𝜀  , betw een 𝒞 and its Cauchy completion in t he monoidal bicategory of profunc- tors, see Example 2.113 . The unit 𝜂 𝑥 , 𝑦 : 𝒞 ( 𝑥 , 𝑦 ) − →  𝑐 ∈ 𝒞 𝒞 ( 𝜄 𝑥 , 𝑐 ) × 𝒞 ( 𝑐 , 𝜄 𝑦 ) is deﬁned b y 𝒞 ( 𝑥 , 𝑦 ) 𝜄 𝑥 , 𝑦 − − − → 𝒞 ( 𝜄 𝑥 , 𝜄 𝑦 )   𝑐 ∈ 𝒞 𝒞 ( 𝜄 𝑥 , 𝑐 ) × 𝒞 ( 𝑐 , 𝜄 𝑦 ) , and for t he counit w e hav e 𝜀 𝑥 , 𝑦 :  𝑐 ∈ 𝒞 𝒞 ( 𝑥 , 𝜄 𝑐 ) × 𝒞 ( 𝜄 𝑐 , 𝑦 ) − → 𝒞 ( 𝑥 , 𝑦 ) [( 𝑓 , 𝑔 )] ↦− → 𝑔 ◦ 𝑓 . Since 𝜄 is full y faithful, it immediately follo ws t hat 𝜂 𝑥 , 𝑦 is an isomorphism, for all 𝑥 , 𝑦 ∈ 𝒞 , and t he counit 𝜀 is one b y [ AEBSJ01 , Theorem 1.1]. 76 3.2. Funct or categories In principle, t he absence of free objects in t he categor y of copreshea v es necessitates a dev elopment of these notions in abstract categorical ter ms; this is discussed extensiv ely for example in [ Pre09 ]. F or our pur poses, t he follo wing characterisation in terms of the Cauch y completion of 𝒞 op suﬃces. Lemma 3.39. Let 𝒞 be a category . For any 𝐹 ∈  𝒞 op , t he following are equivalent: (i) 𝐹 is ﬁnitel y-gener ated projectiv e, (ii) 𝐹 is a direct summand of a ﬁnite direct sum of repr esentable functors, and (iii) t he functor  𝒞 op ( 𝐹 , −) commutes with small colimits. Proof. The equiv alence betw een (i) and (ii) is pro v en in Corollary 10.1.14 of [ Pre09 ]. In order to show t hat (ii) and (iii) are equiv alent, obser v e that the full subcategor y 𝒞 op of  𝒞 op consisting of direct summands of ﬁnite direct sums of representable functors is Cauch y com plete by [ L T22 , Corollary 4.22]. The claim no w follo ws b y proceeding analogous to Proposition 2 of [ BD86 ]. □ Notation 3.40. In view of Lemma 3.39 , w e shall adopt t he follo wing notation to emphasise that 𝑋 ∈ 𝒞 op is a direct summand of a direct sum of representables: 𝑋 𝑛  𝑖 = 1 よ 𝑢 𝑖 , 𝜄 𝑋 𝜋 𝑋 where 𝜋 𝑋 ◦ 𝜄 𝑋 = id 𝑋 , and よ : 𝒞 op − → [( 𝒞 op ) op , V ect ] = [ 𝒞 , V ect ] , 𝑥 ↦− → 𝒞 op (− , 𝑥 ) = 𝒞 ( 𝑥 , −) is the contra variant Y oneda embedding as in Equation ( 2.8.9 ). Proposition 3.41. F or all objects 𝑋  𝑛 𝑖 = 1 よ 𝑢 𝑖 𝜄 𝑋 𝜋 𝑋 and 𝑌  𝑚 𝑗 = 1 よ 𝑣 𝑖 𝜄 𝑌 𝜋 𝑌 , t he right int ernal hom of t he Cauchy completion 𝒞 op of a right closed monoidal category 𝒞 op exis ts and satisﬁes [ 𝑋 , 𝑌 ] 𝑟 [⊕ 𝑛 𝑖 = 1 よ 𝑢 𝑖 , ⊕ 𝑚 𝑗 = 1 よ 𝑣 𝑗 ]  𝑛  𝑖 = 1 𝑚  𝑗 = 1 よ [ 𝑢 𝑖 , 𝑣 𝑗 ] 𝑟 . [ 𝜋 𝑋 , 𝜄 𝑌 ] 𝑟 [ 𝜄 𝑋 , 𝜋 𝑌 ] 𝑟 (3.2.5) Proof. The category 𝒞 op ⊆  𝒞 op is closed under taking tensor products since the Y oneda embedding is strong monoidal and 𝒞 op is the full subcategor y of  𝒞 op consisting of direct summands of ﬁnite direct sums of representables. 77 3. Du al ity t heory for m onoid a l ca t egor ies Let 𝑋 , 𝑌 ∈ 𝒞 op . W rite 𝑋 = colim 𝑖 𝒞 ( 𝑢 𝑖 , −) and 𝑌 = colim 𝑗 𝒞 ( 𝑣 𝑗 , −) as direct summands of ﬁnite direct sums of representables. W e compute [ 𝑋 , 𝑌 ] 𝑟 ( 2.8.6 ) =  𝑎 , 𝑏 V ect ( 𝒞 ( 𝑎 ⊗ − , 𝑏 ) , V ect ( 𝑋 𝑎 , 𝑌 𝑏 )) ( 2.8.1 )   𝑎 V ect ( 𝑋 𝑎 , 𝑌 ( 𝑎 ⊗ −))   𝑎 V ect ( colim 𝑖 𝒞 ( 𝑢 𝑖 , 𝑎 ) , colim 𝑗 𝒞 ( 𝑣 𝑗 , 𝑎 ⊗ −))  lim 𝑖 colim 𝑗  𝑎 V ect ( 𝒞 ( 𝑢 𝑖 , 𝑎 ) , 𝒞 ( 𝑣 𝑗 , 𝑎 ⊗ −))  lim 𝑖 colim 𝑗 𝒞 ( 𝑣 𝑗 , 𝑢 𝑖 ⊗ −)  lim 𝑖 colim 𝑗 𝒞 ([ 𝑢 𝑖 , 𝑣 𝑗 ] 𝑟 , −) . It follo ws that 𝒞 op is right closed monoidal and t hat Equation ( 3.2.5 ) holds. □ Exam ple 3.42. Proposition 3.41 can be understood as a variation of t he fact t hat a direct summand or direct sum of (rigidly) dualisable objects is dualisable. Indeed, suppose 𝒞 op to be a closed monoidal categor y and consider three direct summands 𝑋 𝑈 𝜄 𝑋 𝜋 𝑋 , 𝑌 𝑉 𝜄 𝑌 𝜋 𝑌 , and 𝑍 𝑊 𝜄 𝑊 𝜋 𝑊 of objects in 𝒞 op . The following diagram, whose horizontal arrow s are t he isomor phisms of the tensor– hom adjunction of 𝒞 op , commutes: 𝒞 op ( 𝑋 ∗ 𝑌 , 𝑍 ) 𝒞 op ( 𝑌 , [ 𝑋 , 𝑍 ] 𝑟 ) 𝒞 op ( 𝑈 ∗ 𝑉 , 𝑊 ) 𝒞 op ( 𝑉 , [ 𝑈 , 𝑊 ] 𝑟 ) 𝜙 𝑋 ,𝑌 , 𝑍 𝒞 op ( 𝜋 𝑋 ∗ 𝜋 𝑌 , 𝜄 𝑊 ) 𝒞 op ( 𝜋 𝑉 , [ 𝜋 𝑋 , 𝜄 𝑊 ] 𝑟 ) 𝒞 op ( 𝜄 𝑋 ∗ 𝜄 𝑌 , 𝜋 𝑍 ) 𝜙 𝑈 ,𝑉 ,𝑊 𝒞 op ( 𝜄 𝑉 , [ 𝜄 𝑋 , 𝜋 𝑍 ] 𝑟 ) Thus the unit and counit of t he adjunction 𝑋 ∗ − : 𝒞 ⇄ 𝒞 : [ 𝑋 , −] 𝑟 satisfy 𝜂 ( 𝑋 ) 𝑌 = 𝑌 𝜄 𝑌 − − → 𝑉 𝜂 ( 𝑈 ) 𝑉 − − − → [ 𝑈 , 𝑈 ∗ 𝑉 ] [ 𝜄 𝑋 , 𝜋 𝑋 ∗ 𝜋 𝑌 ] − − − − − − − − → [ 𝑋 , 𝑋 ∗ 𝑉 ] , 𝜀 ( 𝑋 ) 𝑌 = 𝑋 ∗ [ 𝑋 , 𝑌 ] 𝜄 𝑋 ∗[ 𝜋 𝑋 , 𝜄 𝑌 ] − − − − − − − → 𝑈 ∗ [ 𝑈 , 𝑉 ] 𝜀 ( 𝑈 ) 𝑉 − − − → 𝑉 𝜋 𝑌 − − − → 𝑌 . (3.2.6) Corollar y 3.43. Let 𝒞 op be a right closed monoidal category . W e hav e: (i) 𝒞 op is right rigid if and onl y if 𝒞 op is. (ii) 𝒞 op is right tensor r epresent able if and only if 𝒞 op is. (iii) ( 𝒞 op , 𝑑 ) is a right Grot hendieck – V erdier category if and only if ( 𝒞 op , よ 𝑑 ) is. 78 3.2. Funct or categories Proof. That right rigidity and tensor representability of 𝒞 op impl y the same property for 𝒞 op follo ws from the descrip tion of the inter nal hom of 𝒞 op and the units and counits of t he tensor-hom adjunctions, see Equation ( 3.2.6 ). If ( 𝒞 op , 𝑑 ) is a right Grothendieck – V erdier category , t hen its right dualising functor is, up to natural isomor phism, giv en by [− , 𝑑 ] 𝑟 : 𝒞 − → 𝒞 op . A direct computation using Proposition 3.41 show s t hat [− , よ 𝑑 ] 𝑟 : 𝒞 op op − → 𝒞 op is an equiv alence, and therefore  𝒞 op , よ 𝑑  is right Grothendieck – V erdier. By Proposition 3.41 t he right inter nal hom of tw o representable functors よ 𝑥 and よ 𝑦 is [ よ 𝑥 , よ 𝑦 ] 𝑟  よ [ 𝑥 , 𝑦 ] 𝑟 . Thus, t he con v erse of an y of the three statements is a consequence of t he fact t hat, via t he Y oneda embedding, 𝒞 op is equiv alent as a right closed monoidal categor y to t he full subcategor y of 𝒞 op whose objects are representable functors. □ Remark 3.44. The linearisation k 𝒲 of t he tensor representable category 𝒲 discussed in Theorem 3.23 is tensor representable, with left and right dualising functors 𝐿 and 𝑅 , but not rigid. The latter follo ws from t he fact t hat b y the proof of Theorem 3.23 t here exists an object 𝑡 ∈ 𝒲 whose canonical morphism [ 𝑡 , 1 ] 𝑟 ⊗ 𝑡 − → [ 𝑡 , 𝑡 ] 𝑟 is not in v ertible. Since the right dualising functor 𝑅 of 𝒲 is an anti-equivalence, and so 𝐿  𝑅 − 1 , t here also is a tensor representable, but not rigid, structure on k 𝒲 op . One takes t he left dualising functor to be 𝑅 , and the right one to be 𝐿 . By Corollary 3.43 , k 𝒲 op is tensor representable but not rigid. N otice t hat t his is in star k contras t to many cases arising in representation theor y , like modules ov er commutativ e rings or k -algebr as, where rigidity and tensor representability are equiv alent ; see for exam ple [ NW17 , Proposition 2.1] for a slightly more general statement. Since a k -algebr a can be inter preted as a k -linear category wit h one object, t he follo wing proposition ma y be seen as a “man y object” v ersion of t he classical case. Proposition 3.45. Let 𝒞 be a lef t rigid monoidal cat egor y . the following are equi- valent for some 𝑋 ∈  𝒞 op : (i) it has a right rigid dual, (ii) t here exists a D 𝑋 ∈  𝒞 op such t hat 𝑋 ∗ − :  𝒞 op ⇄  𝒞 op : D 𝑋 ∗ − , and (iii) it is ﬁnitel y-gener ated projectiv e. Proof. By deﬁnition, (i) = ⇒ (ii) . T o show that (ii) = ⇒ (iii) , w e assume t hat there exists a D 𝑋 ∈  𝒞 op such that 𝑋 ∗ − :  𝒞 op ⇄  𝒞 op : D 𝑋 ∗ − , and ﬁx a small 79 3. Du al ity t heory for m onoid a l ca t egor ies colimit colim 𝑖 ∈ 𝐼 𝐹 𝑖 ∈  𝒞 op of some diag r am 𝐹 : 𝐼 − →  𝒞 op . F or ev ery 𝑖 ∈ 𝐼 , write 𝜄 𝑖 : 𝐹 𝑖 − → colim 𝑖 ∈ 𝐼 𝐹 𝑖 ∈  𝒞 op for its structure mor phisms. No w consider the commutativ e diagram: colim 𝑖 ∈ 𝐼  𝒞 op ( 𝑋 , 𝐹 𝑖 )  𝒞 op ( 𝑋 , colim 𝑖 ∈ 𝐼 𝐹 𝑖 ) colim 𝑖 ∈ 𝐼  𝒞 op ( 1 , D 𝑋 ∗ 𝐹 𝑖 )  𝒞 op ( 1 , D 𝑋 ∗ colim 𝑖 ∈ 𝐼 𝐹 𝑖 ) colim 𝑖 ∈ 𝐼 (  𝒞 op ( 𝑋 , 𝜄 𝑖 )) colim 𝑖 ∈ 𝐼 𝜙 𝑋 , 𝐹 𝑖 𝜙 𝑋 , colim 𝑖 ∈ 𝐼 𝐹 𝑖 colim 𝑖 ∈ 𝐼 (  𝒞 op ( 1 , D 𝑋 ∗ 𝜄 𝑖 )) Its horizontal arrow s are induced by the univ ersal property of t he colimit and t he v ertical arrow s are due t o the tensor– hom adjunction of  𝒞 op . The functors  𝒞 op ( 1 , −) and D 𝑋 ⊗ − commute with all small colimits; the ﬁrst one due to t he fact t hat  𝒞 op ( 1 , −) is ﬁnitel y-gener ated projectiv e, see Lemma 3.39 , and t he second one due to Da y con v olution being a colimit. Therefore, the horizontal arrow at the bottom is in v ertible. Since t he v ertical arrow s are also in v ertible, the canonical arrow colim 𝑖 ∈ 𝐼  𝒞 op ( 𝑋 , 𝐹 𝑖 ) − →  𝒞 op ( 𝑋 , colim 𝑖 ∈ 𝐼 𝐹 𝑖 ) displa y ed at the top of t he diagram must be an isomorphism. A gain, using Lemma 3.39 , 𝑋 must be ﬁnitely-g enerated projectiv e. Thus, (ii) implies (iii) . Finall y , if 𝑋 ∈  𝒞 op is ﬁnitely-g enerated projectiv e, then it is contained in the Cauchy -completion of 𝒞 op , which is right rigid. Then, by Corollar y 3.43 , so is 𝒞 op . Therefore 𝑋 admits a rigid dual and (iii) = ⇒ (i) . □ 3 . 3 a p p l i c at i o n s We c o ncl ude the chap ter by d iscus sing sev eral examples. W e inv estigate Boolean alg ebras, t heir applications in group and ring t he- ory , and how t he y induce abelian k -linear Grothendieck –V erdier categories. N ext, w e focus on Macke y functors. These are, roughl y speaking, col- lections of v ector spaces indexed b y all subg roups of a ﬁxed ﬁnite g roup, tog ether with morphisms subject to relations resembling t he behaviour of induction, restriction, and conjugation operations, including t he eponymous Macke y identity; see [ Lin76 ; TW95 ]. Finite-dimensional Mackey functors form a Grothendieck – V erdier categor y [ PS07 ]; w e show in Proposition 3.57 that it is rigid if and only if it is semisimple. The last example arises in the study of crossed modules, which — in cat- egorical terms — correspond to strict 2-g roups. The functors from any ﬁnite 80 3.3. Applications strict 2-g roup to vect form an abelian monoidal category that is equiv alent to a direct sum of representation categories of t he isotrop y -group of the monoidal unit of the 2-g roup. In Proposition 3.66 w e prov e t he rigidity of this categor y to be equivalent to t he semisimplicity of a certain g roup algebr a. 3.3.1 Boolean algebr as Deﬁnition 3.46. A lattice consists of a set 𝐿 tog et her wit h tw o associativ e and commutativ e operations ∧ : 𝐿 × 𝐿 − → 𝐿 , ( 𝑎 , 𝑏 ) ↦− → 𝑎 ∧ 𝑏 and ∨ : 𝐿 × 𝐿 − → 𝐿 , ( 𝑎 , 𝑏 ) ↦− → 𝑎 ∨ 𝑏 , called meet and join , which satisfy the absor p tion laws , for all 𝑎 , 𝑏 ∈ 𝐿 : 𝑎 ∨ ( 𝑎 ∧ 𝑏 ) = 𝑎 , 𝑎 ∧ ( 𝑎 ∨ 𝑏 ) = 𝑎 . Remark 3.47. An y lattice ( 𝐿 , ∧ , ∨) deﬁnes a poset via t he relation 𝑎 ≤ 𝑏 ⇐ ⇒ 𝑏 = 𝑎 ∨ 𝑏 ⇐ ⇒ 𝑎 = 𝑎 ∧ 𝑏 , 𝑎 , 𝑏 ∈ 𝐿 . Con v ersely , a poset ( 𝑃 , ≤ ) that admits for an y pair of objects 𝑎 , 𝑏 ∈ 𝑃 a least upper bound 𝑎 ∨ 𝑏 ∈ 𝑃 and a g reates t lo w er bound 𝑎 ∧ 𝑏 ∈ 𝑃 is a lattice. A direct computation show s t hat an element 1 ∈ 𝐿 is maximal wit h respect to t he partial order of t he lattice 𝐿 if and onl y if 1 ∨ 𝑎 = 1 for all 𝑎 ∈ 𝐿 . Analogousl y , 0 ∈ 𝐿 being minimal equates to 0 ∧ 𝑎 = 0 for an y 𝑎 ∈ 𝐿 . Minimal and maximal elements are unique. In case they exist, w e call 𝐿 bounded . Deﬁnition 3.48. A Boolean alg ebra is a bounded lattice ( 𝐿 , ∧ , ∨) satisfying the follo wing distributivity condition for all 𝑎 , 𝑏 , 𝑐 ∈ 𝐿 : 𝑎 ∧ ( 𝑏 ∨ 𝑐 ) = ( 𝑎 ∧ 𝑏 ) ∨ ( 𝑎 ∧ 𝑐 ) , 𝑎 ∨ ( 𝑏 ∧ 𝑐 ) = ( 𝑎 ∧ 𝑏 ) ∨ ( 𝑎 ∧ 𝑐 ) , (3.3.1) and ev ery 𝑎 ∈ 𝐿 admits a complement 𝑎 ⊥ ∈ 𝐿 in t he sense t hat 𝑎 ∨ 𝑎 ⊥ = 1 and 𝑎 ∧ 𝑎 ⊥ = 0 . Remark 3.49. F or an y Boolean algebra ( 𝐿 , ∧ , ∨) , an y of t he tw o distributivity requirements of Equation ( 3.3.1 ) implies t he other . Further , a direct compu- tation sho ws that complements are unique. W e obtain an in v olutiv e map (−) ⊥ : 𝐿 − → 𝐿 t hat maps 𝑎 to 𝑎 ⊥ . The maximal element 1 ∈ 𝐿 is a unit for ∧ : 𝑎 ∧ 1 = 𝑎 ∧ ( 𝑎 ∨ 𝑎 ⊥ ) = 𝑎 , for all 𝑎 ∈ 𝐿 . An analogous computation show s that t he minimal element 0 in a Boolean algebr a satisﬁes 𝑎 ∨ 0 = 𝑎 . 81 3. Du al ity t heory for m onoid a l ca t egor ies Exam ple 3.50. • Centr al idem pot ents : the set 𝐶 . . = { 𝑒 ∈ 𝑍 ( 𝑅 ) | 𝑒 2 = 𝑒 } of centr al idem- potents of a ring 𝑅 is a Boolean algebra when endow ed wit h ∧ : 𝐶 × 𝐶 − → 𝐶 𝑒 ∧ 𝑓 = 𝑒 𝑓 , ∨ : 𝐶 × 𝐶 − → 𝐶 𝑒 ∨ 𝑓 = 𝑒 + 𝑓 − 𝑒 𝑓 , (−) ⊥ : 𝐶 − → 𝐶 𝑒 ⊥ = 1 − 𝑒 . • Annihilator s in semiprime rings : consider a commutativ e semiprime ring 𝑅 with 1 . That is, its Jacobson radical is trivial. In case 𝑅 is furt hermore Artinean, this is equivalent to it being semisimple. The annihilator of an ideal 𝐼 ⊂ 𝑅 is 𝐼 ⊥ . . = { 𝑥 ∈ 𝑅 | 𝑥 𝐼 = 0 } ; it is a radical ideal. W e deﬁne on the set Ann ( 𝑅 ) of annihilators t he maps ∧ : Ann ( 𝑅 ) × Ann ( 𝑅 ) − → Ann ( 𝑅 ) , 𝐼 ∧ 𝐽 = 𝐼 ∩ 𝐽 , ∨ : Ann ( 𝑅 ) × Ann ( 𝑅 ) − → Ann ( 𝑅 ) , 𝐼 ∨ 𝐽 = ( 𝐼 + 𝐽 ) ⊥ . This deﬁnes t he structure of a Boolean algebr a on Ann ( 𝑅 ) . In fact, t he (right) annihilators of a no t necessarily commutativ e ring form a Boolean algebr a if and only if t he ring is semiprime, see [ DT21 ]. • The subgr oup lattice : let 𝐻 ⊆ 𝐺 be a subg roup of a ﬁnite group. W e write 𝐻 ⊥ for the intersection of all maximal subg roups that do not contain 𝐻 . The minimal g roup constructed in t his manner is the Fr attini subgroup Φ ( 𝐺 ) = 𝐺 ⊥ of 𝐺 . Deaconescu, Isaacs, and W all show ed that t he set { Φ ( 𝐺 ) ⊆ 𝐻 ⊆ 𝐺 | 𝐺 = 𝐻 𝐻 ⊥ } , partiall y ordered under inclusion forms a Boolean algebr a, see [ DIW11 ]. Barr show ed in [ Bar79 ] t hat t he duality of Boolean algebr as ﬁts wit hin t he frame w ork of Grothendieck – V erdier categories. Proposition 3.51. Let ( 𝐿 , ∧ , ∨) be a Boolean alg ebra. The associated poset-category ℒ is a left Gro t hendieck – V er dier cat egor y with meet as tensor product, 1 as monoidal unit, and 0 as dualising object. The dualising functor is induced by (−) ⊥ . If 𝐿 is ﬁnite, the category ⟨ ℒ , vect ⟩ of or dinary funct ors from ℒ to t he category of ﬁnite-dimensional k -v ector spaces can be equipped wit h a right Gr ot hendieck – V er dier structur e, with k ℒ (− , 𝑠 ) ∗ as dualising object and dualising functor D 𝑟 : ⟨ ℒ , vect ⟩ op − → ⟨ ℒ , vect ⟩ , 𝐹 ↦− → 𝐹 (− ⊥ ) ∗ . 82 3.3. Applications Proof. A direct consequence of Equation ( 3.3.1 ) is that f or 𝑎 ≤ 𝑐 and 𝑏 ≤ 𝑑 , w e hav e 𝑎 ∧ 𝑏 ≤ 𝑐 ∧ 𝑑 . Thus, since ∧ : 𝐿 × 𝐿 − → 𝐿 is associativ e, commutativ e, and unital, it induces a (symmetric) monoidal structure on ℒ . T o see t hat ( ℒ , 0 ) is a Grothendieck – V erdier category , w e com pute its lef t internal hom. Let us ﬁx a triple of elements 𝑎 , 𝑏 , 𝑐 ∈ 𝐿 . W e ha v e 𝑏 ∧ 𝑎 ≤ 𝑐 ⇐ ⇒ ( 𝑏 ∧ 𝑎 ) ∨ 𝑎 ⊥ ≤ 𝑐 ∨ 𝑎 ⊥ ⇐ ⇒ ( 𝑏 ∨ 𝑎 ⊥ ) ∧ ( 𝑎 ∨ 𝑎 ⊥ ) ≤ 𝑐 ∨ 𝑎 ⊥ ⇐ ⇒ 𝑏 ∨ 𝑎 ⊥ ≤ 𝑐 ∨ 𝑎 ⊥ ⇐ ⇒ 𝑏 ≤ 𝑐 ∨ 𝑎 ⊥ , where t he last equivalence is due to t he fact t hat 𝑏 = 𝑏 ∨ 0 ≤ 𝑏 ∨ 𝑎 ⊥ and 𝑏 ≤ 𝑐 ∨ 𝑎 ⊥ , impl ying t hat 𝑏 ∨ 𝑎 ⊥ ≤ ( 𝑐 ∨ 𝑎 ⊥ ) ∨ 𝑎 ⊥ = 𝑐 ∨ 𝑎 ⊥ . It follo ws t hat [− , = ] ℓ : 𝐿 × 𝐿 − → 𝐿 , [ 𝑎 , 𝑏 ] ℓ . . = ( 𝑎 ∧ 𝑏 ⊥ ) ⊥ = 𝑏 ∨ 𝑎 ⊥ , deﬁnes the lef t inter nal hom of ℒ , and the order rev ersing in v olution [− , 0 ] ℓ = (−) ⊥ : 𝐿 − → 𝐿 induces an equiv alence of categories 𝐷 ℓ : ℒ op − → ℒ . As discussed in Re- mar k 3.13 , ( ℒ , 0 ) is a left Grothendieck – V erdier categor y . By Example 2.82 , w e ha v e an equivalence ⟨ ℒ , vect ⟩  [ k ℒ , vect ] . U nder the assump tion that 𝐿 is ﬁnite, Proposition 3.27 shows that ⟨ ℒ , vect ⟩ has the structure of a right Grothendieck – V erdier category wit h the speciﬁed dualising object and dualising functor . □ Suppose 𝐿 is a ﬁnite Boolean algebr a and ℒ is its poset-category . As discussed in t he ﬁrst steps of t he proof of Lemma 3.35 , the categor y [ k ℒ , vect ] can be identiﬁed with t he ﬁnite-dimensional right modules of the path algebra 𝐴   𝑒 , 𝑓 ∈ 𝐿 k ℒ ( 𝑒 , 𝑓 ) of t he poset associated to 𝐿 . Exam ple 3.52. The partial order on the set 𝐿 . . = { 0 , 𝑎 , 𝑏 , 1 } displa y ed in the follo wing diag r am deﬁnes a Boolean algebr a: 𝑎 0 1 𝑏 𝑣 𝑢 𝑥 𝑦 83 3. Du al ity t heory for m onoid a l ca t egor ies A direct computation show s that its path algebra 𝐴 is isomorphic to a subal- gebr a of t he k -v alued 4 × 4 upper triangular matrices: 𝐴                  0 𝑢 𝑥 𝑧 0 𝑎 0 𝑣 0 0 𝑏 𝑦 0 0 0 1              0 , 𝑢 , 𝑥 , 𝑧 , 𝑎 , 𝑣 , 𝑏 , 𝑦 , 1 ∈ k            . Subalgebr as of upper triangular matrices w ere studied by Thrall in [ Thr48 ] to provide generalisations of quasi-Frobenius algebr as. 9 The follo wing deﬁn- 9 A k -algebr a is quasi-F robenius if all projectiv e right modules are injectiv e. ition gener alises t his property . Deﬁnition 3.53. A k -algebr a is called qf -2 if all indecomposable projectiv e right and all indecomposable projectiv e lef t modules ha v e simple socles. Using the Grothendieck – V erdier structure of Proposition 3.51 , w e will sho w t hat t he pat h algebr a of an y ﬁnite Boolean algebra 𝐿 is qf -2. Hereto w e need the f ollo wing obser v ation. T aking com plements in 𝐿 extends to an anti- algebr a isomor phism 𝜙 : 𝐴 − → 𝐴 ; its pushforwar d (−) 𝜙 : mod- 𝐴 − → 𝐴 -mod is an in v olutiv e equivalence of categories. It maps an y left module 𝑀 to t he right module 𝑀 𝜙 that has the same under lying v ector space and is endow ed with t he action 𝑎 ⊲ 𝑚 . . = 𝑚 ⊳ 𝜙 ( 𝑎 ) for all 𝑚 ∈ 𝑀 and 𝑎 ∈ 𝐴 . Proposition 3.54. Let 𝐴 be the path algebr a of a ﬁnite Boolean algebr a 𝐿 . Then (− 𝜙 ) ∗ : ( mod- 𝐴 ) op − → mod- 𝐴 is an equivalence of cat egories. In particular , 𝑀 ∈ mod- 𝐴 is projectiv e if and onl y if 𝑀 𝜙 ∗ is injective. F urt her , 𝐴 is qf -2, and quasi-F robenius if and only if | 𝐿 | = 1 . Proof. The ﬁrst statement follo ws directly from t he equivalence of categories betw een [ k ℒ , vect ] and mod- 𝐴 giv en in t he proof of Lemma 3.35 , and the deﬁnition of the dualising functor of [ k ℒ , vect ] stated in Proposition 3.51 . In order to show the second claim, w e obser v e t hat t he ﬁnite-dimensional algebr a 𝐴 =  𝑛 ≥ 0 𝐴 𝑛 is g r aded b y the pat h lengt hs. The elements of 𝐿 form a basis of 𝐴 0 and correspond to the primitive idempot ents — non-zero idempo tents, which cannot be written as a sum of tw o non-zero ort hogonal idempo tents. Furthermore, the Jacobson radical of 𝐴 is 𝐽 ( 𝐴 ) =  𝑛 ≥ 1 𝐴 𝑛 . The indecomposable projectiv es of 𝐴 are of the form 𝑒 𝐴 , for 𝑒 ∈ 𝐿 . N ote t hat 𝑒 𝐴 has a v ector space basis b y paths [ 𝑒 , 𝑦 ] for 𝑒 ≤ 𝑦 ≤ 1 . The socle of 𝑒 𝐴 is soc ( 𝑒 𝐴 ) = { 𝑚 ∈ 𝑒 𝐴 | 𝑚 𝐽 ( 𝐴 ) = 0 } = span k { [ 𝑒 , 1 ]} , 84 3.3. Applications which is one-dimensional and t heref ore simple. Using t hat 𝐴 -mod  mod- 𝐴 , it follo ws t hat 𝐴 is a qf -2 algebr a. F or | 𝐿 | = 1 , w e obtain t he trivial (quasi-)Frobenius-alg ebra 𝐴 = k . Other- wise, t here exists an 𝑒 ∈ 𝐿 such that dim 𝑒 𝐴 ≥ 2 . A direct computation show s that soc ( 𝑒 𝐴 )  1 𝐴 . If 𝑒 𝐴 w as injectiv e, the inclusion 1 𝐴  soc ( 𝑒 𝐴 ) ↩ − → 𝑒 𝐴 w ould ha v e a retraction, contradicting t he indecomposability of 𝑒 𝐴 . □ 3.3.2 Mac ke y functor s Let 𝐺 be a fin ite group and write Sp 𝐺 for the categor y of isomorphism classes of spans of ﬁnite 𝐺 -sets. In t his section w e again implicitl y assume all categories and functors to be k -linear . One can show t hat each hom-set is a free and ﬁnitel y-gener ated commutativ e monoid, see for example the discussion preceding Lemma 2.1 of [ TW95 ]. Besides the deﬁnition of Mackey functors sketched at t he beginning of Section 3.3 , t here is a more succinct formulation due to Lindner [ Lin76 ]. Deﬁnition 3.55. Let k be a ﬁeld and 𝐺 a ﬁnite g roup. The categor y Mky k ( 𝐺 ) of Mackey functor s of 𝐺 is giv en by t he k -linear functor categor y [ k Sp 𝐺 , V ect ] . Examples of Mackey functors are plentiful: an y representation of 𝐺 deﬁnes one, see [ Thé95 , Exam ple 53.1] as w ell as [ PS07 , Propostion 10.1]. F or an extensiv e ov erview , w e refer t he reader to [ Thé95 , Chapter 53]. Exam ple 3.56. T o any ﬁnite group 𝐺 , one can associate a ﬁnite-dimensional al- gebr a M 𝐺 — t he Mackey alg ebra of 𝐺 — whose categor y of modules is equivalent to Mky k ( 𝐺 ) , see [ TW95 , Propositions 3.1 and 3.2]. Its ﬁnite-dimensional mod- ules are in correspondence wit h t he (pointwise) ﬁnite-dimensional Mackey functors, which w e denote by mky k ( 𝐺 ) . Giv en that k Sp 𝐺 is symmetric monoidal, Mky k ( 𝐺 ) is closed symmetric monoidal when equipped wit h Da y con v olution as its tensor product. 10 Fur - 10 As such, w e refrain from adding “left” or “right” preﬁxes to t he diﬀerent duality notions when talking about Mackey functors. ther more, k Sp 𝐺 has a ﬁnite dense subcategor y whose objects form a com plete set of representatives of tr ansitiv e 𝐺 -sets. The ar guments of Exam ple 3.26 ma y now be used t o show t hat t he tensor product and inter nal hom of ﬁnite- dimensional Macke y functors is ﬁnite-dimensional, see [ PS07 , Section 9]. The follo wing results gives an alternativ e proof for the sketched argu- ment in [ Bou05 , Lemma 2.2] t hat the classes of rigid and ﬁnitel y-gener ated projectiv e Mackey functors coincide. 85 3. Du al ity t heory for m onoid a l ca t egor ies Proposition 3.57. Let 𝐺 be a ﬁnit e group and suppose k is a ﬁeld. The cat egor y mky k ( 𝐺 ) is a Grot hendieck – V erdier category with mky k (− , 1 ) ∗ as dualising object. Its rigid objects ar e precisel y the ﬁnitel y-gener ated projective Mackey functor s. F urthermor e, mky k ( 𝐺 ) is rigid itself if and only if M 𝐺 is semisimple, which is equiv alent to char k not dividing t he or der of 𝐺 . Proof. The fact t hat mky k ( 𝐺 ) is a Grothendieck – V erdier category w as prov en in [ PS07 , Theorem 9.2]. Alternativ ely , w e ma y use t hat k Sp 𝐺 is a rigid categor y , see [ PS07 , Section 2], and apply Proposition 3.27 to obtain that mky k ( 𝐺 ) is a Grothendieck – V erdier categor y wit h mky k (− , 1 ) ∗ ∈ mky k ( 𝐺 ) as its dualising object. Due to Proposition 3.45 , a Mackey functor is rigid dualisable if and onl y if it is ﬁnitely-g enerated projective. N o w let us assume mky k ( 𝐺 ) is rigid and recall that it is equivalent to the categor y M 𝐺 -mod of ﬁnite-dimensional modules of t he Mackey algebr a. Since M 𝐺 is ﬁnite-dimensional and ev ery object in mky k 𝐺 is projectiv e by Proposition 3.45 , an y submodule of M 𝐺 must ha v e a com plement. In partic- ular , M 𝐺 is semisimple. Con v ersely , in case M 𝐺 is semisimple, all objects of M 𝐺 -mod , and therefore also mky k ( 𝐺 ) , are projectiv e. As they are furt hermore ﬁnitel y-gener ated, Proposition 3.45 implies the rigidity of mky k ( 𝐺 ) . Corollary 14.4 of [ TW95 ], sho ws t hat M 𝐺 is semisimple if char k does not divide | 𝐺 | . On t he other hand, t he categor y Rep k ( 𝐺 ) of ﬁnite-dimensional representations of 𝐺 o v er k is a full subcategor y of mky k ( 𝐺 ) , see [ PS07 , Pro- position 10.1]. Thus, if mky k ( 𝐺 ) is semisimple, so is Rep k ( 𝐺 ) , impl ying t hat char k does not divide | 𝐺 | by Maschke’ s t heorem. □ 3.3.3 Cr ossed modules Anoth er ex ampl e for Grothe ndie ck – Ver dier structu res on abelian k - linear functor categories arises from studying (strict) 2-g roups, which can be identiﬁed with crossed modules. Our exposition follo ws [ W ag21 ]. Deﬁnition 3.58. A strict 2 -group is a (small) groupoid 𝒢 endo w ed with a strict monoidal structure, such that the monoid of objects ( Ob ( 𝒢 ) , ⊗ , 1 ) is a g roup. Ev er y strict 2-group is a rigid categor y ; t he left and right dual of an object 𝑔 ∈ 𝒢 is giv en by its in v erse 𝑔 − 1 ∈ 𝒢 . Let 𝐺 be t he group of objects of 𝒢 and 𝐻 = 𝒢 ( 1 , −) the set of arro ws which start at t he monoidal unit 1 ∈ 𝐺 . W riting 𝑡 : 𝐻 − → 𝐺 for t he tar g et map , w e deﬁne a group structure on 𝐻 via t he multiplication ℎ ′ ℎ . . = ( ℎ ′ ⊗ id 𝑡 ( ℎ ) ) ◦ ℎ , for all ℎ , ℎ ′ ∈ 𝐻 . 86 3.3. Applications F or an y 𝑔 ∈ 𝐺 and ℎ ∈ 𝐻 there is a unique ℎ ′ ∈ 𝐻 such t hat id 𝑔 ⊗ ℎ = ℎ ′ ⊗ id 𝑔 . This induces a map 𝛼 : 𝐺 − → A ut ( 𝐻 ) and turns the quadruple ( 𝐺 , 𝐻 , 𝑡 , 𝛼 ) into a crossed module as deﬁned below . Deﬁnition 3.59. A crossed module is a quadr uple ( 𝐺 , 𝐻 , 𝑡 , 𝛼 ) consisting of tw o groups 𝐺 , 𝐻 and tw o g roup homomor phisms 𝑡 : 𝐻 − → 𝐺 , 𝛼 : 𝐺 − → A ut ( 𝐻 ) , such that for all 𝑔 ∈ 𝐺 and ℎ , 𝑙 ∈ 𝐻 , w e hav e 𝑡 ( 𝛼 ( 𝑔 ) ℎ ) = 𝑔 ◦ 𝑡 ( ℎ ) ◦ 𝑔 − 1 , 𝛼 ( 𝑡 ( 𝑙 )) ℎ = 𝑙 ◦ ℎ ◦ 𝑙 − 1 . (3.3.2) Remark 3.60. An y crossed module ( 𝐺 , 𝐻 , 𝑡 , 𝛼 ) deﬁnes a strict 2 -group 𝒢 , with Ob 𝒢 = 𝐺 , and 𝒢 ( 𝑔 , 𝑔 ′ ) = { ( ℎ , 𝑔 ) ∈ 𝐻 × 𝐺 | 𝑡 ( ℎ ) 𝑔 = 𝑔 ′ } . Composition is giv en by ( ℎ ′ , 𝑡 ( ℎ ) 𝑔 ) ◦ ( ℎ , 𝑔 ) . . = ( ℎ ′ ℎ , 𝑔 ) , and for t he tensor product w e ha v e ( ℎ , 𝑔 ) ⊗ ( ℎ ′ , 𝑔 ′ ) . . = ( ℎ 𝛼 ( 𝑔 ) ℎ ′ , 𝑔 𝑔 ′ ) . The left and right dualising functors coincide; t he y map an y object 𝑏 ∈ 𝐺 to ∨ 𝑏 = 𝑏 − 1 = 𝑏 ∨ . F or a morphism 𝑓 = ( ℎ , 𝑏 ) : 𝑏 − → 𝑐 , w e hav e ∨ 𝑓 = ( 𝛼 ( 𝑐 − 1 ) ℎ , 𝑐 − 1 ) = ( 𝛼 ( 𝑏 − 1 ) ℎ , 𝑐 − 1 ) = 𝑓 ∨ . There is an equivalence of categories betw een crossed modules and strict 2-groups as is shown for example in [ W ag21 , Theorem 1.9.2]. Exam ple 3.61. • Hopf –Galois extensions and skew br aces : the holomorph of a group 𝐻 is the semidirect product 𝐻 ⋊ A ut ( 𝐻 ) . The tw o group homomor phisms 𝑡 : 𝐻 − → A ut ( 𝐻 ) , 𝑡 ( 𝑙 ) ℎ = 𝑙 ℎ 𝑙 − 1 , and 𝛼 = id : A ut ( 𝐻 ) − → A ut ( 𝐻 ) turn ( A ut ( 𝐻 ) , 𝐻 , 𝑡 , id A ut ( 𝐻 ) ) into a crossed module. As explained in t he In- troduction of [ By o24 ] and [ Bac16 , Section 2], holomor phs can be used to classify certain Hopf – Galois extensions as w ell as skew -braces. • Repr esentations of ﬁnite groups : a short exact sequence of g roups 0 𝐶 𝐸 𝐺 0 𝜄 𝑡 such that 𝐶 embeds into t he centre of 𝐸 is called a central ext ension of 𝐺 . Let 𝜅 : 𝐺 − → 𝐸 be a set-t heoretical section of 𝑡 : 𝐸 − → 𝐺 . A direct com putation show s t hat the map 𝛼 : 𝐺 − → A ut ( 𝐸 ) deﬁned b y 𝛼 ( 𝑔 ) 𝑒 . . = 𝜅 ( 𝑔 ) 𝑒 𝜅 ( 𝑔 ) − 1 is independent of the choice of the section and a homomorphism of g roups. The tuple ( 𝐸 , 𝐺 , 𝑡 , 𝛼 ) forms a crossed module, see [ W ag21 , Section 1.3]. 87 3. Du al ity t heory for m onoid a l ca t egor ies As discussed in [ HH92 , Chapter 1], if 𝐺 is a ﬁnite group, t hen an y projectiv e representation 𝜌 : 𝐺 − → PGL C ( 𝑉 ) of 𝐺 can be lifted to a linear representation 𝜚 : 𝐸 − → GL ( 𝑉 ) of a certain central extension 𝐸 of 𝐺 . Projectiv e representations t hemsel v es are studied in the context of representation t heory of semidirect products, see for example [ CSST22 ]. Let ( 𝐺 , 𝐻 , 𝑡 , 𝛼 ) be a crossed module whose associated 2 -group w e denote b y 𝒢 . Using t he results of Section 3.2 , we determine a Grothendieck – V erdier structure on t he categor y ⟨ 𝒢 , vect ⟩ of ordinary functors betw een 𝒢 and vect . Hereto, w e need to analyse the structure of 𝒢 in more detail. By Equa- tion ( 3.3.2 ), the image 𝐾 . . = im 𝑡 ⊂ 𝐺 is a nor mal subg roup of 𝐺 and t he ker nel 𝐿 . . = ker 𝑡 is normal and central in 𝐻 . The latter follo ws from the identity 𝑙 ℎ 𝑙 − 1 = 𝛼 ( 𝑡 ( 𝑙 )) ℎ = 𝛼 ( 𝑒 ) ℎ = ℎ for all 𝑙 ∈ 𝐿 , ℎ ∈ 𝐻 . F or an y 𝑘 = 𝑡 ( ℎ ) ∈ 𝐾 and 𝑙 ∈ 𝐿 , w e g et 𝛼 ( 𝑘 ) 𝑙 = 𝛼 ( 𝑡 ( ℎ )) 𝑙 = ℎ 𝑙 ℎ − 1 = 𝑙 . Hence, there is a unique g roup homomor phism 𝛼 : 𝑄 . . = 𝐺 / 𝑁 − → A ut ( 𝐿 ) , wit h 𝛼 ([ 𝑔 ]) 𝑙 = 𝛼 ( 𝑔 ) 𝑙 ∈ 𝐿 ⊂ 𝐻 . for all 𝑔 ∈ 𝐺 and 𝑙 ∈ 𝐿 . The connected components of 𝒢 are in bĳection wit h the elements of 𝑄 . Giv en an y element 𝑔 ∈ 𝐺 , w e write 𝒢 𝑔 for t he maximal full connected pointed subgroupoid of 𝒢 whose distinguished object is 𝑔 . The set of objects of 𝒢 𝑔 is 𝐾 𝑔 and its morphisms correspond to 𝐻 × 𝐾 𝑔 . Lemma 3.62. Let 𝑔 ∈ 𝐺 and write B 𝐿 for the delooping of 𝐿 with single object 𝑔 . The canonical inclusion Θ 𝑔 : B 𝐿 ↩ − → 𝒢 𝑔 sending 𝑔 to 𝑔 and 𝑙 : 𝑔 − → 𝑔 to ( 𝑙 , 𝑔 ) : 𝑔 − → 𝑔 is essentially surjective and full y fait hful — an equivalence. T o specify a quasi-inv erse, ﬁx a set-t heoretical section 𝜄 : 𝐾 − → 𝐻 of t he sur jectiv e homomor phisms of g roups 𝑡 : 𝐻 − → 𝐾 . Deﬁne the quasi-in v erse Ψ 𝑔 : 𝒢 𝑔 − → B 𝐿 b y mapping each object to 𝑔 and an y morphism ( ℎ , 𝑘 𝑔 ) to 𝜄 (( 𝑡 ( ℎ ) 𝑘 ) − 1 ) ℎ 𝜄 ( 𝑘 ) . Using that ⟨ B 𝐿 , vect ⟩ can be identiﬁed with the category rep ( 𝐿 ) of ﬁnite-dimensional representations of 𝐿 , t he pushforw ards of Θ 𝑔 and Ψ 𝑔 establish an equiv alence of categories Θ ∗ 𝑔 : ⟨ 𝒢 𝑔 , vect ⟩ ⇄ rep ( 𝐿 ) : Ψ ∗ 𝑔 𝐹 ↦− → ( 𝐹 𝑔 , 𝜌 𝐹 ) (where 𝜌 𝐹 ( 𝑙 ) = 𝐹 ( 𝑙 , 𝑔 ) )  𝑘 𝑔 ↦− → 𝑀 ( ℎ 𝑘 , 𝑔 ) ↦− → 𝜌  𝜄 (( 𝑡 ( ℎ ) 𝑘 ) − 1 ) ℎ 𝜄 ( 𝑘 )   ← − [ ( 𝑀 , 𝜌 ) . (3.3.3) 88 3.3. Applications Notation 3.63. Deﬁne rep 𝒢 ( 𝐿 ) . . =  𝑞 ∈ 𝑄 rep ( 𝐿 ) 𝑞 , where rep ( 𝐿 ) 𝑞 = rep ( 𝐿 ) . Giv en some 𝑀 ∈ rep 𝒢 ( 𝐿 ) , w e write 𝑀 𝑞 for its the homogeneous component of degree 𝑞 ∈ 𝑄 . N ote t hat in case 𝐾 has ﬁnite index in 𝐺 , w e hav e ⟨ 𝒢 , vect ⟩   [ 𝑔 ]∈ 𝑄 ⟨ 𝒢 𝑔 , vect ⟩ . Lemma 3.64. Let 𝒢 be a strict 2 -group, and write ( 𝐺 , 𝐻 , 𝑡 , 𝛼 ) for the corr esponding crossed module. If 𝐾 = im 𝑡 ⊂ 𝐺 has ﬁnite-index, t hen rep 𝒢 ( 𝐿 ) ≃ ⟨ 𝒢 , vect ⟩ . The tools dev eloped in Section 3.2 allow us to endow ⟨ 𝒢 , vect ⟩ with the structure of a Grothendieck –V erdier category which w e will transf er to rep 𝒢 ( 𝐿 ) . The tensor product obtained in this manner will per mute the homogeneous components and “twist” the action of 𝐿 b y virtue of 𝛼 . In t his regar d, we introduce the follo wing notation: for any 𝑞 ∈ 𝑄 and 𝑀 ∈ rep ( 𝐿 ) , w e denote b y 𝑀 𝛼 ( 𝑞 ) the representation of 𝐿 whose under l ying v ector space is 𝑀 , endow ed with t he action 𝑙 ▶ 𝑚 = 𝛼 ( 𝑞 ) 𝑙 ⊲ 𝑚 for all 𝑙 ∈ 𝐿 and 𝑚 ∈ 𝑀 . Proposition 3.65. Suppose ( 𝐺 , 𝐻 , 𝑡 , 𝛼 ) is a crossed module wit h 𝐺 and 𝐻 ﬁnite and let 𝒢 be its associated strict 2 -group. The category rep 𝒢 ( 𝐿 ) is a right 𝑟 -cat egory. Its tensor product and dualising functor are deﬁned by the assignments 𝑀 ⊗ 𝑁 . . = ( 𝑀 ⊗ k 𝐿 𝑁 𝛼 ( 𝑝 − 1 ) ) ∈ rep 𝒢 ( 𝐿 ) 𝑝 𝑞 , 𝐷 𝑀 . . =  𝑀 𝛼 ( 𝑝 )  ∗ ∈ rep 𝒢 ( 𝐿 ) 𝑝 − 1 for 𝑝 , 𝑞 ∈ 𝑄 and 𝑀 ∈ rep 𝒢 ( 𝐿 ) 𝑝 , 𝑁 ∈ rep 𝒢 ( 𝐿 ) 𝑞 . Proof. By Proposition 3.27 , ⟨ 𝒢 , vect ⟩  [ k 𝒢 , vect ] can be endow ed wit h t he structure of a right Gro thendieck – V erdier categor y wit h k 𝒢 (− , 1 ) ∗ as dualising object ; write 𝑅 : ⟨ 𝒢 , vect ⟩ op − → ⟨ 𝒢 , vect ⟩ f or its dualising functor . By Lemma 3.64 , there are k -linear equiv alences  Θ : [ k 𝒢 , vect ] ⇄ rep 𝒢 ( 𝐿 ) :  Ψ that are deter mined b y t he pushforw ards of t he equivalences Θ 𝑔 : ⟨ 𝒢 𝑔 , vect ⟩ ⇄ B 𝐿 : Ψ 𝑔 , for each homogeneous component [ k 𝒢 [ 𝑔 ] , vect ] . Using t he formulas of Equa- tion ( 3.3.3 ), w e can therefore explicitl y transfer t he Grothendieck – V erdier structure of [ k 𝒢 , vect ] to rep 𝒢 ( 𝐿 ) . 89 3. Du al ity t heory for m onoid a l ca t egor ies The dualising object is mapped to Θ ( k 𝒢 (− , 1 ) ∗ ) = k 𝐿 ∗  k 𝐿 ∈ rep 𝒢 ( 𝐿 ) [ 1 ] , where we used that k 𝐿 is a Frobenius algebra for t he last equality . The tensor product and right dualising functor of rep 𝒢 ( 𝐿 ) are giv en b y ⊗ . . =  Θ ∗ (  Ψ ×  Ψ ) : rep 𝒢 ( 𝐿 ) × rep 𝒢 ( 𝐿 ) − → rep 𝒢 ( 𝐿 ) , 𝐷 . . =  Θ R  Ψ op : rep 𝒢 ( 𝐿 ) op − → rep 𝒢 ( 𝐿 ) . In order to compute it explicitly , w e consider tw o elements 𝑝 , 𝑞 ∈ 𝑄 as well as two representations ( 𝑀 , 𝜌 ) ∈ rep 𝒢 ( 𝐿 ) 𝑝 and ( 𝑁 , 𝜏 ) ∈ rep 𝒢 ( 𝐿 ) 𝑞 . The Da y con v olution of t he functors corresponding to 𝑀 and 𝑁 is computed via a certain colimit and  Θ , as an equivalence of categories, commutes wit h this colimit. Thus, b y Equation ( 2.8.4 ) and t he deﬁnition of coends, t he homogeneous component ( 𝑀 ⊗ 𝑁 ) 𝑥 of degree 𝑥 ∈ 𝑄 is t he coequaliser of  𝑟 ,𝑠 ∈ 𝑄 k 𝐿 𝑟 ,𝑠 ⊗ k 𝑀 𝑟 ⊗ k 𝑁 𝑠 − 1 𝑥  𝑟 ∈ 𝑄 𝑀 𝑟 ⊗ k 𝑁 𝑟 − 1 𝑥 , where 𝐿 𝑟 ,𝑠 = 𝐿 if 𝑟 = 𝑠 and ∅ other wise. Its parallel mor phisms are 𝑙 ⊗ k 𝑚 ⊗ k 𝑛 ↦− → 𝐹 𝜌 ( 𝑙 ) 𝑚 ⊗ k 𝑛 , 𝑙 ⊗ k 𝑚 ⊗ k 𝑛 ↦− → 𝑚 ⊗ 𝜏 ( 𝑙 ∨ ) 𝑛 = 𝑚 ⊗ 𝜏 ( 𝛼 ( 𝑠 − 1 ) 𝑙 ) 𝑛 , where 𝑙 ∈ 𝐿 𝑟 ,𝑠 , 𝑚 ∈ 𝑀 𝑟 and 𝑛 ∈ 𝑁 𝑠 − 1 𝑥 . In order to compute 𝐷 , w e note that R ( 𝐹 ) = 𝐹 ( ∨ −) ∗ for an y 𝐹 ∈ [ k 𝒢 , vect ] . Thus, giv en 𝑥 ∈ 𝑄 , w e compute 𝐷 𝑀 𝑥 =  𝑀 ∗ 𝑥 − 1 = 𝑝 , { 0 } other wise . Hence, t he action of an y 𝑙 ∈ 𝐿 on 𝑀 ∗ is giv en by  𝜌 ( 𝛼 ( 𝑝 ) 𝑙 )  ∗ : 𝑀 ∗ 𝑝 − → 𝑀 ∗ 𝑝 . □ Proposition 3.66. Let ( 𝐺 , 𝐻 , 𝛼 , 𝑡 ) be a cr ossed module with 𝐺 and 𝐻 ﬁnite and write 𝒢 for its associated s trict 2-group. The category rep 𝒢 ( 𝐿 ) is rigid if and only if char k does not divide t he or der of 𝐿 = k er 𝑡 . Proof. It follo ws from Proposition 3.45 t hat rep 𝒢 ( 𝐿 ) is rigid if and only if all of its objects are ﬁnitely-g enerated and projectiv e. As rep 𝒢 ( 𝐿 ) is a direct sum of ﬁnitel y man y copies of rep ( 𝐿 ) and ev ery object of rep ( 𝐿 ) is ﬁnitel y-gener ated, this is the case if and onl y if all objects of rep ( 𝐿 ) are projectiv e. The latter is equivalent to k 𝐿 being semisimple, which corresponds t o char k and | 𝐿 | being coprime b y Maschke’ s t heorem. □ 90 Ich gehe am Str and, dudududu; ich gehe am Str and, r ududu; ich gehe am Str and, dudududuu; ich gehe am Str and, r uududuu. Spo n ge Bob S chw a mmk op f , S01E18b T W I S T E D C E N T R E S 4 A p ecul iari ty of H opf- cy cl ic coh omology in t he sense of Connes and Mosco vici is the lack of “canonical” coeﬃcients [ CM99 ]. Originally , modular pairs in in v olution w ere considered [ CM00 ]. These consist of a g roup-like ele- ment and a character of the Hopf algebr a under consideration im plementing the square of t he antipode b y t heir respectiv e adjoint actions. Later , Hajac, Khalkhali, Rangipour , and Sommerhäuser obtained a more general source of coeﬃcients in the categor y of anti-Y etter – Drinf eld modules, [ HKRS04 ]. As mentioned in Exam ple 2.42 , anti-Y etter –Drinfeld modules do not form a monoidal categor y , but rather a module category ov er t he Y etter –Drinfeld modules. This is reﬂected by the fact that t he y can be identiﬁed wit h the modules ov er t he anti-Drinfeld double, a comodule alg ebra o v er t he Drinfeld double, see Remar k 2.79 . The special role of pairs in inv olution is captured b y the follo wing theorem due to Hajac and Sommerhäuser . Theorem 4.1 ([ Hal21 , Theorem 3.4]) . F or a ﬁnite-dimensional Hopf alg ebra 𝐻 , t he following st atements are equiv alent: (i) The Hopf algebr a 𝐻 admits a pair in involution. (ii) Ther e exists a one-dimensional anti-Y etter – Drinfeld module over 𝐻 . (iii) The Drinfeld double and anti-Drinfeld double of 𝐻 are isomor phic algebr as. P airs in inv olution are of categorical interest because t he y give rise to piv otal structures on t he Y etter– Drinf eld modules. Using t he t heory of heaps, certain alg ebraic structures equipped with a ternar y operation, w e can lift the Picard g roup of a space into the Picard heap of a categor y . These can be seen as an analogue of the classical pairs in in v olution. Further abstr acting the anti- Y etter – Drinf eld modules into the anti-centre, w e can reformulate Theorem 4.1 in more categorical terms, emphasising t he role of piv otal structures. 91 4. Twist ed ce ntres Theorem 4.23 . In Theorem 6.44 w e shall giv e a monadic interpretation of this result. Let 𝒞 be a rigid category . There is a bĳection between (i) eq uivalence classes of quasi-pivo tal structur es on 𝒞 , (ii) t he Picard heap of the anti-centre, and (iii) isomorphism classes of equiv alences of module categories between t he centre and t he anti-centre. Further inv estig ating the heap structure of t he anti-centre of 𝒞 , w e study the natural injectiv e map 𝜅 : Pic A ( 𝒞 ) − → Piv Z ( 𝒞 ) , see Theorem 4.33 , ﬁrst introduced b y Shimizu [ Shi19 ]. By [ Shi23 a , Theorem 4.1], t his arrow is alwa ys bĳectiv e if 𝒞 is a ﬁnite tensor category . Ho w ev er , w e show t hat this fails to be true in gener al, conﬁrming a conjecture raised in the introduction of ibid . Theorem 4.50 . Ther e exis ts a pivot al structur e on a category 𝒞 that is not induced by an y element of the Picard heap of t he anti-centre of 𝒞 . In ot her wor ds, the map 𝜅 : Pic A ( 𝒞 ) − → Piv Z ( 𝒞 ) is not surjective. 4 . 1 h e a p s Hea ps ca n b e tho ught of as groups without a ﬁxed neutral element. They w ere ﬁrst studied under t he name Schar , [ Prü24 ; Bae29 ]. Recentl y , t heir homological properties w ere studied in [ ESZ21 ], and a gener alisation to w ards a “quantum v ersion” is hinted at in [ Ško07 ]. Heaps are equiv alent to 𝐺 - tor sors — nonemp ty sets wit h a freely transitiv e 𝐺 -action — for a g roup 𝐺 , see for example [ Gir71 ]. W e follow Section 2 of [ Brz20 ] for our exposition. Deﬁnition 4.2. A heap is a set 𝐺 together wit h a ter nary operation ⟨− , = , ≡⟩ : 𝐺 × 𝐺 × 𝐺 − → 𝐺 , which w e call t he heap operation , satisfying a generalised associativity axiom and the Mal’cev identities , which w e think of as unitality axioms: ⟨ 𝑔 , ℎ , ⟨ 𝑖 , 𝑗 , 𝑘 ⟩ ⟩ = ⟨⟨ 𝑔 , ℎ , 𝑖 ⟩ , 𝑗 , 𝑘 ⟩ , f or all 𝑔 , ℎ , 𝑖 , 𝑗 , 𝑘 ∈ 𝐺 , (4.1.1) ⟨ 𝑔 , 𝑔 , ℎ ⟩ = ℎ = ⟨ ℎ , 𝑔 , 𝑔 ⟩ , for all 𝑔 , ℎ ∈ 𝐺 . (4.1.2) There are tw o peculiarities w e w ould like to point out. Firs t, our deﬁnition does intentionall y not ex clude t he emp ty set from being a heap. Second, due to a slightl y diﬀ erent setup, an additional “middle associativity axiom” is required in [ HL17 ]. How ev er , as noted in [ Brz20 , Lemma 2.3], t his is implied b y Equations ( 4.1.1 ) and ( 4.1.2 ). 92 4.1. Heaps Deﬁnition 4.3. A map 𝑓 : 𝐺 − → 𝐻 betw een heaps is a mor phism of heaps if 𝑓  ⟨ 𝑔 , ℎ , 𝑖 ⟩  = ⟨ 𝑓 ( 𝑔 ) , 𝑓 ( ℎ ) , 𝑓 ( 𝑖 )⟩ , for all 𝑔 , ℎ , 𝑖 ∈ 𝐺 . The next lemma can be sho wn analogously to its g roup-theoretical v ersion. Lemma 4.4. A morphism of heaps is an isomorphism if and only if it is bĳective. By forg etting its unit, an y g roup deﬁnes a heap, Con v ersely , an y non-emp ty heap can be turned into a group by choosing a unit, see [ Cer43 ]. Lemma 4.5. Every group ( 𝐺 , · , 𝑒 ) is a heap via ⟨− , = , ≡⟩ : 𝐺 × 𝐺 × 𝐺 − → 𝐺 , ⟨ 𝑔 , ℎ , 𝑖 ⟩ . . = 𝑔 · ℎ − 1 · 𝑖 . A morphism of groups becomes a mor phism of t he induced heaps. Lemma 4.6. A non-empty heap 𝐺 with a ﬁxed element 𝑒 ∈ 𝐺 can be consider ed as a group wit h unit 𝑒 via the multiplication − · 𝑒 = : 𝐺 × 𝐺 − → 𝐺 , 𝑔 · 𝑒 ℎ . . = ⟨ 𝑔 , 𝑒 , ℎ ⟩ . The in verse of an element 𝑔 ∈ 𝐺 with respect to · 𝑒 is giv en by 𝑔 − 1 . . = ⟨ 𝑒 , 𝑔 , 𝑒 ⟩ . A morphism of heaps is a mor phism of the induced groups, provided it maps the ﬁxed element of its sour ce to t he ﬁxed element of its tar get. More generall y , if 𝒢 is a g roupoid and 𝑥 , 𝑦 ∈ 𝒢 , then t he set of mor phisms 𝒢 ( 𝑥 , 𝑦 ) becomes a heap with heap operation giv en by ⟨ 𝑓 , 𝑔 , ℎ ⟩ . . = 𝑓 𝑔 − 1 ℎ . The next example is a special case of this construction, which will pla y a prominent role in this chapter . Exam ple 4.7. Let 𝐹 , 𝐺 : 𝒞 − → 𝒞 be oplax monoidal endofunctors. The set Iso ⊗ ( 𝐹 , 𝐺 ) . . =  oplax monoidal natural isomor phisms from 𝐹 t o 𝐺  bears a heap structure wit h heap operation ⟨− , = , ≡⟩ : Iso ⊗ ( 𝐹 , 𝐺 ) 3 − → Iso ⊗ ( 𝐹 , 𝐺 ) , ⟨ 𝜙 , 𝜓 , 𝜉 ⟩ . . = 𝜙 𝜓 − 1 𝜉 . 93 4. Twist ed ce ntres 4 . 2 p i vo ta l s t r u c t u r e s a n d t w i s t e d c e n t r e s 4.2.1 T wist ed centres and t heir Picar d heaps Reca ll from Ex ampl e 2. 45 t ha t w e ca n tw ist t he r egul ar actio n of a monoidal categor y on itself wit h strong monoidal functors. If 𝒞 is a rigid monoidal category , then w e refer to the centre Z ( 𝐿 𝒞 𝑅 ) as a twist ed centr e , and to Z ( 𝒞 𝑅 ) and Z ( 𝐿 𝒞 ) as right and left twisted centres. Remark 4.8. A natur al descrip tion of twisted centres is obtained from the perspectiv e of bicategories. Giv en a monoidal category 𝒞 , let B 𝒞 denote its delooping ; that is, B 𝒞 is a bicategory wit h a single object • , and B 𝒞 (• , •) ≃ 𝒞 . In t his setting, strong monoidal functors are identiﬁed wit h pseudofunc- tors betw een one-object bicategories, and strong monoidal transformations correspond to pseudo- i c on 𝑠 , see for example [ JY21 , Proposition 4.6.9]. Let P s ps ( B 𝒞 ) be the bicategor y of endopseudofunctors of B 𝒞 , their pseu- donatural transf ormations, and modiﬁcations. Then P s ps ( B 𝒞 )( 𝐿 , 𝑅 ) ≃ Z ( 𝐿 𝒞 𝑅 ) . See [ FH23 , Proposition 3.6] for a proof of t his fact. F or t he rest of t his section, ﬁx a strict rigid categor y 𝒞 , see Theorem 2.72 . The for getful functor from the centre of a twisted bimodule categor y to the underl ying monoidal categor y is faithful, so w e can use a v ariant of the graphical calculus for monoidal categories, as long as w e pa y special attention to t he half-braidings. Whence, w e introduce a colouring scheme to help us keep track of t he various categories. (i) R ed for objects in the right twisted centre Z ( 𝒞 𝑅 ) , (ii) blue for objects in t he lef t twisted centre Z ( 𝐿 𝒞 ) , and (iii) black for objects in t he Drinfeld centre Z ( 𝒞 ) or in 𝒞 . F or example, t he half-braidings of objects 𝑎 ∈ Z ( 𝒞 𝑅 ) and 𝑞 ∈ Z ( 𝐿 𝒞 ) are: 𝑎 𝑅 𝑥 𝑥 𝑎 𝜎 𝑎 , 𝑥 : 𝑎 ⊗ 𝑅 𝑥 − → 𝑥 ⊗ 𝑎 𝑞 𝑥 𝐿 𝑥 𝑞 𝜎 𝑞 , 𝑥 : 𝑞 ⊗ 𝑥 − → 𝐿 𝑥 ⊗ 𝑞 Hypothesis 4.9. In the rest of t his chapter , w e are predominantly interested in twisting wit h the same strict monoidal functor from t he lef t or right. F or the pur pose of brevity , w e t heref ore ﬁx such a functor 𝐿 = 𝑅 : 𝒞 − → 𝒞 and consider the categories 𝐿 𝒞 and 𝒞 𝑅 . 94 4.2. Piv otal structures and twisted centres Suppose w e are giv en three objects ( 𝑎 , 𝜎 𝑎 , − ) ∈ Z ( 𝒞 𝑅 ) , ( 𝑞 , 𝜎 𝑞 , − ) ∈ Z ( 𝐿 𝒞 ) , and ( 𝑥 , 𝜎 𝑥 , − ) ∈ Z ( 𝒞 ) . Proposition 4.10. The tensor product of 𝒞 extends to a lef t action of Z ( 𝒞 ) on Z ( 𝒞 𝑅 ) and a right action of Z ( 𝒞 ) on Z ( 𝐿 𝒞 ) . The half-br aidings are as deﬁned in F igure 4.1 . 𝑞 𝑥 𝑦 𝑥 𝑞 𝐿 𝑦 𝜎 𝑞 ⊗ 𝑥 , 𝑦 : 𝑞 ⊗ 𝑥 ⊗ 𝑦 − → 𝐿 𝑦 ⊗ 𝑞 ⊗ 𝑥 𝑥 𝑎 𝑅 𝑦 𝑎 𝑥 𝑦 𝜎 𝑥 ⊗ 𝑎 , 𝑦 : 𝑥 ⊗ 𝑎 ⊗ 𝑅 𝑦 − → 𝑦 ⊗ 𝑥 ⊗ 𝑎 𝑎 𝑞 𝑦 𝑞 𝑎 𝑦 𝜎 𝑎 ⊗ 𝑞 , 𝑦 : 𝑎 ⊗ 𝑞 ⊗ 𝑦 − → 𝑦 ⊗ 𝑎 ⊗ 𝑞 𝑞 𝑎 𝑅 𝑦 𝑎 𝑞 𝐿 𝑦 𝜎 𝑥 ⊗ 𝑎 , 𝑦 : 𝑥 ⊗ 𝑎 ⊗ 𝑅 𝑦 − → 𝑦 ⊗ 𝑥 ⊗ 𝑎 Figure 4.1: Canonical actions of Z ( 𝒞 ) on Z ( 𝐿 𝒞 ) and Z ( 𝒞 𝑅 ) . Remark 4.11. A direct computation prov es t he categories Z ( 𝒞 op , rev 𝑅 ) and Z ( 𝑅 𝒞 ) op to be equiv alent. Further more, t his identiﬁcation is compatible with the module structure: for all 𝑥 ∈ Z ( 𝒞 op , rev ) and 𝑎 ∈ Z ( 𝒞 op , rev 𝑅 ) , w e ha ve 𝑥 ⊗ op 𝑅 ( 𝑎 ) = 𝑅 ( 𝑎 ) ⊗ 𝑥 and 𝜎 𝑥 ⊗ op 𝑎 , − = 𝜎 𝑎 ⊗ 𝑥 , − . Hence, w e restrict ourselv es to the study of right twisted centres. Recall from Proposition 2.80 t hat, since 𝒞 is a rigid monoidal category , the centre Z ( 𝒞 ) is rigid as w ell. While t he same cannot be said of the twisted centre, t he left dual ∨ 𝑎 of any object ( 𝑎 , 𝜎 𝑎 , − ) ∈ Z ( 𝒞 𝑅 ) can be tur ned into an object of Z ( 𝑅 𝒞 ) if w e equip it wit h t he half-braiding 𝑅 𝑥 𝑥 ∨ 𝑎 ∨ 𝑎 This sugges ts t hat, in analogy wit h Proposition 2.80 , w e may lift t he dualising functor of 𝒞 to t he lev el of twisted centres by interchanging right wit h left twists. A more concep tual descrip tion is pro vided in [ FH23 , Proposition 3.15]. 95 4. Twist ed ce ntres Proposition 4.12. The left dualising funct or ∨ (−) : 𝒞 − → 𝒞 op , rev lifts to a functor between right and left twist ed centres ∨ (−) : Z ( 𝒞 𝑅 ) − → Z ( 𝑅 𝒞 ) op . The half-braidings displa y ed in the right column of Figure 4.1 show t hat ev ery object 𝑎 ∈ Z ( 𝒞 𝑅 ) giv es rise to tw o functors of left Z ( 𝒞 ) -modules: − ⊗ 𝑎 : Z ( 𝒞 ) − → Z ( 𝒞 𝑅 ) and − ⊗ ∨ 𝑎 : Z ( 𝒞 𝑅 ) − → Z ( 𝒞 ) . (4.2.1) Let ( 𝑎 , 𝜎 𝑎 , − ) ∈ Z ( 𝒞 𝑅 ) . In view of Proposition 4.12 , w e use t he follo wing notation for t he ev aluation and coev aluation mor phisms of twisted centres: ev ℓ 𝑎 : ∨ 𝑎 ⊗ 𝑎 − → 1 ∨ 𝑎 𝑎 coev ℓ 𝑎 : 1 − → 𝑎 ⊗ ∨ 𝑎 ∨ 𝑎 𝑎 (4.2.2) Proposition 4.13. Every object 𝑎 ∈ Z ( 𝒞 𝑅 ) induces adjoint Z ( 𝒞 ) -module functor s − ⊗ 𝑎 : Z ( 𝒞 ) ⇄ Z ( 𝒞 𝑅 ) : − ⊗ ∨ 𝑎 . (4.2.3) Proof. W e kno w that for any object ( 𝑎 , 𝜎 𝑎 , − ) ∈ Z ( 𝒞 𝑅 ) , Equation ( 4.2.3 ) is an ordinary adjunction, whose unit 𝜂 and counit 𝜀 are given b y 𝜂 𝑦 . . = id 𝑦 ⊗ coev ℓ 𝑎 : 𝑦 − → 𝑦 ⊗ 𝑎 ⊗ ∨ 𝑎 , f or all 𝑦 ∈ Z ( 𝒞 ) , 𝜀 𝑥 . . = id 𝑥 ⊗ ev ℓ 𝑎 : 𝑥 ⊗ ∨ 𝑎 ⊗ 𝑎 − → 𝑥 , for all 𝑥 ∈ Z ( 𝒞 𝑅 ) . The next diag r am sho ws that 𝜀 𝑥 is a mor phism in Z ( 𝒞 𝑅 ) for ev er y 𝑥 ∈ Z ( 𝒞 𝑅 ) : 𝑥 ∨ 𝑎 𝑎 𝑅 𝑦 𝑦 𝑥 = 𝑥 ∨ 𝑎 𝑎 𝑅 𝑦 𝑦 𝑥 = 𝑥 ∨ 𝑎 𝑎 𝑅 𝑦 𝑦 𝑥 (4.2.4) Furthermore, 𝜀 𝑤 ⊗ 𝑥 = id 𝑤 ⊗ 𝜀 𝑥 for all 𝑤 ∈ Z ( 𝒞 ) . A similar argument show s t hat the unit of t he adjunction is a natural transformation of module functors. □ 96 4.2. Piv otal structures and twisted centres Since t he forg etful functor from the (twisted) centre to its underl ying category is faithful, it is also conser v ativ e 11 , which allows us to characterise 11 A functor 𝐹 : 𝒞 − → 𝒟 is called conservativ e if it reﬂects isomorphisms; i.e., if for all 𝑓 ∈ 𝒞 ( 𝑥 , 𝑦 ) , the fact t hat 𝐹 𝑓 is an isomorphism implies that 𝑓 is one. equiv alences of module categories betw een Z ( 𝒞 ) and right twisted centres. Remark 4.14. Recall from Proposition 2.51 t hat for a monoidal category 𝒞 and a 𝒞 -module category ℳ , there is an equivalence of 𝒞 -module categories Str 𝒞 Mod ( 𝒞 , ℳ ) ∼ − → ℳ , 𝐹 ↦− → 𝐹 1 , − ⊲ 𝑚 ← − [ 𝑚 . In particular , an y functor of lef t module categories 𝐹 : Z ( 𝒞 ) − → Z ( 𝒞 𝑅 ) is naturall y isomorphic to − ⊗ 𝐹 1 : Z ( 𝒞 ) − → Z ( 𝒞 𝑅 ) . By Proposition 2.67 , 𝐹 is an equivalence if and only if 𝐹 1 ∈ 𝒞 is inv ertible, and tw o left Z ( 𝒞 ) -module functors 𝐹 , 𝐺 : Z ( 𝒞 ) − → Z ( 𝒞 𝑅 ) are isomorphic if and only if 𝐹 1  𝐺 1 . Deﬁnition 4.15. An object ( 𝛼 , 𝜎 𝛼 , − ) ∈ Z ( 𝒞 𝑅 ) in is called 𝒞 -in vertible if t he image 𝑈 𝛼 ∈ 𝒞 of 𝛼 under t he forg etful funct or 𝑈 ( 𝑅 ) : Z ( 𝒞 𝑅 ) − → 𝒞 is in v ertible. Notation 4.16. Let ( 𝛼 , 𝜎 ( 𝛼 , −) ) ∈ Z ( 𝒞 𝑅 ) be a 𝒞 -in v ertible element. Analogously to Equation ( 4.2.2 ), w e use the follo wing notation for t he in v erses ev − ℓ and coev − ℓ of ev ℓ and coev ℓ , respectiv ely : ev − ℓ 𝛼 : 1 − → ∨ 𝛼 ⊗ 𝛼 ∨ 𝛼 𝛼 coev − ℓ 𝛼 : 𝛼 ⊗ ∨ 𝛼 − → 1 ∨ 𝛼 𝛼 The notion of heaps allo ws us to deﬁne an alg ebraic structure on t he isomorphism classes of objects implementing module equiv alences betw een the Drinfeld centre Z ( 𝒞 ) and its twisted relativ e Z ( 𝒞 𝑅 ) . In analogy wit h t he Picard g roup, w e call t his t he Picard heap of a twisted centre. Deﬁnition 4.17. The Picard heap of t he right twisted centre Z ( 𝒞 𝑅 ) is t he set of isomorphism classes Pic Z ( 𝒞 𝑅 ) . . =  [ 𝛼 ]   𝛼 ∈ Z ( 𝒞 𝑅 ) is 𝒞 -in v ertible  tog ether wit h t he heap operation deﬁned for [ 𝛼 ] , [ 𝛽 ] , [ 𝛾 ] ∈ Pic Z ( 𝒞 𝑅 ) b y  [ 𝛼 ] , [ 𝛽 ] , [ 𝛾 ]  = [ 𝛼 ⊗ ∨ 𝛽 ⊗ 𝛾 ] . 97 4. Twist ed ce ntres Lemma 4.18. The Picar d heap deﬁned in Deﬁnition 4.17 forms a heap. Proof. The gener alised associativity , see Equation ( 4.1.1 ), follo ws from the associativity of the tensor product of 𝒞 and its compatibility wit h t he “gluing” of half-braidings. T o show that t he Mal’cev identities hold, ﬁx 𝒞 -in v ertible objects 𝛼 , 𝛽 ∈ Z ( 𝒞 𝑅 ) . Proposition 2.80 and Equation ( 4.2.4 ) impl y t hat 𝛼 ⊗ ∨ 𝛼 ⊗ 𝛽 coev ℓ 𝛼 − 1 ⊗ id 𝛽 − − − − − − − − − → 𝛽 and 𝛽 ⊗ ∨ 𝛼 ⊗ 𝛼 id 𝛽 ⊗ e v ℓ 𝛼 − − − − − − → 𝛽 are isomorphisms in Z ( 𝒞 𝑅 ) , hence  [ 𝛼 ] , [ 𝛼 ] , [ 𝛽 ]  = [ 𝛽 ] =  [ 𝛽 ] , [ 𝛼 ] , [ 𝛼 ]  . □ In general, the twisted centre Z ( 𝒞 𝑅 ) does not inherit a monoidal structure from 𝒞 . Lemma 4.18 , how ev er , hints to war ds a slight generalisation where the tensor product is replaced b y a trivalent functor , essentiall y categorifying heaps (without the Mal’cev identities). The w ell-deﬁnedness of this concept w as hinted at in [ Ško07 ] under the name of heapy categories . 4.2.2 Quasi-pivo tality A p a r t icul arl y in tere sting conse quence of o ur pr evio us fin dings arises in the case of 𝑅 . . = ∨∨ (−) ; t he centre of the regular bimodule twisted on the right b y 𝑅 can be underst ood as a gener alisation of anti-Y etter– Drinf eld modules, see [ HKS19 , Theorem 2.3]. As before, ﬁx a strict rigid categor y 𝒞 and consider the twisted bimodules categories 𝒞 ∨∨ (−) and ∨∨ (−) 𝒞 . Notation 4.19. W e write A ( 𝒞 ) . . = Z ( 𝒞 ∨∨ (−) ) and Q ( 𝒞 ) . . = Z ( ∨∨ (−) 𝒞 ) , and call the former t he anti-Drinfeld centr e of 𝒞 . W e ha v e already mentioned the connection betw een t he twisted centre A ( 𝒞 ) and anti-Y etter– Drinfeld modules ov er Hopf algebr as giv en in [ HKS19 ]. The case where 𝒞 is t he categor y of modules o v er a Hopf algebroid was in v estig ated b y Ko walzig in [ K o w24 ]. The counter part Q ( 𝒞 ) of t he gener alised anti-Y etter– Drinfeld modules is less common in t he literature, but pla ys a crucial role in t he monadic in v estigations of Chapters 5 and 6 . Deﬁnition 4.20 ([ Shi23 a , Section 4]) . A quasi-pivo tal structur e on a rigid mo- noidal category 𝒞 is a pair ( 𝛽 , 𝜌 𝛽 ) consisting of an in v ertible object 𝛽 ∈ 𝒞 and a monoidal natural isomor phism 𝜌 𝛽 : (−) ∼ = ⇒ 𝛽 ⊗ ∨∨ (−) ⊗ ∨ 𝛽 . The tuple ( 𝒞 , ( 𝛽 , 𝜌 𝛽 )) will be called a quasi-piv ot al category . 98 4.2. Piv otal structures and twisted centres If 𝒞 is the categor y of ﬁnite-dimensional modules ov er a ﬁnite-dimen- sional Hopf algebr a, quasi-piv otal structures ha v e a w ell-known inter pret- ation — t he y translate to pairs in inv olution. This can be deduced from a slight variation of [ Hal21 , Lemma 5.6], t he main obser v ation being t hat t he in v ertible object 𝛽 of a quasi-pivo tal structure ( 𝛽 , 𝜌 𝛽 ) on 𝒞 corresponds to a character , and t hat 𝜌 𝛽 determines a g roup-lik e element. The fact that 𝜌 𝛽 is a natural transf ormation from t he identity to a conjugate of t he double dual functor is captured by t he character and g roup-lik e implementing t he square of the antipode. W e study a monadic analogue of t his in Section 6.5 . Remark 4.21. Ev er y piv otal category is quasi-piv otal; t he conv erse does not hold. A counterexample are t he ﬁnite-dimensional modules ov er the gener alised T af t algebr as discussed in [ HK19 ]. Any of t hese Hopf algebr as admit pairs in in volution but in gener al neither t he character nor the g roup- like can be trivial. The previous discussion and [ Hal21 , Lemma 5.6] sho w that Mod- 𝐻 is quasi-piv otal but not piv otal — in contras t to its Drinfeld centre Z ( Mod- 𝐻 ) , which admits a pivo tal structure by [ Hal21 , Lemma 5.5]. Let ( 𝛽 , 𝜌 𝛽 ) be a quasi-piv otal structure on 𝒞 and 𝜙 : 𝛽 ′ − → 𝛽 an isomor ph- ism in 𝒞 . Clear l y , t he pair ( 𝛽 ′ , ( 𝜙 − 1 ⊗ id ⊗ ∨ 𝜙 ) ◦ 𝜌 𝛽 ) is another quasi-piv otal structure on 𝒞 . This deﬁnes an equivalence relation and w e write QPiv ( 𝒞 ) . . = { [( 𝛽 , 𝜌 𝛽 )] | ( 𝛽 , 𝜌 𝛽 ) is a quasi-piv otal structure on 𝒞 } for t he set of equiv alence classes of quasi-piv otal structures on 𝒞 . Lemma 4.22. Let 𝒞 be a s trict rigid cat egor y . The Picar d heap Pic A ( 𝒞 ) and the set of equiv alence classes of quasi-piv ot al structur es QPiv ( 𝒞 ) are in bĳection. Proof. Let ( 𝛽 , 𝜌 𝛽 ) be a quasi-piv otal structure on 𝒞 . W riting ev − ℓ 𝛽 . . = ( ev ℓ 𝛽 ) − 1 , w e deﬁne t he half-braiding 𝜌 − 1 𝛽 𝛽 ∨∨ 𝑥 𝛽 𝑥 𝜎 𝛽 , 𝑥 . . = ( 𝜌 − 1 𝛽 , 𝑥 ⊗ 𝛽 ) ◦ ( id ⊗ ev − ℓ 𝛽 ) : 𝛽 ⊗ ∨∨ 𝑥 − → 𝑥 ⊗ 𝛽 99 4. Twist ed ce ntres Then, as 𝜌 𝛽 is monoidal, 𝜎 𝛽 , 𝑥 satisﬁes the hexagon identity . This deﬁnes 𝜙 : QPiv ( 𝒞 ) − → Pic A ( 𝒞 ) , [( 𝛽 , 𝜌 𝛽 )] ↦− → [( 𝛽 , 𝜎 𝛽 , − )] . Con v ersely , let ( 𝛼 , 𝜎 𝛼 , − ) ∈ A ( 𝒞 ) be 𝒞 -in v ertible. From its half-braiding w e obtain a monoidal natural transf ormation 𝛽 𝛼 𝜎 𝛽 , 𝑥 = ( 𝜌 − 1 𝛽 , 𝑥 ⊗ 𝛽 ) ◦ ( id ⊗ ev − ℓ 𝛽 ) : 𝛽 ⊗ ∨∨ 𝑥 − → 𝑥 ⊗ 𝛽 ∨∨ 𝑥 𝑥 Due to t he snake identities, the follo wing map is the inv erse of 𝜙 : 𝜓 : Pic A ( 𝒞 ) − → QPiv ( 𝒞 ) , [( 𝛼 , 𝜎 𝛼 , − )] ↦− → [( 𝛼 , 𝜌 𝛼 )] . □ Theorem 4.23. Let 𝒞 be a strict rigid category . Then ther e is a bĳection between (i) eq uivalence classes of quasi-pivo tal structur es on 𝒞 , (ii) t he Picard heap of A ( 𝒞 ) , and (iii) isomorphism classes of Z ( 𝒞 ) -module equiv alences between Z ( 𝒞 ) and A ( 𝒞 ) . Proof. The equiv alence of (ii) and (iii) follo ws by Remar k 4.14 , and for (i) being equiv alent to (ii) w e in vok e Lemma 4.22 . □ 4.2.3 Pivo tality of the Drinfeld centre In t his se ctio n w e sh all ex amin e t he rel a ti onsh ip betw een pairs in in- v olution and piv otal structures, see for example Remar k 4.21 , from a cat- egorical perspectiv e. Let 𝑎 . . = ( 𝛼 , 𝜎 𝛼 , − ) ∈ A ( 𝒞 ) be 𝒞 -in v ertible and write Ω . . = ( 𝜔 , 𝜎 𝜔 , − ) ∈ Q ( 𝒞 ) for its left dual. Figure 4.2 collects some useful proper- ties of the coev aluation of 𝛼 and its in v erse coev − ℓ 𝛼 . Deﬁnition 4.24. Let 𝑎 . . = ( 𝛼 , 𝜎 𝛼 , − ) ∈ A ( 𝒞 ) be a 𝒞 -in v ertible element wit h lef t dual Ω . . = ( 𝜔 , 𝜎 𝜔 , − ) ∈ Q ( 𝒞 ) . F or an y 𝑥 ∈ Z ( 𝒞 ) , deﬁne t he mor phism 𝜌 𝑎 , 𝑥 b y ∨∨ 𝑥 𝑥 𝜌 𝑎 , 𝑥 . . = ( ∨∨ 𝑥 ⊗ coev − ℓ 𝛼 ) ◦ ( 𝜎 − 1 ∨∨ 𝑥 , 𝛼 ⊗ 𝜔 ) ◦ ( 𝛼 ⊗ 𝜎 𝜔 , 𝑥 ) ◦ ( coev ℓ 𝛼 ⊗ 𝑥 ) : 𝑥 − → ∨∨ 𝑥 100 4.2. Piv otal structures and twisted centres 𝛼 𝛼 𝜔 𝜔 = 𝛼 𝛼 𝜔 𝜔 1 1 = = 𝛼 𝛼 𝜔 𝜔 𝑥 𝑥 𝑥 𝑥 = 𝛼 𝛼 𝜔 𝜔 𝑥 𝑥 𝑥 𝑥 Figure 4.2: Properties of coev ℓ 𝛼 and coev − ℓ 𝛼 . As mentioned in Remar k 4.8 , objects in twisted centres correspond to pseudonatural transf ormations betw een deloopings of monoidal functors. The follo wing lemma show s that if these objects are 𝒞 -in v ertible, then one can reconstruct monoidal natural isomorphisms from t hem. Lemma 4.25 ([ Shi23 a , Section 4.4]) . F or any 𝒞 -in vertible object 𝑎 ∈ A ( 𝒞 ) t he map 𝜌 𝑎 , − : (−) = ⇒ ∨∨ (−) deﬁnes a pivo tal structur e on Z ( 𝒞 ) . Proof. Fix an object 𝑎 . . = ( 𝛼 , 𝜎 𝛼 , − ) ∈ A ( 𝒞 ) , such that 𝛼 is inv ertible in 𝒞 , and write Ω . . = ( 𝜔 , 𝜎 𝜔 , − ) ∈ Q ( 𝒞 ) for its lef t dual. Furthermore, w e assume 𝑥 ∈ Z ( 𝒞 ) to be any object in the Drinf eld centre of 𝒞 . W e note t hat for any 𝑦 ∈ 𝒞 a v ariant of the Y ang –Baxter identity holds: 𝑦 𝑥 𝛼 ∨∨ 𝑥 𝛼 ∨∨ 𝑦 = 𝑦 𝑥 𝛼 ∨∨ 𝑥 𝛼 ∨∨ 𝑦 = 𝑦 𝑥 𝛼 ∨∨ 𝑥 𝛼 ∨∨ 𝑦 = 𝑦 𝑥 𝛼 ∨∨ 𝑥 𝛼 ∨∨ 𝑦 Combining this fact wit h F igure 4.2 , one obtains t hat 𝜌 𝑎 , 𝑥 : 𝑥 − → ∨∨ 𝑥 is a morphism in t he Drinfeld centre of 𝒞 : 𝑦 ∨∨ 𝑥 𝑦 𝑥 𝑦 ∨∨ 𝑥 𝑦 𝑥 = 𝑦 ∨∨ 𝑥 𝑦 𝑥 = 𝑦 ∨∨ 𝑥 𝑦 𝑥 = 101 4. Twist ed ce ntres Since the forg etful functor 𝑈 ( 𝑍 ) : Z ( 𝒞 ) − → 𝒞 is conservativ e and 𝜌 𝑎 , 𝑥 is a composite of isomorphisms in 𝒞 , it is an isomor phism in t he centre Z ( 𝒞 ) . The naturality of t he half-braidings implies t hat 𝜌 𝑎 is natural as w ell: ∨∨ 𝑥 𝑥 𝑓 ∨∨ 𝑥 ∨∨ 𝑓 𝑥 = ∨∨ 𝑥 𝑥 = ∨∨ 𝑓 Lastl y , t he natural isomorphism 𝜌 𝑎 : Id Z ( 𝒞 ) − → ∨∨ (−) being monoidal is established b y t he hexagon identities. ∨∨ 𝑥 𝑥 ⊗ ∨∨ 𝑦 𝑦 = ∨∨ 𝑥 𝑥 ∨∨ 𝑦 𝑦 𝛼 𝛼 𝜔 𝜔 = ∨∨ 𝑥 𝑥 ∨∨ 𝑦 𝑦 = ∨∨ 𝑥 ⊗ ∨∨ 𝑦 𝑥 ⊗ 𝑦 □ Giv en diﬀerent piv otal structures of Z ( 𝒞 ) , induced by 𝒞 -in v ertible objects in A ( 𝒞 ) , it is a priori unclear whether they coincide. The follo wing lemma is a ﬁrst step in t his direction. It show s t hat t he induced piv otal structures only depend on the isomor phism classes of 𝒞 -in v ertible objects in A ( 𝒞 ) . Lemma 4.26. Suppose that 𝑎 1 , 𝑎 2 ∈ A ( 𝒞 ) are two repr esentatives of the equiv alence class [ 𝑎 1 ] = [ 𝑎 2 ] ∈ Pic A ( 𝒞 ) . Then 𝜌 𝑎 1 = 𝜌 𝑎 2 . Proof. Suppose t hat there exists an isomorphism 𝜙 : 𝑎 1 − → 𝑎 2 in t he anti- Drinfeld centre. Then by Figure 4.3 the induced piv otal structures 𝜌 𝑎 1 and 𝜌 𝑎 2 are the same, for an y 𝑥 ∈ Z ( 𝒞 ) . □ Deﬁnition 4.27. W e call an object 𝑥 ∈ Z ( 𝒞 ) symmetric if w e hav e 𝜎 − 1 𝑥 , 𝑦 = 𝜎 𝑦 , 𝑥 , for all 𝑦 ∈ Z ( 𝒞 ) . W e call t he full (symmetric) monoidal subcategor y SZ ( 𝒞 ) of Z ( 𝒞 ) whose objects are symmetric the symmetric or Müger centre of Z ( 𝒞 ) , see [ Müg13 ]. 102 4.2. Piv otal structures and twisted centres ∨∨ 𝑥 𝑥 = ∨∨ 𝑥 𝑥 ∨ 𝜙 ∨ ( 𝜙 − 1 ) 𝜙 − 1 𝜙 = ∨∨ 𝑥 𝑥 ∨ ( 𝜙 − 1 ) ∨ 𝜙 𝜙 𝜙 − 1 𝜔 1 𝜔 1 𝛼 1 𝛼 1 𝛼 2 𝛼 2 𝜔 2 𝜔 2 ∨∨ 𝑥 𝑥 = 𝜔 2 𝛼 2 𝛼 2 𝜔 2 𝜔 1 𝛼 1 𝜔 1 𝛼 1 Figure 4.3: The induced pivo tal structures of 𝜌 𝑎 1 and 𝜌 𝑎 2 coincide. Lemma 4.28. If 𝒞 is a rigid monoidal category , then SZ ( 𝒞 ) is rigid monoidal. Proof. Suppose 𝑥 ∈ Z ( 𝒞 ) to be symmetric and let 𝑦 ∈ Z ( 𝒞 ) . W e compute ∨ 𝑥 ∨ 𝑥 𝑦 𝑦 = ∨ 𝑥 ∨ 𝑥 𝑦 𝑦 = ∨ 𝑥 ∨ 𝑥 𝑦 𝑦 = ∨ 𝑥 ∨ 𝑥 𝑦 𝑦 = ∨ 𝑥 ∨ 𝑥 𝑦 𝑦 𝑥 𝑥 𝑥 This implies t hat 𝜎 − 1 ∨ 𝑥 , 𝑦 = 𝜎 𝑦 , ∨ 𝑥 . Since t he left dual of an y 𝑥 ∈ SZ ( 𝒞 ) ⊆ Z ( 𝒞 ) can be equipped wit h the structure of a right dual and SZ ( 𝒞 ) is a full subcategory of Z ( 𝒞 ) , it must be rigid. □ Let us now consider t he Picard g roup Pic SZ ( 𝒞 ) . It acts on Pic A ( 𝒞 ) via tensoring from t he left, as sho wn in Figure 4.1 . T w o elements 𝑎 , 𝑐 ∈ Pic A ( 𝒞 ) are equiv alent if they are contained in the same orbit; t hat is, [ 𝑎 ] ∼ [ 𝑐 ] ⇐ ⇒ t here exists a [ 𝑏 ] ∈ Pic SZ ( 𝒞 ) such that [ 𝑏 ⊗ 𝑎 ] = [ 𝑐 ] . T o sho w that tw o objects are equiv alent if and only if t he y induce t he same piv otal structure on Z ( 𝒞 ) , w e need tw o technical obser v ations. F irst, an alternate description of symmetric inv ertible objects; and second, a more detailed in v estig ation into t he inv erse of an induced piv otal structure. Lemma 4.29. An inv ertible object ( 𝛽 , 𝜎 𝛽 , − ) ∈ Z ( 𝒞 ) is symmetric if and only if, for all 𝑥 ∈ Z ( 𝒞 ) , it satisﬁes 𝑥 𝑥 𝛽 ∨ 𝛽 ∨ 𝛽 𝛽 = 𝑥 𝑥 (4.2.5) 103 4. Twist ed ce ntres Proof. Let 𝐵 = ( 𝛽 , 𝜎 𝛽 , − ) ∈ Z ( 𝒞 ) be in v ertible and 𝑥 ∈ Z ( 𝒞 ) . The left-hand side of Equation ( 4.2.5 ) can be rephrased as: 𝑥 𝑥 = 𝑥 𝑥 Deﬁne t he mor phism 𝑓 . . = id 𝑥 ⊗ coev ℓ 𝛽 : 𝑥 − → 𝑥 ⊗ 𝛽 ⊗ ∨ 𝛽 and obser v e that Equation ( 4.2.5 ) is identical to 𝑓 − 1 ◦ (( 𝜎 𝛽 , 𝑥 ◦ 𝜎 𝑥 , 𝛽 ) − 1 ⊗ id ∨ 𝛽 ) ◦ 𝑓 = id 𝑥 . This is equiv alent to ( 𝜎 𝛽 , 𝑥 ◦ 𝜎 𝑥 , 𝛽 ) ⊗ id ∨ 𝛽 = id 𝑥 ⊗ 𝛽 ⊗ id ∨ 𝛽 . As the funct or − ⊗ ∨ 𝛽 is conservativ e, the claim follo ws. □ Lemma 4.30. Let 𝑎 . . = ( 𝛼 , 𝜎 𝛼 , − ) ∈ A ( 𝒞 ) be 𝒞 -inv ertible, and write Ω . . = ( 𝜔 , 𝜎 𝜔 , − ) for its dual in Q ( 𝒞 ) . F or any 𝑥 ∈ Z ( 𝒞 ) , the inv erse of 𝜌 𝑎 , 𝑥 is ∨∨ 𝑥 𝑥 𝜔 ∨∨ 𝛼 ∨∨ 𝛼 𝜔 Proof. F or 𝑥 ∈ Z ( 𝒞 ) , the snake identities and a v ariant of F igure 4.2 im ply Figure 4.4 . Thus, for Ω . . = ( ∨ 𝛼 , 𝜎 ∨ 𝛼 , − ) ∈ Z ( 𝒞 ) , w e ha v e 𝜌 𝑎 , 𝑥 ◦ 𝜌 Ω ,𝑥 = id 𝑥 . □ Proposition 4.31. T wo elements [ 𝑎 ] , [ 𝑐 ] ∈ Pic A ( 𝒞 ) induce the same pivot al s truc- tur e on Z ( 𝒞 ) if and only if t here exis ts a [ 𝑏 ] ∈ Pic SZ ( 𝒞 ) such t hat [ 𝑏 ⊗ 𝑎 ] = [ 𝑐 ] . Proof. Let [ 𝑎 ] , [ 𝑐 ] ∈ Pic A ( 𝒞 ) . Suppose there exists a [ 𝑏 ] ∈ Pic SZ ( 𝒞 ) such t hat [ 𝑏 ⊗ 𝑎 ] = [ 𝑐 ] . For an y 𝑥 ∈ Z ( 𝒞 ) , we compute: ∨∨ 𝑥 𝑥 = 𝑥 ∨∨ 𝑥 𝜌 𝑐 , 𝑥 ∨∨ 𝑥 𝑥 = 𝜌 𝑎 , 𝑥 = 𝑥 ∨∨ 𝑥 𝜌 𝑎 , 𝑥 (4.2.6) 104 4.2. Piv otal structures and twisted centres ∨∨ 𝑥 ∨ 𝛼 ∨∨ 𝛼 ∨∨ 𝛼 ∨ 𝛼 ∨∨ 𝑥 ∨ 𝛼 𝛼 𝛼 ∨ 𝛼 = ∨∨ 𝑥 ∨ 𝛼 ∨∨ 𝛼 ∨∨ 𝛼 ∨ 𝛼 ∨ 𝛼 𝛼 𝛼 ∨ 𝛼 ∨∨ 𝑥 = ∨∨ 𝑥 ∨ 𝛼 ∨ 𝛼 𝛼 ∨∨ 𝑥 𝛼 ∨ 𝛼 ∨∨ 𝑥 = ∨∨ 𝑥 ∨∨ 𝑥 = ∨∨ 𝑥 ∨∨ 𝑥 = ∨∨ 𝑥 Figure 4.4: V eriﬁcation of Lemma 4.30 . If con v ersely 𝜌 𝑎 = 𝜌 𝑐 , w e claim t hat 𝑐 ⊗ ∨ 𝑎 is symmetric. By Lemma 4.29 w e ha v e to sho w that for ev ery 𝑥 ∈ Z ( 𝒞 ) the “entwinement” 𝜌 𝑐 ⊗ ∨ 𝑎 of 𝑐 ⊗ ∨ 𝑎 with 𝑥 is t he identity . Indeed, 𝜌 𝑐 ⊗ ∨ 𝑎 , 𝑥 = 𝜌 ∨ 𝑎 , 𝑥 ◦ 𝜌 𝑐 , 𝑥 = 𝜌 − 1 𝑎 , 𝑥 ◦ 𝜌 𝑐 , 𝑥 = id 𝑥 . F or t he ﬁrst equality w e used the hexagon identities as in Equation ( 4.2.6 ) to separate 𝜌 𝑐 ⊗ ∨ 𝑎 , 𝑥 into tw o parts. The second one follo ws from t he description of the in v erse of 𝜌 𝑎 , 𝑥 giv en in Lemma 4.30 . F inally , since id 𝑐 ⊗ ev ℓ 𝑎 : 𝑐 ⊗ ∨ 𝑎 ⊗ 𝑎 − → 𝑐 is an isomorphism in A ( 𝒞 ) , w e ha v e [( 𝑐 ⊗ ∨ 𝑎 ) ⊗ 𝑎 ] = [ 𝑐 ] . □ The isomorphism classes of 𝒞 -in v ertible objects in A ( 𝒞 ) are not just a set, but form t he Picard heap Pic A ( 𝒞 ) . Lemma 4.32. The canonical pr ojection 𝜋 : Pic A ( 𝒞 ) − → Pic A ( 𝒞 )/ Pic SZ ( 𝒞 ) induces a heap structur e on the set of equiv alence classes Pic A ( 𝒞 )/ Pic SZ ( 𝒞 ) . 105 4. Twist ed ce ntres Proof. The claim follo ws from a gener al obser v ation. Let 𝑥 ∈ Z ( 𝒞 ) and 𝑎 ∈ A ( 𝒞 ) . The half-braiding 𝜎 𝑥 , 𝑎 : 𝑥 ⊗ 𝑎 − → 𝑎 ⊗ 𝑥 is an isomor phism in A ( 𝒞 ) : 𝑦 𝑎 𝑥 ∨∨ 𝑦 𝑎 𝑥 = 𝑦 𝑎 𝑥 ∨∨ 𝑦 𝑎 𝑥 𝜎 𝑎 ⊗ 𝑥 , 𝑦 ◦ ( 𝜎 𝑥 , 𝑎 ⊗ ∨∨ 𝑦 ) = ( 𝑦 ⊗ 𝜎 𝑥 , 𝑎 ) ◦ 𝜎 𝑥 ⊗ 𝑎 , 𝑦 Likewise, 𝜎 𝑥 , ∨ 𝑎 : 𝑥 ⊗ ∨ 𝑎 − → ∨ 𝑎 ⊗ 𝑥 is an isomorphism in Q ( 𝒞 ) . F or all [ 𝑎 ] , [ 𝑎 ′ ] , [ 𝑎 ′ ′ ] ∈ Pic A ( 𝒞 ) and [ 𝑏 ] , [ 𝑏 ′ ] , [ 𝑏 ′ ′ ] ∈ Pic SZ ( 𝒞 ) , one calculates 𝜋 (⟨ [ 𝑎 ] , [ 𝑎 ′ ] , [ 𝑎 ′ ′ ] ⟩ ) = 𝜋  [ 𝑎 ⊗ ∨ 𝑎 ′ ⊗ 𝑎 ′ ′ ]  = 𝜋  [ 𝑏 ⊗ ∨ 𝑏 ′ ⊗ 𝑏 ′ ′ ⊗ 𝑎 ⊗ ∨ 𝑎 ′ ⊗ 𝑎 ′ ′ ]  = 𝜋  [ 𝑏 ⊗ 𝑎 ⊗ ∨ ( 𝑏 ′ ⊗ 𝑎 ′ ) ⊗ 𝑏 ′ ′ ⊗ 𝑎 ′ ′ ]  = 𝜋 (⟨ [ 𝑏 ⊗ 𝑎 ] , [ 𝑏 ′ ⊗ 𝑎 ′ ] , [ 𝑏 ′ ′ ⊗ 𝑎 ′ ′ ] ⟩ ) . □ Recall that, by Exam ple 4.7 , the piv otal structures on Z ( 𝒞 ) form a heap. Using Deﬁnition 4.24 , w e can relate t his structure to t hat of t he anti-centre. Theorem 4.33. The morphism of heaps 𝜅 : Pic A ( 𝒞 ) − → Piv Z ( 𝒞 ) , [ 𝑎 ] ↦− → 𝜌 𝑎 induces a unique injective heap morphism 𝜄 : Pic A ( 𝒞 )/ Pic SZ ( 𝒞 ) − → Piv Z ( 𝒞 ) , such t hat the following diagram commutes: Pic A ( 𝒞 ) Piv Z ( 𝒞 ) Pic A ( 𝒞 )/ Pic SZ ( 𝒞 ) 𝜋 𝜅 ∃ ! 𝜄 (4.2.7) Proof. Lemmas 4.25 and 4.26 sho w t hat 𝜅 is w ell-deﬁned. Giv en t hree ele- ments [ 𝑎 ] , [ 𝑏 ] , [ 𝑐 ] ∈ Pic A ( 𝒞 ) , w e compute 𝜅 (⟨[ 𝑎 ] , [ 𝑏 ] , [ 𝑐 ]⟩ ) = 𝜌 𝑎 ⊗ ∨ 𝑏 ⊗ 𝑐 = 𝜌 𝑎 ◦ 𝜌 ∨ 𝑏 𝜌 𝑐 = 𝜌 𝑎 ◦ 𝜌 − 1 𝑏 ◦ 𝜌 𝑐 = ⟨ 𝜌 𝑎 , 𝜌 − 1 𝑏 , 𝜌 𝑐 ⟩ . The second equality follo ws from the hexagon identities as Equation ( 4.2.6 ), and the t hir d one b y Lemma 4.30 . 106 4.2. Piv otal structures and twisted centres Proposition 4.31 states t hat for an y tw o elements [ 𝑎 ] , [ 𝑏 ] ∈ Pic A ( 𝒞 ) w e ha v e 𝜅 ([ 𝑎 ]) = 𝜅 ([ 𝑏 ]) if and onl y if 𝜋 ([ 𝑎 ]) = 𝜋 ([ 𝑏 ]) . It follo ws from Lemma 4.32 that the unique injectiv e map 𝜄 : Pic A ( 𝒞 )/ Pic SZ ( 𝒞 ) − → Piv Z ( 𝒞 ) that lets Diagram ( 4.2.7 ) commute is a morphism of heaps. □ Exam ple 4.34. By [ Shi19 , Theorem 1.1], t he Picar d g roup Pic SZ ( 𝒞 ) of the Müger centre of a ﬁnite tensor categor y 𝒞 o v er an algebr aically closed ﬁeld is trivial. In t his setting, the induced piv otal structures depend onl y on the Picard heap Pic A ( 𝒞 ) and not on a quotient t hereof. On the o t her side of t he spectr um, one might consider the discrete categor y E 𝐺 of an abelian group 𝐺 ; its set of objects is 𝐺 and all morphisms are identities. 12 The category E 𝐺 is rigid monoidal; the tensor product giv en b y 12 In analogy with the delooping B 𝐺 of 𝐺 , the category E 𝐺 is also called the to tal space of t he group. the multiplication of 𝐺 and t he lef t and right duals giv en b y t he respectiv e in v erses. A direct computation sho ws that SZ ( E 𝐺 ) = Z ( E 𝐺 )  E 𝐺 . Since E 𝐺 is skeletal and ev er y object is in vertible, Pic SZ ( E 𝐺 )  𝐺 . As the double dual and identity functor on E 𝐺 coincide, t he same argument implies that Pic A ( E 𝐺 )  𝐺 , whence it follo ws t hat Pic A ( E 𝐺 )/ Pic SZ ( E 𝐺 )  { 1 } . In good cases, all piv otal structures on the centre of 𝒞 are induced b y t he quasi-piv otal structures of 𝒞 . Proposition 4.35 ([ Shi23 a , Theorem 4.1]) . F or a ﬁnite tensor cat egory 𝒞 , the map 𝜄 : Pic A ( 𝒞 ) ⧸ Pic SZ ( 𝒞 ) − → Piv Z ( 𝒞 ) is bĳective. Ho w ev er , in the introduction of [ Shi23 a ] the author states t hat it is no t t o be expected t hat t his holds true in general. In the remainder of t his section, w e will construct an explicit counterexample. Let us sketch our gener al approach: suppose t here is an object 𝑥 ∈ 𝒞 that can be endow ed with tw o diﬀerent half-braidings 𝜎 𝑥 , − and 𝜒 𝑥 , − . Assume further more t hat there is a piv otal structure 𝜁 : Id Z ( 𝒞 ) − → ∨∨ (−) on Z ( 𝒞 ) such that 𝜁 ( 𝑥 , 𝜎 𝑥 , − ) ≠ 𝜁 ( 𝑥 , 𝜒 𝑥 , − ) . If the unit of 𝒞 is t he only in v ertible object, t here is no (quasi-)piv otal structure inducing 𝜁 and therefore 𝜄 cannot be sur jectiv e. Let us deﬁne such a category 𝒞 in terms of g enerators and relations. Our approach is similar to Section 3.1.2 and again follo ws [ Kas98 , Chapter xii ]. Consider t he free monoidal categor y 𝒞 free gener ated b y a single object 𝑥 . 107 4. Twist ed ce ntres Their tensor product is giv en by 𝑥 𝑛 ⊗ 𝑥 𝑚 . . = 𝑥 𝑛 + 𝑚 . The morphisms of 𝒞 free are identities on objects plus the set 𝑀 of generating morphisms: 𝜎 𝑥 , 𝑥 : 𝑥 2 − → 𝑥 2 𝜌 𝑥 : 𝑥 − → 𝑥 ev 𝑥 : 𝑥 2 − → 1 coev 𝑥 : 1 − → 𝑥 2 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 Remark 4.36. By [ Kas98 , Lemma xii .1.2], ev er y mor phism 𝑓 : 𝑥 𝑛 − → 𝑥 𝑚 in 𝒞 free is either t he identity or can be written as 𝑓 = ( id 𝑥 𝑗 𝑙 ⊗ 𝑓 𝑙 ⊗ id 𝑥 𝑖 𝑙 ) ◦ · · · ◦ ( id 𝑥 𝑗 2 ⊗ 𝑓 2 ⊗ id 𝑥 𝑖 2 ) ◦ ( id 𝑥 𝑗 1 ⊗ 𝑓 1 ⊗ id 𝑥 𝑖 1 ) , where 𝑖 1 , 𝑗 1 , . . . , 𝑖 𝑙 , 𝑗 𝑙 ∈ N and 𝑓 1 , . . . , 𝑓 𝑙 ∈ 𝑀 . Such a presentation is not unique, but t he number 𝑙 ∈ N of gener ating morphisms needed to write 𝑓 in such a manner is. W e call it t he degree of 𝑓 and write deg ( 𝑓 ) . . = 𝑙 . Deﬁnition 4.37. The categor y 𝒞 is deﬁned as the quotient of 𝒞 free b y t he relations depicted below . This turns 𝒞 into a piv otal (strict) rigid monoidal category , and allow s us to extend 𝜎 to a symmetric braiding. T o increase readability , w e omit labelling t he strings wit h 𝑥 . = = = = (4.2.8) = = = (4.2.9) = = = = = (4.2.10) 108 4.2. Piv otal structures and twisted centres By [ Kas98 , Proposition x ii .1.4] there is a unique functor 𝑃 : 𝒞 free − → 𝒞 that maps objects to t hemsel v es and the gener ating mor phisms to t heir respectiv e equiv alence classes. Deﬁnition 4.38. Consider a morphism 𝑓 ∈ 𝒞 ( 𝑥 𝑛 , 𝑥 𝑚 ) . A present ation of 𝑓 is a morphism 𝑔 ∈ 𝒞 free ( 𝑥 𝑛 , 𝑦 𝑛 ) such t hat 𝑓 = 𝑃 𝑔 . If t he deg ree of 𝑔 is minimal amongst t he presentations of 𝑓 , w e call it a minimal present ation . Theorem 4.39. The category 𝒞 of Deﬁnition 4.37 is strict rigid and the double dual functor is t he identity . F urther , id 𝑥 , 𝜌 𝑥 : 𝑥 − → 𝑥 can be ext ended to pivo tal structur es, and 𝜎 𝑥 , 𝑥 : 𝑥 2 − → 𝑥 2 may be ext ended to a symmetric br aiding. Proof. The ev aluation and coev aluation morphisms plus t heir snake identities make 𝑥 ∈ 𝒞 , and by extension ev er y object of 𝒞 , its own lef t and right dual. Using Equation ( 4.2.9 ), w e compute ∨ 𝜌 𝑥 = 𝜌 𝑥 = 𝜌 𝑥 ∨ , ∨ 𝜎 𝑥 , 𝑥 = 𝜎 𝑥 , 𝑥 = 𝜎 𝑥 , 𝑥 ∨ , ∨ ev 𝑥 = coev 𝑥 = ev 𝑥 ∨ , ∨ coev 𝑥 = ev 𝑥 = coev 𝑥 ∨ . Hence, 𝒞 is strict rigid and its double dual functor is equal to t he identity . Our candidate for a non-trivial piv otal structure on 𝒞 is 𝜌 : Id 𝒞 − → Id 𝒞 , 𝜌 𝑥 𝑛 . . = 𝜌 𝑥 ⊗ · · · ⊗ 𝜌 𝑥 : 𝑥 𝑛 − → 𝑥 𝑛 , for 𝑛 ∈ N . By construction, this famil y of isomorphisms is compatible wit h t he monoidal structure of 𝒞 , so w e onl y ha v e to prov e naturality , for which it suﬃces t o check the generat ors. Equation ( 4.2.10 ) implies t hat 𝜌 𝑥 2 commutes with 𝜎 𝑥 , 𝑥 . F or the evaluation of 𝑥 ∈ 𝒞 we use the dual of Equation ( 4.2.9 ) to compute ev 𝑥 ◦ 𝜌 𝑥 2 = ev 𝑥 ◦ ( 𝜌 𝑥 ⊗ 𝜌 𝑥 ) = ev 𝑥 ◦ ( ∨ 𝜌 𝑥 ⊗ 𝜌 𝑥 ) = ev 𝑥 ◦ ( id 𝑥 ⊗ 𝜌 2 𝑥 ) = 𝜌 1 ◦ ev 𝑥 . Appl ying the lef t dualising functor , one obtains coev 𝑥 ◦ 𝜌 1 = 𝜌 𝑥 2 ◦ coev 𝑥 , whence 𝜌 deﬁnes a pivo tal structure. Lastl y , 𝜎 𝑥 , 𝑥 implements a symmetry 𝜎 : ⊗ = ⇒ ⊗ op on 𝒞 . Set 𝜎 𝑥 , 𝑥 𝑚 . . = ( id 𝑥 ⊗ 𝜎 𝑥 , 𝑥 𝑚 − 1 ) ◦ ( 𝜎 𝑥 , 𝑥 ⊗ id 𝑥 𝑚 − 1 ) , for 𝑚 ∈ N , and extend this to arbitrary objects b y 𝜎 𝑥 𝑛 , 𝑥 𝑚 . . = ( 𝜎 𝑥 𝑛 − 1 , 𝑚 ⊗ id 𝑥 ) ◦ ( id 𝑥 𝑛 − 1 ⊗ 𝜎 𝑥 , 𝑥 𝑚 ) , for 𝑛 , 𝑚 ∈ N . 109 4. Twist ed ce ntres As t his family of isomor phisms is constructed according to the hexagon axioms, w e onl y ha v e to prov e its naturality . Ag ain, it suﬃces to consider the gener ating morphisms. Equation ( 4.2.10 ) implies that 𝜎 is natural with respect to 𝜌 𝑥 , 𝜎 𝑥 , 𝑥 , and coev 𝑥 . The self-duality of 𝜎 𝑥 , 𝑥 and ∨ coev 𝑥 = ev 𝑥 impl y t he desired commutation betw een 𝜎 and ev 𝑥 . Thus 𝜎 is a braiding on 𝒞 , which is symmetric b y Equation ( 4.2.8 ). □ W e t hink of a generic mor phism of 𝒞 to be of the form 1 2 3 4 5 6 1 2 3 4 5 6 7 8 (4.2.11) This diag r am sugges ts a distinction betw een diﬀerent kinds of mor phisms: there are connectors , which link an input to an output v ertex, closed loops , and half-circles of evaluation and coevaluation-type . Connectors induce a per - mutation on a subset of N . For example, t he permutation arising from Equa- tion ( 4.2.11 ) can be identiﬁed with ( 1 2 )( 3 4 ) . Con v ersely , let 𝑠 . . = 𝑡 𝑖 1 . . . 𝑡 𝑖 𝑙 ∈ 𝑆 𝑛 As usual, 𝑆 𝑛 denotes the symmetric group on { 1 , 2 , . . . , 𝑛 } . be a permutation written as a product of elementary transpositions and set 𝑓 𝑠 . . = 𝑓 𝑡 𝑖 1 . . . 𝑓 𝑡 𝑖 𝑙 : 𝑥 𝑛 − → 𝑥 𝑛 , where 𝑓 𝑡 𝑖 . . = id 𝑥 𝑖 − 1 ⊗ 𝜎 𝑥 , 𝑥 ⊗ id 𝑥 𝑛 −( 𝑖 + 1 ) : 𝑥 𝑛 − → 𝑥 𝑛 , for 1 ≤ 𝑖 ≤ 𝑛 − 1 . As the braiding 𝜎 is symmetric, 𝑓 𝑠 does not depend on t he presentation of 𝑠 . If t he presentation of 𝑠 is minimal, ho w ev er , so is the corresponding one of 𝑓 𝑠 . T o deriv e a nor mal form of the automorphisms of 𝒞 and turn our previ- ousl y explained t houghts into precise mat hematical statements, w e need to study t he “topological” features of the mor phisms in 𝒞 . Remark 4.40. W e recall t he category of tang les 𝒯 , a close relativ e to t he string diagrams arising from 𝒞 , based on [ Kas98 , Chapter xi i .2]. Its objects are ﬁnite sequences in { + , − } and its mor phisms are isotop y classes of oriented tangles. A detailed discussion of tangles is giv en in [ Kas98 , Deﬁnition x .5.1]. F or us, it suﬃces to think of an oriented tangle 𝐿 of type ( 𝑛 , 𝑚 ) as a ﬁnite disjoint union of embeddings of eit her t he unit circle 𝑆 1 or the interv al [ 0 , 1 ] into R 2 × [ 0 , 1 ] , such that 𝜕 𝐿 = 𝐿 ∩  R 2 × { 0 , 1 }  = ( [ 𝑛 ] × { ( 0 , 0 ) }) ∪ ( [ 𝑙 ] × { ( 0 , 1 ) }) , 110 4.2. Piv otal structures and twisted centres where [ 𝑛 ] = { 1 , . . . , 𝑛 } and [ 𝑙 ] = { 1 , . . . , 𝑙 } . The orientation on each of t he connected com ponents of 𝐿 is induced b y t he counter-clockwise orientation of 𝑆 1 and the (ascending) orientation of [ 0 , 1 ] . The tensor product of tangles is giv en by pasting them next to each other . Their composition is implemented, b y appropriate gluing and rescaling. T o distinguish isotop y classes of tangles, one can study t heir images un- der t he projection R 2 × [ 0 , 1 ] − ↠ R × [ 0 , 1 ] . This leads to a combinatorial description of 𝒯 , see for example [ Kas98 , Theorem xii .2.2]. Proposition 4.41. The strict monoidal category 𝒯 is gener ated by the morphisms + − − + + − − + + + + + + + + + ev + : + ⊗ − − → 1 coev + : 1 − → − ⊗ + ev − : − ⊗ + − → 1 coev − : 1 − → + ⊗ − 𝜏 + , + : + ⊗ + − → + ⊗ + 𝜏 − 1 + , + : + ⊗ + − → + ⊗ + They are subject to the following r elations: + + = + + + + = − − = − − − − = + + + + = + + + + + + + + = + + + + + + + + + + + + = = − − − − − − − − = − − − − − − − − 111 4. Twist ed ce ntres − + + + − + − + − + = − − = − + + + − + − + − + = − − = = + + + + + + = The connection betw een tangles and t he categor y 𝒞 is attained b y appl y- ing [ Kas98 , Proposition x ii .1.4] in a similar w a y as w e did in Lemma 3.21 . Lemma 4.42. Ther e exists a strict monoidal funct or 𝑆 : 𝒯 − → 𝒞 t hat is uniquel y determined by 𝑆 (+) = 𝑥 = 𝑆 (−) and 𝑆 ( e v ± ) = ev 𝑥 , 𝑆 ( coe v ± ) = coev 𝑥 , 𝑆 ( 𝜏 ± + , + ) = 𝜎 𝑥 , 𝑥 . W e now want to lif t t he morphisms of 𝒞 to 𝒯 . Hereto w e w ant to “trivialise” the gener ator 𝜌 𝑥 , 𝑥 : 𝑥 − → 𝑥 . Set 𝒞 /⟨ 𝜌 𝑥 ⟩ to be the categor y obtained from 𝒞 b y identifying 𝜌 𝑥 with id 𝑥 . The projection functor Pr : 𝒞 − → 𝒞 /⟨ 𝜌 𝑥 ⟩ allo ws us to deﬁne an equivalence relation on the mor phisms of 𝒞 : 𝑓 ∼ 𝑔 ⇐ ⇒ Pr ( 𝑓 ) = Pr ( 𝑔 ) . F or example t he endomor phisms  : 1 − → 1 and •  : 1 − → 1 of the monoidal unit of 𝒞 w ould be equivalent with respect to t his relation. Proposition 4.43. Every 𝑓 ∈ A ut 𝒞 ( 𝑥 𝑛 , 𝑥 𝑛 ) can be uniquel y written as 𝑓 = 𝑓 𝑠 ◦ 𝑓 𝜙 , wher e 𝑓 𝑠 : 𝑥 𝑛 − → 𝑥 𝑛 is the automorphism induced by a permutation 𝑠 ∈ 𝑆 𝑛 and 𝑓 𝜙 = 𝜌 𝜙 1 𝑥 ⊗ · · · ⊗ 𝜌 𝜙 𝑛 𝑥 , with 𝜙 1 , . . . , 𝜙 𝑛 ∈ Z 2 . F urthermor e, if a minimal pr esentation 𝑠 . . = 𝑡 𝑖 1 . . . 𝑡 𝑖 𝑙 is ﬁxed, t he resulting pr esentation of 𝑓 is minimal as well. Proof. F or any 𝑓 ∈ A ut 𝒞 ( 𝑥 𝑛 ) there exists another automorphism 𝑔 ∈ A ut 𝒞 ( 𝑥 𝑛 ) , such that Pr 𝑓 = Pr 𝑔 and 𝑔 has a presentation in which no copies of 𝜌 occur . By proceeding analogous to [ Kas98 , Lemma x .3.3], w e construct a tangle 𝐿 𝑔 out of 𝑔 such t hat 𝑆 ( 𝐿 𝑔 ) = 𝑔 , and it is isotopic to a tangle 𝐿 ′ 𝑔 whose images of its connected com ponents under t he projection R 2 × [ 0 , 1 ] − → R × [ 0 , 1 ] are 112 4.2. Piv otal structures and twisted centres either closed loops, half-circles of ev aluation or coev aluation-type, or straight lines. W rite 𝐿 triv 𝑛 for a tangle t hat projects to 𝑛 parallel straight lines: { ( 𝑘 , 𝑡 ) | 𝑡 ∈ [ 0 , 1 ] and 𝑘 ∈ { 1 , . . . , 𝑛 } } . Since 𝑔 is in v ertible by assump tion, w e can lif t its in v erse 𝑔 − 1 to a tangle 𝐿 𝑔 − 1 with [ 𝐿 𝑔 ] ◦ [ 𝐿 𝑔 − 1 ] = [ 𝐿 triv 𝑛 ] = [ 𝐿 𝑔 − 1 ] ◦ [ 𝐿 𝑔 ] . This equation implies t hat 𝐿 ′ 𝑔 could not hav e contained any loops or half-circles. In o ther w ords 𝑔 = 𝑓 𝑠 , where 𝑓 𝑠 is t he morphism obtained from t he permutation 𝑠 ∈ 𝑆 𝑛 , induced b y the projection of 𝐿 ′ 𝑔 onto R × [ 0 , 1 ] . Due to t he naturality of 𝜎 𝑥 , 𝑥 , the equiv alence betw een 𝑓 and 𝑔 implies 𝑓 = 𝑓 𝑠 ◦ 𝑓 𝜙 , wit h 𝑓 𝜙 being a tensor product of identities and copies of 𝜌 𝑥 . Consequentially , a minimal representation of 𝑠 induces a minimal representation of 𝑓 . □ The ﬁrs t step in sho wing that the 𝜄 deﬁned in Theorem 4.33 cannot be sur jectiv e is to prov e t hat t he Pic A ( 𝒞 ) contains at most tw o elements. Corollar y 4.44. The only quasi-pivo tal structur es on the category 𝒞 of Deﬁni- tion 4.37 ar e id : Id 𝒞 − → Id 𝒞 and 𝜌 : Id 𝒞 − → Id 𝒞 . Proof. The onl y in v ertible object of 𝒞 is its monoidal unit, which im plies t hat an y quasi-piv otal structure on 𝒞 is pivo tal. By Proposition 4.43 , these are determined by their value on 𝑥 and A ut 𝒞 ( 𝑥 ) = { id 𝑥 , 𝜌 𝑥 } . □ Let us no w f ocus on the v arious w a ys in which we can equip an object 𝑦 ∈ 𝒞 with a half-br aiding. Our classiﬁcation of automor phisms in 𝒞 allo ws us to easil y v erify t hat on 𝑥 ∈ 𝒞 there are four diﬀerent half-braidings, which are determined by 𝜎 ◦ , ◦ 𝑥 , 𝑥 : 𝑥 2 − → 𝑥 2 𝜎 ◦ , • 𝑥 , 𝑥 : 𝑥 2 − → 𝑥 2 𝜎 • , • 𝑥 , 𝑥 : 𝑥 2 − → 𝑥 2 𝜎 • , ◦ 𝑥 , 𝑥 : 𝑥 2 − → 𝑥 2 The fact that t hese braidings are distinguished by the appearances of 𝜌 on the respectiv e strings motiv ates our next deﬁnition. Deﬁnition 4.45. Let 𝑓 . . = 𝑓 𝑠 𝑓 𝜙 : 𝑥 𝑛 − → 𝑥 𝑛 be an automorphism in 𝒞 . Its char acteris tic sequence is 𝜙 . . = ( 𝜙 1 , . . . , 𝜙 𝑛 ) ∈ ( Z 2 ) 𝑛 with 𝑓 𝜙 = 𝜌 𝜙 1 𝑥 ⊗ · · · ⊗ 𝜌 𝜙 𝑛 𝑥 . 113 4. Twist ed ce ntres Indeed, it is the interpla y between instances of 𝜌 and t he underl ying permutation t hat deter mine whet her an automorphism 𝜒 𝑦 , 𝑥 : 𝑦 ⊗ 𝑥 − → 𝑥 ⊗ 𝑦 can be lifted to a half-braiding. Lemma 4.46. Any automorphism 𝜒 𝑦 , 𝑥 : 𝑦 ⊗ 𝑥 − → 𝑥 ⊗ 𝑦 ext ends t o a half- br aiding on 𝑦 if and only if ther e exists an 𝑓 ∈ A ut 𝒞 ( 𝑦 ) with char acteris tic se- quence ( 𝜙 1 , . . . , 𝜙 𝑛 ) and underl ying permutation 𝑠 ∈ 𝑆 𝑛 , suc h t hat 𝑠 2 ( 𝑖 ) = 𝑖 and 𝜙 𝑠 ( 𝑖 ) = 𝜙 𝑖 for all 1 ≤ 𝑖 ≤ 𝑛 , and 𝜒 𝑦 , 𝑥 = 𝜎 𝑦 , 𝑥 ◦ ( 𝑓 ⊗ 𝜌 𝑗 𝑥 ) for an integ er 𝑗 ∈ Z 2 . Proof. Assume 𝜒 𝑦 , 𝑥 : 𝑦 ⊗ 𝑥 − → 𝑥 ⊗ 𝑦 to induce a half-br aiding on 𝑦 . . = 𝑥 𝑛 . Due to Proposition 4.43 , w e can write 𝜒 𝑦 , 𝑥 = 𝜎 𝑦 , 𝑥 ◦ ( 𝑓 ⊗ 𝜌 𝑗 𝑥 ) , where 𝑓 : 𝑦 − → 𝑦 is an automorphism of 𝑦 and 𝑗 ∈ Z 2 . Let 𝜙 = ( 𝜙 1 , . . . , 𝜙 𝑛 ) be the characteristic sequence of 𝑓 and 𝑠 ∈ 𝑆 𝑛 its underl ying per mutation. W rite 𝑓 𝑠 : 𝑦 − → 𝑦 for the mor phism induced by 𝑠 and set 𝑓 𝜙 = 𝜌 𝜙 1 𝑥 ⊗ · · · ⊗ 𝜌 𝜙 𝑛 𝑥 , 𝑓 𝑠 − 1 ( 𝜙 ) = 𝜌 𝜙 𝑠 − 1 ( 1 ) 𝑥 ⊗ · · · ⊗ 𝜌 𝜙 𝑠 − 1 ( 𝑛 ) 𝑥 . Using that 𝑓 = 𝑓 𝑠 ◦ 𝑓 𝜙 , the naturality of 𝜒 𝑦 , − , and Equation ( 4.2.9 ), w e compute 𝑥 𝑦 𝑦 = 𝑥 = ev 𝑥 ⊗ 𝑥 𝑛 − 1 𝑥 𝑛 − 1 𝑦 ev 𝑥 ⊗ 𝑥 𝑛 − 1 𝑥 𝑛 − 1 𝑦 𝜙 𝑗 𝑓 𝑠 𝑓 𝜙 𝜙 𝑗 𝑓 𝑠 𝑓 𝜙 𝑥 = ev 𝑥 ⊗ 𝑥 𝑛 − 1 𝑥 𝑛 − 1 𝑦 𝑓 𝑠 𝑓 𝜙 𝑓 𝑠 𝑓 𝑠 (− 1 ) )( 𝜙 ) 𝑦 = 𝑦 𝑓 𝑠 𝑓 𝜙 𝑓 𝑠 𝑓 𝑠 (− 1 ) )( 𝜙 ) This is equiv alent to 𝑠 being an inv olution and 𝜙 being in variant under 𝑠 . Con v ersely , let 𝜒 𝑦 , 𝑥 = 𝜎 𝑦 , 𝑥 ◦ ( 𝑓 ⊗ 𝜌 𝑗 𝑥 ) : 𝑦 ⊗ 𝑥 − → 𝑥 ⊗ 𝑦 , where 𝑓 is an automorphism satisfying t he assump tions of t he lemma. W e extend it to a famil y of automorphisms 𝜒 𝑦 , − : 𝑦 ⊗ − − → − ⊗ 𝑦 according to t he hexagon axioms and v erify its naturality on t he gener ators of 𝒞 . F or 𝜌 𝑥 and 𝜎 𝑥 , 𝑥 this is immediate consequence of t heir respectiv e naturality conditions. T o pro v e the commutation relations betw een 𝜒 𝑦 , − , coev 𝑥 , and ev 𝑥 , argue as abo v e. □ The previous lemma sev erel y restricts the number of possibilities in which an automorphism of 𝒞 can lift to the centre Z ( 𝒞 ) . 114 4.2. Piv otal structures and twisted centres Corollar y 4.47. Consider an object 𝑥 𝑛 ∈ 𝒞 equipped with two half-braidings 𝜒 𝑥 𝑛 , 𝑥 = 𝜎 𝑥 𝑛 , 𝑥 ◦ (( 𝑓 𝑠 ◦ 𝑓 𝜙 ) ⊗ 𝜌 𝑗 𝑥 ) , 𝜃 𝑥 𝑛 , 𝑥 = 𝜎 𝑥 𝑛 , 𝑥 ◦ (( 𝑓 𝑡 ◦ 𝑓 𝜓 ) ⊗ 𝜌 𝑘 𝑥 ) . If 𝑔 = 𝑔 𝑟 ◦ 𝑔 𝜆 ∈ A ut 𝒞 ( 𝑥 𝑛 ) lifts to a morphism 𝑔 : ( 𝑥 𝑛 , 𝜒 𝑥 𝑛 , − ) − → ( 𝑥 𝑛 , 𝜃 𝑥 𝑛 , − ) of objects in t he centre of 𝒞 , then 𝜙 𝑖 ◦ 𝜆 𝑠 𝑟 ( 𝑖 ) = 𝜓 𝑟 ( 𝑖 ) ◦ 𝜆 𝑟 ( 𝑖 ) for all 1 ≤ 𝑖 ≤ 𝑛 . Proof. For 𝑔 = 𝑓 𝑟 ◦ 𝑓 𝜆 ∈ A ut 𝒞 ( 𝑥 𝑛 ) to lif t to t he centre it must satisfy 𝜎 𝑥 𝑛 , 𝑥 ◦ (( 𝑓 𝑠 ◦ 𝑓 𝜙 ◦ 𝑔 ) ⊗ 𝜌 𝑗 𝑥 ) = 𝜒 𝑥 𝑛 , 𝑥 ◦ ( 𝑔 ⊗ id 𝑥 ) = ( id 𝑥 ⊗ 𝑔 ) ◦ 𝜃 𝑥 𝑛 , 𝑥 = 𝜎 𝑥 𝑛 , 𝑥 ◦ (( 𝑔 ◦ 𝑓 𝑡 ◦ 𝑓 𝜓 ) ⊗ 𝜌 𝑘 𝑥 ) . This im plies that 𝑓 𝑠 ◦ 𝑓 𝜙 ◦ 𝑔 = 𝑔 ◦ 𝑓 𝑡 ◦ 𝑓 𝜓 , and therefore 𝜙 𝑠 ( 𝑖 ) ◦ 𝜆 𝑠 𝑟 ( 𝑖 ) = 𝜆 𝑟 ( 𝑖 ) ◦ 𝜓 𝑟 𝑡 ( 𝑖 ) for all 1 ≤ 𝑖 ≤ 𝑛 . Since Z 2 is abelian and 𝜙 𝑠 ( 𝑖 ) = 𝜙 𝑖 as w ell as 𝜓 𝑡 ( 𝑖 ) = 𝜓 𝑖 , t he claim follo ws. □ In view of Lemma 4.46 , w e state a reﬁned v ersion of Deﬁnition 4.45 . Deﬁnition 4.48. Consider an object 𝑦 = ( 𝑥 𝑛 , 𝜒 𝑥 𝑛 , 𝑥 ) ∈ Z ( 𝒞 ) whose half- braiding is deﬁned b y 𝜒 𝑥 𝑛 , 𝑥 = 𝜎 𝑥 𝑛 , 𝑥 ◦ ( 𝑓 ⊗ 𝜌 𝑗 𝑥 ) for an integer 𝑗 ∈ Z 2 . W e call the characteristic sequence 𝜙 of 𝑓 the signature of 𝑦 . W e now construct a piv otal structure on t he centre of 𝒞 that diﬀers from the lif ts of id and 𝜌 from 𝒞 to Z ( 𝒞 ) . Theorem 4.49. The Drinfeld centre Z ( 𝒞 ) of 𝒞 admits a pivot al structur e 𝜁 wit h 𝜁 ( 𝑥 , 𝜎 ◦ , ◦ 𝑥 , − ) = id 𝑥 , 𝜁 ( 𝑥 , 𝜎 ◦ , • 𝑥 , − ) = id 𝑥 , 𝜁 ( 𝑥 , 𝜎 • , ◦ 𝑥 , − ) = 𝜌 𝑥 , 𝜁 ( 𝑥 , 𝜎 • , • 𝑥 , − ) = 𝜌 𝑥 . Proof. For an y object 𝑦 ∈ Z ( 𝒞 ) , deﬁne 𝜁 𝑦 = 𝜌 𝜙 1 𝑥 ⊗ · · · ⊗ 𝜌 𝜙 𝑛 𝑥 , where 𝜙 = ( 𝜙 1 , . . . , 𝜙 𝑛 ) is the signature of 𝑦 . Since the signature of a tensor product of objects in Z ( 𝒞 ) is giv en b y concaten- ating t he signatures of its components, t his deﬁnes a family of isomorphisms 𝜁 : Id Z ( 𝒞 ) − → Id Z ( 𝒞 ) that is compatible wit h t he monoidal structure. It remains to prov e t he naturality of 𝜁 , which can be veriﬁed by considering all possible lif ts of identities and generat ors of 𝒞 to its Drinfeld centre. For id 𝑥 , 𝜌 𝑥 : 𝑥 − → 𝑥 and 𝜎 𝑥 , 𝑥 : 𝑥 2 − → 𝑥 2 , this follo ws from Corollar y 4.47 . T o 115 4. Twist ed ce ntres study t he coev aluation of 𝑥 , ﬁx a half-braiding 𝜒 𝑥 2 , − : 𝑥 2 ⊗ − − → − ⊗ 𝑥 2 on 𝑥 2 . Due to Lemma 4.46 , it is deter mined b y 𝜒 𝑥 2 , 𝑥 = 𝜎 𝑥 2 , 𝑥 ◦  ( 𝜎 𝑖 𝑥 , 𝑥 ◦ ( 𝜌 𝑗 𝑥 ⊗ 𝜌 𝑘 𝑥 )) ⊗ 𝜌 𝑙 𝑥  , where 𝑖 , 𝑗 , 𝑘 , 𝑙 ∈ Z 2 . N o w suppose coev 𝑥 : 1 − → 𝑥 2 lifts to a morphism in Z ( 𝒞 ) , where 𝑥 2 is equipped with this half-braiding. Equation ( 4.2.9 ) and t he self-duality of 𝜎 𝑥 , 𝑥 impl y t hat 𝜎 𝑥 , 𝑥 ◦ coev 𝑥 = coev 𝑥 and ev 𝑥 ◦ 𝜎 𝑥 , 𝑥 = ev 𝑥 ; w e compute 𝑥 𝑥 = 𝑥 𝑥 = 𝑥 𝑥 𝜌 𝑗 𝜌 𝑘 = 𝑥 𝑥 𝜌 𝑗 𝜌 𝑘 = 𝑥 𝑥 𝜌 𝑗 𝜌 𝑘 Therefore, 𝑗 = 𝑘 and 𝜁 ( 𝑥 2 , 𝜒 𝑥 2 , − ) = id 2 𝑥 or 𝜁 ( 𝑥 2 , 𝜒 𝑥 2 , − ) = 𝜌 2 𝑥 , impl ying naturality . A similar argument for t he ev aluation of 𝑥 concludes t he proof. □ By Corollar y 4.44 , the Picard heap of A ( 𝒞 ) can ha v e at most tw o elements. Ho w ev er , the abo v e t heorem constructs a third piv otal structure on Z ( 𝒞 ) ; t his implies our desired result. Theorem 4.50. The pivo tal structur e 𝜁 of Z ( 𝒞 ) is not induced by the Picar d heap of A ( 𝒞 ) . In particular , t he map 𝜄 : Pic A ( 𝒞 )/ Pic SZ ( 𝒞 ) − → Piv Z ( 𝒞 ) of The- or em 4.33 is not surjective. 116 [D]ie Monade, das un v ollkommenste aller W esen. Arthur Sc hopen ha ue r ; Die K unst, Recht zu behalten M O N A D I C TA N N A K A – K R E I N R E C O N S T RU C T I O N 5 Bim onads a nd H opf monads are a v ast gener alisation of bialgebr as and Hopf algebr as. They naturall y arise in t he s tudy of (rigid) monoidal categories and topological quantum ﬁeld t heories, see amongst others [ KL01 ; Moe02 ; B V07 ; BL V11 ; TV17 ]. Recall t hat the deﬁnition of a bialgebr a object necessaril y requires a braided monoidal category as a base, in order to write down what it means f or the multiplication to be a morphism of comonoids. How ev er , in gener al t he categor y of endofunctors is not braided, so t he naïv e notion of bialgebras does not gener alise to the monadic setting and needs to be adjusted. One possible wa y of ov ercoming this problem w as introduced and studied b y Moerdĳk under t he name Hopf monad , [ Moe02 ]. 13 Here, t he 13 As remarked in [ Moe02 ], this concept is strictl y dual to that of monoidal comonads, which are studied in [ Boa95 ]. structure mor phisms of an oplax monoidal functor ser v e as a substitute of the comultiplication and counit. There are other , sometimes non-equiv alent, notions of Hopf monad, see [ Boa95 ; MW11 ]. W e follo w [ Moe02 ], wit h a slight terminology change due to [ B V07 ; BL V11 ]. This deﬁnition aims to gener alise the paradigmatic example of t he free – f orgetful adjunction of a Hopf algebra. Thus, equipping a monad wit h the preﬁxes “bi-” or “Hopf” refers to additional structure or properties put on its Eilenberg –Moore categor y . More gener ally , a monadic inter pretation of module categories was giv en b y A guiar and Chase under t he name comodule monad , [ A C12 ]. A comodule monad o v er a bimonad generalises the notion of a comodule alg ebra ov er a bialgebr a, see Example 5.14 belo w . The main result of t his chap ter extends t he reconstruction results of [ A C12 , Proposition 4.1.2] and [ TV17 , Lemma 7.10]: Theorem 5.28 . Let 𝒞 and 𝒟 be monoidal categories, and suppose that ℳ and 𝒩 ar e right 𝒞 - and 𝒟 -module categories, respectiv ely . Let 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 be an oplax mo- noidal adjunction. Lifts of an adjunction 𝐺 : ℳ ⇄ 𝒩 : 𝑉 to a comodule adjunction ar e in bĳection with lifts of 𝑉 : 𝒩 − → ℳ to a str ong comodule functor . From the proof of this result one immediately obtains an analogue of Kell y’s doctrinal adjunction result, [ Kel7 4 ], for 𝒞 -module categories. 117 5. Monadi c T annak a–K rein reconst r uc tion Porism 5.29 . An adjunction 𝐺 : ℳ ⇄ 𝒩 : 𝑉 of 𝒞 -module categories yields a bĳec- tion between oplax 𝒞 -module structur es on 𝐹 and lax 𝒞 -module s tructures on 𝑈 . W e then obtain a T annaka –Krein reconstruction result for comodule mon- ads in the spirit of [ Moe02 , Theorem 7.1] and [ McC02 , Corollar y 3.13]. Theorem 5.31 . Let 𝐵 be a bimonad on the monoidal cat egory 𝒞 , and 𝑇 a monad on a right 𝒞 -module category ℳ . Coactions of 𝐵 on 𝑇 are in bĳection with right actions of 𝒞 𝐵 on ℳ 𝑇 , such t hat 𝑈 𝑇 is a strict comodule functor over 𝑈 𝐵 . W e can subsequently appl y these results in order to study the com parison functor of Section 2.2.1 . Namel y , t his functor alw a ys inherits the strong module structure of t he “strong adjoint”; i.e., t he right adjoint for oplax 𝒞 -module adjunctions, and the lef t one in t he lax case. Proposition 5.35 . Let ℳ and 𝒩 be left 𝒞 -module categories. • The comparison functor 𝐾 𝑉 𝐺 of an oplax 𝒞 -module adjunction 𝐺 : ℳ ⇄ 𝒩 : 𝑉 is a str ong 𝒞 -module functor . • The comparison functor 𝐾 𝑉 𝐺 of a lax 𝒞 -module adjunction 𝐺 : ℳ ⇄ 𝒩 : 𝑉 is a str ong 𝒞 -module functor . 5 . 1 b i m o n a d s Deﬁnition 5.1. A bimonad on a monoidal category 𝒞 consists of an oplax mo- noidal endofunctor 𝐵 on 𝒞 , as w ell as oplax monoidal natural transf or mations 𝜇 : 𝐵 2 = ⇒ 𝐵 , 𝜂 : Id 𝒞 = ⇒ 𝐵 such that ( 𝐵 , 𝜇 , 𝜂 ) is a monad on 𝒞 . A mor phism of bimonads is a natural transformation 𝑓 : 𝐵 = ⇒ 𝐻 betw een bimonads that is oplax monoidal as w ell as a morphism of monads. Let us make Deﬁnition 5.1 explicit. In order for a monad ( 𝐵 , 𝜇 , 𝜂 ) to be a bimonad, there has to exist a natural transf ormation 𝐵 2 : 𝐵 ◦ ⊗ = ⇒ 𝐵 ⊗ 𝐵 and a mor phism 𝐵 0 : 𝐵 1 − → 1 , such t hat all diag r ams in Figure 5.1 commute. Further , what it means for 𝜇 and 𝜂 to be oplax monoidal transformations in the diag r ammatic calculus of Section 2.7.1 is displa yed in Figure 5.2 . Remark 5.2. W e can also express Deﬁnition 5.1 in more bicategorical languag e. Then, a bimonad ma y be deﬁned as a monad 14 in t he bicategor y O plMon opl 14 A monoid in the monoidal categor y O plMon opl ( 𝒞 , 𝒞 ) . where objects are monoidal categories, 1-cells are oplax monoidal functors, and 2-cells are oplax monoidal transf ormations. 118 𝐵 ( 𝑥 ⊗ 𝑦 ⊗ 𝑧 ) 𝐵 𝑥 ⊗ 𝐵 ( 𝑦 ⊗ 𝑧 ) 𝐵 1 ⊗ 𝐵 𝑥 𝐵 𝑥 𝐵 𝑥 ⊗ 𝐵 1 𝐵 ( 𝑥 ⊗ 𝑦 ) ⊗ 𝐵 𝑧 𝐵 𝑥 ⊗ 𝐵 𝑦 ⊗ 𝐵 𝑧 𝐵 𝑥 𝐵 2; 𝑥 , 𝑦 ⊗ 𝑧 𝑥 ⊗ 𝐵 2; 𝑦 ,𝑧 𝐵 2; 𝑥 ⊗ 𝑦 , 𝑧 𝐵 2; 𝑥 , 𝑦 ⊗ 𝑧 𝐵 2; 𝑥 , 1 𝐵 2;1 , 𝑥 id 𝑥 𝐵 0 ⊗ 𝐵 𝑥 𝐵 𝑥 ⊗ 𝐵 0 𝐵 2 ( 𝑥 ⊗ 𝑦 ) 𝐵 2 1 𝐵 ( 𝐵 𝑥 ⊗ 𝐵 𝑦 ) 𝐵 ( 𝑥 ⊗ 𝑦 ) 𝐵 1 𝐵 2 𝑥 ⊗ 𝐵 2 𝑦 𝐵 𝑥 ⊗ 𝐵 𝑦 1 𝜇 𝑥 ⊗ 𝑦 𝐵 2; 𝑥 , 𝑦 𝐵 𝐵 2; 𝑥 , 𝑦 𝐵 2; 𝐵 𝑥 , 𝐵𝑦 𝜇 𝑥 ⊗ 𝜇 𝑦 𝜇 1 𝐵 0 𝐵 𝐵 0 𝑥 ⊗ 𝑦 𝐵 ( 𝑥 ⊗ 𝑦 ) 1 𝐵 1 𝐵 𝑥 ⊗ 𝐵 𝑦 1 𝜂 𝑥 ⊗ 𝑦 𝐵 2; 𝑥 , 𝑦 𝜂 𝑥 ⊗ 𝜂 𝑦 id 1 𝜂 1 𝐵 0 Figure 5.1: Explicit conditions for 𝐵 to be a bimonad. = 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 = 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 = 𝐵 𝐵 = Figure 5.2: Conditions for the multiplication and unit of a bimonad to be oplax monoidal natural transf ormations. 5. Monadi c T annak a–K rein reconst r uc tion Exam ple 5.3. Let 𝐴 ∈ 𝒞 be an object in a monoidal category . In Example 2.96 w e sa w t hat t he functor 𝐴 ⊗ − is a monad on 𝒞 if and onl y if 𝐴 is an algebr a object in 𝒞 . This relationship extends to bialgebr as: let 𝐵 ∈ 𝒞 be an algebra object, and suppose 𝑇 . . = 𝐵 ⊗ − is t he associated monad on 𝒞 . Then 𝑇 is a bimonad if and only if 𝐵 is a bialgebra. More precisely , starting wit h a bialgebr a structure ( Δ , 𝜀 ) on 𝐵 , set 𝑇 2 : 𝐵 ⊗ − Δ ⊗ − − − − − → 𝐵 ⊗ 𝐵 ⊗ − and 𝑇 0 : 𝐵 ⊗ − 𝜀 ⊗ − − − − − → − . This turns 𝑇 into a bimonad. Con v ersely , assume ( 𝑇 , 𝑇 2 , 𝑇 0 ) is an oplax monoidal functor . One deﬁnes a bialgebr a structure on 𝐵 by Δ : 𝐵 = 𝐵 ⊗ 1 ⊗ 1 𝑇 2;1 , 1 − − − → 𝐵 ⊗ 1 ⊗ 𝐵 ⊗ 1 = 𝐵 ⊗ 𝐵 , 𝜀 : 𝐵 = 𝐵 ⊗ 1 𝑇 0 − − → 1 . The fact t hat t hese arrows satisfy all of t he required identities follo ws from a str aightforw ard calculation. Exam ple 5.4. The intricate inter pla y betw een monads and adjunctions of Ex- ample 2.11 transcends to monoidal categories and bimonads. Giv en an oplax monoidal adjunction 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 , t he monad 𝑈 𝐹 : 𝒞 − → 𝒞 is a bimonad, whose comultiplication is deﬁned for ev er y 𝑥 , 𝑦 ∈ 𝒞 as t he composition 𝑈 𝐹 ( 𝑥 ⊗ 𝑦 ) 𝑈 𝐹 2; 𝑥 , 𝑦 − − − − − → 𝑈 ( 𝐹 𝑥 ⊗ 𝐹 𝑦 ) 𝑈 2; 𝐹 𝑥 , 𝐹 𝑦 − − − − − → 𝑈 𝐹 𝑥 ⊗ 𝑈 𝐹 𝑦 . Its counit is giv en by 𝑈 𝐹 1 𝑈 𝐹 0 − − − → 𝑈 1 𝑈 0 − − → 1 . The follo wing statement is a special case of [ Kel74 , Theorem 1.2]. Proposition 5.5. Given monoidal categories 𝒞 and 𝒟 , as well as an adjunction 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 , t here is a bĳection between oplax monoidal structur es on 𝐹 and lax monoidal structur es on 𝑈 . The proof of [ Kel7 4 , Theorem 1.2] is giv en constructiv ely in terms of mates. Giv en an oplax monoidal s tructure ( 𝐹 2 , 𝐹 0 ) on 𝐹 , t he lax monoidal natural transf ormation 𝑈 2 of 𝑈 is, for all 𝑥 , 𝑦 ∈ 𝒟 , giv en by 𝑈 𝑥 ⊗ 𝑈 𝑦 𝜂 𝑈 𝑥 ⊗ 𝑈 𝑦 − − − − − → 𝑈 𝐹 ( 𝑈 𝑥 ⊗ 𝑈 𝑦 ) 𝑈 𝐹 2; 𝑈 𝑥 ,𝑈 𝑦 − − − − − − − → 𝑈 ( 𝐹𝑈 𝑥 ⊗ 𝐹 𝑈 𝑦 ) 𝑈 ( 𝜀 𝑥 ⊗ 𝜀 𝑦 ) − − − − − − − → 𝑈 ( 𝑥 ⊗ 𝑦 ) , where 𝜂 and 𝜀 are the unit and counit of the adjunction. Constructing an oplax monoidal structure on 𝐹 from a lax monoidal structure on 𝑈 is similar . 120 5.1. Bimonads Let us no w highlight an important special case of Proposition 5.5 . Lemma 5.6 ([ TV17 , Lemma 7.10]) . Any adjunction between two monoidal cat- egories is oplax monoidal if and onl y if the right adjoint is str ong monoidal. Exam ple 5.7. Let k be a ﬁeld, and 𝐵 ∈ V ect a bialgebr a. Then, by Example 5.4 , the monad 𝐵 ⊗ k − : V ect − → V ect is a bimonad, and hence the adjunction 𝐵 ⊗ k − : V ect ⇄ 𝐵 -Mod : 𝑈 is oplax monoidal, where 𝑈 : 𝐵 -Mod − → V ect is the canonical f orgetful functor t hat simpl y forg ets t he 𝐵 -module structure. By Lemma 5.6 , w e obtain t hat 𝑈 is ev en strong monoidal. More generall y , let 𝐵 : 𝒞 − → 𝒞 be t he bimonad arising from an oplax monoidal adjunction 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 . As the forg etful functor 𝑈 𝐵 : 𝒞 𝐵 − → 𝒞 is strict monoidal, the Eilenberg –Moore adjunction 𝐹 𝐵 ⊣ 𝑈 𝐵 is, oplax monoidal. The follo wing result is due to [ Kel7 4 ], see also [ B V07 , Theorem 2.6]. Lemma 5.8. Let 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 be an oplax monoidal adjunction. Then t he com- parison functor 𝐾 𝑈 𝐹 : 𝒟 − → 𝒞 𝑈 𝐹 of the bimonad 𝑈 𝐹 is strong monoidal, and 𝑈 𝑈 𝐹 𝐾 𝑈 𝐹 = 𝑈 and 𝐾 𝑈 𝐹 𝐹 = 𝐹 𝑈 𝐹 as str ong respectiv el y oplax monoidal functor s. The question to which extent t he monoidal structure on 𝒞 𝐵 is unique w as answ ered by Moerdĳk [ Moe02 , Theorem 7.1] and McCr udden [ McC02 , Corollary 3.13]. In particular , t his kind of T annaka –Krein reconstruction giv es another justiﬁcation for the name “bimonad”. Proposition 5.9. Let ( 𝐵 , 𝜇 , 𝜂 ) be a monad on a monoidal category 𝒞 . Ther e ex- ists a one-to-one correspondence between bimonad structur es on 𝐵 and monoidal structur es on 𝒞 𝐵 such t hat the for getful functor 𝑈 𝐵 is strict monoidal. Sketch of proof. Giv en a bimonad ( 𝐵 , 𝜇 , 𝜂 , 𝐵 2 , 𝐵 0 ) : 𝒞 − → 𝒞 , as w ell as tw o modules ( 𝑚 , ∇ 𝑚 ) and ( 𝑛 , ∇ 𝑛 ) ∈ 𝒞 𝐵 , w e set ( 𝑚 , ∇ 𝑚 ) ⊗ ( 𝑛 , ∇ 𝑛 ) . . =  𝑚 ⊗ 𝑛 , 𝐵 ( 𝑚 ⊗ 𝑛 ) 𝐵 2; 𝑚 , 𝑛 − − − − → 𝐵 𝑚 ⊗ 𝐵 𝑛 ∇ 𝑚 ⊗ ∇ 𝑛 − − − − − → 𝑚 ⊗ 𝑛  . Moreo v er , deﬁne ∇ 1 : 𝐵 1 𝐵 0 − − → 1 . The coassociativity and counitality of t he comultiplication of 𝐵 impl y t hat t he abov e cons truction implements a mono- idal structure on 𝒞 𝐵 , parallel to t hat on t he modules ov er a bialgebr a. Con v ersely , let ( 𝐵 , 𝜇 , 𝜂 ) be a monad on the monoidal categor y 𝒞 . Suppose 𝒞 𝑇 is monoidal such that the forg etful functor 𝑈 𝑇 is strict monoidal. Consider the free modules ( 𝐵 𝑚 , 𝜇 𝑚 ) and ( 𝐵 𝑛 , 𝜇 𝑛 ) , and denote t heir tensor product b y ( 𝐵 𝑚 , 𝜇 𝑚 ) ⊗ ( 𝐵 𝑛 , 𝜇 𝑛 ) = ( 𝐵 𝑚 ⊗ 𝐵 𝑛 , 𝜇 𝑚 · 𝜇 𝑛 : 𝐵 ( 𝐵 𝑚 ⊗ 𝐵 𝑛 ) − → 𝐵 𝑚 ⊗ 𝐵 𝑛 ) . 121 5. Monadi c T annak a–K rein reconst r uc tion The comultiplication of 𝐵 is then giv en b y 𝐵 ( 𝑚 ⊗ 𝑛 ) 𝐵 ( 𝜂 𝑚 ⊗ 𝜂 𝑛 ) − − − − − − − → 𝐵 ( 𝐵 𝑚 ⊗ 𝐵 𝑛 ) 𝜇 𝑚 · 𝜇 𝑛 − − − − → 𝐵 𝑚 ⊗ 𝐵 𝑛 . F or the counit, take t he action 𝐵 1 − → 1 of t he unit of 𝒞 𝐵 . □ 5 . 2 h o p f m o n a d s We can now i n corpo ra t e r igid ity int o t he frame w ork of Section 5.1 . Deﬁnition 5.10 ([ BL V11 , Section 2.6]) . Let 𝐻 be a bimonad on a monoidal category 𝒞 . The right fusion operat or 𝐻 rf of 𝐻 is t he natural transf ormation 𝐻 rf ; 𝑥 , 𝑦 : 𝐻 ( 𝐻 𝑥 ⊗ 𝑦 ) 𝐻 2; 𝐻 𝑥 ,𝑦 − − − − − → 𝐻 2 𝑥 ⊗ 𝐻 𝑦 𝜇 𝑥 ⊗ 𝐻 𝑦 − − − − − → 𝐻 𝑥 ⊗ 𝐻 𝑦 , for 𝑥 , 𝑦 ∈ 𝒞 . The bimonad 𝐻 is called right Hopf if its right fusion operator is in v ertible. Left fusion operators and lef t Hopf monads are deﬁned dually , and a bimonad is said to be a Hopf monad if it is both lef t and right Hopf. Theorem 5.11 ([ BL V11 , Theorem 3.6]) . Let 𝒞 be a right closed monoidal cat egor y and let 𝐻 be a bimonad on 𝒞 . Then 𝐻 is a right Hopf monad if and onl y if 𝒞 𝐻 is right closed and t he for getful functor 𝑈 𝐻 : 𝒞 𝐻 − → 𝒞 is right closed. A Hopf monad 𝐻 : 𝒞 − → 𝒞 can be deﬁned on an y monoidal categor y . If 𝒞 is rigid t hen one may use [ BL V11 , Lemma 3.4 and Theorem 3.6] to obtain a rigid — not just closed — monoidal structure on the categor y of 𝐻 -modules. This is reﬂected b y the existence of tw o natural transformations 𝑠 ℓ : 𝐻 ( ∨ 𝐻 ) = ⇒ ∨ 𝐻 and 𝑠 𝑟 : 𝐻 ( 𝐻 ∨ ) = ⇒ 𝐻 ∨ , for all 𝑥 ∈ 𝒞 , (5.2.1) called t he left and right antipode of 𝐻 . In Exam ple 2.4 of [ B V12 ] it is explained ho w these generalise the antipode of a Hopf algebra; an analogous result to Example 5.3 holds in the Hopf case. 5 . 3 ( co ) m o d u l e m o na d s Alongs ide t he th eor y o f comod ule m on ad s of A guiar and Chase, [ A C12 ], w e will also dev elop t he special case of (op)lax 𝒞 -module monads in this section. The latter will pla y an important role in Chapters 8 and 9 . 122 5.3. (Co)module monads Deﬁnition 5.12 ([ A C12 , Deﬁnition 3.3.1]) . Let ( 𝐹 , 𝐹 2 , 𝐹 0 ) : 𝒞 − → 𝒟 be an oplax monoidal functor , ℳ a right 𝒞 -module, and 𝒩 a right 𝒟 -module category . A (right) comodule funct or over 𝐹 15 is a pair ( 𝐺 , 𝐺 a ) consisting of a functor 15 Alternativel y , w e will say that 𝐺 is a (right) 𝐹 -comodule functor . 𝐺 : ℳ − → 𝒩 tog ether wit h a natural transf ormation 𝐺 a ; 𝑚 , 𝑥 : 𝐺 ( 𝑚 ⊳ 𝑥 ) − → 𝐺 𝑚 ⊳ 𝐹 𝑥 , for all 𝑥 ∈ 𝒞 and 𝑚 ∈ ℳ , called t he coaction of 𝐺 , which is coassociative and counital , in t he sense t hat the follo wing diagrams commute for all 𝑥 , 𝑦 ∈ 𝒞 and 𝑚 ∈ ℳ : 𝐺 (( 𝑚 ⊳ 𝑥 ) ⊳ 𝑦 ) 𝐺 ( 𝑚 ⊳ ( 𝑥 ⊗ 𝑦 )) 𝐺 ( 𝑚 ⊳ 1 ) 𝐺 𝑚 𝐺 ( 𝑚 ⊳ 𝑥 ) ⊳ 𝐹 𝑦 𝐺 𝑚 ⊳ 𝐹 ( 𝑥 ⊗ 𝑦 ) 𝐺 𝑚 ⊳ 𝐹 1 𝐺 𝑚 ⊳ 1 ( 𝐺 𝑚 ⊳ 𝐹 𝑥 ) ⊳ 𝐹 𝑦 𝐺 𝑚 ⊳ ( 𝐹 𝑥 ⊗ 𝐹 𝑦 )  𝐺 a ; 𝑚 ⊳ 𝑥 , 𝑦 𝐺 a ; 𝑚 , 𝑥 ⊗ 𝑦  𝐺 a ; 𝑚 , 1 𝐺 a ; 𝑚 , 𝑥 ⊳ 𝐹 𝑦 𝐺 𝑚 ⊳ 𝐹 2; 𝑥 , 𝑦 𝐺 𝑚 ⊳ 𝐹 0   A comodule functor is called str ong if its coaction is an isomor phism, and strict if it is the identity . Exam ple 5.13. A right comodule functor ov er t he identity functor Id 𝒞 of a monoidal categor y 𝒞 is nothing more t han an oplax 𝒞 -module functor in the sense of Deﬁnition 2.47 , considering right instead of left 𝒞 -module categories. Exam ple 5.14 ([ A C12 , Section 6.1]) . If 𝐵 is a bialgebr a o v er a commutative ring k , then 𝐵 ⊗ k − : k -Mod − → k -Mod becomes an oplax monoidal functor . Let 𝐴 be a right 𝐵 -comodule algebra with coaction 𝜈 : 𝐴 − → 𝐴 ⊗ k 𝐵 . Recall that t he subalgebr a of 𝐵 -coinv ariants of 𝐴 is giv en by 𝐴 co 𝐵 . . = { 𝑎 ∈ 𝐴 | 𝜈 ( 𝑎 ) = 𝑎 ⊗ 1 } . Suppose t hat 𝐶 is a subalgebra of 𝐴 co 𝐵 . Then 𝜈 is a map of 𝐶 - 𝐶 -bimodules, which turns 𝐴 ⊗ 𝐶 − : 𝐶 -Mod − → 𝐶 -Mod into a right comodule funct or ov er 𝐵 ⊗ k − . The action of k -Mod on 𝐶 -Mod is giv en b y tensoring ov er k . Recall t he constructions of t he centre and twisted centre from Sections 2.4.4 and 4.2.1 . The canonical functor 𝑈 ( 𝑍 ) : Z ( 𝒞 ) − → 𝒞 that forg ets t he half braiding is strict monoidal. Similar ly , giv en a strong monoidal funct or 𝐿 , the for getful functor 𝑈 ( 𝐿 ) : Z ( 𝐿 𝒞 ) − → 𝒞 is a strict comodule functor ov er 𝑈 ( 𝑍 ) . 123 5. Monadi c T annak a–K rein reconst r uc tion Remark 5.15. The diagrammatic calculus of Section 2.7.1 can be adap ted to t he setting of comodule functors. In addition to combining sheets using tensor products, w e no w additionally consider actions t o do so as w ell. In order to keep track of which splitting occurred, functors betw een t he module categories shall be coloured blue. F or example, a coaction will be dra wn as: 𝐹 𝐺 𝐺 The coassociativity and counitality of 𝐺 a is displa y ed in Figure 5.3 ; notice the analogous nature of t he diag r ams to Figure 2.4 . 𝐺 𝐺 𝐺 𝐺 = = 𝐺 𝐺 𝐹 𝐹 𝐺 𝐹 𝐺 𝐹 Figure 5.3: Coassociativity and counitality conditions of the coaction of a comodule functor . Deﬁnition 5.16. Suppose that 𝐶 , 𝐺 : ℳ − → 𝒩 are comodule functors o v er 𝐵 , 𝐹 : 𝒞 − → 𝒟 . A comodule (natural) tr ansformation from 𝐶 to 𝐺 comprises a pair of natural tr ansformations 𝜙 : 𝐶 = ⇒ 𝐺 and 𝜓 : 𝐵 = ⇒ 𝐹 , such t hat t he follo wing diag r am commutes for all 𝑥 ∈ 𝒞 and 𝑚 ∈ ℳ : 𝐶 ( 𝑚 ⊳ 𝑥 ) 𝐺 ( 𝑚 ⊳ 𝑥 ) 𝐶 𝑚 ⊳ 𝐵 𝑥 𝐺 𝑚 ⊳ 𝐹 𝑥 𝜙 𝑚 ⊳ 𝑥 𝐺 a ; 𝑚 , 𝑥 𝐶 a ; 𝑚 , 𝑥 𝜙 𝑚 ⊳𝜓 𝑥 (5.3.1) W e call ( 𝜙 , 𝜓 ) a mor phism of comodule functor s if 𝐵 = 𝐹 and 𝜓 = id 𝐵 . 124 5.3. (Co)module monads Exam ple 5.17. If 𝒞 = 𝒟 and 𝐵 = 𝐹 = Id 𝒞 , then 𝐶 and 𝐺 are oplax 𝒞 -module functors, see Example 5.13 . A comodule natural transformation from 𝐶 to 𝐺 is exactl y a 𝒞 -module transf ormation in t he sense of Deﬁnition 2.48 . Giv en a comodule natural transf ormation ( 𝜙 : 𝐺 = ⇒ 𝐶 , 𝜓 : 𝐵 = ⇒ 𝐹 ) , the graphical v ersion of Diag r am ( 5.3.1 ) is displa yed in our next picture, where the blue dot represents 𝜙 and t he black dot represents 𝜓 . 𝐺 𝐹 𝐶 = 𝐺 𝐹 𝐶 The notion of a morphism of comodule functors might seem restrictiv e; ho w ev er , it is actuall y suﬃcient to only consider arrow s of t his form. Exam ple 5.18. Let the pair 𝜙 : 𝐶 = ⇒ 𝐺 and 𝜓 : 𝐵 = ⇒ 𝐹 be a comodule transf ormation. W e can view 𝜙 : 𝐶 = ⇒ 𝐺 as a mor phism of comodule functors ov er 𝐹 if we equip 𝐶 wit h a new coaction. It is giv en b y 𝐶 ( 𝑚 ⊳ 𝑥 ) 𝐶 a ; 𝑚 , 𝑥 − − − − − → 𝐶 𝑚 ⊳ 𝐵 𝑥 𝐶 𝑚 ⊳𝜓 𝑥 − − − − − → 𝐶 𝑚 ⊳ 𝐹 𝑥 , for all 𝑥 ∈ 𝒞 and 𝑚 ∈ ℳ . Thus, b y suitabl y altering the in v olv ed coactions, comodule natural trans- formations and mor phisms of comodule functors can be identiﬁed. Let 𝐵 : 𝒞 − → 𝒞 be a bimonad and ℳ a module categor y ov er 𝒞 . The unit 𝜂 : Id 𝒞 = ⇒ 𝐵 implements a coaction on Id ℳ : ℳ = ⇒ ℳ via id 𝑚 ⊳ 𝜂 𝑥 : Id ℳ ( 𝑚 ⊳ 𝑥 ) − → Id ℳ 𝑚 ⊳ 𝐵 𝑥 , for all 𝑥 ∈ 𝒞 , 𝑚 ∈ ℳ . Using t he multiplication 𝜇 : 𝐵 2 = ⇒ 𝐵 , we can equip t he composition 𝐶 𝐺 of tw o comodule functors 𝐶 , 𝐺 : ℳ − → ℳ with a comodule structure: ( 𝐶 𝐺 ) a : 𝐶 𝐺 (− ⊳ = ) 𝐶 𝐺 a − − − → 𝐶 ( 𝐺 (−) ⊳ 𝐵 ( = )) 𝐶 a − − → 𝐶 𝐺 (−) ⊳ 𝐵 2 ( = ) id ⊳𝜇 − − − → 𝐶 𝐺 (−) ⊳ 𝐵 ( = ) . Due to t he associativity and unitality of t he multiplication of 𝐵 , in t his wa y the categor y of comodule endofunctors on ℳ o v er 𝐵 becomes monoidal. 125 5. Monadi c T annak a–K rein reconst r uc tion Deﬁnition 5.19. Consider a bimonad 𝐵 : 𝒞 − → 𝒞 and a module categor y ℳ o v er 𝒞 . A comodule monad ov er 𝐵 on ℳ comprises a comodule endofunctor ( 𝐶 , 𝐶 a ) : ℳ − → ℳ tog ether with comodule mor phisms 𝜇 : 𝐶 2 = ⇒ 𝐶 and 𝜂 : Id ℳ = ⇒ 𝐶 such t hat ( 𝐶 , 𝜇 , 𝜂 ) is a monad. A mor phism of comodule monads is a natural transformation of comodule functors 𝑓 : 𝐶 = ⇒ 𝐺 that is also a mor phism of monads. The conditions for t he multiplication and unit of a comodule monad 𝐶 on ℳ o v er a bimonad 𝐵 on 𝒞 to be mor phisms of comodule functors is giv en in Figure 5.4 . N otice how t he conditions are analogous to Figure 5.2 . 𝐶 𝐵 𝐶 = 𝐶 𝐵 𝐶 𝐶 𝐶 = 𝐵 𝐶 𝐵 𝐶 Figure 5.4: Conditions for the multiplication and unit of a comodule monad to be morphisms of comodule functors. Exam ple 5.20. An oplax 𝒞 -module monad is a comodule monad ov er t he identity monad on 𝒞 . Alter nativ ely , it is monoid in t he categor y Oplax 𝒞 Mod ( ℳ , ℳ ) of oplax 𝒞 -module endofunctors on a (lef t) 𝒞 -module category ℳ . Analogousl y , one can deﬁne t he notion of a lax 𝒞 -module comonad . Exam ple 5.21. Consider t he poset ( R , ≤ ) . It is a monoidal category with addition as tensor product and 0 ∈ R as unit. In [ HL18 ], Hasega wa and Lema y deﬁned a bimonad using t he ceiling function. Let 𝐻 : R − → R , 𝑥 ↦− → ⌈ 𝑥 ⌉ . . = min { 𝑛 ∈ Z | 𝑛 ≥ 𝑥 } . As ⌈ 𝑥 + 𝑦 ⌉ ≤ ⌈ 𝑥 ⌉ + ⌈ 𝑦 ⌉ for all 𝑥 , 𝑦 ∈ R and ⌈ 0 ⌉ = 0 , the functor 𝐻 is oplax monoidal. The comultiplication and counit are giv en by the unique arrows 𝐻 0 : 𝐻 0 = 0 − → 0 and 𝐻 2; 𝑥 , 𝑦 : 𝐻 ( 𝑥 + 𝑦 ) − → 𝐻 𝑥 + 𝐻 𝑦 , 126 5.3. (Co)module monads for all 𝑥 , 𝑦 ∈ R . The idempotence of t he ceiling function implies that t he identity natural transf or mation deﬁnes a multiplication 𝜇 : 𝐻 2 = ⇒ 𝐻 . Its unit corresponds to { 𝑥 − → 𝐻 𝑥 } 𝑥 ∈ R . Giv en a number 𝑧 in t he half-open inter v al [ 0 , 1 ) , let 𝐶 ( 𝑧 ) : R − → R , 𝑥 ↦− → min { 𝑧 + 𝑛 | 𝑛 ∈ Z , 𝑧 + 𝑛 ≥ 𝑥 } . In case 𝑧 = 0 , w e ha v e 𝐶 ( 𝑧 ) = 𝐻 . Ot herwise 𝐶 ( 𝑧 ) 0 = 𝑧 > 0 and 𝐶 ( 𝑧 ) cannot be oplax monoidal. N onetheless, 𝐶 ( 𝑧 ) ( 𝑥 + 𝑦 ) ≤ 𝐶 ( 𝑧 ) 𝑥 + ⌈ 𝑦 ⌉ = 𝐶 ( 𝑧 ) 𝑥 + 𝐻 𝑦 holds, and the unique natural arro w 𝐶 ( 𝑧 ) a ; 𝑥 , 𝑦 : 𝐶 ( 𝑧 ) ( 𝑥 + 𝑦 ) − → 𝐶 ( 𝑧 ) 𝑥 + 𝐻 𝑦 , for all 𝑥 , 𝑦 ∈ R , deﬁnes a coaction of 𝐻 on 𝐶 . Again, w e ha v e t hat ( 𝐶 ( 𝑧 ) ) 2 = 𝐶 ( 𝑧 ) is idempotent and 𝑥 ≤ 𝐶 ( 𝑧 ) 𝑥 , for all 𝑥 ∈ R . Thus, it is a comodule monad ov er 𝐻 . Exam ple 5.22. Let ( 𝒱 , ⊗ , 1 ) be a closed symmetric monoidal categor y . A 𝒱 -category 𝒞 is said to be copower ed ov er 𝒱 , see [ Kel05 , Section 3.7], if t here exists a functor − · = : 𝒞 × 𝒱 − → 𝒞 , such t hat for all 𝑐 ∈ 𝒞 w e hav e 𝑐 · − : 𝒱 ⇄ 𝒞 : 𝒞 ( 𝑐 , −) . One obtains 𝑐 · ( 𝑣 ⊗ 𝑤 )  ( 𝑐 · 𝑣 ) · 𝑤 b y the Y oneda lemma: 𝒞  𝑐 · ( 𝑣 ⊗ 𝑤 ) , 𝑥   𝒱  𝑣 ⊗ 𝑤 , 𝒞 ( 𝑐 , 𝑥 )   𝒱  𝑤 , 𝒱 ( 𝑣 , 𝒞 ( 𝑐 , 𝑥 ))   𝒱  𝑤 , 𝒞 (( 𝑐 · 𝑣 ) , 𝑥 )   𝒞  ( 𝑐 · 𝑣 ) · 𝑤 , 𝑥  . In fact, t his tur ns 𝒞 into a 𝒱 -module category , and t heref ore t he identity monad on 𝒞 into a comodule monad ov er Id 𝒱 with trivial coaction. Suppose t hat 𝐵 . . = 𝑈 𝐹 is a bimonad on 𝒱 . The unit 𝜂 : Id 𝒱 = ⇒ 𝐵 is a morphism of bimonads and w e ma y extend t he coaction of Id 𝒞 to 𝛿 : − · = − · 𝜂 − − − → − · 𝐵 = , turning 𝐵 into a 𝒞 -module monad. The following proposition connects t he notion of a 𝒞 -module monad to that of an algebra object in 𝒞 in the sense of Section 2.6 . Proposition 5.23. Let 𝒞 be a monoidal category . There is a bĳection between str ong 𝒞 -module monads on 𝒞 and alg ebra objects in 𝒞 . 127 5. Monadi c T annak a–K rein reconst r uc tion Proof. By Proposition 2.51 , t here is an equiv alence of left module categories Str 𝒞 Mod ( 𝒞 , 𝒞 ) ∼ − → 𝒞 rev , 𝑇 ↦− → 𝑇 1 , − ⊗ 𝐴 ← − [ 𝐴 , and b y Example 2.96 , 𝐴 is an algebra object if and only if − ⊗ 𝐴 is a monad. In particular , if ( 𝐴 , 𝜏 , 𝜐 ) is an algebra object in 𝒞 , t hen (− ⊗ 𝐴 , − ⊗ 𝜏 , − ⊗ 𝜐 ) becomes a strong 𝒞 -module monad on 𝒞 . The strong 𝒞 -module structure is giv en by the associator of 𝒞 : (− ⊗ 𝐴 ) a ; 𝑋 . . = 𝑋 ⊗ (− ⊗ 𝐴 ) ∼ − → ( 𝑋 ⊗ −) ⊗ 𝐴 , for all 𝑋 ∈ 𝒞 . If ( 𝑇 , 𝜇 , 𝜂 ) is a strong 𝒞 -module monad, then evaluating at the monoidal unit yields an algebr a object in 𝒞 , wit h multiplication 𝜇 1 and unit 𝜂 1 . □ In particular , the categor y of right 𝐴 -modules and t he Eilenberg –Moore category of − ⊗ 𝐴 coincide as 𝒞 -module categories. Remark 5.24. Let 𝐵 : 𝒞 − → 𝒞 be a bimonad and ( 𝐶 , 𝐶 a ) : ℳ − → ℳ a comod- ule monad o v er it. The coaction of 𝐶 allo ws us to deﬁne an action ⊳ : ℳ 𝐶 × 𝒞 𝐵 − → ℳ 𝐶 . F or an y tw o modules ( 𝑚 , ∇ 𝑚 ) ∈ ℳ 𝐶 and ( 𝑥 , ∇ 𝑥 ) ∈ 𝒞 𝐵 , it is giv en by ( 𝑚 , ∇ 𝑚 ) ⊳ ( 𝑥 , ∇ 𝑥 ) . . =  𝑚 ⊳ 𝑥 , (∇ 𝑚 ⊳ ∇ 𝑥 ) ◦ 𝐶 a ; 𝑚 , 𝑥  . The axioms of the coaction of 𝐵 on 𝐶 translate precisely to t he compatibility of the action of 𝒞 𝐵 on ℳ 𝐶 with t he tensor product and unit of 𝒞 𝐵 . Deﬁnition 5.25. Let 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 be an oplax monoidal adjunction. Suppose that 𝐺 : ℳ ⇄ 𝒩 : 𝑉 is an adjunction such that 𝐺 is an 𝐹 -comodule functor , and 𝑉 is a 𝑈 -comodule functor . The pair ( 𝐺 ⊣ 𝑉 , 𝐹 ⊣ 𝑈 ) is a comodule adjunction if the follo wing identities hold: 𝑚 ⊳ 𝑥 𝑉 𝐺 ( 𝑚 ⊳ 𝑥 ) 𝐺𝑉 ( 𝑚 ⊳ 𝑥 ) 𝐺 ( 𝑉 𝑚 ⊳ 𝑈 𝑥 ) 𝑉 𝐺 𝑚 ⊳ 𝑈 𝐹 𝑥 𝑉 ( 𝐺 𝑚 ⊳ 𝐹 𝑥 ) 𝑚 ⊳ 𝑥 𝐺𝑉 𝑚 ⊳ 𝐹 𝑈 𝑥 𝜂 ( 𝐺 ⊣ 𝑉 ) 𝑚 ⊳ 𝑥 𝑉 𝐺 a ; 𝑚 , 𝑥 𝑉 a ; 𝐺 𝑚 , 𝐹 𝑥 𝜂 ( 𝐺 ⊣ 𝑉 ) 𝑚 ⊳𝜂 ( 𝐹 ⊣ 𝑈 ) 𝑥 𝜀 ( 𝐺 ⊣ 𝑉 ) 𝑚 ⊳ 𝑥 𝐺𝑉 a ; 𝑚 , 𝑥 𝐺 a ; 𝑉 𝑚 ,𝑈 𝑥 𝜀 ( 𝐺 ⊣ 𝑉 ) 𝑚 ⊳ 𝜀 ( 𝐹 ⊣ 𝑈 ) 𝑥 128 5.3. (Co)module monads 𝐶 𝐺 = 𝐹 𝐺 𝑉 𝑈 𝑉 𝑈 𝐺 = 𝐺 𝑉 𝑉 Figure 5.5: String diagrammatic conditions for a comodule adjunction. Exam ple 5.26. If 𝐹 = 𝑈 = Id ℳ for a (left) 𝒞 -module category ℳ , t hen w e sa y t hat 𝐺 ⊣ 𝑉 is an oplax 𝒞 -module adjunction . More explicitly , we ha v e t hat 𝐺 and 𝑉 are oplax 𝒞 -module functors such that the unit and counit of the adjunction are 𝒞 -module transf ormations. Analogousl y to t his, one can deﬁne the notion of a lax 𝒞 -module adjunction . Figure 5.5 makes plain t hat t he conditions required by Deﬁnition 5.25 are analogous to t hose stated in Figure 2.5 . Exam ple 5.27. The philosophy that monads and adjunctions are tw o sides of the same coin extends t o comodule functors. Suppose t hat w e ha v e an oplax monoidal adjunction 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 and o v er it a comodule adjunction 𝐺 : ℳ ⇄ 𝒩 : 𝑉 . By [ A C12 , Proposition 4.3.1], t he bimonad 𝐵 . . = 𝑈 𝐹 admits a coaction on the monad 𝐶 . . = 𝑉 𝐺 ; for all 𝑚 ∈ ℳ and 𝑥 ∈ 𝒞 it is giv en b y 𝑉 𝐺 ( 𝑚 ⊳ 𝑥 ) 𝑉 𝐺 a ; 𝑚 , 𝑥 − − − − − → 𝑉 ( 𝐺 𝑚 ⊳ 𝐹 𝑥 ) 𝑉 a ; 𝐺 𝑚 , 𝐹 𝑥 − − − − − − → 𝑉 𝐺 𝑚 ⊳ 𝑈 𝐹 𝑥 = 𝐶 𝑚 ⊳ 𝐵 𝑥 . 5.3.1 Recons truction for comodule monads The ne xt th eore m off ers a w a y to reconstruct comodule monads from their module categories. It extends [ A C12 , Proposition 4.1.2] and [ TV17 , Lemma 7.10]. Recall Deﬁnition 2.38 , and note t hat one can analogousl y deﬁne lifts of adjunctions to comodule adjunctions. Theorem 5.28. Let 𝒞 and 𝒟 be monoidal categories, and suppose that ℳ and 𝒩 ar e right 𝒞 - and 𝒟 -module categories, respectiv ely . Let 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 be an oplax mo- noidal adjunction. Lifts of an adjunction 𝐺 : ℳ ⇄ 𝒩 : 𝑉 to a comodule adjunction ar e in bĳection with lifts of 𝑉 : 𝒩 − → ℳ to a str ong 𝑈 -comodule functor . 129 5. Monadi c T annak a–K rein reconst r uc tion Proof. Let 𝐺 ⊣ 𝑉 be a comodule adjunction o v er 𝐹 ⊣ 𝑈 . Deﬁne t he in verse 𝑉 − 1 a : 𝑉 − ⊳ 𝑈 = = ⇒ 𝑉 (− ⊳ = ) of 𝑉 a b y 𝑉 𝑈 𝑉 (5.3.2) Using that 𝐺 and 𝑉 are part of a comodule adjunction, a str aightfor w ard computation pro v es 𝑉 − 1 a ◦ 𝑉 a = id 𝑉 (− ⊳ = ) : 𝑉 𝑉 𝑉 𝑉 = 𝑉 𝑉 = A similar str ategy can be used to show t hat 𝑉 a ◦ 𝑉 − 1 a = id 𝑉 − ⊳ 𝑈 = . Thus, 𝑉 a is a natural isomor phism and therefore 𝑉 is a strong comodule functor . Con v ersely , suppose that 𝑉 : 𝒩 − → ℳ is a strong comodule functor ov er 𝑈 . Deﬁne an arrow 𝐺 a : 𝐺 (− ⊳ = ) = ⇒ 𝐺 (−) ⊳ 𝐹 ( = ) b y 𝐺 𝐹 𝐺 (5.3.3) By [ TV17 , Lemma 7.10], the comultiplication and counit of 𝐹 are giv en by 𝐹 ( 𝑥 ⊗ 𝑦 ) 𝐹 ( 𝜂 𝑥 ⊗ 𝜂 𝑦 ) − − − − − − → 𝐹 ( 𝑈 𝐹 𝑥 ⊗ 𝑈 𝐹 𝑦 ) 𝐹𝑈 − 2; 𝐹 𝑥 , 𝐹 𝑦 − − − − − − − → 𝐹 𝑈 ( 𝐹 𝑥 ⊗ 𝐹 𝑦 ) 𝜀 𝐹 𝑥 ⊗ 𝐹 𝑥 − − − − → 𝐹 𝑥 ⊗ 𝐹 𝑦 , 𝐹 1 𝐹𝑈 − 0 − − − → 𝐹 𝑈 1 𝜀 1 − − → 1 , for all 𝑥 , 𝑦 ∈ 𝒞 . N ote that, g r aphicall y , 𝐹 2 looks just like Diag r am ( 5.3.3 ), wit h black strings taking the place of blue ones. W e will show t hat 𝐺 is a comodule functor ov er 𝐹 . Figure 5.6 show s that 𝐺 a : 𝐺 (− ⊳ = ) = ⇒ 𝐺 (−) ⊳ 𝐹 ( = ) is coassociativ e in t he sense of Figure 5.3 . 130 5.3. (Co)module monads 𝐺 𝐺 𝐹 𝐹 𝐺 𝐹 𝐹 𝐺 𝐺 𝐹 𝐹 = = = 𝐺 𝐺 𝐹 𝐹 𝐺 Figure 5.6: The coassociativity condition of 𝐺 . The fact t hat 𝐺 a is counital follo ws from 𝐺 𝐺 = 𝐺 𝐺 𝐺 𝐺 = 𝐺 𝐺 = A str aightforwar d computation prov es t hat t he unit of the adjunction 𝐺 ⊣ 𝑉 satisﬁes t he axioms displa yed in Figure 5.5 : 𝐹 𝑈 𝐺 𝑉 = = 𝐹 𝑈 𝐺 𝑉 𝐹 𝑈 𝐺 𝑉 A similar argument for t he counit show s t hat 𝐺 ⊣ 𝑉 is a comodule adjunction. T o see t hat t hese constructions are in v erse to each other , ﬁrst suppose that w e ha v e a comodule adjunction 𝐺 ⊣ 𝑉 . By utilising 𝑉 − 1 a as giv en in Diagram ( 5.3.2 ), we obtain another coaction 𝜆 on 𝐺 , see Diag r am ( 5.3.3 ). N o w , 131 5. Monadi c T annak a–K rein reconst r uc tion a direct computation sho ws t hat 𝐺 a = 𝜆 : 𝐺 𝐹 𝐺 𝐺 𝐹 𝐺 = The con v erse is clear: t he map t hat associates a comodule adjunction 𝐺 ⊣ 𝑉 to an y strong comodule structure on 𝑉 preserves the coaction of 𝑉 . □ The proof of Theorem 5.28 yields an analogue of Proposition 5.5 , which w e formulate in t he language of 𝒞 -module functors. P or ism 5.29. Let ℳ and 𝒩 be 𝒞 - and 𝒟 -module categories, respectivel y . F or an adjunction 𝐹 : ℳ ⇄ 𝒩 : 𝑈 , t her e is a bĳection between oplax 𝒞 -module s tructures on 𝐹 and lax 𝒞 -module structur es on 𝑈 . Corollar y 5.30. Let ℳ be a 𝒞 -module category , and let 𝑇 be a monad on ℳ . Then t here exis ts a bĳective correspondence  oplax 𝒞 -module monad structures on 𝑇  ↔  𝒞 -module category structur es on ℳ 𝑇 such that 𝑈 𝑇 is a s trict module functor  { 𝑇 (− ⊲ = ) 𝑇 a − − − → − ⊲ 𝑇 ( = ) } ↦− → { 𝑇 (− ⊲ = ) 𝑇 a − − − → − ⊲ 𝑇 ( = ) id ⊲ act − − − − → − ⊲ = } { 𝑈 𝑇 𝐹 𝑇 (− ⊲ = ) 𝑈 𝑇 a ◦ 𝑈 𝑇 𝐹 𝑇 a − − − − − − − → − ⊲ 𝑈 𝑇 𝐹 𝑇 ( = ) } ← − [ { 𝑈 𝑇 (− ⊲ = ) 𝑈 𝑇 a − − → − ⊲ 𝑈 𝑇 ( = ) } Theorem 5.28 also yields a description of a comodule monad coaction in terms of its Eilenberg –Moore adjunction; i.e., a T annaka – Krein reconstruction result in the spirit of Proposition 5.9 . Theorem 5.31. Let 𝐵 be a bimonad on the monoidal cat egory 𝒞 , and 𝑇 a monad on a right 𝒞 -module category ℳ . Coactions of 𝐵 on 𝑇 are in bĳection with right actions of 𝒞 𝐵 on ℳ 𝑇 such t hat 𝑈 𝑇 is a strict comodule functor over 𝑈 𝐵 . Proof. Suppose 𝒞 𝐵 acts from the right on ℳ 𝑇 such that 𝑈 𝑇 is a strict comodule functor . Due to Theorem 5.28 , 𝑇 = 𝑈 𝑇 𝐹 𝑇 is a comodule monad via the coaction 𝑇 a : 𝑇 (− ⊳ = ) 𝑈 𝑇 𝐹 𝑇 a − − − − → 𝑈 𝑇 ( 𝐹 𝑇 (−) ⊳ 𝐹 𝐵 ( = )) 𝑈 𝑇 a − − − → 𝑇 (−) ⊳ 𝐵 ( = ) , 132 5.3. (Co)module monads which is equal to 𝑈 𝑇 𝐹 𝑇 a , as 𝑈 𝑇 is a strict 𝑈 𝐵 -comodule functor . Con v ersely , if 𝑇 is a comodule monad, t hen ℳ 𝑇 becomes a right 𝒞 𝐵 -module category , wit h action giv en as in Remar k 5.24 . As 𝑇 a and the action of 𝒞 𝐵 on ℳ 𝑇 determine the coaction 𝐹 𝑇 a of 𝐹 𝑇 uniquel y , the tw o constructions are in v erse to each other by Theorem 5.28 . □ The follo wing result is dual to Theorem 5.28 for t he special case of 𝒞 - module adjunctions, where w e study strong 𝒞 -module structures on t he left rather than t he right adjoint. Proposition 5.32. Let 𝒞 be a monoidal category , and suppose that ℳ and 𝒩 are left 𝒞 -module categories. Given an adjunction 𝐹 : ℳ ⇄ 𝒩 : 𝑈 , ther e is a one-to-one corr espondence between lif ts of 𝐹 to a str ong 𝒞 -module funct or , and lifts of 𝐹 ⊣ 𝑈 to a lax 𝒞 -module adjunction. Proof. Suppose that 𝐹 : ℳ ⇄ 𝒩 : 𝑈 is a lax 𝒞 -module adjunction. Deﬁne the in v erse of t he action 𝐹 a : − ⊲ 𝐹 ( = ) = ⇒ 𝐹 (− ⊲ = ) b y 𝐹 (− ⊲ = ) 𝐹 ( 𝜂 ⊲ = ) − − − − − → 𝐹 ( 𝑈 𝐹 (−) ⊲ = ) 𝐹𝑈 a − − − → 𝐹 𝑈 ( 𝐹 (−) ⊲ = ) 𝜀 − − → 𝐹 (−) ⊲ = . Con v ersely , giv en an adjunction 𝐹 : ℳ ⇄ 𝒩 : 𝑈 such that 𝐹 is a strong 𝒞 - module functor , deﬁne 𝑈 (−) ⊲ = 𝜂 − − → 𝑈 𝐹 ( 𝑈 (−) ⊲ = ) 𝑈 𝐹 − 1 a − − − − → 𝑈 ( 𝐹 𝑈 (−) ⊲ = ) 𝑈 ( 𝜀 ⊲ = ) − − − − − → 𝑈 (− ⊲ = ) . Reading the string diag r ams in the proof of Theorem 5.28 upside down v eriﬁes t he necessar y t he coherence conditions, and t hat t hese tw o construc- tions are in v erses of each other . □ W e obtain a v ersion of Corollar y 5.30 for t he Kleisli categor y of a monad. Corollar y 5.33. Let ℳ be a lef t 𝒞 -module category , and suppose 𝑇 to be a monad on ℳ . Ther e is a bĳection between lax 𝒞 -module monad s tructur es on 𝑇 , and 𝒞 -module category structur es on ℳ 𝑇 , such t hat 𝐹 𝑇 is a strict 𝒞 -module functor . Proof. Let ℳ 𝑇 be a 𝒞 -module categor y such that 𝐹 𝑇 is a strict 𝒞 -module functor . Then 𝑇 . . = 𝑈 𝑇 𝐹 𝑇 : ℳ − → ℳ is a lax 𝒞 -module monad b y Proposition 5.32 . Con v ersely , if 𝑇 is a lax 𝒞 -module monad, ℳ 𝑇 becomes a 𝒞 -module category as follo ws: for 𝑥 ∈ 𝒞 and 𝑚 , 𝑛 ∈ ℳ , set 𝑥 ⊲ ℳ 𝑇 𝑚 . . = 𝑥 ⊲ ℳ 𝑚 on objects, and for a mor phism 𝑓 : 𝑚 − → 𝑇 𝑛 in ℳ 𝑇 , w e deﬁne t he action of 𝑥 ∈ 𝒞 b y 𝑥 ⊲ ℳ 𝑇 𝑓 . . = 𝑥 ⊲ 𝑚 𝑥 ⊲ 𝑓 − − − → 𝑥 ⊲ 𝑇 𝑛 𝑇 a ; 𝑥 ,𝑛 − − − → 𝑇 ( 𝑥 ⊲ 𝑛 ) . 133 5. Monadi c T annak a–K rein reconst r uc tion It is easy to check t hat t hese assignments deﬁne a 𝒞 -module structure, for which 𝐹 𝑇 is a strict 𝒞 -module functor . The 𝒞 -module structure of ℳ 𝑇 and t he lax module monad s tructure on 𝑇 uniquel y determine t he 𝒞 -module structure of 𝐹 𝑇 . Hence, 𝐹 − 1 𝑇 ; a is giv en as in Proposition 5.32 and the tw o constructions are in v erse of each other . □ Using t hese insights, w e ma y now clarify t he structure of comparison functors associated to comodule adjunctions. Proposition 5.34. Consider a comodule adjunction 𝐺 : ℳ ⇄ 𝒩 : 𝑉 over an oplax monoidal adjunction 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 , and denot e the associated comodule monad and bimonad by 𝐶 . . = 𝑉 𝐺 : ℳ − → ℳ and 𝐵 . . = 𝑈 𝐹 : 𝒞 − → 𝒞 , respectiv ely . Then the comparison functor 𝐾 𝐶 : 𝒩 − → ℳ 𝐶 is a str ong comodule functor over 𝐾 𝐵 : 𝒟 − → 𝒞 𝐵 , and t he following identities of comodule functors hold: 𝑈 𝐶 𝐾 𝐶 = 𝑉 and 𝐾 𝐶 𝐺 = 𝐹 𝐶 . Proof. W e proceed analogousl y to [ B V07 , Theorem 2.6]. For any 𝑛 ∈ 𝒩 w e ha v e 𝐾 𝐶 𝑛 = ( 𝑉 𝑛 , 𝑉 𝜀 𝑛 ) and a direct computation show s that t he coaction of 𝑉 lif ts to a coaction of 𝐾 𝐶 . That is, 𝑈 𝐶 𝐾 𝐶 a ; 𝑛 , 𝑦 = 𝑉 a ; 𝑛 , 𝑦 , for all 𝑛 ∈ 𝒩 and 𝑦 ∈ 𝒟 . Using that 𝑈 𝐶 : ℳ 𝐶 − → ℳ is a fait hful and conser v ativ e functor , one obser v es that 𝐾 𝐶 becomes a strong comodule functor in t his manner . Further more, as 𝑈 𝐶 is a s trict comodule functor , the coactions of 𝑈 𝐶 𝐾 𝐶 and 𝑉 coincide. Lastl y , w e compute t he follo wing, for an y 𝑥 ∈ 𝒞 and 𝑚 ∈ ℳ : ( 𝐾 𝐶 𝐺 ) a ; 𝑚 , 𝑥 = ( 𝑈 𝐶 𝐾 𝐶 𝐺 ) a ; 𝑚 , 𝑥 = ( 𝑉 𝐺 ) a ; 𝑚 , 𝑥 = 𝐶 a ; 𝑚 , 𝑥 = ( 𝑈 𝐶 𝐹 𝐶 ) a ; 𝑚 , 𝑥 = 𝐹 𝐶 a ; 𝑚 , 𝑥 . □ Proposition 5.35. Let ℳ and 𝒩 be left 𝒞 -module categories. • Let 𝐹 : ℳ ⇄ 𝒩 : 𝑈 be an oplax 𝒞 -module adjunction and consider t he cor - r esponding 𝒞 -module category s tructur e on ℳ 𝑇 of Cor ollar y 5.30 . Then for 𝑇 . . = 𝑈 𝐹 t he comparison functor 𝐾 𝑇 is a str ong 𝒞 -module functor . • Let 𝐹 : ℳ ⇄ 𝒩 : 𝑈 be a lax 𝒞 -module adjunctions and consider the corr es- ponding 𝒞 -module category structur e on ℳ 𝑇 of Corollary 5.33 . Then for 𝑇 . . = 𝑈 𝐹 t he comparison functor 𝐾 𝑇 is a str ong 𝒞 -module functor . 134 5.3. (Co)module monads Proposition 5.36. Let ℳ be a 𝒞 -module cat egory , 𝑇 : ℳ − → ℳ an oplax 𝒞 -module monad, and 𝑆 : ℳ − → ℳ a right adjoint to 𝑇 . Then the isomorphism ℳ 𝑇  ℳ 𝑆 of Proposition 2.25 is a strict 𝒞 -module isomor phism, where ℳ 𝑇 is endowed with the 𝒞 -module structur e of Proposition 5.30 , and similar l y for ℳ 𝑆 . Proof. N ote t hat, by Theorem 5.28 t he comonad 𝑆 : ℳ − → ℳ comes equipped with t he follo wing lax 𝒞 -module structure, for all 𝑥 ∈ 𝒞 and 𝑚 ∈ ℳ : 𝑆 a ; 𝑥 , 𝑚 : 𝑥 ⊲ 𝑆 𝑚 𝜂 𝑥 ⊲ 𝑆𝑚 − − − − → 𝑆𝑇 ( 𝑥 ⊲ 𝑆 𝑚 ) 𝑆𝑇 a ; 𝑥 ,𝑆 𝑚 − − − − − → 𝑆 ( 𝑥 ⊲ 𝑇 𝑆 𝑚 ) 𝑆 ( 𝑥 ⊲𝜀 𝑚 ) − − − − − → 𝑆 ( 𝑥 ⊲ 𝑚 ) . Further , the isomor phism 𝐿 : ℳ 𝑇 ∼ − → ℳ 𝑆 of Proposition 2.25 is giv en by  𝑚 , ∇ 𝑚 : 𝑇 𝑚 − → 𝑚  ↦− →  𝑚 , 𝑚 𝜂 𝑚 − − − → 𝑆𝑇 𝑚 𝑆 ∇ 𝑚 − − − → 𝑆 𝑚  . W e ha v e to pro v e that 𝐿 ( 𝑥 ⊲ ( 𝑚 , ∇ 𝑚 )) = 𝑥 ⊲ 𝐿 (( 𝑚 , ∇ 𝑚 )) , for all 𝑥 ∈ 𝒞 and ( 𝑚 , ∇ 𝑚 ) ∈ ℳ 𝑇 . This is equiv alent to t he equality of  𝑥 ⊲ 𝑚 , 𝑥 ⊲ 𝑚 𝜂 𝑥 ⊲ 𝑚 − − − → 𝑆𝑇 ( 𝑥 ⊲ 𝑚 ) 𝑆𝑇 a ; 𝑥 ,𝑚 − − − − − → 𝑆 ( 𝑥 ⊲ 𝑇 𝑚 ) 𝑆 ( 𝑥 ⊲ ∇ 𝑚 ) − − − − − − → 𝑆 ( 𝑥 ⊲ 𝑚 )  and  𝑥 ⊲ 𝑚 , 𝑥 ⊲ 𝑚 𝑥 ⊲𝜂 𝑚 − − − → 𝑥 ⊲ 𝑆𝑇 𝑚 𝑥 ⊲ 𝑆 ∇ 𝑚 − − − − − → 𝑥 ⊲ 𝑆 𝑚 𝜂 𝑥 ⊲ 𝑆𝑚 − − − − → 𝑆𝑇 ( 𝑥 ⊲ 𝑆 𝑚 ) 𝑆 ( 𝑇 a ) 𝑥 ,𝑆 ( 𝑚 ) − − − − − − − → 𝑆 ( 𝑥 ⊲ 𝑇 𝑆 𝑚 ) 𝑆 ( 𝑥 ⊲𝜀 𝑚 ) − − − − − → 𝑆 ( 𝑥 ⊲ 𝑚 )  . This is evidenced by the following commutativ e diag r am: 𝑥 ⊲ 𝑚 𝑆𝑇 ( 𝑥 ⊲ 𝑚 ) 𝑆 ( 𝑥 ⊲ 𝑇 𝑚 ) 𝑥 ⊲ 𝑆 𝑇 𝑚 𝑆𝑇 ( 𝑥 ⊲ 𝑆𝑇 𝑚 ) 𝑆 ( 𝑥 ⊲ 𝑇 𝑆𝑇 𝑚 ) 𝑆 ( 𝑥 ⊲ 𝑇 𝑚 ) 𝑥 ⊲ 𝑆 𝑚 𝑆𝑇 ( 𝑥 ⊲ 𝑆 𝑚 ) 𝑆 ( 𝑥 ⊲ 𝑇 𝑆 𝑚 ) 𝑆 ( 𝑥 ⊲ 𝑚 ) 𝜂 𝑥 ⊲ 𝑚 𝑥 ⊲𝜂 𝑚 𝑆𝑇 a ; 𝑥 ,𝑚 𝑆𝑇 ( 𝑥 ⊲𝜂 𝑚 ) 𝑆 ( 𝑥 ⊲ 𝑇 𝜂 𝑚 ) 𝜂 𝑥 ⊲ 𝑆𝑇 𝑚 𝑥 ⊲ 𝑆 ∇ 𝑚 𝑆𝑇 a ; 𝑥 ,𝑆𝑇 𝑚 𝑆𝑇 ( 𝑥 ⊲ 𝑆 ∇ 𝑚 ) 𝑆 ( 𝑥 ⊲𝜀 𝑇 𝑚 ) 𝑆 ( 𝑥 ⊲ 𝑇 𝑆 ∇ 𝑚 ) 𝑆 ( 𝑥 ⊲ ∇ 𝑚 ) 𝜂 𝑥 ⊲ 𝑆𝑚 𝑆𝑇 a ; 𝑥 ,𝑆 𝑚 𝑆 ( 𝑥 ⊲𝜀 𝑚 ) nat 𝜂 nat 𝑇 a adj nat 𝜂 nat 𝑇 a nat 𝜀 □ In an analogous wa y t o Proposition 5.36 , one obtains t he follo wing result. Proposition 5.37. Let ℳ be a 𝒞 -module category , 𝑇 : ℳ − → ℳ a lax 𝒞 -module monad, and 𝐿 : ℳ − → ℳ a left adjoint to 𝑇 . Then t he isomorphism ℳ 𝑇  ℳ 𝐿 of Proposition 2.26 is a 𝒞 -module isomorphism, where ℳ 𝑇 and ℳ 𝐿 ar e endowed with t he 𝒞 -module structur e of Corollary 5.33 . 135 [Der W ert dieser Arbeit] wird umso grösser sein, je besser die Gedanken ausgedrückt sind. Je mehr der Nag el auf den K opf getroﬀen ist.— Hier bin ich mir bewusst, w eit hinter dem Möglichen zur ück geblieben zu sein. Einfach dar um, w eil meine Kraft zur Bew ältigung der A ufgabe zu g ering ist. Lud wig Wi ttgenste in ; T ractatus Logico-Philosophicus M O N A D I C T W I S T E D C E N T R E S 6 The ant i-Ye tter– Dri nfel d m odul es of a ﬁnite-dimensional Hopf algebr a are a module categor y ov er the Y etter– Drinf eld modules. Subsequently , they are implemented by a comodule algebr a ov er t he Drinfeld double, see [ HKRS04 ]. As explained in Section 4.2 , we ﬁnd ourselv es in a similar situation. The anti-Drinfeld centre, our replacement of the anti-Y etter–Drinfeld modules, is a module category ov er the Drinfeld centre. Replacing ﬁnite-dimensional v ector spaces b y a rigid, possibly piv otal, category 𝒞 , and t he underl ying Hopf algebr a with a Hopf monad 𝐻 on 𝒞 , t his section serv es to study a Hopf monad 𝐷 ( 𝐻 ) : 𝒞 − → 𝒞 and o v er it a comodule monad 𝑄 ( 𝐻 ) : 𝒞 − → 𝒞 , which realise t he centre and its twisted cousin as their respectiv e modules. Br uguières and V irelizier ga v e a transparent description of 𝐷 ( 𝐻 ) in [ B V12 ] b y extending results of Da y and Street, [ DS07 ]. If 𝐻 is the identity functor , one deﬁnes t he centr al Hopf monad D ( 𝒞 𝐻 ) on 𝒞 𝐻 , wit h Z ( 𝒞 𝐻 ) as its Eilenberg –Moore categor y . As an application of Beck’s theor y of distributiv e la ws, one obtains t he Drinfeld double 𝐷 ( 𝐻 ) : 𝒞 − → 𝒞 . W e appl y the same techniques to deﬁne the anti-double 𝑄 ( 𝐻 ) of 𝐻 , whose modules are isomorphic to t he “dual” of t he anti-Drinfeld centre Q ( 𝒞 𝐻 ) . This approach is best summarised by Figure 6.1 . W e obtain a monadic v ersion of Theorem 4.1 . Theorem 6.44 . Let 𝒞 be a rigid monoidal category , and suppose that 𝐻 : 𝒞 − → 𝒞 is a Hopf monad that admits a double 𝐷 ( 𝐻 ) and anti-double 𝑄 ( 𝐻 ) . The following st atements are equivalent: (i) t he monoidal unit 1 ∈ 𝒞 lif ts to a module over 𝑄 ( 𝐻 ) , (ii) t here is an isomor phism of comodule monads 𝐷 ( 𝐻 )  𝑄 ( 𝐻 ) , and (iii) t here is an isomor phism of monads 𝑄 ( 𝐻 )  𝐷 ( 𝐻 ) . If 𝒞 is pivo tal with pivot al structur e 𝜙 , any of the above stat ements hold if and onl y if 𝐻 and 𝜙 admit a pair in involution. 137 6. Monadi c t wisted cen tres Z ( 𝒞 𝐻 ) Q ( 𝒞 𝐻 ) 𝒞 𝐻 𝒞 𝐻 D 𝒞 𝐻 Q 𝒞 𝒞 𝐷 ( 𝐻 ) 𝒞 𝑄 ( 𝐻 ) 𝑈 ( 𝑍 ) 𝐾 D 𝑈 ( 𝑄 ) action 𝐾 Q 𝐹 ( 𝑄 ) 𝐹 ( 𝑍 ) 𝐹 D 𝐹 Q 𝑈 𝐻 𝑈 D Σ ( 𝐷 ( 𝐻 )) 𝑈 Q Σ ( 𝑄 ( 𝐻 )) 𝐹 𝐻 𝐹 𝑄 ( 𝐻 ) 𝐹 𝐷 ( 𝐻 ) 𝑈 𝐷 ( 𝐻 ) 𝑈 𝑄 ( 𝐻 ) action Figure 6.1: A cob w eb of adjunctions, monads, and v arious v ersions of the centre and anti-centre. From Theorem 6.44 we can deduce how piv otal structures on 𝒞 𝐻 arise from module mor phisms betw een t he central Hopf monad D and t he anti- central comodule monad Q . Corollar y 6.45 . Let 𝒞 be a rigid monoidal category . If 𝒞 admits a central Hopf monad D ( 𝒞 ) and an anti-central comodule monad Q ( 𝒞 ) , then it is pivot al if and onl y if D ( 𝒞 )  Q ( 𝒞 ) as monads. 6 . 1 c ro s s p r o d u c t s a n d d i s t r i b u t i v e l a w s The Ho pf mo n ad ic de scri ption of the Drinfeld centre Z ( 𝒞 𝐻 ) , for a rigid category 𝒞 and Hopf monad 𝐻 on 𝒞 , is achiev ed as a tw o-step process: one ﬁrst ﬁnds a suitable monad on 𝒞 𝐻 , which is t hen “lifted” to a monad on 𝒞 . W e shall review t his lif ting process based on [ B V12 , Sections 3 and 4]. Deﬁnition 6.1. Let 𝐻 : 𝒞 − → 𝒞 be a monad and 𝑇 : 𝒞 𝐻 − → 𝒞 𝐻 a functor . The cross product 𝑇 ⋊ 𝐻 of 𝑇 b y 𝐻 is t he endofunctor 𝑈 𝐻 𝑇 𝐹 𝐻 : 𝒞 − → 𝒞 . If ( 𝑇 , 𝜇 , 𝜂 ) is a monad, t hen the cross product 𝑇 ⋊ 𝐻 inherits this structure: the multiplication and unit are giv en b y Here 𝜀 and 𝜂 are the unit and counit of the adjunction 𝐹 𝐻 ⊣ 𝑈 𝐻 . 𝜇 ( 𝑇 ⋊ 𝐻 ) : 𝑈 𝐻 𝑇 𝐹 𝐻 𝑈 𝐻 𝑇 𝐹 𝐻 𝑈 𝐻 𝑇 𝜀 𝑇 𝐹 𝐻 − − − − − − − → 𝑈 𝐻 𝑇 𝑇 𝐹 𝐻 𝑈 𝐻 𝜇 ( 𝑇 ) 𝐹 𝐻 − − − − − − − → 𝑈 𝐻 𝑇 𝐹 𝐻 and 𝜂 ( 𝑇 ⋊ 𝐻 ) : Id 𝒞 𝜂 − − → 𝑈 𝐻 𝐹 𝐻 𝑈 𝐻 𝜂 ( 𝑇 ) − − − − − → 𝑈 𝐻 𝑇 𝐹 𝐻 . 138 6.1. Cross products and distributiv e la ws Giv en tw o bimonads 𝐻 : 𝒞 − → 𝒞 and 𝐵 : 𝒞 𝐻 − → 𝒞 𝐻 , iterativ ely applying the comultiplication and counit of 𝑈 𝐻 , 𝐵 , and 𝐹 𝐻 yields a bimonad structure on 𝐵 ⋊ 𝐻 : 𝒞 − → 𝒞 . The comultiplication is giv en b y 𝑈 𝐻 2; 𝑇 𝐹 𝐻 (−) ,𝑇 𝐹 𝐻 ( = ) ◦ 𝑈 𝐻 𝑇 2; 𝐹 𝐻 (−) ,𝐹 𝐻 ( = ) ◦ 𝑈 𝐻 𝑇 𝐹 𝐻 2; − , = , and for t he counit w e hav e 𝜀 ( 𝑈 𝐻 ) ◦ 𝑈 𝐻 𝜀 ( 𝑇 ) ◦ 𝑈 𝐻 𝑇 𝜀 ( 𝐹 𝐻 ) . Similar considerations impl y t he follo wing result. Lemma 6.2. Let 𝐻 : 𝒞 − → 𝒞 and 𝐵 : 𝒞 𝐻 − → 𝒞 𝐻 be bimonads which r espectivel y coact on the comodule monads 𝐺 : ℳ − → ℳ and 𝐶 : ℳ 𝐺 − → ℳ 𝐺 . The cross product 𝐶 ⋊ 𝐺 : ℳ − → ℳ is a comodule monad over 𝐵 ⋊ 𝐻 via t he coaction 𝑈 𝐺 a ; 𝐶 𝐹 𝐺 (−) ,𝐵 𝐹 𝐻 ( = ) ◦ 𝑈 𝐺 𝐶 a ; 𝐹 𝐺 (−) ,𝐹 𝐻 ( = ) ◦ 𝑈 𝐺 𝐶 𝐹 𝐺 a ; − , = . Remark 6.3. Let 𝐻 : 𝒞 − → 𝒞 and 𝐵 : 𝒞 𝐻 − → 𝒞 𝐻 be monads. The question under which conditions the modules 𝒞 𝐵 ⋊ 𝐻 of 𝐵 ⋊ 𝐻 are isomor phic to ( 𝒞 𝐻 ) 𝐵 is closel y related to the theor y of distributiv e la ws of Section 2.2.2 . W e ma y apply the formal theor y t hereof, see [ Str72 ], to the bicategory O plMon opl of monoidal categories, oplax monoidal functors, and oplax monoidal natural transf orm- ations, to obtain a description of bimonads and oplax monoidal distributiv e law s — oplax monoidal natural transformations Λ : 𝐻 𝐵 − → 𝐵 𝐻 betw een bi- monads 𝐻 , 𝐵 : 𝒞 − → 𝒞 t hat are moreov er distributiv e law s, see [ McC02 ]. Suppose Λ : 𝐻 𝐵 − → 𝐵 𝐻 to be an oplax monoidal distributiv e la w . The comultiplication and counit of t he underl ying functor 𝐵 𝐻 : 𝒞 − → 𝒞 turn t he composite 𝐵 ◦ Λ 𝐻 into a bimonad. Thus, comodule monads can be intrinsically described in the bicategor y t hat has • as objects, pairs ( ℳ , 𝒞 ) consisting of a right module categor y ℳ o v er a monoidal category 𝒞 ; • as 1-morphisms, pairs ( 𝐺 , 𝐹 ) of a comodule functor 𝐺 o v er an oplax monoidal functor 𝐹 ; and • as 2-morphisms, pairs ( 𝜙 , 𝜓 ) t hat form a comodule transf ormation. The subsequent results arise immediatel y from [ Str72 ]. 139 6. Monadi c t wisted cen tres Deﬁnition 6.4. Let 𝐺 , 𝐶 : ℳ − → ℳ be tw o comodule monads o v er t he bi- monads 𝐻 , 𝐵 : 𝒞 − → 𝒞 , respectiv ely . A comodule distributiv e law is a pair of distributiv e la ws Ω : 𝐺 𝐶 − → 𝐶 𝐺 and Λ : 𝐻 𝐵 − → 𝐵 𝐻 such that ( Λ , Ω ) is a comodule natural transf ormation. Proposition 6.5. Consider two comodule monads 𝐺 , 𝐶 : ℳ − → ℳ over the bimon- ads 𝐻 , 𝐵 : 𝒞 − → 𝒞 . There exis ts a bĳective correspondence between: (i) comodule distributiv e law s  𝐺 𝐶 Ω − − − → 𝐶 𝐺 , 𝐻 𝐵 Λ − − − → 𝐵 𝐻  ; and (ii) lifts of 𝐵 to a bimonad  𝐵 : 𝒞 𝐻 − → 𝒞 𝐻 tog ether wit h lifts of 𝐶 to a comodule monad  𝐶 : ℳ 𝐺 − → ℳ 𝐺 over  𝐵 , such t hat 𝐵𝑈 𝐻 = 𝑈 𝐻  𝐵 as oplax monoidal functor s and 𝐶 𝑈 𝐺 = 𝑈 𝐺  𝐶 as comodule functors. Let  𝐺 𝐶 Ω − − − → 𝐶 𝐺 , 𝐻 𝐵 Λ − − − → 𝐵 𝐻  be a comodule distributiv e la w . The coactions of 𝐺 and 𝐶 tur n 𝐶 ◦ Ω 𝐺 into a comodule monad ov er 𝐵 ◦ Λ 𝐻 . Lemma 6.6. Suppose Ω : 𝐺 𝐶 − → 𝐶 𝐺 and Λ : 𝐻 𝐵 − → 𝐵 𝐻 to form a comodule dis- tributive law . Then t here are equiv alences ( 𝒞 𝐻 )  𝐵 Λ ≃ 𝒞 𝐵 ◦ Λ 𝐻 as monoidal categories, and ( ℳ 𝐺 )  𝐶 Ω ≃ ℳ 𝐶 ◦ Ω 𝐺 as 𝒞 𝐵 ◦ Λ 𝐻 -module categories. In fact, by [ B V12 , Section 4.5], the previous results transcend to t he Hopf monadic setting. Let 𝐵 , 𝐻 : 𝒞 − → 𝒞 be Hopf monads. If Λ : 𝐻 𝐵 − → 𝐵 𝐻 is an oplax monoidal distributiv e la w , then 𝐵 ◦ Λ 𝐻 : 𝒞 − → 𝒞 and the lif t  𝐵 Λ : 𝒞 𝐻 − → 𝒞 𝐻 are Hopf monads as w ell. 6 . 2 c e n t r a l i s a b l e f u nc to r s a n d t h e c e n t r a l b i m o n a d The con s truct ion of th e d oubl e of a Hopf monad 𝐻 : 𝒞 − → 𝒞 giv en in [ B V12 ] relies on explicitly describing t he lef t dual of the forg etful functor 𝑈 ( 𝑍 ) : Z ( 𝒞 𝐻 ) − → 𝒞 𝐻 , which is realised as a coend. Deﬁnition 6.7. Let 𝒞 be a rigid categor y and 𝑇 : 𝒞 − → 𝒞 an endofunctor . W e call 𝑇 centralisable if t he follo wing coend exists for all 𝑥 ∈ 𝒞 : 𝑍 𝑇 ( 𝑥 ) . . =  𝑦 ∈ 𝒞 ∨ 𝑇 𝑦 ⊗ 𝑥 ⊗ 𝑦 . Remark 6.8. An y centralisable functor 𝑇 : 𝒞 − → 𝒞 admits a univer sal coaction 𝜒 𝑥 , 𝑦 . . = ( id 𝑇 𝑦 ⊗ copr 𝑦 , 𝑥 ) ◦ ( coev 𝑙 𝑇 𝑦 ⊗ id 𝑥 ⊗ 𝑦 ) , for 𝑥 , 𝑦 ∈ 𝒞 , which is natural in both v ariables. W e call t he pair ( 𝑍 𝑇 , 𝜒 ) a centraliser of 𝑇 . 140 6.2. Centralisable functors and the central bimonad Remark 6.9. Graphicall y , w e represent t he univ ersal coaction as follo ws: 𝑥 𝑦 𝑇 𝑦 𝑍 𝑇 𝑥 𝜒 𝑥 , 𝑦 : 𝑥 ⊗ 𝑦 − → 𝑇 𝑦 ⊗ 𝑍 𝑇 𝑥 F or all 𝑓 : 𝑥 − → 𝑥 ′ and 𝑔 : 𝑦 − → 𝑦 ′ , the naturality condition equates to 𝑔 = 𝑇 𝑔 𝑓 𝑍 𝑇 𝑓 𝑥 𝑦 𝑥 𝑦 𝑇 ( 𝑦 ′ ) 𝑍 𝑇 ( 𝑥 ′ ) 𝑇 ( 𝑦 ′ ) 𝑍 𝑇 ( 𝑥 ′ ) 𝜒 𝑥 ′ , 𝑦 ′ ◦ ( 𝑓 ⊗ 𝑔 ) = ( 𝑇 𝑔 ⊗ 𝑍 𝑇 𝑓 ) ◦ 𝜒 𝑥 , 𝑦 U niv ersal coactions hav e a certain univ ersal property t hat will be vital in the rest of t his chapter — the extended fact orisation property . In particular , it pro vides us with a potent tool for constructing bi- and comodule monads. Lemma 6.10 ([ B V12 , Lemma 5.4]) . Let ( 𝑍 𝑇 , 𝜒 ) be t he centraliser of a funct or 𝑇 : 𝒞 − → 𝒞 and suppose that 𝐿 , 𝑅 : 𝒟 − → 𝒞 are two functors. F or any 𝑛 ∈ N , 𝑦 ∈ 𝒟 , and 𝑥 1 , . . . , 𝑥 𝑛 ∈ 𝒞 , and any natural transf ormation 𝜙 𝑦 , 𝑥 1 , .. . , 𝑥 𝑛 : 𝐿 𝑦 ⊗ 𝑥 1 ⊗ · · · ⊗ 𝑥 𝑛 − → 𝑇 𝑥 1 ⊗ · · · ⊗ 𝑇 𝑥 𝑛 ⊗ 𝑅 𝑦 , t here exis ts a unique natural transf ormation 𝜈 : 𝑍 𝑛 𝑇 𝐿 = ⇒ 𝑅 , satisfying 𝑇 𝑦 1 𝑇 𝑦 𝑛 𝑅 𝑥 . . . 𝜙 𝑥 , 𝑦 1 , ... , 𝑦 𝑛 𝑦 𝑛 𝑦 1 𝐿 𝑥 . . . = 𝑇 𝑦 1 𝑇 𝑦 𝑛 𝑅 𝑥 . . . 𝑦 𝑛 𝑦 1 𝐿 𝑥 . . . 𝜈 𝑥 Exam ple 6.11. Let 𝑇 : 𝒞 − → 𝒞 be an oplax monoidal functor wit h centraliser ( 𝑍 𝑇 , 𝜒 ) . For all 𝑥 ∈ 𝒞 , deﬁne t he unit of 𝑍 𝑇 b y 𝜂 ( 𝑍 𝑇 ) 𝑥 : 𝑥 𝜒 𝑥 , 1 − − − − → 𝑇 1 ⊗ 𝑍 𝑇 𝑥 𝑇 0 ⊗ 𝑍 𝑇 𝑥 − − − − − − → 𝑍 𝑇 𝑥 . 141 6. Monadi c t wisted cen tres By Lemma 6.10 , we can deriv e a unique multiplication 𝜇 ( 𝑍 𝑇 ) : 𝑍 2 𝑇 = ⇒ 𝑍 𝑇 from the comultiplication of 𝑇 . 𝑇 2 , 𝑥 , 𝑦 𝑇 𝑦 𝑇 𝑤 𝑍 𝑇 𝑥 𝑥 𝑦 ⊗ 𝑤 𝜙 = 𝑥 𝑦 𝑧 𝜇 𝑍 𝑇 𝑥 𝑇 𝑦 𝑇 𝑤 𝑍 𝑇 𝑥 The follo wing result is due to Da y and Street, [ DS07 , pp. 191 –192], see also [ B V12 , Theorem 5.6] for a proof. Lemma 6.12. The centr aliser ( 𝑍 𝑇 , 𝜒 ) of an oplax monoidal endofunctor 𝑇 on 𝒞 is a monad wit h multiplication and unit as given in Example 6.11 . In t he proof of Lemma 6.12 giv en in [ B V12 , Theorem 5.6], t he authors further consider 𝑇 : 𝒞 − → 𝒞 to be equipped wit h a Hopf monad structure and sho w that in this case 𝑍 𝑇 is a Hopf monad as w ell. The extended factorisation property giv en in Lemma 6.10 reconstructs a comultiplication on 𝑍 𝑇 from a tw ofold application of t he univ ersal coaction and the multiplication of 𝑇 : ( 𝑍 𝑇 ) 2; 𝑥 , 𝑦 𝑍 𝑇 𝑦 𝑍 𝑇 𝑥 𝑇 𝑤 𝑤 𝑥 ⊗ 𝑦 𝑥 𝑦 𝑤 𝜇 𝑇 𝑤 𝑍 𝑇 𝑦 𝑍 𝑇 𝑥 𝑇 𝑤 = (6.2.1) Likewise, the unit of 𝑇 induces a counit on 𝑍 𝑇 via 𝑇 𝑥 = 𝑇 𝑥 ⊗ 1 𝑥 = 1 ⊗ 𝑥 𝜂 𝑇 𝑥 = ( 𝑍 𝑇 ) 0 1 𝑥 1 𝑥 A direct com putation v eriﬁes t hat t he centraliser 𝑍 𝑇 is a bimonad as w ell. For the construction of left and right antipodes, see [ BV12 , Theorem 5.6]. Remark 6.13. W e t hink of Z ( 𝐻 𝒞 ) as t he centre of an oplax bimodule categor y as stated in Remar k 2.46 , see also [ B V07 , Section 5.5]. Objects in Z ( 𝐻 𝒞 ) are pairs 142 6.2. Centralisable functors and the central bimonad ( 𝑥 , 𝜎 𝑥 , − ) , where 𝑥 ∈ 𝒞 and 𝜎 𝑥 , − : 𝑥 ⊗ − = ⇒ 𝐻 (−) ⊗ 𝑥 is a natural transformation, satisfying for all 𝑥 , 𝑦 , 𝑧 ∈ 𝒞 ( 𝐻 2; 𝑦 , 𝑧 ⊗ 𝑥 ) ◦ 𝜎 𝑥 , 𝑦 ⊗ 𝑧 = ( 𝐻 𝑦 ⊗ 𝜎 𝑥 , 𝑧 ) ◦ ( 𝜎 𝑥 , 𝑦 ⊗ 𝑧 ) , ( 𝐻 0 ⊗ 𝑥 ) ◦ 𝜎 𝑥 , 1 = id 𝑥 . Analogous to t he centres studied before, the arrow s in Z ( 𝐻 𝒞 ) are those mor ph- isms of 𝒞 that commute with the half-braidings. As sho wn in [ B V12 , Pro- position 5.9], the structure mor phisms of a Hopf monad 𝐻 : 𝒞 − → 𝒞 can be used to deﬁne a rigid structure on Z ( 𝐻 𝒞 ) . For example, t he tensor product of ( 𝑥 , 𝜎 𝑥 , − ) , ( 𝑦 , 𝜎 𝑦 , − ) ∈ Z ( 𝐻 𝒞 ) is 𝑥 ⊗ 𝑦 ∈ 𝒞 , tog et her with t he half-braiding 𝑤 𝑥 𝑦 𝑥 𝑦 𝜇 𝐻 𝑤 𝐻 𝑥 F or a treatment of centres twisted by (op)lax monoidal functors, see [ FH23 ]. Since centralisers of Hopf monads are Hopf monads themselv es, their modules implement t he twisted centres of Remar k 6.13 as a rigid category . This is pro v en in [ BV12 , Theorem 5.12 and Corollary 5.14]. Proposition 6.14. Suppose 𝐻 : 𝒞 − → 𝒞 to be a centralisable Hopf monad. The modules 𝒞 𝑍 𝐻 of its centr aliser ( 𝑍 𝐻 , 𝜒 ) are isomorphic as a rigid category to Z ( 𝐻 𝒞 ) . Appl ying t he abov e proposition to t he identity functor Id : 𝒞 − → 𝒞 , w e ob- tain a Hopf monadic description of t he Drinfeld centre Z ( 𝒞 ) of a rigid category 𝒞 . The terminology of our next deﬁnition is due to Shimizu, see [ Shi17 ]. Deﬁnition 6.15. Let 𝒞 be a monoidal categor y , and let t he identity functor on 𝒞 be centralisable with centraliser ( 𝑍 , 𝜒 ) . The central Hopf monad of 𝒞 is D ( 𝒞 ) . . = 𝑍 Id : 𝒞 − → 𝒞 . W e denote its Eilenberg– Moore b y 𝒞 D . An important step in proving Proposition 6.14 is deter mining an in v erse to t he comparison functor 𝐾 𝑍 𝑇 : Z ( 𝑇 𝒞 ) − → 𝒞 𝑍 𝑇 . This construction will also pla y a substantial role in our monadic descrip tion of the anti-Drinfeld centre, so w e recall it in its full generality . Let 𝑇 : 𝒞 − → 𝒞 be a centralisable oplax monoidal endofunctor with ( 𝑍 𝑇 , 𝜒 ) as its centraliser . T o ev er y 𝑍 𝑇 -module 143 6. Monadi c t wisted cen tres ( 𝑚 , ∇ 𝑚 ) , w e associate a half-braiding 𝜎 𝑚 , − : 𝑚 ⊗ − = ⇒ 𝑇 (−) ⊗ 𝑚 . For an y 𝑥 ∈ 𝒞 , it is giv en by t he composition 𝜎 𝑚 , 𝑥 : 𝑚 ⊗ 𝑥 𝜒 𝑚 , 𝑥 − − − − → 𝑇 𝑥 ⊗ 𝑍 𝑇 𝑥 𝑇 𝑥 ⊗∇ 𝑚 − − − − − → 𝑇 𝑥 ⊗ 𝑚 . (6.2.2) This yields a functor 𝐸 𝑍 𝑇 : 𝒞 𝑍 𝑇 − → Z ( 𝑇 𝒞 ) , which is t he identity on mor phisms and on objects is giv en by 𝐸 𝑍 𝑇 ( 𝑚 , ∇ 𝑚 ) = ( 𝑚 , 𝜎 𝑚 , − ) , for all ( 𝑚 , ∇ 𝑀 ) ∈ 𝒞 𝑍 𝑇 . (6.2.3) Con v ersely , to ev er y object ( 𝑚 , 𝜎 𝑚 , − ) ∈ Z ( 𝑇 𝒞 ) w e ma y assign a 𝑍 𝑇 -module, whose action ∇ 𝑚 is, due to Lemma 6.10 , uniquely deﬁned by 𝑇 𝑥 𝑚 𝑚 𝑥 = 𝑇 𝑥 𝑚 𝑥 ∇ 𝑚 𝑚 This yields the comparison functor 𝐾 𝑍 𝑇 : Z ( 𝑇 𝒞 ) − → 𝒞 𝑍 𝑇 . Remark 6.16. Suppose 𝑇 : 𝒞 − → 𝒞 to be a centralisable oplax monoidal endofunctor with ( 𝑍 𝑇 , 𝜒 ) as its centraliser . Denote the free functor of t he Eilenberg– Moore adjunction of 𝑍 𝑇 b y 𝐹 𝑍 𝑇 : 𝒞 − → 𝒞 𝑍 𝑇 . The composition 𝒞 𝐹 𝑍 𝑇 − − − → 𝒞 𝑍 𝑇 𝐸 𝑍 𝑇 − − − → Z ( 𝑇 𝒞 ) deﬁnes a left adjoint of t he forg etful functor 𝑈 ( 𝑇 ) : Z ( 𝑇 𝒞 ) − → 𝒞 . In fact, t he central adjunction 𝐹 ( 𝑇 ) ⊣ 𝑈 ( 𝑇 ) is monadic. Proposition 6.17 ([ B V12 , Theorem 5.12]) . Let ( 𝑍 𝑇 , 𝜒 ) be a centraliser of t he oplax monoidal endofunctor 𝑇 : 𝒞 − → 𝒞 . The comparison functor 𝐾 𝑍 𝑇 : Z ( 𝑇 𝒞 ) − → 𝒞 𝑍 𝑇 is an isomorphism of categories with inv erse 𝐸 𝑍 𝑇 : 𝒞 𝑍 𝑇 − → Z ( 𝑇 𝒞 ) . 6 . 3 c e n t r a l i s e r s a n d c o m o d u l e m o n a d s We wi ll now ap pl y t he me thod s of Br uguières and V irelizier t o twisted centres, for t he purpose of obtaining a comodule monad t hat implements the anti-Drinfeld centre. Hereto, w e need a gener alised v ersion of t he concept of modules o v er a monad. Our approach is based on [ MW11 ]. 144 6.3. Centralisers and comodule monads Deﬁnition 6.18. Let ( 𝐵 , 𝜇 , 𝜂 ) : 𝒞 − → 𝒞 be a bimonad and 𝐹 : 𝒞 − → 𝒟 an oplax monoidal functor . An oplax monoidal right action of 𝐵 on 𝐹 is an oplax natural transf ormation 𝛼 : 𝐹 𝐵 = ⇒ 𝐹 , such t hat t he follo wing diag r ams commute: 𝐹 𝐵 𝐵 𝐹 𝐵 𝐹 𝐹 𝐵 𝐹 𝐵 𝐹 𝐹 𝐹 𝜇 𝛼 𝛼 𝐵 𝛼 𝐹 𝜂 id 𝐹 𝛼 Similar l y , one deﬁnes oplax monoidal left actions. Remark 6.19. Recall from Remar k 5.2 t hat a bimonad 𝐵 is a monoid in the monoidal category O plMon opl ( 𝒞 , 𝒞 ) of oplax monoidal endofunctors on 𝒞 , with oplax monoidal natural transformations betw een them. There is an ob vious right action of this category on O plMon opl ( 𝒞 , 𝒟 ) giv en b y com position of functors. In this context, an oplax monoidal right action of a bimonad 𝐵 : 𝒞 − → 𝒞 is t he same as an O plMon opl ( 𝒞 , 𝒟 ) -module of 𝐵 . A prime example of an oplax monoidal left action is giv en b y the forg etful functor 𝑈 𝐵 : 𝒞 𝐵 − → 𝒞 of a bimonad 𝐵 : 𝒞 − → 𝒞 tog ether wit h the action ∇ = 𝑈 𝐵 𝜀 : 𝐵𝑈 𝐵 − → 𝑈 𝐵 , see Remar k 2.30 . Hypothesis 6.20. T o keep our notation concise, in t he follo wing w e ﬁx an oplax monoidal functor 𝐿 : 𝒞 − → 𝒞 with an oplax right action 𝛼 : 𝐿 𝐵 = ⇒ 𝐿 b y a bimonad 𝐵 : 𝒞 − → 𝒞 and assume t hat 𝐿 and 𝐵 are centralisable. Their centralisers will be denoted by ( 𝑍 𝐿 , 𝜉 ) and ( 𝑍 𝐵 , 𝜒 ) , respectiv ely . W e t hink of Z ( 𝐵 𝒞 ) as a more gener al v ersion of t he Drinfeld centre which is supposed t o act on Z ( 𝐿 𝒞 ) from the right. T o em phasise this, and in line with t he colouring scheme of Section 4.2 , w e use black for objects in 𝒞 or its gener alised Drinfeld centre and blue for objects in Z ( 𝐿 𝒞 ) . Consider objects ( 𝑚 , 𝜎 𝑚 , − ) ∈ Z ( 𝐿 𝒞 ) and ( 𝑥 , 𝜎 𝑥 , − ) ∈ Z ( 𝐵 𝒞 ) . The action of 𝐵 on 𝐿 and t he half-braidings of 𝑚 and 𝑥 yield a natural transf ormation 𝜎 𝑚 ⊗ 𝑥 , 𝑦 : 𝑚 ⊗ 𝑥 ⊗ 𝑦 − → 𝐿 𝑦 ⊗ 𝑚 ⊗ 𝑥 , which w e can depict g r aphicall y b y 𝑦 𝑚 𝑥 𝑚 𝑥 𝛼 𝑦 𝐿 𝑦 (6.3.1) 145 6. Monadi c t wisted cen tres Lemma 6.21. The centre Z ( 𝐵 𝒞 ) acts on Z ( 𝐿 𝒞 ) from t he right by tensoring the un- der lying objects and g luing tog ether t he half-braidings as in Equation ( 6.3.1 ). Wit h r espect t o t his action, the for getful functor 𝑈 ( 𝐿 ) : Z ( 𝐿 𝒞 ) − → 𝒞 is a s trict comodule functor over 𝑈 ( 𝐵 ) : Z ( 𝐵 𝒞 ) − → 𝒞 . Proof. W e proceed as in [ B V12 , Proposition 5.9]. Fix objects ( 𝑚 , 𝜎 𝑚 , − ) ∈ Z ( 𝐿 𝒞 ) and ( 𝑥 , 𝜎 𝑥 , − ) ∈ Z ( 𝐵 𝒞 ) . The com patibility of t he half-braiding of 𝑚 ⊗ 𝑥 with the unit of 𝒞 is a short computation: 𝑚 ⊗ 𝑥 1 𝐿 0 1 𝑚 ⊗ 𝑥 = 𝑚 𝐿 0 1 𝑚 𝛼 1 𝑥 𝑥 1 = 𝑚 𝐿 0 1 𝑚 𝑥 𝑥 1 𝐵 0 = 𝑚 𝑚 𝑥 𝑥 Similar l y , w e v erify t he hexagon axiom: 𝐿 2; 𝑦 ,𝑤 𝑚 ⊗ 𝑥 𝑦 ⊗ 𝑤 𝑚 ⊗ 𝑥 𝐿𝑤 𝐿 𝑦 = 𝐿 2; 𝑦 ,𝑤 𝑚 𝑚 𝐿𝑤 𝐿 𝑦 𝛼 𝑦 ⊗ 𝑤 𝑥 𝑦 ⊗ 𝑤 𝑥 = 𝐿 2; 𝑦 ,𝑤 𝑚 𝑚 𝐿𝑤 𝐿 𝑦 𝑥 𝑦 ⊗ 𝑤 𝑥 𝛼 𝑦 𝛼 𝑤 𝐵 2; 𝑦 ,𝑤 = 𝑚 𝑚 𝐿𝑤 𝐿 𝑦 𝑥 𝑦 𝑥 𝛼 𝑦 𝛼 𝑤 𝑤 The compatibility of t he action 𝛼 : 𝐿 𝐵 = ⇒ 𝐿 with t he multiplication and unit of 𝐵 asserts t hat Z ( 𝐿 𝒞 ) is a right Z ( 𝐵 𝒞 ) -module categor y . By construction, for all ( 𝑚 , 𝜎 𝑚 , − ) ∈ Z ( 𝐿 𝒞 ) and ( 𝑥 , 𝜎 𝑥 , − ) ∈ Z ( 𝐵 𝒞 ) w e ha v e 𝑈 ( 𝐿 )  ( 𝑚 , 𝜎 𝑚 , − ) ⊳ ( 𝑥 , 𝜎 𝑥 , − )  = 𝑚 ⊗ 𝑥 = 𝑈 ( 𝐿 ) ( 𝑚 , 𝜎 𝑚 , − ) ⊗ 𝑈 ( 𝐵 ) ( 𝑥 , 𝜎 𝑥 , − ) . Thus, 𝑈 ( 𝐿 ) is a strict comodule functor ov er 𝑈 ( 𝐵 ) . □ Notation 6.22. W e extend our colouring scheme to univ ersal coactions: 𝐿 𝑦 𝑍 𝐿 𝑥 𝑥 𝑦 𝐵 𝑦 𝑍 𝐵 𝑥 𝑥 𝑦 𝜉 𝑥 , 𝑦 : 𝑥 ⊗ 𝑦 − → 𝐿 𝑦 ⊗ 𝑍 𝐿 𝑥 𝜒 𝑥 , 𝑦 : 𝑥 ⊗ 𝑦 − → 𝐵 𝑦 ⊗ 𝑍 𝐵 𝑥 146 6.3. Centralisers and comodule monads Remark 6.23. The identiﬁcation of 𝒞 𝑍 𝐵 and 𝒞 𝑍 𝐿 with t he gener alised Drinfeld centre and its twisted cousin sugges t that 𝑍 𝐿 is a comodule monad o v er 𝑍 𝐵 . In analogy with Equation ( 6.2.1 ) w e deﬁne the coaction 𝑍 𝐿 ; a of 𝑍 𝐿 b y 𝑥 𝐿𝑤 𝑍 𝐿 𝑥 𝛼 𝑤 𝑦 𝑤 𝑍 𝐵 𝑦 = 𝐿𝑤 𝑥 ⊗ 𝑦 𝑤 𝑍 𝐿 𝑥 𝑤 𝑍 𝐿 ; a ; 𝑥 ,𝑦 (6.3.2) Proposition 6.24. Let 𝛼 : 𝐿 𝐵 = ⇒ 𝐿 be an oplax monoidal right action of a bimonad 𝐵 : 𝒞 − → 𝒞 on an oplax monoidal functor 𝐿 : 𝒞 − → 𝒞 . Suppose furt hermore that t he centr alisers ( 𝑍 𝐿 , 𝜉 ) of 𝐿 and ( 𝑍 𝐵 , 𝜒 ) of 𝐵 exis t. The coaction of Equation ( 6.3.2 ) turns 𝑍 𝐿 into a comodule monad over 𝑍 𝐵 such that 𝒞 𝑍 𝐿 is isomorphic as a right module category over 𝒞 𝑍 𝐵 to Z ( 𝐿 𝒞 ) . Proof. By Remar k 6.16 and Proposition 6.17 , w e ha v e monadic adjunctions 𝐹 ( 𝐵 ) : 𝒞 ⇄ Z ( 𝐵 𝒞 ) : 𝑈 ( 𝐵 ) and 𝐹 ( 𝐿 ) : 𝒞 ⇄ Z ( 𝐿 𝒞 ) : 𝑈 ( 𝐿 ) that, due t o [ B V12 , Remar k 5.13], giv e rise to the bimonad 𝑍 𝐵 and monad 𝑍 𝐿 . Lemma 6.21 sho ws t hat 𝑈 ( 𝐿 ) is a strict comodule functor ov er 𝑈 ( 𝐵 ) . By Theorem 5.31 , see also R emar k 5.24 and Example 5.27 , 𝑍 𝐿 is a comodule monad o v er 𝑍 𝐵 ; the coaction 𝜆 : 𝑍 𝐿 (− ⊗ = ) − → 𝑍 𝐿 (−) ⊗ 𝑍 𝐵 ( = ) implementing the action of 𝒞 𝑍 𝐵 on 𝒞 𝑍 𝐿 is for all 𝑥 , 𝑦 ∈ 𝒞 giv en by 𝜆 𝑥 , 𝑦 : 𝑍 𝐿 ( 𝑥 ⊗ 𝑦 ) 𝑍 𝐿 ( 𝜂 ( 𝑍 𝐿 ) 𝑥 ⊗ 𝜂 ( 𝑍 𝐵 ) 𝑦 ) − − − − − − − − − − − → 𝑍 𝐿 ( 𝑍 𝐿 𝑥 ⊗ 𝑍 𝐵 𝑦 ) ∇ 𝑍 𝐿 𝑥 ⊗ 𝑍 𝐵 𝑦 − − − − − − − → 𝑍 𝐿 𝑥 ⊗ 𝑍 𝐵 𝑦 , where ∇ is deﬁned as in Equation ( 2.3.1 ). By using the relation betw een univ ersal coactions and half-braidings, explained in Equation ( 6.2.2 ), and appl ying the hexagon identity w e compute Figure 6.2 . The uniqueness of univ ersal coactions implies t hat 𝜆 = 𝑍 𝐿 ; a . It remains to show that 𝒞 𝑍 𝐿 and Z ( 𝐿 𝒞 ) are isomor phic as modules ov er 𝒞 𝑍 𝐵 . By Proposition 5.34 , the comparison functor 𝐾 𝑍 𝐵 : Z ( 𝐵 𝒞 ) − → 𝒞 𝑍 𝐵 is strong monoidal and 𝐾 𝑍 𝐿 : Z ( 𝐿 𝒞 ) − → 𝒞 𝑍 𝐿 is a strong comodule functor o v er it. Furthermore, due to Proposition 6.17 , both 𝐾 𝑍 𝐵 and 𝐾 𝑍 𝐿 admit in v erses 𝐸 𝑍 𝐵 : 𝒞 𝑍 𝐵 − → Z ( 𝐵 𝒞 ) and 𝐸 𝑍 𝐿 : 𝒞 𝑍 𝐿 − → Z ( 𝐿 𝒞 ) . 147 6. Monadi c t wisted cen tres 𝜆 𝑥 , 𝑦 𝑍 𝐿 𝑥 𝑍 𝐵 𝑦 𝐿𝑤 𝑥 ⊗ 𝑦 𝑤 = 𝑍 𝐿 ( 𝜂 𝑍 𝐿 𝑥 ⊗ 𝜂 𝑍 𝐵 𝑦 ) 𝑍 𝐿 𝑥 𝑍 𝐵 𝑦 𝐿𝑤 𝑥 ⊗ 𝑦 𝑤 ∇ 𝑍 𝐿 𝑥 ⊗ 𝑍 𝐵 𝑦 = 𝜎 𝑍 𝐿 𝑥 ⊗ 𝑍 𝐵 𝑦 ,𝑤 𝑍 𝐿 𝑥 𝑍 𝐵 𝑦 𝐿𝑤 𝑥 ⊗ 𝑦 𝑤 𝜂 𝑍 𝐿 𝑥 ⊗ 𝜂 𝑍 𝐵 𝑦 = 𝜎 𝑍 𝐿 𝑥 ,𝐵 𝑤 𝑍 𝐿 𝑥 𝑍 𝐵 𝑦 𝐿𝑤 𝑥 𝑤 𝜂 𝑍 𝐿 𝑥 𝛼 𝑤 𝜎 𝑍 𝐵 𝑦 ,𝑤 𝑦 𝜂 𝑍 𝐵 𝑦 = 𝑍 𝐿 𝑥 𝑍 𝐵 𝑦 𝐿𝑤 𝑥 𝑤 𝛼 𝑤 𝑦 = 𝑍 𝐿 𝑥 𝑍 𝐵 𝑦 𝐿𝑤 𝑥 ⊗ 𝑦 𝑤 𝑍 𝐿 ; a ; 𝑥 ,𝑦 Figure 6.2: The arrow s 𝜆 and 𝑍 𝐿 ; a satisfy t he same univ ersal property . Using that 𝐸 𝑍 𝐵 is monoidal, w e identify t he right action of Z ( 𝐵 𝒞 ) on Z ( 𝐿 𝒞 ) with a right action ◀ : Z ( 𝐿 𝒞 ) × 𝒞 𝑍 𝐵 − → Z ( 𝐿 𝒞 ) of 𝒞 𝑍 𝐵 b y setting Z ( 𝐿 𝒞 ) × 𝒞 𝑍 𝐵 id × 𝐸 𝑍 𝐵 − − − − − → Z ( 𝐿 𝒞 ) × Z ( 𝐵 𝒞 ) ⊳ − − − → Z ( 𝐿 𝒞 ) . F or an y 𝑚 ∈ Z ( 𝐿 𝒞 ) and 𝑥 ∈ Z ( 𝐿 𝒞 ) w e ha v e 𝐾 𝑍 𝐿 ( 𝑚 ◀ 𝑥 ) = 𝐾 𝑍 𝐿 ( 𝑚 ⊳ 𝐸 𝑍 𝐵 𝑥 ) 𝐾 𝑍 𝐿 a ; 𝑚 , 𝐸 𝑍 𝐵 𝑥 − − − − − − − → 𝐾 𝑍 𝐿 𝑚 ⊳ 𝐾 𝑍 𝐵 𝐸 𝑍 𝐵 𝑥 = 𝐾 𝑍 𝐿 𝑚 ⊳ 𝑥 , and hence 𝐾 𝑍 𝐿 : Z ( 𝐿 𝒞 ) − → 𝒞 𝑍 𝐿 is an isomorphism of module categories. □ Exam ple 6.25. Let 𝒞 be a rigid monoidal categor y , and let ( 𝑍 𝐿 , 𝜉 ) and ( 𝑍 𝐵 , 𝜒 ) be the centr alisers of ∨∨ (−) : 𝒞 − → 𝒞 and Id 𝒞 , respectiv ely . Then t here exists a trivial right action of Id 𝒞 on ∨∨ (−) : id 𝑥 : ∨∨ ( Id 𝒞 ( 𝑥 )) − → ∨∨ 𝑥 , for all 𝑥 ∈ 𝒞 . This action tur ns 𝑍 𝐿 into a comodule monad o v er 𝑍 𝐵 , and its modules 𝒞 𝑍 𝐿 become isomorphic to Q ( 𝒞 ) as a 𝒞 𝑍 𝐵 -module category . Recall that, by Remar k 4.11 , w e can identify Q ( 𝒞 ) with A ( 𝒞 op , rev ) op . Deﬁnition 6.26. Assume ∨∨ (−) , Id 𝒞 : 𝒞 − → 𝒞 to admit centr alisers ( 𝑍 𝐿 , 𝜉 ) and ( 𝑍 𝐵 , 𝜒 ) . W e call Q ( 𝒞 ) . . = 𝑍 𝐿 the anti-central comodule monad of 𝒞 . 148 6.4. The Drinfeld and anti-Drinfeld double of a Hopf monad 6 . 4 t h e d r i n f e l d a n d a n t i - d r i n f e l d d o u b l e o f a h o p f m o n a d We ar e now ab le to un t angl e t he relationship betw een the v arious diﬀerent categories and adjunctions in Figure 6.1 . Hypothesis 6.27. For t he rest of t his chapter , ﬁx a Hopf monad 𝐻 on a rigid category 𝒞 , tog et her wit h an oplax monoidal endofunctor 𝐿 on 𝒞 𝐻 , a bimonad 𝐵 on 𝒞 𝐻 , an oplax monoidal right action 𝛼 : 𝐿 𝐵 = ⇒ 𝐵 , and assume t hat the cross products 𝐵 ⋊ 𝐻 and 𝐿 ⋊ 𝐻 ha v e centralisers ( 𝑍 𝐵 ⋊ 𝐻 , 𝜈 ) and ( 𝑍 𝐿 ⋊ 𝐻 , 𝜏 ) . The f ollowing obser v ation — which follo ws b y a s traightf or w ard calcu- lation — extends the action of 𝐵 on 𝐿 to an action of the respectiv e cross products. Lemma 6.28. The oplax monoidal right action 𝛼 : 𝐿 𝐵 = ⇒ 𝐵 induces an oplax monoidal action of 𝐵 ⋊ 𝐻 on 𝐿 ⋊ 𝐻 by 𝐹 𝐻 𝑈 𝐻 𝐿 𝛼 𝐹 𝐻 𝑈 𝐻 𝐵 𝐹 𝐻 𝑈 𝐻 𝐿 The next result is a v ariant of [ B V12 , Theorem 7.4]. Theorem 6.29. The functor s 𝐵 , 𝐿 : 𝒞 𝐻 − → 𝒞 𝐻 admit centraliser s ( 𝑍 𝐵 , 𝜒 ) , ( 𝑍 𝐿 , 𝜉 ) , such t hat 𝑍 𝐵 lifts 𝑍 𝐵 ⋊ 𝐻 as a bimonad and 𝑍 𝐿 lifts 𝑍 𝐿 ⋊ 𝐻 as a comodule monad. Proof. By [ B V12 , Theorem 7.4(a)], there are centralisers ( 𝑍 𝐿 , 𝜉 ) and ( 𝑍 𝐵 , 𝜒 ) of 𝐿 and 𝐵 t hat, for all ( 𝑥 , ∇ 𝑥 ) , ( 𝑦 , ∇ 𝑦 ) ∈ 𝒞 𝐻 , satisfy 𝑈 𝐻 𝑍 𝐿 ( 𝑥 , ∇ 𝑥 ) = 𝑍 𝐿 ⋊ 𝐻 𝑥 , 𝑈 𝐻 𝜉 ( 𝑥 , ∇ 𝑥 ) , ( 𝑦 , ∇ 𝑦 ) = ( 𝑈 𝐻 𝐿 ∇ 𝑦 ⊗ 𝑍 𝐿 ⋊ 𝐻 𝑥 ) ◦ 𝜏 𝑥 , 𝑦 , 𝑈 𝐻 𝑍 𝐵 ( 𝑥 , ∇ 𝑥 ) = 𝑍 𝐵 ⋊ 𝐻 𝑥 , 𝑈 𝐻 𝜒 ( 𝑥 , ∇ 𝑥 ) , ( 𝑦 , ∇ 𝑦 ) = ( 𝑈 𝐻 𝐵 ∇ 𝑦 ⊗ 𝑍 𝐵 ⋊ 𝐻 𝑥 ) ◦ 𝜈 𝑥 , 𝑦 . The second and third part of ibid state t hat 𝑍 𝐿 is a lift of the monad 𝑍 𝐿 ⋊ 𝐻 and 𝑍 𝐵 is a lift of t he bimonad 𝑍 𝐵 ⋊ 𝐻 . It remains for us to show t hat t he coactions of 𝑍 𝐿 and 𝑍 𝐿 ⋊ 𝐻 are compatible wit h the for getful functor 𝑈 𝐻 : 𝒞 𝐻 − → 𝒞 . Fixing objects ( 𝑥 , ∇ 𝑥 ) , ( 𝑦 , ∇ 𝑦 ) ∈ 𝒞 𝐻 and 𝑤 ∈ 𝒞 , this f ollows from Figure 6.3 . The uniqueness property of univ ersal coactions as giv en in Lemma 6.10 then implies that 𝑈 𝐻 𝑍 𝐿 ; a ; ( 𝑥 , ∇ 𝑥 ) , ( 𝑦 , ∇ 𝑦 ) = 𝑍 𝐿 ⋊ 𝐻 ; a ; 𝑥 , 𝑦 . Since 𝑈 𝐻 : 𝒞 𝐻 − → 𝒞 is a strict comodule functor , the claim follow s. □ 149 6. Monadi c t wisted cen tres ( 𝐿 ⋊ 𝐻 ) 𝑤 𝑍 𝐿 ⋊ 𝐻 𝑥 𝑍 𝐵 ⋊ 𝐻 𝑦 𝑈 𝐻 𝑍 𝐿 ; a ; 𝑥, 𝑦 𝜏 𝑈 𝐻 ( 𝑥 ⊗ 𝑦 ) 𝑤 = ( 𝐿 ⋊ 𝐻 ) 𝑤 𝑍 𝐿 ⋊ 𝐻 𝑥 𝑍 𝐵 ⋊ 𝐻 𝑦 𝑈 𝐻 𝑍 𝐿 ; a ; 𝑥, 𝑦 𝜏 𝑈 𝐻 ( 𝑥 ⊗ 𝑦 ) 𝑤 𝑈 𝐻 𝐿 𝜀 𝐹 𝐻 𝑤 𝑈 𝐻 𝐿𝐹 𝐻 𝜂 𝐻 𝑤 = ( 𝐿 ⋊ 𝐻 ) 𝑤 𝑍 𝐿 ⋊ 𝐻 𝑥 𝑍 𝐵 ⋊ 𝐻 𝑦 𝑈 𝐻 𝑍 𝐿 ; a ; 𝑥, 𝑦 𝜏 𝑈 𝐻 ( 𝑥 ⊗ 𝑦 ) 𝑤 𝑈 𝐻 𝐿 𝜀 𝐹 𝐻 𝑤 𝜂 𝐻 𝑤 = ( 𝐿 ⋊ 𝐻 ) 𝑤 𝑍 𝐿 ⋊ 𝐻 𝑥 𝑍 𝐵 ⋊ 𝐻 𝑦 𝑈 𝐻 𝑍 𝐿 ; a ; 𝑥, 𝑦 𝑈 𝐻 𝜉 𝑈 𝐻 ( 𝑥 ⊗ 𝑦 ) 𝑤 𝜂 𝐻 𝑤 = ( 𝐿 ⋊ 𝐻 ) 𝑤 𝑍 𝐿 ⋊ 𝐻 𝑥 𝑍 𝐵 ⋊ 𝐻 𝑦 𝑈 𝐻 𝜉 𝑈 𝐻 𝑥 𝑈 𝐻 𝛼 𝐹 𝐻 𝑤 𝜂 𝐻 𝑤 𝑈 𝐻 𝜒 𝑤 𝑈 𝐻 𝑦 = ( 𝐿 ⋊ 𝐻 ) 𝑤 𝑍 𝐿 ⋊ 𝐻 𝑥 𝑍 𝐵 ⋊ 𝐻 𝑦 𝜏 𝑈 𝐻 𝑥 𝛼 𝐻 𝑤 𝜁 𝑤 𝑈 𝐻 𝑦 ( 𝐿 ⋊ 𝐻 ) 𝑤 𝑍 𝐿 ⋊ 𝐻 𝑥 𝑍 𝐵 ⋊ 𝐻 𝑦 𝑄 𝐻 ; a ; 𝑥 , 𝑦 𝜏 𝑈 𝐻 ( 𝑥 ⊗ 𝑦 ) 𝑤 = Figure 6.3: The coactions of 𝑍 𝐿 and 𝑍 𝐿 ⋊ 𝐻 are compatible with the forg etful functor . Remark 6.30. The pre vious t heorem tog ether wit h Lemma 6.2 im ply t hat w e obtain a comodule monad 𝐷 ( 𝐿 , 𝐻 ) . . = 𝑍 𝐿 ⋊ 𝐻 o v er 𝐷 ( 𝐵 , 𝐻 ) . . = 𝑍 𝐵 ⋊ 𝐻 . The correspondence between lifts and monads giv en in Proposition 6.5 yields a unique comodule distributiv e law  𝐻 𝑍 𝐿 ⋊ 𝐻 Ω − − → 𝑍 𝐿 ⋊ 𝐻 𝐻 , 𝐻 𝑍 𝐵 ⋊ 𝐻 Λ − − → 𝑍 𝐵 ⋊ 𝐻 𝐻  , such that 𝐷 ( 𝐿 , 𝐻 ) = 𝑍 𝐿 ⋊ 𝐻 ◦ Ω 𝐻 and 𝐷 ( 𝐵 , 𝐻 ) = 𝑍 𝐵 ⋊ 𝐻 ◦ Λ 𝐻 . Deﬁnition 6.31. W e call 𝐷 ( 𝐵 , 𝐻 ) and 𝐷 ( 𝐿 , 𝐻 ) of Remar k 6.30 t he double and twist ed double of t he pairs ( 𝐵 , 𝐻 ) and ( 𝐿 , 𝐻 ) , respectivel y . 150 6.5. P airs in in v olution for Hopf monads The relationship betw een doubles and gener alised Drinfeld centres is studied in [ B V12 , Proposition 7.5 and Theorem 7.6]. Our next result uses t he same techniques to pro v e how twisted doubles parameterise twisted centres. Theorem 6.32. The twist ed double 𝐷 ( 𝐿 , 𝐻 ) is a comodule monad over 𝐷 ( 𝐵 , 𝐻 ) , and 𝒞 𝐷 ( 𝐿 , 𝐻 ) is isomorphic to Z ( 𝐿 𝒞 𝐻 ) as a 𝒞 𝐷 ( 𝐵 , 𝐻 ) -module category . Proof. Since 𝑍 𝐿 is a lift of 𝑍 𝐿 ⋊ 𝐻 as a comodule monad, the twisted double 𝐷 ( 𝐿 , 𝐻 ) is a comodule monad ov er 𝐷 ( 𝐵 , 𝐻 ) . By Lemma 6.6 , this implies the exis tence of an isomorphism 𝐼 : 𝒞 𝐷 ( 𝐿 , 𝐻 ) ∼ − →  𝒞 𝐻  𝑍 𝐿 of 𝒞 𝐷 ( 𝐵 , 𝐻 ) -module categories. Due to t he proof of Proposition 6.24 , t he comparison functor 𝐾 𝑍 𝐿 : Z ( 𝐿 𝒞 𝐻 ) − →  𝒞 𝐻  𝑍 𝐿 implements an isomorphism of module categories and the statement follo ws by considering 𝒞 𝐷 ( 𝐿 , 𝐻 ) 𝐼 − − − → ( 𝒞 𝐻 ) 𝑍 𝐿 𝐸 𝑍 𝐿 − − − → Z ( 𝐿 𝒞 𝐻 ) . □ The follo wing deﬁnition can be unders tood as an extension of t he notion of the anti-Drinfeld double giv en by [ HKRS04 ] to t he monadic framew or k. Deﬁnition 6.33. Let 𝐻 be a Hopf monad on a rigid categor y 𝒞 . F or 𝐵 . . = Id 𝒞 𝐻 and 𝐿 . . = ∨∨ (−) : 𝒞 𝐻 − → 𝒞 𝐻 , w e call 𝐷 ( 𝐻 ) . . = 𝐷 ( 𝐵 , 𝐻 ) and 𝑄 ( 𝐻 ) . . = 𝐷 ( 𝐿 , 𝐻 ) the Drinfeld and anti-Drinfeld double of 𝐻 , respectiv el y . 6 . 5 pa i r s i n i n vo l u t i o n f o r h o p f m o na d s We no w c ons ider a Ho pf mo n ad that admits a double and anti-double, and dev elop the notion of pairs in inv olution in this setting. Classically , these consist of a g roup-lik e and character of a Hopf algebr a t hat implement t he square of its antipode b y their adjoint actions. Deﬁnition 6.34. Let 𝐻 be a Hopf monad on a rigid monoidal category 𝒞 . A char acter of 𝐻 is an 𝐻 -algebr a 𝛽 . . = ( 1 , ∇ 𝛽 ) ∈ 𝒞 𝐻 , whose underl ying object is the monoidal unit 1 ∈ 𝒞 . Remark 6.35. Explicitl y , Deﬁnition 6.34 sa ys t hat a g roup-like 𝑔 of 𝐻 satisﬁes 𝐻 2; 𝑥 , 𝑦 ◦ 𝑔 𝑥 ⊗ 𝑦 = 𝑔 𝑥 ⊗ 𝑔 𝑦 and 𝐻 0 ◦ 𝑔 1 = id 1 , for all 𝑥 , 𝑦 ∈ 𝒞 . The characters Char ( 𝐻 ) of a Hopf monad 𝐻 on 𝒞 form a monoid and, b y [ B V07 , Lemma 3.21], t he set Gr ( 𝐻 ) of group-likes bears a group structure. 151 6. Monadi c t wisted cen tres Exam ple 6.36. The g roup-lik es of a Hopf monad 𝐻 act on it b y conjugation. W e recall t his construction based on [ B V07 , Section 1.4]. Giv en a natural transf or mation 𝑔 : Id 𝒞 − → 𝐻 , deﬁne t he left and right regular action of 𝑔 on 𝐻 to be t he natural transformations deﬁned by 𝐿 𝑔 : 𝐻 𝑔 𝐻 − − → 𝐻 2 𝜇 ( 𝐻 ) − − − → 𝐻 and 𝑅 𝑔 : 𝐻 𝐻 𝑔 − − → 𝐻 2 𝜇 ( 𝐻 ) − − − → 𝐻 . Deﬁnition 6.37. Ev er y pair ( 𝑔 ∈ Gr ( 𝐻 ) , 𝛽 ∈ Char ( 𝐻 )) of a Hopf monad 𝐻 : 𝒞 − → 𝒞 giv es rise to natural transf ormations Recall the deﬁnition of 𝐻 3 from Remar k 2.35 . A d 𝑔 . . = 𝐿 𝑔 ◦ 𝑅 𝑔 − 1 : 𝐻 = ⇒ 𝐻 , A d 𝛽 . . = (∇ 𝛽 ⊗ 𝐻 (−) ⊗ ∇ ∨ 𝛽 ) ◦ 𝐻 3;1 , − , 1 : 𝐻 = ⇒ 𝐻 , called the adjoint actions of 𝑔 and 𝛽 on 𝐻 , respectiv ely . T o deﬁne pairs in inv olution, w e need an analogue of t he square of t he antipode of a Hopf algebr a. This notion is dev eloped in [ B V07 , Section 7.3]. Deﬁnition 6.38. Suppose 𝜙 : Id 𝒞 − → ∨∨ (−) to be a piv otal structure on 𝒞 and let 𝐻 : 𝒞 − → 𝒞 be a Hopf monad. The squar e of the antipode of 𝐻 is a natural transf ormation 𝑆 2 : 𝐻 = ⇒ 𝐻 , giv en for all 𝑥 ∈ 𝒞 b y 𝑆 2 𝑥 . . = 𝜙 − 1 𝐻 𝑥 ◦ 𝑠 𝑙 ∨ 𝐻 𝑥 ◦ 𝐻 ∨ 𝑠 𝑙 𝑥 ◦ 𝐻 𝜙 𝑥 , where 𝑠 𝑙 is t he lef t antipode of 𝐻 , see Equation ( 5.2.1 ) and [ B V07 , Section 3.3]. Analogous to t he Hopf algebr aic case, w e state the follo wing: Deﬁnition 6.39. Let 𝐻 : 𝒞 − → 𝒞 be a Hopf monad, and 𝜙 : Id 𝒞 − → ∨∨ (−) a piv otal s tructure. A pair in inv olution ( 𝑔 , 𝛽 ) ∈ PI 𝜙 𝐻 of 𝐻 and 𝜙 consists of a group-like 𝑔 ∈ Gr ( 𝐻 ) and a character 𝛽 ∈ Char ( 𝐻 ) , such t hat for all 𝑥 ∈ 𝒞 𝐴 𝑑 𝑔 , 𝑥 = 𝐴 𝑑 𝛽 , 𝑥 ◦ 𝑆 2 𝑥 . T o prov e that pairs in in v olution correspond to certain piv otal structures on the Drinfeld centre of 𝒞 𝐻 , w e need tw o technical results. The ﬁrst one is classical; for a proof see for example [ BV07 , Lemmas 1.2 and 1.3]. Lemma 6.40. Let 𝐻 be a monad with canonical for g etful functor 𝑈 𝐻 : 𝒞 𝐻 − → 𝒞 . If 𝐹 , 𝐺 : 𝒞 − → 𝒟 are funct ors, for some category 𝒟 , then t here is a canonical bĳection (−) ♯ : Nat ( 𝐹 , 𝐺 𝐻 ) − → N at ( 𝐹𝑈 𝐻 , 𝐺𝑈 𝐻 ) , 𝛼 ↦− → 𝛼 ♯ , wher e 𝛼 ♯ ( 𝑚 , ∇ 𝑚 ) . . = 𝐺 ∇ 𝑚 ◦ 𝛼 𝑚 . 152 6.5. P airs in in v olution for Hopf monads The next lemma is a v ariant of [ B V07 , Lemma 7.5]. Lemma 6.41. Let 𝜙 : Id 𝒞 − → ∨∨ (−) be a pivo tal structur e on 𝒞 and 𝐻 : 𝒞 − → 𝒞 a Hopf monad. F or any group-like 𝑔 ∈ Gr ( 𝐻 ) and char acter 𝛽 ∈ Char ( 𝐻 ) the following are equiv alent: (i) t he mor phisms 𝑔 and 𝛽 form a pair in involution of 𝐻 and 𝜙 ; and (ii) t he arrow 𝜙 𝑔 ♯ ∈ Nat ( 𝑈 𝐻 , ∨∨ (−) ◦ 𝑈 𝐻 ) lifts to Nat ( Id 𝒞 𝐻 , 𝛽 ⊗ ∨∨ (−) ⊗ ∨ 𝛽 ) . Proof. Consider a module ( 𝑚 , ∇ 𝑚 ) ∈ 𝒞 𝐻 . By [ B V07 , Theorem 3.8(a)] and t he deﬁnition of 𝑆 2 , the action on ∨∨ 𝑚 is giv en b y ∇ ∨∨ 𝑚 = ∨∨ ∇ 𝑚 ◦ 𝑠 𝑙 ∨∨ 𝐻 𝑚 ◦ 𝐻 ( ∨ 𝑠 𝑙 𝑚 ) = 𝜙 𝑚 ◦ ∇ 𝑚 ◦ 𝑆 2 𝑚 ◦ 𝐻 ( 𝜙 − 1 𝑚 ) , and therefore w e ha v e ∇ 𝛽 ⊗ ∨∨ 𝑚 ⊗ ∨ 𝛽 = (∇ 𝛽 ⊗ ∇ ∨∨ 𝑚 ⊗ ∇ ∨ 𝛽 ) 𝐻 3;1 ,𝑚 , 1 = (∇ 𝛽 ⊗ 𝜙 𝑚 ∇ 𝑚 𝑆 2 𝑚 𝐻 ( 𝜙 − 1 𝑚 ) ⊗ ∇ ∨ 𝛽 ) 𝐻 3;1 ,𝑚 , 1 . By deﬁnition, 𝜙 𝑔 ♯ lifts to a natural transf or mation from Id 𝒞 𝐻 to 𝛽 ⊗ ∨∨ (−) ⊗ ∨ 𝛽 if and onl y if for an y 𝐻 -module ( 𝑚 , ∇ 𝑚 ) w e ha v e ( 𝜙 𝑔 ♯ ) 𝑚 ∇ 𝑚 = ∇ 𝛽 ⊗ ∨∨ 𝑚 ⊗ ∨ 𝛽 𝐻  ( 𝜙 𝑔 ♯ ) 𝑚  . (6.5.1) Let us now successiv ely simplify both sides of t his equation. Using t he naturality of 𝑔 : Id 𝒞 = ⇒ 𝐻 , t he fact that ∇ 𝑚 is an action, and t he deﬁnition of 𝑔 ♯ as giv en in Lemma 6.40 , w e can rewrite t he left hand side as ( 𝜙 𝑔 ♯ ) 𝑚 ∇ 𝑚 = 𝜙 𝑚 ∇ 𝑚 𝑔 𝑚 ∇ 𝑚 = 𝜙 𝑚 ∇ 𝑚 𝐻 (∇ 𝑚 ) 𝑔 𝐻 𝑚 = 𝜙 𝑚 ∇ 𝑚 𝜇 ( 𝐻 ) 𝑚 𝑔 𝐻 𝑚 . Similar l y , w e simplify the right-hand side to ∇ 𝛽 ⊗ ∨∨ 𝑚 ⊗ ∨ 𝛽 𝐻  ( 𝜙 𝑔 ♯ ) 𝑚  = (∇ 𝛽 ⊗ 𝜙 𝑚 ∇ 𝑚 𝑆 2 𝑚 𝐻 ( 𝜙 − 1 𝑚 ) ⊗ ∇ ∨ 𝛽 ) 𝐻 3;1 ,𝑚 , 1 𝐻  ( 𝜙 𝑔 ♯ ) 𝑚  = (∇ 𝛽 ⊗ 𝜙 𝑚 ∇ 𝑚 𝑆 2 𝑚 𝐻 ( 𝜙 − 1 𝑚 ) 𝐻  ( 𝜙 𝑔 ♯ ) 𝑚  ⊗ ∇ ∨ 𝛽 ) 𝐻 3;1 ,𝑚 , 1 = (∇ 𝛽 ⊗ 𝜙 𝑚 ∇ 𝑚 𝑆 2 𝑚 𝐻 (∇ 𝑚 𝑔 𝑚 ) ⊗ ∇ ∨ 𝛽 ) 𝐻 3;1 ,𝑚 , 1 = (∇ 𝛽 ⊗ 𝜙 𝑚 ∇ 𝑚 𝐻 (∇ 𝑚 𝑔 𝑚 ) 𝑆 2 𝑚 ⊗ ∇ ∨ 𝛽 ) 𝐻 3;1 ,𝑚 , 1 = 𝜙 𝑚 ∇ 𝑚 𝐻 (∇ 𝑚 𝑔 𝑚 )(∇ 𝛽 ⊗ id 𝐻 𝑚 ⊗ ∇ ∨ 𝛽 ) 𝐻 3;1 ,𝑚 , 1 𝑆 2 𝑚 = 𝜙 𝑚 ∇ 𝑚 𝜇 ( 𝐻 ) 𝑚 𝐻 ( 𝑔 𝑚 ) Ad 𝛽 , 𝑚 𝑆 2 𝑚 . 153 6. Monadi c t wisted cen tres Using the fact that 𝜙 is an isomorphism, Equation ( 6.5.1 ) can be restated as ∇ 𝑚 𝜇 ( 𝐻 ) 𝑚 𝑔 𝐻 𝑚 = ∇ 𝑚 𝜇 ( 𝐻 ) 𝑚 𝐻 ( 𝑔 𝑚 ) Ad 𝛽 , 𝑚 𝑆 2 𝑚 ⇐ ⇒ ∇ 𝑚 𝐿 𝑔 , 𝑚 = ∇ 𝑚 𝑅 𝑔 , 𝑚 A d 𝛽 , 𝑚 𝑆 2 𝑚 By Lemma 6.40 , the abov e is equivalent to 𝐿 𝑔 , 𝑚 = 𝑅 𝑔 , 𝑚 A d 𝛽 , 𝑚 𝑆 2 𝑚 . W e conclude the proof by multiplying both sides with 𝑅 𝑔 − 1 , 𝑚 . □ Lemma 6.41 leads to an identiﬁcation of pairs in in v olution of 𝐻 and 𝜙 with certain quasi-piv otal structures on 𝒞 𝐻 . Proposition 6.42. Let 𝐻 be a Hopf monad on 𝒞 and 𝜙 : Id 𝒞 − → ∨∨ (−) a piv ot al structur e on 𝒞 . Then ( 𝐻 , 𝜙 ) admits a pair in involution if and only if t her e exists a quasi-piv ot al structur e on 𝒞 𝐻 whose under lying inv ertible object is a char acter . Proof. W e proceed analogous to [ B V07 , Proposition 7.6]. Suppose ( 𝑔 , 𝛽 ) ∈ PI 𝜙 𝐻 . By Lemma 6.41 , t he natural transformation 𝜙 𝑔 ♯ lifts to a natural isomorphism 𝜌 𝛽 , 𝑥 : 𝑥 − → 𝛽 ⊗ ∨∨ 𝑥 ⊗ ∨ 𝛽 , for all 𝑥 ∈ 𝒞 𝐻 . Since 𝜙 is monoidal b y deﬁnition, and 𝑔 ♯ is monoidal b y virtue of 𝑔 being a group-like — see [ B V07 , Lemma 3.20] — w e obtain a quasi-piv otal structure 𝜌 𝛽 : Id 𝒞 𝐻 − → 𝛽 ⊗ ∨∨ (−) ⊗ ∨ 𝛽 . On t he other hand, let ( 𝛽 , 𝜌 𝛽 ) be a quasi-piv otal structure, for 𝛽 ∈ Char ( 𝐻 ) . Since the forg etful functor 𝑈 𝐻 is strong monoidal and t hus 𝑈 𝐻 ( 𝛽 ⊗ ∨∨ (−) ⊗ ∨ 𝛽 ) = 𝑈 𝐻 ( ∨∨ (−)) = ∨∨ ( 𝑈 𝐻 (−)) , there exists a monoidal natural transf ormation 𝜙 − 1 𝑈 𝐻 𝑥 ◦ 𝑈 𝐻 ( 𝜌 𝛽 , 𝑥 ) : 𝑈 𝐻 𝑥 − → 𝑈 𝐻 𝑥 , for all 𝑥 ∈ 𝒞 𝐻 . Appl y [ B V07 , Lemma 3.20] to obtain a unique g roup-lik e 𝑔 ∈ Gr ( 𝐻 ) , wit h 𝑔 ♯ = 𝜙 − 1 𝑈 𝐻 ( 𝑥 ) ◦ 𝑈 𝐻 ( 𝜌 𝛽 , 𝑥 ) . As 𝜙 𝑔 ♯ = 𝑈 𝐻 ( 𝜌 𝛽 ) lifts to t he quasi-piv otal structure ( 𝛽 , 𝜌 𝛽 ) on 𝒞 𝐻 , Lemma 6.41 implies that ( 𝑔 , 𝛽 ) ∈ PI 𝜙 𝐻 . □ 154 6.5. P airs in in v olution for Hopf monads Let us no w study a variant of [ B V07 , Lemma 2.9]. Proposition 6.43. Let 𝐶 , 𝐷 : ℳ − → ℳ be two comodule monads over a bimonad 𝐵 : 𝒞 − → 𝒞 . Ther e is a bĳective corr espondence between morphisms of comodule monads 𝑓 : 𝐷 = ⇒ 𝐶 and strict module functor s 𝐹 : ℳ 𝐶 − → ℳ 𝐷 wit h 𝑈 𝐷 𝐹 = 𝑈 𝐶 . Proof. As shown in [ B V07 , Lemma 1.7], any functor 𝐹 : ℳ 𝐶 − → ℳ 𝐷 with 𝑈 𝐷 𝐹 = 𝑈 𝐶 is induced by a unique mor phism of monads 𝑓 : 𝐷 − → 𝐶 . That is, 𝐹 is t he identity on mor phisms and on objects it is deﬁned by 𝐹 ( 𝑚 , ∇ 𝑚 ) = ( 𝑚 , ∇ 𝑚 𝑓 𝑚 ) , for all ( 𝑚 , ∇ 𝑚 ) ∈ 𝓂 𝐶 . It remains to show t hat 𝑓 is a morphism of comodules if and only if 𝐹 is a strict 𝒞 𝐵 -module functor . Let ( 𝑚 , ∇ 𝑚 ) ∈ 𝓂 𝐶 and ( 𝑥 , ∇ 𝑥 ) ∈ 𝒞 𝐵 . W e compute 𝐹  ( 𝑚 , ∇ 𝑚 ) ⊳ ( 𝑥 , ∇ 𝑥 )  = ( 𝑚 ⊳ 𝑥 , (∇ 𝑚 ⊳ ∇ 𝑥 ) ◦ 𝐶 a ; 𝑚 , 𝑥 ◦ 𝑓 𝑚 ⊳ 𝑥 ) , 𝐹 ( 𝑚 , ∇ 𝑚 ) ⊳ ( 𝑥 , ∇ 𝑥 ) = ( 𝑚 ⊳ 𝑥 , (∇ 𝑚 ⊳ ∇ 𝑥 ) ◦ ( 𝑓 𝑚 ⊳ 𝐵 𝑥 ) ◦ 𝐷 a ; 𝑚 , 𝑥 ) . A ccording to [ BV07 , Lemma 1.4], these modules coincide if and only if 𝐶 a ; 𝑚 , 𝑥 ◦ 𝑓 𝑚 ⊳ 𝑥 = ( 𝑓 𝑚 ⊳ 𝐵 𝑥 ) ◦ 𝐷 a ; 𝑚 , 𝑥 , which is exactl y the condition for 𝑓 to be a comodule mor phism. □ The abo v e result implies t he desired monadic v ersion of Theorem 4.1 . Theorem 6.44. Let 𝒞 be a rigid monoidal category , and suppose that 𝐻 : 𝒞 − → 𝒞 is a Hopf monad that admits a double 𝐷 ( 𝐻 ) and anti-double 𝑄 ( 𝐻 ) . The following st atements are equivalent: (i) t he monoidal unit 1 ∈ 𝒞 lif ts to a module over 𝑄 ( 𝐻 ) ; (ii) t here is an isomor phism of comodule monads 𝐷 ( 𝐻 )  𝑄 ( 𝐻 ) ; and (iii) t here is an isomor phism of monads 𝑄 ( 𝐻 )  𝐷 ( 𝐻 ) . Additionall y , if 𝒞 is pivot al with pivo tal structur e 𝜙 , any of the above st atements hold if and onl y if 𝐻 and 𝜙 admit a pair in in volution. Proof. (i) = ⇒ (ii) : Suppose 𝜔 ∈ Q ( 𝒞 𝐻 ) with 𝑈 𝑄 ( 𝐻 ) 𝜔 = 1 . As sho wn in Equation ( 4.2.1 ), this induces a functor of module categories 𝜔 ⊗ − : 𝒞 𝐷 ( 𝐻 ) − → 𝒞 𝑄 ( 𝐻 ) . 155 6. Monadi c t wisted cen tres Since 𝑈 𝑄 ( 𝐻 ) 𝜔 = 1 ∈ 𝒞 , w e can apply Proposition 6.43 , and obtain t hat 𝑄 ( 𝐻 ) and 𝐷 ( 𝐻 ) are isomor phic as comodule monads. It immediatel y follo ws that (ii) implies (iii) ; w e proceed with (iii) = ⇒ (i) : consider an isomor phism of monads 𝑔 : 𝑄 ( 𝐻 ) = ⇒ 𝐷 ( 𝐻 ) . It giv es rise to a functor 𝐺 : 𝒞 𝐷 ( 𝐻 ) − → 𝒞 𝑄 ( 𝐻 ) that, on objects, is deﬁned by 𝐺 ( 𝑚 , ∇ 𝑚 ) = ( 𝑚 , ∇ 𝑚 𝑔 𝑚 ) , for all ( 𝑚 , ∇ 𝑚 ) ∈ 𝒞 𝐷 ( 𝐻 ) . W e compose 𝐺 with 𝐸 𝑄 ( 𝐻 ) : 𝒞 𝑄 ( 𝐻 ) − → Q ( 𝒞 𝐻 ) — t he in v erse of t he com parison functor deﬁned in Equation ( 6.2.3 ) — and see t hat t here exists an object 1 ( 𝑄 ) . . = 𝐸 𝑄 ( 𝐻 ) 𝐺 ( 1 ) ∈ Q ( 𝒞 𝐻 ) , whose underl ying object is t he unit of 𝒞 . N o w let ( 𝒞 , 𝜙 ) be piv otal. By Lemma 4.22 , lifts of 1 ∈ 𝒞 to the dual of the anti-centre Q ( 𝒞 𝐻 ) are in cor - respondence with quasi-piv otal structures ( 𝛽 , 𝜌 𝛽 ) , where 𝛽 ∈ Char ( 𝐻 ) . By Proposition 6.42 , such a quasi-piv otal structure exists if and onl y if t here exists a pair in in v olution for 𝐻 and 𝜙 . □ As a corollary , w e can determine whet her a categor y is piv otal in ter ms of monad isomorphisms betw een t he central and anti-central monad. For a cat- egory 𝒞 , recall Deﬁnition 6.15 of its centr al Hopf monad, and Deﬁnition 6.26 of its anti-central comodule monad. Corollar y 6.45. Let 𝒞 be a rigid monoidal category . If 𝒞 admits a central Hopf monad D ( 𝒞 ) and an anti-central comodule monad Q ( 𝒞 ) , then it is pivot al if and onl y if D ( 𝒞 )  Q ( 𝒞 ) as monads. Proof. W e consider t he identity Id 𝒞 : 𝒞 − → 𝒞 as a Hopf monad. Its Drinfeld and anti-Drinfeld double are 𝐷 ( Id 𝒞 ) = D ( 𝒞 ) ⋊ Id 𝒞 and 𝑄 ( Id 𝒞 ) = Q ( 𝒞 ) ⋊ Id 𝒞 . From here it follow s t hat 𝐷 ( Id 𝒞 ) = D ( 𝒞 ) , and similarl y 𝑄 ( Id 𝒞 ) = Q ( 𝒞 ) . The proof is concluded b y Theorem 6.44 . □ 156 The struggle itself to ward t he heights is enough to ﬁll a man ’s heart. One must imagine Sisyphus happy . Alb ert Camu s ; The Myth of Sisyphus D U O I DA L R - M A T R I C E S 7 Duo id al ca t egor ies w ere i ntrodu ced in [ AM10 ] under the name 2-monoidal categories , in order to study bilax monoidal functors and v arious constructions on linear species. They gener alise both braided monoidal categories, by considering tw o monoidal structures t hat are connected b y a non-in v ertible interchange la w , as well as t he 2-fold monoidal categories of [ BFSV03 ] where the tw o tensor products are assumed to share a unit. Duoidal categories ha v e since been used to study higher-dimensional Hopf t heory [ BCZ13 ; BS13 ; AHLF18 ; LFV20 ; Böh21 ], and hav e also found applications in v arious other ﬁelds of mathematics, [ GLF16 ; SS22 ; Rom24 ; T or24 ]. The aim of this chapter is to generalise a reconstruction-type result for R- matrices on bimonads, [ B V07 , Proposition 8.5], which in turn gener alises the classical theory of R-matrices f or bialg ebras. The f ormer has the additional ad- v antage of not requiring a braided monoidal base category , as bimonads — in contras t with bialgebras — ma y be deﬁned on an y monoidal category . Theorem 7.21 . Let 𝒟 be a category with monoidal structur es ◦ and • , and 𝑇 a monad on 𝒟 t hat has a ◦ -oplax monoidal and a • -oplax monoidal structur e. Then quasitriangular structur es on 𝑇 are in bĳection with duoidal structur es on 𝒟 𝑇 . This result can be seen as a gener alisation of the double comonoidal monads of [ AM10 , Section 7], analogous to how R-matrices for bialgebr as gener alise cocommutativ e bialgebr as. In Section 7.3 w e study the relationship betw een normal duoidal and linear l y distributiv e categories from this point of view . W e see non-planar linear l y dis tributiv e categories ℒ as an analogue of preduoidal categories, in t he sense t hat the additional structure trivialises in t he monoidal case, see Example 7.27 . Equipping ℒ with a planar structure, w e can relate double comonoidal monads to linearl y distributiv e monads in t he sense of [ P as12 ]. 157 7. Duoi d a l R -m a tr ices 7 . 1 d u o i da l c at e g o r i e s We sh all loos el y follow the nomenclature of [ BM12 , Deﬁnition 3], who introduced the ter m duoidal category , and the notation of [ BCZ13 ]. Deﬁnition 7.1 ([ AM10 , Deﬁnition 6.1]) . A duoidal categor y is a quintuple ( 𝒟 , ◦ , ⊥ , • , 1 ) , consisting of the follo wing data: • monoidal categories ( 𝒟 , ◦ , ⊥) and ( 𝒟 , • , 1 ) ; • a not-necessaril y in v ertible natural transf ormation 𝜁 : ( 𝑥 • 𝑦 ) ◦ ( 𝑎 • 𝑏 ) = ⇒ ( 𝑥 ◦ 𝑎 ) • ( 𝑦 ◦ 𝑏 ) , called the middle inter chang e law ; • three mor phisms 𝜈 : ⊥ − → ⊥ • ⊥ , 𝜛 : 1 ◦ 1 − → 1 , 𝜄 : ⊥ − → 1; This data has to satisfy t he follo wing relations: • ( 1 , 𝜛 , 𝜄 ) is a monoid in ( 𝒟 , ◦ , ⊥) ; • (⊥ , 𝜈 , 𝜄 ) is a comonoid in ( 𝒟 , • , 1 ) ; • the follo wing diagrams commute, witnessing associativity : (( 𝑥 • 𝑦 ) ◦ ( 𝑎 • 𝑏 )) ◦ ( 𝑐 • 𝑑 ) ( 𝑥 • 𝑦 ) ◦ (( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 )) (( 𝑥 ◦ 𝑎 ) • ( 𝑦 ◦ 𝑏 )) ◦ ( 𝑐 • 𝑑 ) ( 𝑥 • 𝑦 ) ◦ (( 𝑎 ◦ 𝑐 ) • ( 𝑏 ◦ 𝑑 )) (( 𝑥 ◦ 𝑎 ) ◦ 𝑐 ) • (( 𝑦 ◦ 𝑏 ) ◦ 𝑑 ) ( 𝑥 ◦ ( 𝑎 ◦ 𝑐 )) • ( 𝑦 ◦ ( 𝑏 ◦ 𝑑 )) 𝛼 id ◦ 𝜁 𝜁 𝜁 ◦ id 𝜁 𝛼 • 𝛼 (7.1.1) (( 𝑥 • 𝑎 ) • 𝑐 ) ◦ (( 𝑦 • 𝑏 ) • 𝑑 ) ( 𝑥 • ( 𝑎 • 𝑐 )) ◦ ( 𝑦 • ( 𝑏 • 𝑑 )) (( 𝑥 • 𝑎 ) ◦ ( 𝑦 • 𝑏 )) • ( 𝑐 ◦ 𝑑 ) ( 𝑥 ◦ 𝑦 ) • (( 𝑎 • 𝑐 ) ◦ ( 𝑏 • 𝑑 )) (( 𝑥 ◦ 𝑦 ) • ( 𝑎 ◦ 𝑏 )) • ( 𝑐 ◦ 𝑑 ) ( 𝑥 ◦ 𝑦 ) • (( 𝑎 ◦ 𝑏 ) • ( 𝑐 ◦ 𝑑 )) 𝛼 ◦ 𝛼 𝜁 id • 𝜁 𝜁 𝜁 • id 𝛼 (7.1.2) 158 7.1. Duoidal categories • the follo wing diagrams commute, witnessing unitality : ⊥ ◦ ( 𝑎 • 𝑏 ) (⊥ ◦ ⊥) ◦ ( 𝑎 • 𝑏 ) ( 𝑎 • 𝑏 ) ◦ ⊥ ( 𝑎 • 𝑏 ) ◦ (⊥ • ⊥) 𝑎 • 𝑏 (⊥ ◦ 𝑎 ) • (⊥ ◦ 𝑏 ) 𝑎 • 𝑏 ( 𝑎 ◦ ⊥) • ( 𝑏 ◦ ⊥) ( 1 • 𝑎 ) ◦ ( 1 • 𝑏 ) ( 1 ◦ 1 ) • ( 𝑎 ◦ 𝑏 ) ( 𝑎 • 1 ) ◦ ( 𝑏 • 1 ) ( 𝑎 ◦ 𝑏 ) • ( 1 ◦ 1 ) 𝑎 ◦ 𝑏 1 • ( 𝑎 ◦ 𝑏 ) 𝑎 ◦ 𝑏 ( 𝑎 ◦ 𝑏 ) • 1 𝜆 𝜆 ◦ 𝜆 𝜁 𝜛 • id 𝜆 𝜆 ◦ 𝜆 𝜁 id • 𝜛 𝜈 ◦ id 𝜁 𝜆 𝜆 − 1 • 𝜆 − 1 id ◦ 𝜈 𝜁 𝜆 𝜆 − 1 • 𝜆 − 1 (7.1.3) By abuse of notation, w e shall of ten call 𝒟 a duoidal categor y , lea ving the rest of t he data implicit. Deﬁnition 7.2. A duoidal category 𝒟 is called normal if ⊥  1 . N ote t hat explicitly requiring the exis tence of 𝜄 : ⊥ − → 1 in Deﬁnition 7.1 is not strictl y necessar y , as it ma y be deriv ed from t he other speciﬁed data: 𝜄 : ⊥ 𝜆 −⊥ − − → ⊥ ◦ ⊥ 𝜆 − 1 ◦ 𝜌 − 1 − − − − − − → ( 1 • ⊥) ◦ (⊥ • 1 ) 𝜁 − − → ( 1 ◦ ⊥) • (⊥ ◦ 1 ) 𝜆 ⊥ • 𝜌 ⊥ − − − − → 1 • 1 𝜆 1 − − → 1 . Exam ple 7.3. By [ AM10 , Proposition 6.10], a braided monoidal categor y ( 𝒞 , ⊗ , 1 , 𝜎 ) yields a duoidal categor y ( 𝒞 , ⊗ , 1 , ⊗ , 1 ) with structure morphisms 𝜁 . . = ( 𝑎 ⊗ 𝑏 ) ⊗ ( 𝑐 ⊗ 𝑑 )  𝑎 ⊗ ( 𝑏 ⊗ 𝑐 ) ⊗ 𝑑 𝑎 ⊗ 𝜎 𝑏 , 𝑐 ⊗ 𝑑 − − − − − − → 𝑎 ⊗ ( 𝑐 ⊗ 𝑏 ) ⊗ 𝑑  ( 𝑎 ⊗ 𝑐 ) ⊗ ( 𝑏 ⊗ 𝑑 ) , N ote in particular that w e hav e 𝜌 1 = 𝜆 1 . 𝜛 . . = 1 ⊗ 1 𝜆 − − → 1 , 𝜈 . . = 1 𝜆 − 1 − − − → 1 ⊗ 1 , 𝜄 . . = 1 id 1 − − − → 1 . Exam ple 7.4. The con v erse of Example 7.3 also holds. If 𝒟 is a duoidal category , such t hat the interchange la w and structure morphisms are iso- morphisms, t hen [ AM10 , Proposition 6.11] yields a br aiding on ( 𝒟 , ◦ , ⊥) and ( 𝒟 , • , 1 ) , such that t he y become isomorphic as braided monoidal categories, and the interchange law arises from the braiding. N ote, how ev er , t hat there exist non-trivial duoidal structures on a mono- idal categor y ( 𝒞 , ⊗ , 1 ) . Recall t he deﬁnition of t he categor y 𝐻 𝐻 𝒴𝒟 of (left-lef t) Y etter – Drinf eld modules from Example 2.54 . If t he Hopf algebra 𝐻 does not admit an inv ertible antipode, t hen 𝐻 𝐻 𝒴𝒟 is lax braided — t he Y etter– Drinfeld braiding is a non-inv ertible natural transf or mation that satisﬁes the braid equations. This yields a duoidal structure on ( 𝒞 , ⊗ , 1 , ⊗ , 1 ) that is not braided. 159 7. Duoi d a l R -m a tr ices There are v arious equivalent deﬁnitions of duoidal categories. F or ex- ample, as pseudomonoids in t he monoidal 2-categor y of monoidal categor - ies, oplax monoidal functors, and oplax monoidal natural transf ormations, see [ AM10 , Proposition 6.72] and [ GLF16 , Deﬁnition 1]. In particular , t his means that • is a lax monoidal and that ◦ is an oplax monoidal functor ; 16 from 16 This extends to normal duoidal categories, in which ◦ : 𝒟 × 𝒟 − → 𝒟 and ⊥ : 1 − → 𝒟 are normal oplax monoidal functors, where 1 is the terminal category . this characterisation, one may obtain a coher ence result, see [ Lew72 ], [ AM10 , Section 6.2], and [ MP22 , Theorem 5.9]. Proposition 7.5. Any etc diagr am in a duoidal category commutes. Loosel y speaking, an etc diagram is a formal diagram 𝐹 : 𝒥 − → 𝒟 in the sense of [ MP22 , p. 20], consisting of only structure mor phisms of t he duoidal category , such t hat for all 𝑗 ∈ 𝒥 the object 𝐹 𝑗 is not isomor phic to an y of t he tw o units. W e refer to [ MP22 , Deﬁnition 5.8 and Theorem 5.9] for a precise deﬁnition and a proof of Proposition 7.5 . A counterexample in t he case of a formal diag r am wit h parallel arrow s 1 • 1 ⇒ 1 is given in [ R om23 , Proposition 3.1.6 and Example 3.1.7]. Remark 7.6. The tensor product and unit being nor mal monoidal functors, normal duoidal categories admit an analogue of the w ell-known coherence result for braided monoidal categories [ JS93 ]. That is, any formal diag r am comprised only of the structure morphisms in a nor mal duoidal category commutes, see [ MP22 , Theorem 5.18]. 7.1.1 Double opmonoidal monads Deﬁnition 7.7 ([ AM10 , Deﬁnition 6.25]) . Suppose that 𝒟 is a duoidal categor y . A bimonoid in 𝒟 is a quintuple ( 𝐵 , 𝜇 , 𝜂 , Δ , 𝜀 ) , consisting of a monoid ( 𝐵 , 𝜇 , 𝜂 ) in ( 𝒟 , ◦ , ⊥) , and a comonoid ( 𝐵 , Δ , 𝜀 ) in ( 𝒟 , • , 1 ) , such that t he follo wing compatibility conditions are satisﬁed: 𝐵 ◦ 𝐵 𝐵 𝐵 • 𝐵 ( 𝐵 • 𝐵 ) ◦ ( 𝐵 • 𝐵 ) ( 𝐵 ◦ 𝐵 ) • ( 𝐵 ◦ 𝐵 ) Δ ◦ Δ 𝜁 𝜇 • 𝜇 𝜇 Δ 𝐵 ◦ 𝐵 1 ◦ 1 ⊥ 𝐵 ⊥ 𝐵 1 ⊥ • ⊥ 𝐵 • 𝐵 𝐵 1 𝜀 ◦ 𝜀 𝜛 𝜇 𝜀 𝜂 Δ 𝜈 𝜂 • 𝜂 𝜄 𝜂 𝜀 160 7.1. Duoidal categories Proposition 7.8 ([ BS13 ]) . F or a monoid 𝑏 in a duoidal category ( 𝒟 , ◦ , ⊥ , • , 1 ) t here is a bĳective correspondence between bimonoid s tructur es on 𝑏 , and bimonad structur es on the monad 𝑏 ◦ − on ( 𝒟 , • , 1 ) . Exam ple 7.9. A bimonoid in a braided monoidal category 𝒞 is the same as a bimonoid in the duoidal category 𝒞 from Exam ple 7.3 . In this wa y one reco v ers t he fact t hat an object 𝑏 ∈ 𝒞 is a bimonoid if and onl y if t he induced monad 𝑏 ⊗ − is a bimonad on 𝒞 . Exam ple 7.10. Let k be a commutativ e ring, and 𝐴 a commutativ e k -algebr a. In [ AM10 , Example 6.18] it is shown t hat t he categor y of 𝐴 -bimodules is duoidal, with t he tw o tensor products giv en by 𝑀 • 𝑁 . . = 𝑀 ⊗ 𝐴 𝑁 . . = 𝑀 ⊗ 𝑘 𝑁 ⧸ ⟨ 𝑚 𝑎 ⊗ 𝑛 − 𝑚 ⊗ 𝑎 𝑛 ⟩ , and 𝑀 ◦ 𝑁 . . = 𝑀 ⊗ 𝐴 ⊗ 𝑘 𝐴 𝑁 . . = 𝑀 ⊗ 𝑘 𝑁 ⧸ ⟨ 𝑎 𝑚 𝑏 ⊗ 𝑛 − 𝑚 ⊗ 𝑎 𝑛 𝑏 ⟩ , for all 𝑎 ∈ 𝐴 , 𝑚 ∈ 𝑀 , and 𝑛 ∈ 𝑁 . Further more, from [ AM10 , Example 6.44] w e know t hat a bimonoid in t his duoidal categor y is an 𝐴 -bialgebroid in t he sense of Ra v enel, see [ Ra v86 , Deﬁnition A1.1.1]. In this setting, Proposition 7.8 reco v ers a special case of [ Szl03 , Theorems 5.1 and 5.4]. Deﬁnition 7.11 ([ AHLF18 , Section 7]) . A double opmonoidal monad on a duoidal category 𝒟 consists of a monad ( 𝑇 , 𝜇 , 𝜂 ) on 𝒟 , together wit h a bimonad structures ( 𝑇 , 𝐵 • 2 , 𝐵 • 0 ) on ( 𝒟 , • , 1 ) and ( 𝑇 , 𝐵 ◦ 2 , 𝐵 ◦ 0 ) on ( 𝒟 , ◦ , ⊥) , such t hat the follo wing diag r ams commute: 𝑇 ( 1 ◦ 1 ) 𝑇 1 𝑇 ⊥ 𝑇 (⊥ • ⊥) 𝑇 ⊥ 𝑇 1 𝑇 1 ◦ 𝑇 1 𝑇 ⊥ • 𝑇 ⊥ 1 ◦ 1 1 ⊥ ⊥ • ⊥ ⊥ 1 𝑇 𝜛 𝑇 ◦ 2 , 1 , 1 𝑇 • 0 𝑇 𝜈 𝑇 ◦ 0 𝑇 • 2 , ⊥ , ⊥ 𝑇 𝜄 𝑇 ◦ 0 𝑇 • 0 𝑇 • 0 ◦ 𝑇 • 0 𝑇 ◦ 0 • 𝑇 ◦ 0 𝜛 𝜈 𝜄 (7.1.4) 𝑇 (( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 )) 𝑇 (( 𝑎 ◦ 𝑐 ) • ( 𝑏 ◦ 𝑑 )) 𝑇 ( 𝑎 • 𝑏 ) ◦ 𝑇 ( 𝑐 • 𝑑 ) 𝑇 ( 𝑎 ◦ 𝑐 ) • 𝑇 ( 𝑏 ◦ 𝑑 ) ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ ( 𝑇 𝑐 • 𝑇 𝑑 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) 𝑇 𝜁 𝑎 ,𝑏 ,𝑐 , 𝑑 𝑇 ◦ 2 , 𝑎 • 𝑏 , 𝑐 • 𝑑 𝑇 • 2 , 𝑎 , 𝑏 ◦ 𝑇 • 2 , 𝑐 , 𝑑 𝑇 • 2 , 𝑎 ◦ 𝑐 , 𝑏 ◦ 𝑑 𝑇 ◦ 2 , 𝑎 , 𝑐 • 𝑇 ◦ 2 , 𝑏 , 𝑑 𝜁 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐 ,𝑇 𝑑 (7.1.5) 161 7. Duoi d a l R -m a tr ices Exam ple 7.12. Let ( 𝒞 , ⊗ , 1 ) be a braided monoidal category wit h braiding 𝜎 , seen as a duoidal categor y as in Example 7.3 . A bimonad 𝐵 on ( 𝒞 , ⊗ , 1 ) that additionall y satisﬁes the equation 𝐵 2 ◦ 𝐵 𝜎 = 𝜎 ◦ 𝐵 2 is a double opmonoidal monad on 𝒞 , where t he tw o oplax monoidal structures are t he same, and the commutativity of Diagram ( 7.1.4 ) amounts to the fact t hat t he monoidal structure morphisms of 𝒞 lift to t he category of 𝐵 -algebr as; see Proposition 5.9 . Exam ple 7.13. For a bialgebr a 𝐵 in ( V ect , ⊗ , k ) , the endofunctor 𝐵 ⊗ − is a double opmonoidal monad in ( V ect , ⊗ , k , ⊗ , k ) . For 𝑈 , 𝑉 , 𝑊 , 𝑋 ∈ V ect , as w ell as 𝑏 ∈ 𝐵 , 𝑢 ∈ 𝑈 , 𝑣 ∈ 𝑉 , 𝑤 ∈ 𝑊 , and 𝑥 ∈ 𝑋 , Diag r am ( 7.1.5 ) simpliﬁes to 𝑏 ( 1 ) ⊗ 𝑢 ⊗ 𝑏 ( 3 ) ⊗ 𝑤 ⊗ 𝑏 ( 2 ) ⊗ 𝑣 ⊗ 𝑏 ( 4 ) ⊗ 𝑥 = 𝑏 ( 1 ) ⊗ 𝑢 ⊗ 𝑏 ( 2 ) ⊗ 𝑤 ⊗ 𝑏 ( 3 ) ⊗ 𝑣 ⊗ 𝑏 ( 4 ) ⊗ 𝑥 , which is equiv alent to 𝑏 ( 1 ) ⊗ 𝑏 ( 2 ) = 𝑏 ( 2 ) ⊗ 𝑏 ( 1 ) ; i.e., 𝐵 has to be cocommutativ e. As in the case of R-matrices for bialgebr as and bimonads, requiring t hat the interchang e morphism of a duoidal category 𝒟 lifts to the category of modules is a rather strong condition. Proposition 7.14 ([ AHLF18 , Theorem 7.2]) . Let 𝒟 be a duoidal category and 𝑇 : 𝒞 − → 𝒞 a monad. Then the structur e morphisms and inter chang e law of 𝒟 lif t to 𝒟 𝑇 if and onl y if 𝑇 is a double opmonoidal monad. In particular , if 𝑇 is a double opmonoidal monad, then 𝒟 𝑇 is a duoidal category . 7 . 2 r- m at r i c e s Inst ead of the s itu ation o f Proposition 7.14 , w e are instead interested in studying which additional s tructure one can im pose on 𝑇 such t hat 𝒟 𝑇 becomes duoidal, where t he interchange morphism is instead giving b y “twisting” t hat of 𝒟 . This gener alises so-called R-matrices for bialgebr as and bimonads [ Kas98 , P art ii ] and [ B V07 , Section 8.2]. Proposition 7.15 ([ B V07 , Theorem 8.5]) . Let 𝐵 be a bimonad on t he monoidal category 𝒞 . Braidings on 𝒞 𝑇 ar e in bĳective correspondence with R-matrices on 𝐵 . A crucial feature of R-matrices for bimonads is t hat t he y can be deﬁned on not necessarily braided monoidal categories. Our deﬁnition of duoidal R-matrices incorporates this feature. Deﬁnition 7.16. A categor y 𝒟 is called preduoidal if it is equipped wit h tw o monoidal structures (◦ , ⊥) and (• , 1 ) . 162 7.2. R -matrices Deﬁnition 7.17. Let 𝒟 be a preduoidal category . A monad 𝑇 on 𝒟 that is equipped with tw o bimonad structures o v er ( 𝒟 , ◦ , ⊥) and ( 𝒟 , • , 1 ) is called a separ atel y opmonoidal monad on 𝒟 . Deﬁnition 7.18. Let 𝒟 be a preduoidal category and 𝑇 a separatel y opmon- oidal monad on 𝒟 . An R-matrix on 𝑇 consists of a natural transformation 𝑅 . . = { 𝑅 𝑎 , 𝑏 , 𝑐 , 𝑑 : ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) = ⇒ ( 𝑇 𝑎 • 𝑇 𝑐 ) ◦ ( 𝑇 𝑏 • 𝑇 𝑑 ) } 𝑎 , 𝑏 , 𝑐 , 𝑑 ∈ 𝒟 , as w ell as mor phisms of 𝑇 -alg ebras 𝜈 : (⊥ , 𝑇 ◦ 0 ) − → (⊥ , 𝑇 ◦ 0 )•(⊥ , 𝑇 ◦ 0 ) , 𝜛 : ( 1 , 𝑇 • 0 ) ◦ ( 1 , 𝑇 • 0 ) − → ( 1 , 𝑇 • 0 ) , 𝜄 : (⊥ , 𝑇 ◦ 0 ) − → ( 1 , 𝑇 • 0 ) , such t hat ( 1 , 𝜛 , 𝜄 ) is a monoid in ( 𝒟 𝑇 , ◦ , ⊥) ; t he tuple (⊥ , 𝜈 , 𝜄 ) is a comonoid in ( 𝒟 𝑇 , • , 1 ) ; and t he follo wing diagrams commute for all 𝑎 , 𝑏 , 𝑐 , 𝑑 , 𝑥 , 𝑦 ∈ 𝒟 : ⊥ ◦ ( 𝑎 • 𝑏 ) (⊥ • ⊥) ◦ ( 𝑎 • 𝑏 ) ( 𝑎 • 𝑏 ) ◦ ⊥ ( 𝑎 • 𝑏 ) ◦ (⊥ • ⊥) ( 𝑇 ⊥ ◦ 𝑇 𝑎 ) • ( 𝑇 ⊥ ◦ 𝑇 𝑏 ) ( 𝑇 𝑎 ◦ 𝑇 ⊥) • ( 𝑇 𝑏 ◦ 𝑇 ⊥) 𝑎 • 𝑏 (⊥ ◦ 𝑎 ) • (⊥ ◦ 𝑏 ) 𝑎 • 𝑏 ( 𝑎 ◦ ⊥) • ( 𝑏 ◦ ⊥) 𝜈 ◦ id 𝜆 𝑅 id ◦ 𝜈 𝜆 𝑅 ( 𝑇 ◦ 0 ◦ 𝛼 )•( 𝑇 ◦ 0 ◦ 𝛽 ) ( 𝛼 ◦ 𝑇 ◦ 0 )•( 𝛽 ◦ 𝑇 ◦ 0 ) 𝜆 − 1 • 𝜆 − 1 𝜆 − 1 • 𝜆 − 1 (7.2.1) ( 1 • 𝑎 ) ◦ ( 1 • 𝑏 ) ( 𝑇 1 ◦ 𝑇 1 ) • ( 𝑇 𝑎 ◦ 𝑇 𝑏 ) ( 𝑎 • 1 ) ◦ ( 𝑏 • 1 ) ( 𝑇 𝑎 ◦ 𝑇 𝑏 ) • ( 𝑇 1 ◦ 𝑇 1 ) ( 1 ◦ 1 ) • ( 𝑎 ◦ 𝑏 ) ( 𝑎 ◦ 𝑏 ) • ( 1 ◦ 1 ) 𝑎 ◦ 𝑏 1 • ( 𝑎 ◦ 𝑏 ) 𝑎 ◦ 𝑏 ( 𝑎 ◦ 𝑏 ) • 1 𝑅 𝜆 ◦ 𝜆 ( 𝑇 • 0 ◦ 𝑇 • 0 )•( 𝛼 ◦ 𝛽 ) 𝑅 𝜆 ◦ 𝜆 ( 𝛼 ◦ 𝛽 )•( 𝑇 • 0 ◦ 𝑇 • 0 ) 𝜛 • id id • 𝜛 𝜆 𝜆 (7.2.2) 𝑇 (( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 )) 𝑇 (( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 )) 𝑇 ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • 𝑇 ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) 𝑇 ( 𝑎 • 𝑏 ) ◦ 𝑇 ( 𝑐 • 𝑑 ) ( 𝑇 2 𝑎 ◦ 𝑇 2 𝑐 ) • ( 𝑇 2 𝑏 ◦ 𝑇 2 𝑑 ) ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ ( 𝑇 𝑐 • 𝑇 𝑑 ) ( 𝑇 2 𝑎 ◦ 𝑇 2 𝑐 ) • ( 𝑇 2 𝑏 ◦ 𝑇 2 𝑑 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) 𝑇 𝑅 𝑎 ,𝑏 , 𝑐 ,𝑑 𝑇 • 2 ,𝑇 𝑎 ◦ 𝑇 𝑐 ,𝑇 𝑏 ◦ 𝑇 𝑑 𝑇 ◦ 2 ,𝑇 𝑎 ,𝑇 𝑐 • 𝑇 ◦ 2 ,𝑇 𝑏 ,𝑇 𝑑 ( 𝜇 𝑎 ◦ 𝜇 𝑐 )•( 𝜇 𝑏 ◦ 𝜇 𝑑 ) 𝑇 ◦ 2 , 𝑎 • 𝑏 ,𝑐 • 𝑑 𝑇 • 2 , 𝑎 ,𝑏 ◦ 𝑇 • 2 , 𝑐 , 𝑑 𝑅 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐 ,𝑇 𝑑 ( 𝜇 𝑎 ◦ 𝜇 𝑐 )•( 𝜇 𝑏 ◦ 𝜇 𝑑 ) (7.2.3) 163 7. Duoi d a l R -m a tr ices ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) ◦ ( 𝑥 • 𝑦 ) ( 𝑎 • 𝑏 ) ◦ (( 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑑 ◦ 𝑇 𝑦 )) (( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 )) ◦ ( 𝑥 • 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 ( 𝑇 𝑐 ◦ 𝑇 𝑥 )) • ( 𝑇 𝑏 ◦ 𝑇 ( 𝑇 𝑑 ◦ 𝑇 𝑦 )) ( 𝑇 ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) ◦ 𝑇 𝑥 ) • ( 𝑇 ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) ◦ 𝑇 𝑦 ) ( 𝑇 𝑎 ◦ ( 𝑇 2 𝑐 ◦ 𝑇 2 𝑥 )) • ( 𝑇 𝑏 ◦ ( 𝑇 2 𝑑 ◦ 𝑇 2 𝑦 )) (( 𝑇 2 𝑎 ◦ 𝑇 2 𝑐 ) ◦ 𝑇 𝑥 ) • (( 𝑇 2 𝑏 ◦ 𝑇 2 𝑑 ) ◦ 𝑇 𝑦 ) (( 𝑇 𝑎 ◦ 𝑇 𝑐 ) ◦ 𝑇 𝑥 ) • (( 𝑇 𝑏 ◦ 𝑇 𝑑 ) ◦ 𝑇 𝑦 ) ( 𝑇 𝑎 ◦ ( 𝑇 𝑐 ◦ 𝑇 𝑥 )) • ( 𝑇 𝑏 ◦ ( 𝑇 𝑑 ◦ 𝑇 𝑦 )) id ◦ 𝑅 𝑐 ,𝑑 , 𝑥, 𝑦 𝑅 𝑎 ,𝑏 , 𝑐, 𝑑 ◦ id 𝑅 𝑎 ,𝑏 ,𝑇 𝑐 ◦ 𝑇 𝑥 ,𝑇 𝑑 ◦ 𝑇 𝑦 ( 𝑇 𝑎 ◦ 𝑇 ◦ 2 ,𝑇 𝑐 ,𝑇 𝑥 )•( 𝑇 𝑏 ◦ 𝑇 ◦ 2 ,𝑇 𝑑 ,𝑇 𝑦 ) ( 𝑇 ◦ 2 ,𝑇 𝑎 ,𝑇 𝑐 ◦ 𝑇 𝑥 )•( 𝑇 ◦ 2 ,𝑇 𝑏 ,𝑇 𝑑 ◦ 𝑇 𝑦 ) 𝑅 𝑇 𝑎 ◦ 𝑇 𝑐 ,𝑇 𝑏 ◦ 𝑇 𝑑 ,𝑥 , 𝑦 ( 𝑇 𝑎 ◦ 𝜇 𝑐 ◦ 𝜇 𝑥 )•( 𝑇 𝑏 ◦ 𝜇 𝑑 ◦ 𝜇 𝑦 ) ( 𝜇 𝑎 ◦ 𝜇 𝑐 ◦ 𝑇 𝑥 )•( 𝜇 𝑏 ◦ 𝜇 𝑑 ◦ 𝑇 𝑦 )  (7.2.4) (( 𝑥 • 𝑎 ) • 𝑐 ) ◦ (( 𝑦 • 𝑏 ) • 𝑑 ) ( 𝑥 • ( 𝑎 • 𝑐 )) ◦ ( 𝑦 • ( 𝑏 • 𝑑 )) ( 𝑇 ( 𝑥 • 𝑎 ) ◦ 𝑇 ( 𝑦 • 𝑏 )) • ( 𝑇 𝑐 ◦ 𝑇 𝑑 ) ( 𝑇 𝑥 ◦ 𝑇 𝑦 ) • ( 𝑇 ( 𝑎 • 𝑐 ) ◦ 𝑇 ( 𝑏 • 𝑑 )) (( 𝑇 𝑥 • 𝑇 𝑎 ) ◦ ( 𝑇 𝑦 • 𝑇 𝑏 )) • ( 𝑇 𝑐 ◦ 𝑇 𝑑 ) ( 𝑇 𝑥 ◦ 𝑇 𝑦 ) • (( 𝑇 𝑎 • 𝑇 𝑐 ) ◦ ( 𝑇 𝑏 • 𝑇 𝑑 )) (( 𝑇 2 𝑥 ◦ 𝑇 2 𝑦 ) • ( 𝑇 2 𝑎 ◦ 𝑇 2 𝑏 )) • ( 𝑇 𝑐 ◦ 𝑇 𝑑 ) ( 𝑇 𝑥 ◦ 𝑇 𝑦 ) • (( 𝑇 2 𝑎 ◦ 𝑇 2 𝑏 ) • ( 𝑇 2 𝑐 • 𝑇 2 𝑑 )) (( 𝑇 𝑥 ◦ 𝑇 𝑦 ) • ( 𝑇 𝑎 ◦ 𝑇 𝑏 )) • ( 𝑇 𝑐 ◦ 𝑇 𝑑 ) ( 𝑇 𝑥 ◦ 𝑇 𝑦 ) • (( 𝑇 𝑎 ◦ 𝑇 𝑏 ) • ( 𝑇 𝑐 • 𝑇 𝑑 )) 𝑅 𝑥 • 𝑎 , 𝑐, 𝑦 • 𝑏 ,𝑑 𝑇 • 2 , 𝑥 , 𝑎 ◦ 𝑇 • 2 , 𝑦 ,𝑏 • id 𝑅 𝑇 𝑥 ,𝑇 𝑎 ,𝑇 𝑦 ,𝑇 𝑏 • id ( 𝜇 𝑥 ◦ 𝜇 𝑦 )•( 𝜇 𝑎 ◦ 𝜇 𝑏 )• id  𝑅 𝑥 , 𝑎 • 𝑐, 𝑦 ,𝑏 • 𝑑 id •( 𝑇 • 2 , 𝑎 , 𝑐 ◦ 𝑇 • 2 , 𝑏 ,𝑑 ) id • 𝑅 𝑇 𝑎 ,𝑇 𝑐 ,𝑇 𝑏 ,𝑇 𝑑 id •(( 𝜇 𝑎 ◦ 𝜇 𝑏 )•( 𝜇 𝑐 • 𝜇 𝑑 ))  (7.2.5) A quasitriangular opmonoidal monad is one equipped wit h an R-matrix. Exam ple 7.19. Let ( 𝒞 , ⊗ , 1 ) be a strict monoidal categor y , and 𝑇 a bimonad on 𝒞 . Let 𝑅 be an R-matrix on 𝑇 in t he sense of [ B V07 , Section 8.2], and deﬁne 𝑆 . . = { 𝜂 𝑎 ⊗ 𝑅 𝑏 , 𝑐 ⊗ 𝜂 𝑑 : 𝑎 ⊗ 𝑏 ⊗ 𝑐 ⊗ 𝑑 − → 𝑇 𝑎 ⊗ 𝑇 𝑐 ⊗ 𝑇 𝑏 ⊗ 𝑇 𝑑 } 𝑎 , 𝑏 , 𝑐 , 𝑑 ∈ 𝒞 . Then 𝑆 , together with 𝜈 , 𝜛 , and 𝜄 being t he identity , is an R-matrix on 𝑇 , seen as a separatel y opmonoidal monad on t he preduoidal categor y 𝒞 . Diagrams ( 7.2.1 ) and ( 7.2.2 ) commute because ( 𝛽 ⊗ 𝛼 ) 𝑅 𝑎 , 𝑏 is a braiding b y [ B V07 , Theorem 8.5]. Diagram ( 7.2.3 ) follo ws b y Figure 7.1 , where 𝑇 3 is deﬁned as in Remar k 2.35 . The other diagrams are pro v ed similarl y . By [ B V07 , Example 8.4] w e also obtain that ev ery R-matrix on a bialgebr a 𝐵 yields an R-matrix on 𝐵 ⊗ − in t he sense of Deﬁnition 7.18 . Remark 7.20. No te that t he conv erse of Example 7.19 is not necessarily true. Let 𝒞 be a monoidal categor y seen as a preduoidal categor y , and assume t hat 𝑇 is a separatel y opmonoidal monad on 𝒞 where t he tw o oplax monoidal 164 𝑇 ( 𝑎 ⊗ 𝑏 ⊗ 𝑐 ⊗ 𝑑 ) 𝑇 ( 𝑇 𝑎 ⊗ 𝑇 𝑐 ⊗ 𝑇 𝑏 ⊗ 𝑇 𝑑 ) 𝑇 ( 𝑇 𝑎 ⊗ 𝑇 𝑐 ) ⊗ 𝑇 ( 𝑇 𝑏 ⊗ 𝑇 𝑑 ) 𝑇 2 𝑎 ⊗ 𝑇 ( 𝑇 𝑐 ⊗ 𝑇 𝑏 ) ⊗ 𝑇 2 𝑑 𝑇 ( 𝑎 ⊗ 𝑏 ) ⊗ 𝑇 ( 𝑐 ⊗ 𝑑 ) 𝑇 𝑎 ⊗ 𝑇 ( 𝑏 ⊗ 𝑐 ) ⊗ 𝑇 𝑑 𝑇 𝑎 ⊗ 𝑇 𝑏 ⊗ 𝑇 𝑐 ⊗ 𝑇 𝑑 𝑇 𝑎 ⊗ 𝑇 ( 𝑇 𝑐 ⊗ 𝑇 𝑏 ) ⊗ 𝑇 𝑑 𝑇 2 𝑎 ⊗ 𝑇 2 𝑐 ⊗ 𝑇 2 𝑏 ⊗ 𝑇 2 𝑑 𝑇 𝑎 ⊗ 𝑇 2 𝑐 ⊗ 𝑇 2 𝑏 ⊗ 𝑇 𝑑 𝑇 𝑎 ⊗ 𝑇 𝑏 ⊗ 𝑇 𝑐 ⊗ 𝑇 𝑑 𝑇 𝑎 ⊗ 𝑇 2 𝑐 ⊗ 𝑇 2 𝑏 ⊗ 𝑇 𝑑 𝑇 2 𝑎 ⊗ 𝑇 2 𝑐 ⊗ 𝑇 2 𝑏 ⊗ 𝑇 2 𝑑 𝑇 𝑎 ⊗ 𝑇 𝑐 ⊗ 𝑇 𝑏 ⊗ 𝑇 𝑑 𝑇 𝑎 ⊗ 𝑇 𝑐 ⊗ 𝑇 𝑏 ⊗ 𝑇 𝑑 𝑇 ( 𝜂 𝑎 ⊗ 𝑅 𝑏 , 𝑐 ⊗ 𝜂 𝑑 ) 𝑇 2 , 𝑎 ⊗ 𝑏 , 𝑐 ⊗ 𝑑 𝑇 3 , 𝑎 , 𝑏 ⊗ 𝑐 , 𝑑 𝑇 2 ,𝑇 𝑎 ⊗ 𝑇 𝑐 ,𝑇 𝑏 ⊗ 𝑇 𝑑 𝑇 3 ,𝑇 𝑎 ,𝑇 𝑐 ⊗ 𝑇 𝑏 ,𝑇 𝑑 𝑇 2 ,𝑇 𝑎 ,𝑇 𝑐 ⊗ 𝑇 2 ,𝑇 𝑏 ,𝑇 𝑑 𝜇 𝑎 ⊗ id ⊗ 𝜇 𝑑 id ⊗ 𝑇 2 ,𝑇 𝑐 ,𝑇 𝑏 ⊗ id 𝑇 2 , 𝑎 , 𝑏 ⊗ 𝑇 2 , 𝑐 , 𝑑 𝑇 𝜂 𝑎 ⊗ 𝑇 𝑅 𝑏 , 𝑐 ⊗ 𝑇 𝜂 𝑑 id ⊗ 𝑇 2 , 𝑏 , 𝑐 ⊗ id id ⊗ 𝑇 𝑅 𝑏 , 𝑐 ⊗ id id ⊗ 𝑇 2 ,𝑇 𝑐 ,𝑇 𝑏 ⊗ id 𝜇 𝑎 ⊗ id ⊗ 𝜇 𝑑 𝜇 𝑎 ⊗ 𝜇 𝑐 ⊗ 𝜇 𝑏 ⊗ 𝜇 𝑑 id ⊗ 𝜇 𝑐 ⊗ 𝜇 𝑏 ⊗ id id ⊗ 𝑅 𝑇 𝑏 ,𝑇 𝑐 ⊗ id 𝜂 𝑇 𝑎 ⊗ 𝑅 𝑇 𝑏 ,𝑇 𝑐 ⊗ 𝜂 𝑇 𝑑 id ⊗ 𝜇 𝑐 ⊗ 𝜇 𝑏 ⊗ id 𝜇 𝑎 ⊗ 𝜇 𝑐 ⊗ 𝜇 𝑏 ⊗ 𝜇 𝑑 ≡ nat 𝑇 3 coassoc coassoc monad ≡ [ B V07 , (57)] monad Figure 7.1: V eriﬁcation that 𝑆 satisﬁes Diagram ( 7.2.3 ). 7. Duoi d a l R -m a tr ices structures are t he same. Then an R-matrix on 𝑇 does not necessarily yield an R-matrix in the sense of [ B V07 , Section 8.2], since we do not require 𝑅 to be ∗ -in vertible 17 , which b y [ B V07 , Theorem 8.5] corresponds bĳectiv el y to the 17 A natural transf ormation 𝑅 : ⊗ = ⇒ 𝑇 ⊗ op 𝑇 is called ∗ -invertible if there exists an “in v erse” natural transf ormation 𝑅 − 1 : ⊗ op = ⇒ 𝑇 ⊗ 𝑇 , such t hat ( 𝜇 ⊗ 𝜇 ) ◦ 𝑅 − 1 ◦ 𝑅 is equal to 𝜂 ⊗ 𝜂 , and similar ly for t he other direction. braiding on 𝒞 𝑇 being in v ertible. By Theorem 7.21 below , t he R-matrices of Deﬁnition 7.18 correspond to duoidal structures on 𝒞 𝑇 . Since the tw o tensor products on 𝒞 𝑇 agree, by arguments analogous to t hose in [ AM10 , Section 6.3], t his forces t he inter- change la w to come from a lax br aiding. How ev er , there is no a priori reason for t his morphism to be in v ertible, see Example 7.4 . 7.2.1 F rom R -matrices to duoidal structur es and back Thi s se ctio n cont ain s th e ma in re sul t of the chapter , which can be seen as an analogue of [ B V07 , Theorem 8.5], and a non-cocommutativ e counter part to [ AHLF18 , Theorem 7.2]. Theorem 7.21. Let 𝒟 be a preduoidal category and suppose that 𝑇 is a separ atel y opmonoidal monad on 𝒟 . F or all 𝑇 -alg ebras ( 𝑎 , 𝛼 ) , ( 𝑏 , 𝛽 ) , ( 𝑐 , 𝛾 ) , and ( 𝑑 , 𝛿 ) , a qua- sitriangular structur e on 𝑇 yields an inter chang e law 𝜉 . . = (( 𝛼 ◦ 𝛾 ) • ( 𝛽 ◦ 𝛿 )) 𝑅 𝑎 , 𝑏 , 𝑐 , 𝑑 : ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) − → ( 𝑎 ◦ 𝑐 ) • ( 𝑏 ◦ 𝑑 ) on 𝒞 𝑇 . Con versel y , an inter chang e law 𝜉 on 𝒟 𝑇 gives rise to an R-matrix 𝑅 . . = 𝜉 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐 ,𝑇 𝑑 (( 𝜂 𝑎 • 𝜂 𝑏 ) ◦ ( 𝜂 𝑐 • 𝜂 𝑑 )) : ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) − → ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) on 𝑇 . These constructions are mutually inv erse to each ot her . W e split up t he proof of Theorem 7.21 into its respectiv e directions. Giv en these results, t he rest of t he proof is straightf orwar d. Proof. Combining Propositions 7.22 and 7.23 belo w , it is lef t to pro v e t hat the constructions are mutually inv erse. In one direction w e calculate (( 𝛼 ◦ 𝛾 ) • ( 𝛽 ◦ 𝛿 )) 𝜉 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐 ,𝑇 𝑑 (( 𝜂 𝑎 • 𝜂 𝑏 ) ◦ ( 𝜂 𝑐 • 𝜂 𝑑 )) = (( 𝛼 ◦ 𝛾 ) • ( 𝛽 ◦ 𝛿 ))(( 𝜂 𝑎 ◦ 𝜂 𝑐 ) • ( 𝜂 𝑏 ◦ 𝜂 𝑑 )) 𝜉 𝑎 , 𝑏 , 𝑐 , 𝑑 naturality of 𝜉 = (( 𝛼 𝜂 𝑎 ◦ 𝛾 𝜂 𝑐 ) • ( 𝛽𝜂 𝑏 ◦ 𝛿 𝜂 𝑑 )) 𝜉 𝑎 , 𝑏 , 𝑐 , 𝑑 functoriality of • and ◦ = 𝜉 𝑎 , 𝑏 , 𝑐 , 𝑑 monadicity of 𝑇 ; and for t he con verse w e hav e (( 𝜇 𝑎 ◦ 𝜇 𝑐 ) • ( 𝜇 𝑏 ◦ 𝜇 𝑑 )) 𝑅 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐 ,𝑇 𝑑 (( 𝜂 𝑎 • 𝜂 𝑏 ) ◦ ( 𝜂 𝑐 • 𝜂 𝑑 )) = (( 𝜇 𝑎 ◦ 𝜇 𝑐 ) • ( 𝜇 𝑏 ◦ 𝜇 𝑑 ))(( 𝜂 𝑇 𝑎 ◦ 𝜂 𝑇 𝑐 ) • ( 𝜂 𝑇 𝑏 ◦ 𝜂 𝑇 𝑑 )) 𝑅 𝑎 , 𝑏 , 𝑐 , 𝑑 = 𝑅 𝑎 , 𝑏 , 𝑐 , 𝑑 . □ 166 7.2. R -matrices Proposition 7.22. Let 𝒟 be a preduoidal category and 𝑇 a quasitriangular sep- ar atel y opmonoidal monad on 𝒟 with R-matrix ( 𝑅 , 𝜈 , 𝜛 , 𝜄 ) . Then 𝒞 𝑇 is a duoidal category , with structur e morphisms 𝜈 , 𝜛 , and 𝜄 , and inter chang e law 𝜉 . . = (( 𝛼 ◦ 𝛾 ) • ( 𝛽 ◦ 𝛿 )) 𝑅 𝑎 , 𝑏 , 𝑐 , 𝑑 : ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) − → ( 𝑎 ◦ 𝑐 ) • ( 𝑏 ◦ 𝑑 ) for all ( 𝑎 , 𝛼 ) , ( 𝑏 , 𝛽 ) , ( 𝑐 , 𝛾 ) , and ( 𝑑 , 𝛿 ) ∈ 𝒟 𝑇 . Proof. The claim t hat 𝜉 ∈ 𝒟 𝑇 (( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) , ( 𝑎 ◦ 𝑐 ) • ( 𝑏 ◦ 𝑑 )) — t hat is, 𝜉 is a morphism of 𝑇 -algebr as — f ollow s from Diagram ( 7.2.3 ), as seen in Figure 7.2 . Diagram ( 7.1.1 ) follo ws by the commutativity of Figure 7.3 , where w e ha v e left out the respectiv e associat ors f or readability ; see Proposition 7.5 . The proof of Diagram ( 7.1.2 ) is analogous. Lastl y , Diag r ams ( 7.2.1 ) and ( 7.2.2 ) immediatel y impl y Diag r am ( 7.1.3 ). □ 𝑇 (( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 )) 𝑇 (( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 )) 𝑇 (( 𝑎 ◦ 𝑐 ) • ( 𝑏 ◦ 𝑑 )) 𝑇 ( 𝑎 • 𝑏 ) ◦ 𝑇 ( 𝑐 • 𝑑 ) 𝑇 ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • 𝑇 ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) 𝑇 ( 𝑎 ◦ 𝑐 ) • 𝑇 ( 𝑏 ◦ 𝑑 ) ( 𝑇 2 𝑎 ◦ 𝑇 2 𝑐 ) • ( 𝑇 2 𝑏 ◦ 𝑇 2 𝑑 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ ( 𝑇 𝑐 • 𝑇 𝑑 ) ( 𝑇 2 𝑎 ◦ 𝑇 2 𝑐 ) • ( 𝑇 2 𝑏 ◦ 𝑇 2 𝑑 ) ( 𝑎 ◦ 𝑐 ) • ( 𝑏 ◦ 𝑑 ) ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) 𝑇 𝑅 𝑎 ,𝑏 , 𝑐 ,𝑑 𝑇 𝜉 𝑎 ,𝑏 , 𝑐 ,𝑑 𝑇 ◦ 2 , 𝑎 • 𝑏 ,𝑐 • 𝑑 𝑇 (( 𝛼 ◦ 𝛾 )•( 𝛽 ◦ 𝛿 )) 𝑇 • 2 ,𝑇 𝑎 ◦ 𝑇 𝑐 ,𝑇 𝑏 ◦ 𝑇 𝑑 𝑇 • 2 , 𝑎 ◦ 𝑐 ,𝑏 ◦ 𝑑 𝑇 • 2 , 𝑎 ,𝑏 ◦ 𝑇 • 2 , 𝑐 , 𝑑 𝑇 ◦ 2 ,𝑇 𝑎 ,𝑇 𝑐 • 𝑇 ◦ 2 ,𝑇 𝑏 ,𝑇 𝑑 𝑇 ◦ 2 , 𝑎 , 𝑐 • 𝑇 ◦ 2 , 𝑏 ,𝑑 ( 𝑇 𝛼 ◦ 𝑇 𝛾 )•( 𝑇 𝛽 ◦ 𝑇 𝛿 ) ( 𝜇 𝑎 ◦ 𝜇 𝑐 )•( 𝜇 𝑏 ◦ 𝜇 𝑑 ) ( 𝛼 ◦ 𝛾 )•( 𝛽 ◦ 𝛿 ) ( 𝛼 ◦ 𝛾 )•( 𝛽 ◦ 𝛿 ) 𝑅 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐 ,𝑇 𝑑 ( 𝛼 • 𝛽 )◦( 𝛾 • 𝛿 ) ( 𝜇 𝑎 • 𝜇 𝑏 )◦( 𝜇 𝑐 • 𝜇 𝑑 ) ( 𝑇 𝛼 • 𝑇 𝛽 )◦( 𝑇 𝛾 • 𝑇 𝛿 ) 𝜉 𝑎 ,𝑏 , 𝑐 ,𝑑 𝑅 𝑎 ,𝑏 , 𝑐 ,𝑑 ( 𝛼 ◦ 𝛾 )•( 𝛽 ◦ 𝛿 ) ( 7.2.3 ) nat ( 𝑇 ◦ 2 • 𝑇 ◦ 2 ) 𝑇 • 2 action nat 𝑅 action Figure 7.2: Proof that 𝜉 is a morphism of 𝑇 -algebras. 167 ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) ◦ ( 𝑥 • 𝑦 ) ( 𝑎 • 𝑏 ) ◦ (( 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑑 ◦ 𝑇 𝑦 )) ( 𝑎 • 𝑏 ) ◦ (( 𝑐 ◦ 𝑥 ) • ( 𝑑 ◦ 𝑦 )) ( 𝑇 𝑎 ◦ 𝑇 ( 𝑐 ◦ 𝑥 )) • ( 𝑇 𝑏 ◦ 𝑇 ( 𝑑 ◦ 𝑦 )) ( 𝑇 𝑎 ◦ 𝑇 ( 𝑇 𝑐 ◦ 𝑇 𝑥 )) • ( 𝑇 𝑏 ◦ 𝑇 ( 𝑇 𝑑 ◦ 𝑇 𝑦 )) ( 𝑇 𝑎 ◦ 𝑇 2 𝑐 ◦ 𝑇 2 𝑥 ) • ( 𝑇 𝑏 ◦ 𝑇 2 𝑑 ◦ 𝑇 2 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ◦ 𝑇 𝑦 ) (( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 )) ◦ ( 𝑥 • 𝑦 ) ( 𝑇 ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) ◦ 𝑇 𝑥 ) • ( 𝑇 ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) ◦ 𝑇 𝑦 ) (( 𝑎 ◦ 𝑐 ) • ( 𝑏 ◦ 𝑑 )) ◦ ( 𝑥 • 𝑦 ) ( 𝑇 2 𝑎 ◦ 𝑇 2 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 2 𝑏 ◦ 𝑇 2 𝑑 ◦ 𝑇 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ◦ 𝑇 𝑦 ) ( 𝑇 ( 𝑎 ◦ 𝑐 ) ◦ 𝑇 𝑥 ) • ( 𝑇 ( 𝑏 ◦ 𝑑 ) ◦ 𝑇 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ◦ 𝑇 𝑦 ) ( 𝑎 ◦ 𝑐 ◦ 𝑥 ) • ( 𝑏 ◦ 𝑑 ◦ 𝑦 ) id ◦ 𝑅 𝑐 , 𝑑 ,𝑥 , 𝑦 𝑅 𝑎 ,𝑏 ,𝑐 , 𝑑 ◦ id id ◦(( 𝛾 ◦ 𝜒 )•( 𝛿 ◦ 𝜔 )) 𝑅 𝑎 ,𝑏 ,𝑇 𝑐 ◦ 𝑇 𝑥 ,𝑇 𝑑 ◦ 𝑇 𝑦 𝑅 𝑎 ,𝑏 ,𝑐 ◦ 𝑥 , 𝑑 ◦ 𝑦 ( 𝑇 𝑎 ◦ 𝑇 ◦ 2 , 𝑐 , 𝑥 )•( 𝑇 𝑏 ◦ 𝑇 ◦ 2 , 𝑑 , 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 ◦ 2 ,𝑇 𝑐 ,𝑇 𝑥 )•( 𝑇 𝑏 ◦ 𝑇 ◦ 2 ,𝑇 𝑑,𝑇 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 𝛼 ◦ 𝑇 𝜉 )•( 𝑇 𝑏 ◦ 𝑇 𝜈 ◦ 𝑇 𝜔 ) ( 𝑇 𝑎 ◦ 𝜇 𝑐 ◦ 𝜇 𝑥 )•( 𝑇 𝑏 ◦ 𝜇 𝑑 ◦ 𝜇 𝑦 ) ( 𝛼 ◦ 𝛾 ◦ 𝜒 )•( 𝛽 ◦ 𝛿 ◦ 𝜔 ) 𝑅 𝑇 𝑎 ◦ 𝑇 𝑐 ,𝑇 𝑏 ◦ 𝑇 𝑑 , 𝑥 , 𝑦 (( 𝛼 ◦ 𝛾 )•( 𝛽 ◦ 𝛿 ))◦ id ( 𝑇 ◦ 2 ,𝑇 𝑎 ,𝑇 𝑐 ◦ 𝑇 𝑥 )•( 𝑇 ◦ 2 ,𝑇 𝑏 ,𝑇 𝑑 ◦ 𝑇 𝑦 ) 𝑅 𝑎 ◦ 𝑐 , 𝑏 ◦ 𝑑, 𝑥 , 𝑦 ( 𝜇 𝑎 ◦ 𝜇 𝑐 ◦ 𝑇 𝑥 )•( 𝜇 𝑏 ◦ 𝜇 𝑑 ◦ 𝑇 𝑦 ) ( 𝑇 𝛼 ◦ 𝑇 𝛾 ◦ 𝑇 𝑥 )•( 𝑇 𝛽 ◦ 𝑇 𝛿 ◦ 𝑇 𝑦 ) ( 𝛼 ◦ 𝛾 ◦ 𝜒 )•( 𝛽 ◦ 𝛿 ◦ 𝜔 ) ( 𝑇 ◦ 2 , 𝑎 , 𝑐 ◦ 𝑇 𝑥 )•( 𝑇 ◦ 2 , 𝑏 , 𝑑 ◦ 𝑇 𝑦 ) ( 𝛼 ◦ 𝛾 ◦ 𝜒 )•( 𝛽 ◦ 𝛿 ◦ 𝜔 ) nat ( 7.2.4 ) action nat action Figure 7.3: Proof that 𝜉 satisﬁes Diag r am ( 7.1.1 ). 7.2. R -matrices Proposition 7.23. Let 𝒟 be a preduoidal category , 𝑇 a separ atel y opmonoidal monad on 𝒟 , and suppose t hat 𝒟 𝑇 is a duoidal category with inter chang e law 𝜉 𝑎 , 𝑏 , 𝑐 , 𝑑 : ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) − → ( 𝑎 ◦ 𝑐 ) • ( 𝑏 ◦ 𝑑 ) . Then t he structur e mor phisms of 𝒟 𝑇 , tog ether with 𝑅 . . = 𝜉 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐 ,𝑇 𝑑 (( 𝜂 𝑎 • 𝜂 𝑏 ) ◦ ( 𝜂 𝑐 • 𝜂 𝑑 )) : ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) − → ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) yield an R-matrix for 𝑇 . Proof. Firs t, let us v erify t hat 𝑅 satisﬁes Diag r am ( 7.2.3 ). Let 𝑎 , 𝑏 , 𝑐 , 𝑑 ∈ 𝒟 𝑇 ; then t he claim follo ws from t he commutativity of Figure 7.4 . The fact t hat Diagram ( 7.2.4 ) holds is due to Figure 7.5 , and Diag r am ( 7.2.5 ) is similar . 𝑇 (( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 )) 𝑇 (( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ ( 𝑇 𝑐 • 𝑇 𝑑 )) 𝑇 (( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 )) 𝑇 ( 𝑎 • 𝑏 ) ◦ 𝑇 ( 𝑐 • 𝑑 ) 𝑇 ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ 𝑇 ( 𝑇 𝑐 • 𝑇 𝑑 ) 𝑇 ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • 𝑇 ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ ( 𝑇 𝑐 • 𝑇 𝑑 ) ( 𝑇 2 𝑎 • 𝑇 2 𝑏 ) ◦ ( 𝑇 2 𝑐 • 𝑇 2 𝑑 ) ( 𝑇 2 𝑎 ◦ 𝑇 2 𝑐 ) • ( 𝑇 2 𝑏 ◦ 𝑇 2 𝑑 ) ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ ( 𝑇 𝑐 • 𝑇 𝑑 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) ( 𝑇 2 𝑎 • 𝑇 2 𝑏 ) ◦ ( 𝑇 2 𝑐 • 𝑇 2 𝑑 ) ( 𝑇 2 𝑎 ◦ 𝑇 2 𝑐 ) • ( 𝑇 2 𝑏 ◦ 𝑇 2 𝑑 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) 𝑇 (( 𝜂 𝑎 • 𝜂 𝑏 )◦( 𝜂 𝑐 • 𝜂 𝑑 )) 𝑇 ◦ 2; 𝑎 • 𝑏 ,𝑐 • 𝑑 𝑇 𝜉 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐,𝑇 𝑑 𝑇 ◦ 2; 𝑇 𝑎 • 𝑇 𝑏 ,𝑇 𝑐 • 𝑇 𝑑 𝑇 • 2; 𝑇 𝑎 ◦ 𝑇 𝑐 ,𝑇 𝑏 ◦ 𝑇 𝑑 𝑇 • 2; 𝑎 , 𝑏 ◦ 𝑇 • 2; 𝑐 , 𝑑 𝑇 • 2; 𝑇 𝑎 ,𝑇 𝑏 ◦ 𝑇 • 2; 𝑇 𝑐 ,𝑇 𝑑 𝑇 ◦ 2; 𝑇 𝑎 ,𝑇 𝑐 • 𝑇 ◦ 2; 𝑇 𝑏,𝑇 𝑑 ( 𝑇 𝜂 𝑎 • 𝑇 𝜂 𝑏 )◦( 𝑇 𝜂 𝑐 • 𝑇 𝜂 𝑑 ) ( 𝜂 𝑇 𝑎 • 𝜂 𝑇 𝑏 )◦( 𝜂 𝑇 𝑐 • 𝜂 𝑇 𝑑 ) ( 𝜇 𝑎 • 𝜇 𝑏 )◦( 𝜇 𝑐 • 𝜇 𝑑 ) ( 𝜇 𝑎 ◦ 𝜇 𝑐 )•( 𝜇 𝑏 ◦ 𝜇 𝑑 ) 𝜉 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐,𝑇 𝑑 ( 𝜂 𝑇 𝑎 ◦ 𝜂 𝑇 𝑐 )•( 𝜂 𝑇 𝑏 ◦ 𝜂 𝑇 𝑑 ) 𝜉 𝑇 2 𝑎 ,𝑇 2 𝑏 ,𝑇 2 𝑐 ,𝑇 2 𝑑 ( 𝜇 𝑎 ◦ 𝜇 𝑐 )•( 𝜇 𝑏 ◦ 𝜇 𝑑 ) nat ( 𝑇 • 2 ◦ 𝑇 • 2 ) 𝑇 ◦ 2 nat 𝜉 𝑇 monad 𝜉 morphism of ( free ) 𝑇 - algebras 𝑇 monad Figure 7.4: The map 𝑅 satisﬁes Diagram ( 7.2.3 ). It is lef t to show t he commutativity of Diagrams ( 7.2.1 ) and ( 7.2.2 ). For example, the ﬁrst diagram in the former follo ws by the commutativity of ⊥ ◦ ( 𝑎 • 𝑏 ) (⊥ • ⊥) ◦ ( 𝑎 • 𝑏 ) ( 𝑇 ⊥ • 𝑇 ⊥) ◦ ( 𝑇 𝑎 • 𝑇 𝑏 ) 𝑎 • 𝑏 (⊥ ◦ 𝑎 ) • (⊥ ◦ 𝑏 ) (⊥ ◦ 𝑎 ) • (⊥ ◦ 𝑏 ) ( 𝑇 ⊥ ◦ 𝑇 𝑎 ) • ( 𝑇 ⊥ ◦ 𝑇 𝑏 ) 𝜈 ◦ id 𝜆 ( 𝜂 ⊥ • 𝜂 ⊥ )◦( 𝜂 𝑎 • 𝜂 𝑏 ) 𝜉 ⊥ , ⊥ , 𝑎 ,𝑏 𝜉 𝑇 ⊥ ,𝑇 ⊥ ,𝑇 𝑎 ,𝑇 𝑏 𝜆 − 1 • 𝜆 − 1 ( 𝜂 ⊥ ◦ 𝜂 𝑎 )•( 𝜂 ⊥ ◦ 𝜂 𝑏 ) ( 𝑇 ◦ 0 ◦ 𝛼 )•( 𝑇 ◦ 0 ◦ 𝛽 ) ( 7.1.3 ) nat 𝜉 𝑇 ◦ - bimonad and the other diag r ams are similar . □ 169 ( 𝑎 • 𝑏 ) ◦ ( 𝑐 • 𝑑 ) ◦ ( 𝑥 • 𝑦 ) ( 𝑎 • 𝑏 ) ◦ ( 𝑇 𝑐 • 𝑇 𝑑 ) ◦ ( 𝑇 𝑥 • 𝑇 𝑦 ) ( 𝑎 • 𝑏 ) ◦ (( 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑑 ◦ 𝑇 𝑦 )) ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ ( 𝑇 ( 𝑇 𝑐 ◦ 𝑇 𝑥 ) • 𝑇 ( 𝑇 𝑑 ◦ 𝑇 𝑦 )) ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ ( 𝑇 𝑐 • 𝑇 𝑑 ) ◦ ( 𝑥 • 𝑦 ) (( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 )) ◦ ( 𝑥 • 𝑦 ) ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ ( 𝑇 𝑐 • 𝑇 𝑑 ) ◦ ( 𝑇 𝑥 • 𝑇 𝑦 ) ( 𝑇 𝑎 • 𝑇 𝑏 ) ◦ (( 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑑 ◦ 𝑇 𝑦 )) (( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 )) ◦ ( 𝑇 𝑥 • 𝑇 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 ( 𝑇 𝑐 ◦ 𝑇 𝑥 )) • ( 𝑇 𝑏 ◦ 𝑇 ( 𝑇 𝑑 ◦ 𝑇 𝑦 )) ( 𝑇 ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) • 𝑇 ( 𝑇 𝑏 ◦ 𝑇 𝑑 )) ◦ ( 𝑇 𝑥 • 𝑇 𝑦 ) ( 𝑇 ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) ◦ 𝑇 𝑥 ) • ( 𝑇 ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) ◦ 𝑇 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ◦ 𝑇 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ◦ 𝑇 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 2 𝑐 ◦ 𝑇 2 𝑥 ) • ( 𝑇 𝑏 ◦ 𝑇 2 𝑑 ◦ 𝑇 2 𝑦 ) ( 𝑇 2 𝑎 ◦ 𝑇 2 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 2 𝑏 ◦ 𝑇 2 𝑑 ◦ 𝑇 𝑦 ) ( 𝑇 𝑎 ◦ 𝑇 𝑐 ◦ 𝑇 𝑥 ) • ( 𝑇 𝑏 ◦ 𝑇 𝑑 ◦ 𝑇 𝑦 ) id ◦( 𝜂 𝑐 • 𝜂 𝑑 )◦( 𝜂 𝑥 • 𝜂 𝑦 ) ( 𝜂 𝑎 • 𝜂 𝑏 )◦( 𝜂 𝑐 • 𝜂 𝑑 )◦ id id ◦ 𝜉 𝑇 𝑐 ,𝑇 𝑑,𝑇 𝑥 ,𝑇 𝑦 ( 𝜂 𝑎 • 𝜂 𝑏 )◦ id ( 𝜂 𝑎 • 𝜂 𝑏 )◦( 𝜂 𝑇 𝑐 ◦ 𝑇 𝑥 • 𝜂 𝑇 𝑑 ◦ 𝑇 𝑦 ) ( 𝜂 𝑎 • 𝜂 𝑏 )◦ id 𝜉 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 ( 𝑇 𝑐 ◦ 𝑇 𝑥 ) ,𝑇 ( 𝑇 𝑑 ◦ 𝑇 𝑦 ) 𝜉 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐,𝑇 𝑑 ◦ id id ◦( 𝜂 𝑥 • 𝜂 𝑦 ) id ◦( 𝜂 𝑥 • 𝜂 𝑦 ) ( 𝜂 𝑇 𝑎 ◦ 𝑇 𝑐 • 𝜂 𝑇 𝑏 ◦ 𝑇 𝑑 )◦( 𝜂 𝑥 • 𝜂 𝑦 ) id ◦ 𝜉 𝑇 𝑐 ,𝑇 𝑑,𝑇 𝑥 ,𝑇 𝑦 𝜉 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐,𝑇 𝑑 ◦ id id ◦( 𝜂 𝑇 𝑐 ◦ 𝑇 𝑥 • 𝜂 𝑇 𝑑 ◦ 𝑇 𝑦 ) ( 7.1.1 ) 𝜉 𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐 ◦ 𝑇 𝑥 ,𝑇 𝑑 ◦ 𝑇 𝑦 ( 𝜂 𝑇 𝑎 ◦ 𝑇 𝑐 • 𝜂 𝑇 𝑏 ◦ 𝑇 𝑑 )◦ id 𝜉 𝑇 𝑎 ◦ 𝑇 𝑐 ,𝑇 𝑏 ◦ 𝑇 𝑑 ,𝑇 𝑥 ,𝑇 𝑦 ( 𝑇 𝑎 ◦ 𝑇 ◦ 2 ,𝑇 𝑐 ,𝑇 𝑥 )•( 𝑇 𝑏 ◦ 𝑇 ◦ 2 ,𝑇 𝑑 ,𝑇 𝑦 ) 𝜉 𝑇 ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) ,𝑇 ( 𝑇 𝑏 ◦ 𝑇 𝑑 ) ,𝑇 𝑥 ,𝑇 𝑦 ( 𝑇 ◦ 2 ,𝑇 𝑎 ,𝑇 𝑐 ◦ 𝑇 𝑥 )•( 𝑇 ◦ 2 ,𝑇 𝑏 ,𝑇 𝑑 ◦ 𝑇 𝑦 ) ( 𝜂 𝑇 𝑎 ◦ 𝑇 𝑐 ◦ id )•( 𝜂 𝑇 𝑏 ◦ 𝑇 𝑑 ◦ id ) ( id ◦ 𝜂 𝑇 𝑐 ◦ 𝑇 𝑥 )•( id ◦ 𝜂 𝑇 𝑑 ◦ 𝑇 𝑦 ) ( id ◦ 𝜂 𝑇 𝑐 ◦ 𝜂 𝑇 𝑥 )•( id ◦ 𝜂 𝑇 𝑑 ◦ 𝜂 𝑇 𝑦 ) ( 𝜂 𝑇 𝑎 ◦ 𝜂 𝑇 𝑐 ◦ id )•( 𝜂 𝑇 𝑏 ◦ 𝜂 𝑇 𝑑 ◦ id ) ( 𝑇 𝑎 ◦ 𝜇 𝑐 ◦ 𝜇 𝑥 )•( 𝑇 𝑏 ◦ 𝜇 𝑑 ◦ 𝜇 𝑦 ) ( 𝜇 𝑎 ◦ 𝜇 𝑐 ◦ 𝑇 𝑥 )•( 𝜇 𝑏 ◦ 𝜇 𝑑 ◦ 𝑇 𝑦 ) ◦ functor ◦ functor ◦ functor ◦ functor ◦ functor nat 𝜉 nat 𝜉 𝑇 ◦ - bimonad 𝑇 ◦ - bimonad 𝑇 monad 𝑇 monad Figure 7.5: The R-matrix satisﬁes Diagram ( 7.2.4 ). 7.3. Linear l y distributiv e monads 7 . 3 l i n e a r l y d i s t r i b u t i v e m o na d s The norma l du oid al ca te gori es of Deﬁnition 7.2 ha v e connections to linear logic: in [ GLF16 , p. 7] it is sho wn t hat ev ery nor mal duoidal categor y 𝒟 has the structure of a linear ly dis tributive cat egory ; see [ CS97 ]. In that case, the linear distribut ors are giv en by 𝜕 ℓ ℓ : 𝑎 ◦ ( 𝑏 • 𝑐 )  ( 𝑎 • 1 ) ◦ ( 𝑏 • 𝑐 ) 𝜁 − − → ( 𝑎 ◦ 𝑏 ) • ( 1 • 𝑐 )  ( 𝑎 ◦ 𝑏 ) • 𝑐 , 𝜕 ℓ 𝑟 : 𝑎 ◦ ( 𝑏 • 𝑐 )  ( 1 • 𝑎 ) ◦ ( 𝑏 • 𝑐 ) 𝜁 − − → ( 1 ◦ 𝑏 ) • ( 𝑎 ◦ 𝑐 )  𝑏 • ( 𝑎 ◦ 𝑐 ) , 𝜕 𝑟 ℓ : ( 𝑏 • 𝑐 ) ◦ 𝑎  ( 𝑏 • 𝑐 ) ◦ ( 𝑎 • 1 ) 𝜁 − − → ( 𝑏 ◦ 𝑎 ) • ( 𝑐 ◦ 1 )  ( 𝑏 ◦ 𝑎 ) • 𝑐 . 𝜕 𝑟 𝑟 : ( 𝑏 • 𝑐 ) ◦ 𝑎  ( 𝑏 • 𝑐 ) ◦ ( 1 • 𝑎 ) 𝜁 − − → ( 𝑏 ◦ 1 ) • ( 𝑐 ◦ 𝑎 )  𝑏 • ( 𝑐 ◦ 𝑎 ) . (7.3.1) Remark 7.24. By [ MP22 , Theorem 5.18], nor mal duoidal categories satisfy a much strong er form of coherence, so structures on them require few er axioms to be fully speciﬁed. If 𝑇 is a double opmonoidal monad on a normal duoidal category ( 𝒟 , ◦ , ⊥ , • , 1 ) , t hen t he follo wing diagram commutes: 1 𝑇 1 1 ⊥ 𝑇 ⊥ ⊥ 𝜂 1  𝑇 • 0 𝑇 (  )  𝜂 ⊥ 𝑇 ◦ 0 bimonad nat 𝜂 ( 7.1.4 ) bimonad In particular , 𝑇 • 0 and 𝑇 ◦ 0 are conjugations of each other b y isomor phisms: ( 𝑇 1 𝑇 • 0 − − − → 1 ) = ( 𝑇 1 𝑇 (  ) − − − − → 𝑇 ⊥ 𝑇 ◦ 0 − − − → ⊥  − 1 − − − → 1 ) . In this setting, Diag r am ( 7.1.4 ) automaticall y holds. For simplicity , assume 𝒟 to be strict, and write 𝑇 0 . . = 𝑇 • 0 = 𝑇 ◦ 0 . Then w e for example ha v e 𝑇 ( 1 ◦ 1 ) 𝑇 1 𝑇 1 ◦ 𝑇 1 𝑇 1 1 ◦ 1 1 𝑇 𝜛 𝑇 ◦ 2 , 1 , 1 𝜛 𝑇 0 𝑇 0 ◦ 𝑇 0 𝑇 (  )  𝑇 0 ◦ id 𝑇 0 coherence coherence (7.3.2) 171 7. Duoi d a l R -m a tr ices Remark 7.25. Sometimes, one considers onl y so-called non-planar linearl y distributiv e categories, see [ CS97 , Section 2.1]. These are categories in which onl y 𝜕 ℓ ℓ and 𝜕 𝑟 𝑟 of Equation ( 7.3.1 ) exist. What w e call a linear l y distributiv e category is referred to as a planar linearl y distributiv e categor y in ibid . Conditions for a comonad to lif t the (non-planar) linear distributiv e struc- ture of its base category to its category of coalgebras w ere deﬁned in [ P as12 , Proposition 2.1]. For t he con v enience of t he reader , t he next proposition expresses this relation in ter ms of monads. Proposition 7.26. Let ( ℒ , ⊗ , ⊙ ) be a non-planar linear l y distributiv e category , and suppose t hat the monad 𝑇 on ℒ is separat ely opmonoidal. If the diagrams 𝑇 ( 𝑎 ⊗ ( 𝑏 ⊙ 𝑐 )) 𝑇 𝑎 ⊗ 𝑇 ( 𝑏 ⊙ 𝑐 ) 𝑇 𝑎 ⊗ ( 𝑇 𝑏 ⊙ 𝑇 𝑐 ) 𝑇 (( 𝑎 ⊗ 𝑏 ) ⊙ 𝑐 ) 𝑇 ( 𝑎 ⊗ 𝑏 ) ⊙ 𝑇 𝑐 ( 𝑇 𝑎 ⊗ 𝑇 𝑏 ) ⊙ 𝑇 𝑐 𝑇 𝜕 𝑙 𝑇 ⊗ 2 , 𝑎 , 𝑏 ⊙ 𝑐 𝑇 𝑎 ⊗ 𝑇 ⊙ 2 , 𝑏 , 𝑐 𝑇 ⊙ 2 , 𝑎 ⊗ 𝑏 , 𝑐 𝑇 ⊗ 2 , 𝑎 , 𝑏 ⊙ 𝑇 𝑐 𝜕 𝑙 (7.3.3) 𝑇 (( 𝑏 ⊙ 𝑐 ) ⊗ 𝑎 ) 𝑇 ( 𝑏 ⊙ 𝑐 ) ⊗ 𝑇 𝑎 ( 𝑇 𝑏 ⊙ 𝑇 𝑐 ) ⊗ 𝑇 𝑎 𝑇 ( 𝑏 ⊙ ( 𝑐 ⊗ 𝑎 )) 𝑇 𝑏 ⊙ 𝑇 ( 𝑐 ⊗ 𝑎 ) 𝑇 𝑏 ⊙ ( 𝑇 𝑐 ⊗ 𝑇 𝑎 ) 𝑇 𝜕 𝑟 𝑇 ⊗ 2 , 𝑏 ⊙ 𝑐 , 𝑎 𝑇 ⊙ 2 , 𝑏 , 𝑐 ⊗ 𝑇 𝑎 𝑇 ⊙ 2 , 𝑏 , 𝑐 ⊗ 𝑎 𝑇 𝑏 ⊙ 𝑇 ⊗ 2 , 𝑐 , 𝑎 𝜕 𝑟 (7.3.4) commute for all 𝑇 -alg ebr as 𝑎 , 𝑏 , and 𝑐 , then ℒ 𝑇 is non-planar linear ly distributiv e. Exam ple 7.27. Setting ⊗ = ⊙ , e v er y monoidal category 𝒞 is a linear l y dis- tributiv e categor y . The linear distributors are giv en b y t he associator and its in v erse. Hence, a bimonad 𝐵 on 𝒞 satisﬁes all assump tions of Proposition 7.26 . Diagrams ( 7.3.3 ) and ( 7.3.4 ) reduce to t he coassociativity of 𝐵 2 . Lifting t he interchange morphism of a nor mal duoidal category is more in v olv ed than lifting onl y t he non-planar linear distributors, much like lifting the preduoidal structure is easier t han lifting t he entire duoidal structure. Exam ple 7.28. Let 𝒞 be a br aided monoidal category , which is normal duoidal b y Example 7.3 . As such, the linear distributor 𝜕 ℓ ℓ is the isomor phism 𝜕 ℓ ℓ : 𝑥 ⊗ ( 𝑦 ⊗ 𝑧 )  𝑥 ⊗ ( 1 ⊗ 𝑦 ) ⊗ 𝑧 𝑥 ⊗ 𝜎 1 , 𝑦 ⊗ 𝑧 − − − − − − − → 𝑥 ⊗ ( 𝑦 ⊗ 1 ) ⊗ 𝑧  𝑥 ⊗ ( 𝑦 ⊗ 𝑧 ) , 172 7.3. Linear l y distributiv e monads and 𝜕 𝑟 𝑟 is similar . By Proposition 7.26 , this s tructure lifts to 𝒞 ( 𝐵 ⊗ −) , which is equal to t he categor y of 𝐵 -modules on 𝒞 . Analogously to Example 7.27 , Diagrams ( 7.3.3 ) and ( 7.3.4 ) reduce to t he coassociativity of Δ . How ev er , it is not tr ue t hat t he modules o v er an arbitrary bialgebr a are braided monoidal; see for example [ EGNO15 , Example 8.3.5]. In other w ords, t he planar structure 𝜕 ℓ 𝑟 : 𝑥 ⊗ ( 𝑦 ⊗ 𝑧 )  1 ⊗ ( 𝑥 ⊗ 𝑦 ) ⊗ 𝑧 1 ⊗ 𝜎 𝑥 , 𝑦 ⊗ 𝑧 − − − − − − − → 1 ⊗ ( 𝑦 ⊗ 𝑥 ) ⊗ 𝑧  𝑦 ⊗ ( 𝑥 ⊗ 𝑧 ) , 𝜕 𝑟 ℓ : ( 𝑥 ⊗ 𝑦 ) ⊗ 𝑧  𝑥 ⊗ ( 𝑦 ⊗ 𝑧 ) ⊗ 1 𝑥 ⊗ 𝜎 𝑦 , 𝑧 ⊗ 1 − − − − − − − → 𝑥 ⊗ ( 𝑧 ⊗ 𝑦 ) ⊗ 1  ( 𝑥 ⊗ 𝑧 ) ⊗ 𝑦 , does not lift to t he category of 𝐵 -modules. As stated in t he introduction, planar duoidal categories also capture and gener alise t he no tion of a br aiding, much like duoidal categories do. The follo wing is a straightf or w ard reformulation of Proposition 7.26 . Proposition 7.29. Let ( ℒ , ⊗ , ⊙ ) be a linear ly distributive category wit h a separat ely opmonoidal monad 𝑇 on it. If, in addition t o Diagrams ( 7.3.3 ) and ( 7.3.4 ), the following diagrams commute for all 𝑇 -alg ebras 𝑎 , 𝑏 , and 𝑐 : 𝑇 ( 𝑎 ⊗ ( 𝑏 ⊙ 𝑐 )) 𝑇 𝑎 ⊗ 𝑇 ( 𝑏 ⊙ 𝑐 ) 𝑇 𝑎 ⊗ ( 𝑇 𝑏 ⊙ 𝑇 𝑐 ) 𝑇 ( 𝑏 ⊙ ( 𝑎 ⊗ 𝑐 )) 𝑇 𝑏 ⊙ 𝑇 ( 𝑎 ⊗ 𝑐 ) 𝑇 𝑏 ⊙ ( 𝑇 𝑎 ⊗ 𝑇 𝑐 ) 𝑇 ⊗ 2; 𝑎 ,𝑏 ⊙ 𝑐 𝑇 𝜕 ℓ 𝑟 𝑇 𝑎 ⊗ 𝑇 ⊙ 2; 𝑏 , 𝑐 𝜕 ℓ 𝑟 𝑇 ⊙ 2; 𝑏 , 𝑎 ⊗ 𝑐 𝑇 𝑏 ⊙ 𝑇 ⊗ 2; 𝑎 ,𝑐 (7.3.5) 𝑇 (( 𝑎 ⊙ 𝑏 ) ⊗ 𝑐 ) 𝑇 ( 𝑎 ⊙ 𝑏 ) ⊗ 𝑇 𝑐 ( 𝑇 𝑎 ⊙ 𝑇 𝑏 ) ⊗ 𝑇 𝑐 𝑇 ( 𝑎 ⊙ ( 𝑐 ⊗ 𝑏 )) 𝑇 𝑎 ⊙ 𝑇 ( 𝑐 ⊗ 𝑏 ) 𝑇 𝑎 ⊙ ( 𝑇 𝑐 ⊗ 𝑇 𝑏 ) 𝑇 ⊗ 2; 𝑎 ⊙ 𝑏 ,𝑐 𝑇 𝜕 𝑟 ℓ 𝑇 ⊙ 2; 𝑎 ,𝑏 ⊗ 𝑇 𝑐 𝜕 ℓ 𝑟 𝑇 ⊙ 2; 𝑎 ,𝑐 ⊗ 𝑏 𝑇 𝑎 ⊙ 𝑇 ⊗ 2; 𝑐 , 𝑏 (7.3.6) t hen ℒ 𝑇 is linear ly distributiv e. Exam ple 7.30. Let 𝐵 ∈ V ect be a bialgebr a. F ocusing on t he planar linear distribut or 𝜕 ℓ 𝑟 , for all 𝑏 ∈ 𝐵 , 𝑥 ∈ 𝑎 , 𝑦 ∈ 𝑏 , and 𝑧 ∈ 𝑐 , Diagram ( 7.3.5 ) becomes 𝑏 ( 1 ) ⊗ 𝑦 ⊗ 𝑏 ( 2 ) ⊗ 𝑥 ⊗ 𝑏 ( 3 ) ⊗ 𝑧 = 𝑏 ( 2 ) ⊗ 𝑦 ⊗ 𝑏 ( 1 ) ⊗ 𝑥 ⊗ 𝑏 ( 3 ) ⊗ 𝑧 , which is easil y seen to be equivalent to 𝑏 ( 2 ) ⊗ 𝑏 ( 1 ) = 𝑏 ( 1 ) ⊗ 𝑏 ( 2 ) . 173 7. Duoi d a l R -m a tr ices Thus, linearl y dis tributiv e monads seem to be connected to t he double opmonoidal monads of Section 7.1.1 . Proposition 7.31. Let ( 𝒟 , • , ◦ , 1 ) be a normal duoidal category . Then double op- monoidal monads on 𝒟 ar e linear distributiv e bimonads on 𝒟 . Proof. Let 𝑇 be a cocommutativ e bimonad on 𝒟 as a duoidal categor y . Then the lef t-left linear distributor 𝜕 ℓ ℓ is giv en by 𝑎 ◦ ( 𝑏 • 𝑐 )  ( 𝑎 • 1 ) ◦ ( 𝑏 • 𝑐 ) 𝜁 − − → ( 𝑎 ◦ 𝑏 ) • ( 1 ◦ 𝑐 )  ( 𝑎 ◦ 𝑏 ) • 𝑐 . N o w , Diag r am ( 7.3.3 ) is satisﬁed b y t he commutativity of Figure 7.6 ; Dia- gram ( 7.3.4 ) is similar . Diag r am ( 7.3.5 ) is satisﬁed b y 𝑇 ( 𝑎 ◦ ( 𝑏 • 𝑐 )) 𝑇 𝑎 ◦ 𝑇 ( 𝑏 • 𝑐 ) 𝑇 𝑎 ◦ ( 𝑇 𝑏 • 𝑇 𝑐 ) 𝑇 (( 1 • 𝑎 ) ◦ ( 𝑏 • 𝑐 )) 𝑇 ( 1 • 𝑎 ) ◦ 𝑇 ( 𝑏 • 𝑐 ) ( 𝑇 1 • 𝑇 𝑎 ) ◦ ( 𝑇 𝑏 • 𝑇 𝑐 ) ( 1 • 𝑇 𝑎 ) ◦ ( 𝑇 𝑏 • 𝑇 𝑐 ) 𝑇 (( 1 ◦ 𝑏 ) • ( 𝑎 ◦ 𝑐 )) 𝑇 ( 1 ◦ 𝑏 ) • 𝑇 ( 𝑎 ◦ 𝑐 ) ( 𝑇 1 ◦ 𝑇 𝑏 ) • ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) ( 1 ◦ 𝑇 𝑏 ) • ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) 𝑇 ( 𝑏 • ( 𝑎 ◦ 𝑐 )) 𝑇 𝑏 • 𝑇 ( 𝑎 ◦ 𝑐 ) 𝑇 𝑏 • ( 𝑇 𝑎 ◦ 𝑇 𝑐 ) 𝑇 ◦ 2; 𝑎 ,𝑏 • 𝑐 id ◦ 𝑇 • 2; 𝑏 ,𝑐 𝑇 ◦ 2;1 • 𝑎 ,𝑏 • 𝑐 𝑇 𝜉 1 , 𝑎 ,𝑏 ,𝑐 𝑇 • 2;1 , 𝑎 ◦ 𝑇 • 2; 𝑏 ,𝑐 ( 7.1.5 ) ( 𝑇 0 • id )◦ id 𝜉 𝑇 1 ,𝑇 𝑎 ,𝑇𝑏 ,𝑇 𝑐 𝜉 1 ,𝑇 𝑎 ,𝑇 𝑏 ,𝑇 𝑐 𝑇 • 2;1 ◦ 𝑏 ,𝑎 ◦ 𝑐 𝑇 ◦ 2;1 , 𝑏 • 𝑇 ◦ 2; 𝑎 , 𝑐 𝑇 bimonad ( 𝑇 0 ◦ id )• id 𝑇 • 2; 𝑏 ,𝑎 ◦ 𝑐 id • 𝑇 ◦ 2; 𝑎 , 𝑐 𝑇 bimonad nat 𝜉 where w e ha v e assumed t he normal duoidal structure to be strict for ease of readability . Diag r am ( 7.3.6 ) follo ws similar l y . □ 174 𝑇 ( 𝑎 ◦ ( 𝑏 • 𝑐 )) 𝑇 𝑎 ◦ 𝑇 ( 𝑏 • 𝑐 ) 𝑇 𝑎 ◦ ( 𝑇 𝑏 • 𝑇 𝑐 ) 𝑇 (( 𝑎 • 1 ) ◦ ( 𝑏 • 𝑐 )) 𝑇 ( 𝑎 • 1 ) ◦ 𝑇 ( 𝑏 • 𝑐 ) ( 𝑇 𝑎 • 𝑇 1 ) ◦ ( 𝑇 𝑏 • 𝑇 𝑐 ) ( 𝑇 𝑎 • 1 ) ◦ ( 𝑇 𝑏 • 𝑇 𝑐 ) 𝑇 (( 𝑎 ◦ 𝑏 ) • ( 1 ◦ 𝑐 )) 𝑇 ( 𝑎 ◦ 𝑏 ) • 𝑇 ( 1 ◦ 𝑐 ) ( 𝑇 𝑎 ◦ 𝑇 𝑏 ) • ( 𝑇 1 ◦ 𝑇 𝑐 ) ( 𝑇 𝑎 ◦ 𝑇 𝑏 ) • ( 1 ◦ 𝑇 𝑐 ) 𝑇 (( 𝑎 ◦ 𝑏 ) • (⊥ ◦ 𝑐 )) 𝑇 ( 𝑎 ◦ 𝑏 ) • 𝑇 (⊥ ◦ 𝑐 ) ( 𝑇 𝑎 ◦ 𝑇 𝑏 ) • ( 𝑇 ⊥ ◦ 𝑇 𝑐 ) ( 𝑇 𝑎 ◦ 𝑇 𝑏 ) • (⊥ ◦ 𝑇 𝑐 ) 𝑇 (( 𝑎 ◦ 𝑏 ) • 𝑐 ) 𝑇 ( 𝑎 ◦ 𝑏 ) • 𝑇 𝑐 ( 𝑇 𝑎 ◦ 𝑇 𝑏 ) • 𝑇 𝑐 𝑇 (  ) 𝑇 𝜁 𝑎 , 1 ,𝑏 , 𝑐 𝑇 • 2 , 𝑎 ◦ 𝑏 , 𝑐 𝑇 ◦ 2 , 𝑎 , 𝑏 • 𝑇 𝑐 𝑇 ◦ 2 , 𝑎 , 𝑏 • 𝑐 𝑇 𝑎 ◦ 𝑇 • 2 , 𝑏 , 𝑐  𝜁 𝑇 𝑎 , 1 ,𝑇 𝑏 ,𝑇 𝑐 𝜁 𝑇 𝑎 ,𝑇 1 ,𝑇 𝑏 ,𝑇 𝑐 𝑇 ◦ 2 , 𝑎 • 1 , 𝑏 • 𝑐 𝑇 • 2 , 𝑎 , 1 ◦ 𝑇 • 2 , 𝑏 , 𝑐 𝑇 • 2 , 𝑎 ◦ 𝑏 , 1 ◦ 𝑐 𝑇 ◦ 2 , 𝑎 , 𝑏 • 𝑇 ◦ 2 , 1 , 𝑐 ( 7.1.5 ) ( 𝑇 𝑎 • 𝑇 • 0 )◦( 𝑇 𝑏 • 𝑇 𝑐 ) ( 𝑇 𝑎 ◦ 𝑇 𝑏 )•( 𝑇 • 0 ◦ 𝑇 𝑐 ) 𝑇 (  )◦ 𝑇 ( 𝑏 • 𝑐 ) 𝑇 (  ) 𝑇 ( 𝑎 ◦ 𝑏 )• 𝑇 (  ) 𝑇 ( 𝑎 ◦ 𝑏 )• 𝑇 (  ) 𝑇 (  )   𝑇 ◦ 2 , 𝑎 , 𝑏 • 𝑇 ◦ 2 , ⊥ , 𝑐 ( 𝑇 𝑎 ◦ 𝑇 𝑏 )•( 𝑇 ◦ 0 ◦ 𝑇 𝑐 )  − 1 ( 7.3.2 ) nat 𝑇 ◦ 2 nat 𝑇 • 2 ( 𝑇 ,𝑇 • 2 ,𝑇 • 0 ) bimonad nat 𝜁 ( 𝑇 ,𝑇 ◦ 2 ,𝑇 ◦ 0 ) bimonad Figure 7.6: The left-left linear distributor satisﬁes Diagram ( 7.3.3 ). Der Mensch, das tapfers te und leidgew ohnteste Thier , v er neint an sich nicht das Leiden: er will es, er sucht es selbst auf, vor ausgesetzt, dass man ihm einen Sinn dafür aufzeigt, ein Dazu des Leidens. Fri edric h Nie tzsche ; Zur Genealogie der Moral. I N F I N I T E A N D N O N - R I G I D R E C O N S T R U C T I O N 8 The go a l o f thi s cha pter is to gener alise the follo wing t heorem: Theorem A ([ Ost03 , Theorem 1]) . Let 𝒞 be a ﬁnite tensor category and let ℳ be a ﬁnite abelian 𝒞 -module category , such that the evaluation functor − ⊲ 𝑚 : 𝒞 − → ℳ is exact, for all 𝑚 ∈ ℳ . Then ther e exis ts an alg ebra object 𝐴 ∈ 𝒞 such t hat t her e is an equiv alence of 𝒞 -module categories mod 𝒞 ( 𝐴 ) ≃ ℳ . This theorem makes sev eral assump tions on 𝒞 and ℳ : • Finiteness assump tions: 𝒞 and ℳ are required to be ﬁnite abelian. • The monoidal structure − ⊗ = of 𝒞 and t he action functor − ⊲ = of ℳ are both assumed to be exact in both variables. • Rigidity assump tions on 𝒞 ; i.e., that all of its objects admit left and right duals with respect to its monoidal structure. W e will gener alise Theorem A in a wa y that g reatl y relaxes t he ﬁrst tw o kinds of assumptions, and remo v es t he t hir d one altogether . As mentioned before, b y [ DSPS19 , Example 2.20], t his reconstruction cannot solely rely on algebr a objects in 𝒞 , and w e hav e to approach Theorem A from a more monadic point of view . Our main result will be t he follo wing, and presents a certain classiﬁcation of right exact lax 𝒞 -module monads. Theorems 8.48 to 8.50 . Assume t hat 𝒞 and ℳ hav e enough projectiv es (injectives) and t hat ther e is an object ℓ ∈ ℳ such that: • t here is a right adjoint ⌊ ℓ , −⌋ (left adjoint ⌈ ℓ , −⌉ ) to − ⊲ ℓ ; • for 𝑥 ∈ 𝒞 projective (injective), the object 𝑥 ⊲ ℓ is projectiv e (injective); • any projectiv e (injective) object of ℳ is a direct summand of an object of the form 𝑥 ⊲ ℓ , for 𝑥 projectiv e (injective). 177 8. Infi nite and n on -rig id re c on s tructi on Let 𝑇 be the monad ⌊ ℓ , − ⊲ ℓ ⌋ . Then ℳ ≃ 𝒞 𝑇 , and t he 𝒞 -module structur e of t he cat egory of 𝑇 -modules is extended from the Kleisli category . F urthermor e, t his ext ends to a bĳection { ( ℳ , ℓ ) as above } ⧸ ( ℳ ≃ 𝒩 ) ↔  Right exact lax 𝒞 -module monads on 𝒞  / ( 𝒞 𝑇 ≃ 𝒞 𝑆 ) ( ℳ , ℓ ) ↦− → ⌊ ℓ , − ⊲ ℓ ⌋ ( 𝒞 𝑇 , 𝑇 1 ) ← − [ 𝑇 In t he case of tensor categories, this statement ma y on the surface seem inapplicable: a tensor categor y t hat is not ﬁnite will g enerall y not ha v e enough projectiv es or injectiv es. Ho w ev er , since t he categor y under lying a tensor category is equiv alent to t he category of ﬁnite-dimensional comodules for a coalgebra, the ind-completion of a tensor category has enough injectiv es. Further , since multitensor categories can be realised as categories of compact objects in locall y ﬁnitel y presentable categories, w e ma y use an appropriate v ariant of t he special adjoint functor theorem, see Propositions 2.133 and 2.134 , characterising when the internal cohom exists f or an object in Ind ( ℳ ) . As such, we are able to “pull back” the appropriate v ersions of Theorems 8.56 and 8.57 to t he under lying categories. Theorem 8.59 . Let 𝒞 be a multitensor category , and let ℳ be an abelian 𝒞 -module category such that the Ind ( 𝒞 ) -module category Ind ( ℳ ) admits a coclosed Ind ( 𝒞 ) - injective Ind ( 𝒞 ) -cogener ator . Then ther e is a coalgebr a object 𝐶 in Ind ( 𝒞 ) such t hat Ind ( ℳ ) ≃ Comod Ind ( 𝒞 ) 𝐶 . F urther , ℳ is t he category of compact 𝐶 -comodule objects.. Consider t he special case where a tensor categor y 𝒞 admits a ﬁbre functor to t he category vect of ﬁnite-dimensional vect or spaces, and is thus mon- oidall y equivalent to t he categor y 𝐻 vect of ﬁnite-dimensional comodules o v er a ﬁnite-dimensional Hopf algebr a 𝐻 , see [ Ulb90 ]. Theorem A realises a ﬁnite 𝒞 -module categor y ℳ as t he categor y of ﬁnite-dimensional modules o v er a ﬁnite-dimensional 𝐻 -comodule algebr a. Since 𝐻 and 𝐴 are ﬁnite- dimensional, t he categor y of 𝐴 -modules is equivalent as an 𝐻 vect -module category to the categor y of comodules for the 𝐻 -comodule coalgebr a 𝐴 ∗ . In this sense, t he follo wing Hopf-t heoretic corollar y of the abo v e theorem about ind-completions is an immediate gener alisation of Theorem A . 178 8.1. Extending module structures Corollar y 8.60 . Let 𝐻 be a Hopf algebr a and let 𝒞 = 𝐻 vect , meaning that ther e is a monoidal equiv alence Ind ( 𝒞 ) ≃ 𝐻 V ect . Let ℳ be an abelian 𝒞 -module category such t hat Ind ( ℳ ) admits a coclosed Ind ( 𝒞 ) -injective Ind ( 𝒞 ) -cogener ator . Then ther e exists a (possibly inﬁnite-dimensional) 𝐻 -comodule coalgebr a 𝐶 , such t hat ther e is an equivalence Ind ( ℳ ) ≃ Comod 𝐻 𝐶 of Ind ( 𝒞 ) -module categories, r estricting to a 𝒞 -module equivalence ℳ ≃ comod 𝐻 𝐶 . Since w e do not assume rigidity for our most g eneral t heorems, w e in- stead in v oke the bicategorical Y oneda lemma to realise the objects of 𝒞 as the 𝒞 -module endofunctors of 𝒞 itself. Then, we view t he lax 𝒞 -module endofunctors of 𝒞 as one appropriate gener alisation of objects in 𝒞 , and the lax 𝒞 -module monads as t he corresponding g eneralisation of algebr a objects in 𝒞 . Using the reconstruction results of Section 5.3 , the right adjoint of a (strong) 𝒞 -module functor is canonically lax, and t hus t he (co)monads w e study are canonically (op)lax 𝒞 -module functors. 8 . 1 e x t e n d i n g m o d u l e s t r u c t u r e s An a ddit ional d iffic ul ty we encoun ter is t hat while the Kleisli categor y of a lax 𝒞 -module monad is naturall y a 𝒞 -module categor y , t he same is not true for t he Eilenberg– Moore categor y . In t his section, w e shall in v estigate under which conditions the 𝒞 -module structure of the former lif ts essen- tiall y uniquel y to t he latter . More precisely , w e ﬁrst establish uniqueness in Theorem 8.9 , and then complement t hat wit h an existence result of such an extension in Theorem 8.25 , under t he assump tion of right exactness (left ex- actness for comonads) of the (co)monad, using so-called Linton coequalisers and multicategorical techniques similar to t hose in [ AHLF18 ]. Hypothesis 8.1. From now on until t he end of t his thesis, w e implicitl y assume all categories and functors to be k -linear , for a ﬁeld k . The fact t hat w e only consider right exact lax 𝒞 -module monads is exem- pliﬁed b y the follo wing fundamental fact. Proposition 8.2 ([ BZBJ18 , Proposition 3.2]) . Let 𝒜 be abelian, 𝑇 a right exact monad on 𝒜 , and 𝑆 a left exact comonad on 𝒜 . Then 𝒜 𝑇 and 𝒜 𝑆 ar e abelian. Proposition 8.3. Let 𝒜 be a locall y ﬁnite abelian category , and let 𝑆 be a left exact, ﬁnitary comonad on Ind ( 𝒜 ) . Then Ind ( 𝒜 ) 𝑆 also is of the form Ind ( ℰ ) , for a locally ﬁnite abelian category ℰ . F urther , ℰ can be chosen to be t he category of compact 179 8. Infi nite and n on -rig id re c on s tructi on objects in Ind ( 𝒜 ) 𝑆 , and it can be char acterised as the objects sent to compact objects under t he for getful functor 𝐹 𝑆 : Ind ( 𝒜 ) 𝑆 − → Ind ( 𝒜 ) . In particular , if 𝑆 is a left exact comonad on 𝒜 , we have Ind ( 𝒜 ) Ind ( 𝑆 ) ≃ Ind ( 𝒜 𝑆 ) and 𝒜 𝑆 is locall y ﬁnite abelian. Proof. Let 𝐷 be the k -coalgebr a such t hat 𝒜 ≃ 𝐷 vect . Then w e also hav e t hat Ind ( 𝒜 ) ≃ 𝐷 V ect . In particular , 𝑆 is a comonad on 𝐷 V ect . By Proposition 2.109 , there is a 𝐷 - 𝐷 -bicomodule 𝐶 such t hat 𝑆  𝐶 □ 𝐷 − . Similar l y to [ T ak77 , Remar k 2.4], under this isomor phism t he comonad structure on 𝑆 corresponds to maps 𝐶 − → 𝐶 □ 𝐷 𝐶 and 𝐶 − → 𝐷 , which endo w 𝐶 with t he structure of a k -coalgebr a, together wit h a coalgebr a morphism 𝐶 − → 𝐷 . Formall y , a 𝑆 -comodule in 𝐷 V ect is a 𝐷 -comodule tog et her wit h a 𝐶 -comodule structure t hat restricts to t he giv en 𝐷 -comodule along t he coalgebr a mor phism 𝐶 − → 𝐷 . Mor phisms of 𝑆 -comodules in 𝐷 V ect are precisel y 𝐶 -comodule morphisms, and so we ha ve Ind ( 𝒜 ) 𝑆 ≃ ( 𝐷 V ect ) 𝑆 ≃ 𝐶 V ect . (8.1.1) Thus w e ma y set ℰ = 𝐶 vect . The characterisation of compact objects in ( 𝐷 V ect ) 𝑆 in ter ms of t he images of the functor 𝐹 𝑆 follo ws immediately from observing that under the equiv alence of Equation ( 8.1.1 ), t he functor 𝐹 𝑆 is simpl y t he restriction functor 𝐶 V ect − → 𝐷 V ect . The second part of the statement follow s by obser ving t hat 𝒜 𝑆 is the cat- egory of compact objects in Ind ( 𝒜 ) Ind ( 𝑆 ) , by t he ﬁrst part of the statement, and by obser ving t hat 𝒜 𝑆 is also t he categor y of compact objects in Ind ( 𝒜 𝑆 ) . Since both Ind ( 𝒜 𝑆 ) and Ind ( 𝒜 ) Ind ( 𝑆 ) are locally ﬁnitely presentable, the equi- v alence betw een t heir respectiv e categories of compact objects establishes an equiv alence betw een t he categories t hemsel v es. □ Remark 8.4. Suppose that 𝑇 is a monad on t he category 𝒞 . For 𝑇 -modules ( 𝑥 , 𝛼 ) and ( 𝑦 , 𝛽 ) , notice t hat t here is a bĳection 𝒞 𝑇 ( 𝑇 𝑥 , 𝑦 )  𝒞 ( 𝑥 , 𝑦 ) ( 𝑓 : 𝑇 𝑥 − → 𝑦 ) ↦− → ( 𝑓 ◦ 𝜂 𝑥 ) ( 𝛽 ◦ 𝑇 𝑔 ) ← − [ ( 𝑔 : 𝑥 − → 𝑦 ) . (8.1.2) This induces an isomor phism 𝒞 𝑇 ( 𝑇 (−) , = )  𝒞 (− , = ) . Using t he deﬁnition of 𝜄 from Equation ( 2.2.2 ), t he lef t-hand side is also obtained from the functor 𝒞 𝑇 よ − − → [( 𝒞 𝑇 ) op , V ect ] 𝜄 ∗ − − → [ 𝒞 𝑇 op , V ect ] . 180 8.1. Extending module structures The analogue of Proposition 8.3 for ﬁnite abelian categories is simpler . Proposition 8.5. Let 𝑇 be a right exact monad on a ﬁnite abelian category 𝒜 . Then 𝒜 𝑇 is a ﬁnite abelian category . F urther , any projective object in 𝒜 𝑇 is a direct summand of one of the form 𝜄 ( 𝑃 ) , wher e 𝜄 : 𝒜 𝑇 − → 𝒜 𝑇 is the canonical embedding of Equation ( 2.2.2 ) and 𝑃 ∈ 𝒜 -proj . Denoting the category of objects of t his form by Kl 𝑝 ( 𝑇 ) , we obtain an equivalence 𝒜 𝑇 -proj ≃ Kl 𝑝 ( 𝑇 ) between the projectiv es of 𝒜 𝑇 and t he Cauchy completion of Kl 𝑝 ( 𝑇 ) . Proof. Let 𝐴 be a ﬁnite-dimensional k -algebr a such t hat 𝒜 ≃ 𝐴 -mod . Then there is an 𝐴 - 𝐴 -bimodule 𝐵 such t hat 𝑇  𝐵 ⊗ 𝐴 − . Similarl y to the proof of Proposition 8.3 , monad structures on 𝑇 correspond to pairs consisting of a ﬁnite-dimensional k -algebr a structure on 𝐵 and an algebr a homomor phism 𝐴 − → 𝐵 . Under this correspondence we ha v e 𝒜 𝑇 ≃ 𝐵 -mod . The latter category is clearl y a ﬁnite abelian categor y . N o w suppose that 𝑃 ∈ 𝒜 -proj . Appl ying Remar k 8.4 , one obtains an isomorphism 𝒜 𝑇 ( 𝜄 ( 𝑃 ) , −)  𝒜 ( 𝑃 , −) , proving t he exactness of t he lef t-hand side, and t hus projectivity of 𝜄 ( 𝑃 ) . F or ( 𝑋 , 𝛼 ) ∈ 𝒜 𝑇 , using t he fact that 𝒜 has enough projectiv es, w e ma y ﬁx an epimor phism 𝑞 : 𝑄 − ↠ 𝑋 in 𝒜 , where 𝑄 ∈ 𝒜 -proj . For the latter claim, notice that t here is a composite epimorphism 𝜄 ( 𝑃 ) 𝜄 ( 𝑞 ) − − → → 𝜄 ( 𝑋 ) = 𝑇 𝑋 𝛼 − − → → 𝑋 , the ﬁrst part of which is epic b y right exactness of 𝜄 . □ Lemma 8.6. Let 𝑇 be a monad on 𝒜 . The embedding 𝒜 𝑇 ↩ − → [ 𝒜 op 𝑇 , V ect ] can be cor estrict ed to an embedding 𝒜 𝑇 ↩ − → Fin co ( 𝒜 𝑇 ) into t he ﬁnite cocompletion of 𝒜 𝑇 . Proof. Let ( 𝑥 , 𝛼 ) and ( 𝑦 , 𝛽 ) be tw o 𝑇 -alg ebras. Recall t hat t he coequaliser 𝑇 2 𝑥 𝑇 𝑥 𝑥 𝜇 𝑥 𝑇 𝛼 𝛼 in 𝒜 𝑇 is an absolute coequaliser 18 in 𝒜 ; thus, its image under t he Y oneda 18 A coequaliser is called absolute if it is preserved by an y functor ; see [ P ar69 ] for details. embedding よ yields a coequaliser 𝒜 (− , 𝑇 2 𝑥 ) 𝒜 (− , 𝑇 𝑥 ) 𝒜 (− , 𝑥 ) . ( 𝜇 𝑥 ) ∗ ( 𝑇 𝛼 ) ∗ 𝛼 ∗ 181 8. Infi nite and n on -rig id re c on s tructi on P assing under t he isomor phism of Equation ( 8.1.2 ), one obtains a coequaliser 𝒜 𝑇 ( 𝑇 (−) , 𝑇 2 𝑥 ) 𝒜 𝑇 ( 𝑇 (−) , 𝑇 𝑥 ) 𝒜 𝑇 ( 𝑇 (−) , 𝑥 ) , ( 𝜇 𝑥 ) ∗ ( 𝑇 𝛼 ) ∗ 𝛼 ∗ which, in turn, is isomor phic to 𝒜 𝑇 (− , 𝑇 𝑥 ) 𝒜 𝑇 (− , 𝑥 ) 𝒜 𝑇 ( 𝑇 (−) , 𝑥 ) . ( 𝜇 𝑥 ) ∗ ( 𝑇 𝛼 ) ∗ 𝛼 ∗ This pro v es t he result. □ Proposition 8.7. Let 𝑇 be a right exact monad on an abelian category 𝒜 . The inclu- sion functor 𝒜 𝑇 ↩ − → Fin co ( 𝒜 𝑇 ) has a lef t adjoint, and t he counit of the adjunction is a natural isomorphism. F urther , the left adjoint F in co ( 𝒜 𝑇 ) − → 𝒜 𝑇 is the right exact extension of the inclusion 𝜄 : 𝒜 𝑇 ↩ − → 𝒜 𝑇 . Proof. By Proposition 8.2 , t he 𝒜 𝑇 is ﬁnitely cocomplete. Similar l y to [ KS06 , Proposition 6.3.1], there is a functor Fin co ( 𝒜 𝑇 ) − → 𝒜 𝑇 that extends t he func- tor Id : 𝒜 𝑇 − → 𝒜 𝑇 , deﬁned b y sending some 𝑥 =  colim 𝑖 𝑥 𝑖 in Fin co ( 𝒜 𝑇 ) to colim 𝑖 𝑥 𝑖 in 𝒜 𝑇 , where  colim 𝑖 denotes t he formall y added colimit. The functor Fin co ( 𝒜 𝑇 ) − → 𝒜 𝑇 is left adjoint to the inclusion 𝒜 𝑇 ↩ − → Fin co ( 𝒜 𝑇 ) . On the other hand, t here is an adjunction [ 𝒜 op 𝑇 , V ect ] [( 𝒜 𝑇 ) op , V ect ] , 𝜄 ! 𝜄 ∗ ⊣ (8.1.3) where 𝜄 ! is t he left Kan extension of よ 𝒜 𝑇 ◦ 𝜄 along よ 𝒜 𝑇 , see for example [ Str24 a , Corollary 3.3]. Equation ( 8.1.3 ) restricts to an adjunction Fin co ( 𝒜 𝑇 ) Fin co ( 𝒜 𝑇 ) , 𝜄 ! 𝜄 ∗ ⊣ since, b y deﬁnition, 𝜄 ! sends ﬁnite colimits of representables to ﬁnite colimits of representables, and 𝜄 ∗ has t he same property , since it preser v es colimits, and, b y Lemma 8.6 , sends representables to ﬁnite colimits of representables. Thus w e ha v e t he follo wing commutativ e diag r am 𝒜 𝑇 Fin co ( 𝒜 𝑇 ) 𝒜 𝑇 Fin co ( 𝒜 𝑇 ) 𝒜 𝑇 𝜄 ∗ 𝜄 ! 𝜄 ⊣ ⊣ 182 8.1. Extending module structures This realises t he embedding 𝒜 𝑇 ↩ − → Fin co ( 𝒜 𝑇 ) as t he composition of tw o right adjoints, and thus as a right adjoint. Lastl y , t he composite 𝒜 𝑇 − → 𝒜 𝑇 − → F in co ( 𝒜 𝑇 ) − → 𝒜 𝑇 is naturall y isomorphic to 𝜄 : 𝒜 𝑇 ↩ − → 𝒜 𝑇 , which prov es t he latter statement. □ Remark 8.8. In other w ords, Proposition 8.7 sa ys t hat for a right exact monad 𝑇 the categor y 𝒜 𝑇 is a reﬂectiv e subcat egor y — one where the inclusion functor has a left adjoint — of Fin co ( 𝒜 𝑇 ) . The next result prov es t he essential uniqueness of the module structure on the Eilenberg –Moore categor y , under t he assump tion that it is “induced” from the Kleisli categor y . Theorem 8.9. Let 𝒞 𝑇 denot e t he Kleisli category for a monad 𝑇 on 𝒞 , equipped wit h a ﬁxed lef t 𝒞 -module structur e. Equip 𝒞 𝑇 wit h two lef t 𝒞 -module cat egory structur es 𝒞 𝑇 1 and 𝒞 𝑇 2 , such that t he inclusion 𝜄 : 𝒞 𝑇 ↩ − → 𝒞 𝑇 gives str ong 𝒞 -module functors 𝒞 𝑇 − → 𝒞 𝑇 1 and 𝒞 𝑇 − → 𝒞 𝑇 2 . Then t here is an equiv alence 𝒞 𝑇 1 ≃ 𝒞 𝑇 2 of left 𝒞 -module categories. Proof. Consider t he follo wing diag r am: 𝒞 𝑇 Fin co ( 𝒞 𝑇 ) 𝒞 𝑇 1 𝒞 𝑇 2 𝜄 𝜄 𝜄 1 ! 𝜄 2 ! 𝐺 1 𝐺 2 ⊣ ⊣ In particular , by Proposition 5.32 , in t his diagram ev ery functor is a lax 𝒞 - module functor , and ev er y adjunction is a 𝒞 -module adjunction. W e claim that 𝜄 2 ! 𝐺 1 : 𝒞 𝑇 1 − → 𝒞 𝑇 2 and 𝜄 1 ! 𝐺 2 : 𝒞 𝑇 2 − → 𝒞 𝑇 1 deﬁne mutuall y quasi-in v erse equiv alences of 𝒞 -module categories. Indeed, 𝜄 1 ! 𝐺 2 𝜄 2 ! 𝐺 1 𝜀 2 𝜀 1 − − − → Id 𝒞 𝑇 1 is a 𝒞 -module isomorphism. Similar l y , t here is a 𝒞 -module isomorphism Id 𝒞 𝑇 2  𝜄 2 ! 𝐺 1 𝜄 1 ! 𝐺 2 . □ 183 8. Infi nite and n on -rig id re c on s tructi on 8.1.1 Ext endable monads Deﬁnition 8.10. W e call a lax 𝒞 -module monad 𝑇 on a 𝒞 -module categor y ℳ ext endable if the category ℳ 𝑇 is a 𝒞 -module categor y , for which t he embedding 𝜄 : ℳ 𝑇 ↩ − → ℳ 𝑇 is a strong 𝒞 -module functor . If 𝑇 is extendable, w e refer to t he essentially unique 𝒞 -module structure on ℳ 𝑇 coming from ℳ 𝑇 as the extended 𝒞 -module structure on ℳ 𝑇 . Analogousl y to Deﬁnition 8.10 , one deﬁnes extendable oplax 𝒞 -module comonads and extended module structures on t heir comodules. Proposition 8.11. Any str ong 𝒞 -module monad ( 𝑇 , 𝜇 , 𝜂 ) : ℳ − → ℳ is extendable. Proof. Since 𝑇 is in particular a lax 𝒞 -module monad, by Proposition 5.35 , t he comparison functor 𝐾 𝑇 : ℳ 𝑇 − → ℳ is a strong 𝒞 -module functor . Similar l y , since 𝑇 is in particular an oplax 𝒞 -module monad, again b y Proposition 5.35 , w e conclude t hat also the comparison functor 𝐾 𝑇 : ℳ − → ℳ 𝑇 is a strong 𝒞 -module functor . Thus, 𝜄 = 𝐾 𝑇 𝐾 𝑇 is a strong 𝒞 -module functor . □ Corollar y 8.12. If 𝒞 is lef t rigid, then any lax 𝒞 -module monad is extendable. Proof. By Proposition 2.73 , a lax 𝒞 -module monad is automaticall y a strong 𝒞 -module monad. The result follo ws by Proposition 8.11 . □ 8.1.2 Semisim ple monads In t his s ecti on we w an t to take a look at semisimple monads — those whose category of modules is semisimple. As it tur ns out, these monads are alwa ys extendable. Recall t he deﬁnition of t he Cauch y completion of a category from Section 3.2.1 . The follo wing result follo ws immediately from t he fact t hat for a monad 𝑇 : 𝒞 − → 𝒞 , t he forg etful functor 𝑈 𝑇 : 𝒞 𝑇 − → 𝒞 creates limits. Lemma 8.13. The Eilenberg– Moor e category of a monad on a Cauchy complet e category is itself Cauchy complete. Proposition 8.14. Let 𝑇 : 𝒜 − → 𝒜 be a right exact monad on an abelian category 𝒜 . The category 𝒜 𝑇 is semisimple if and only if 𝒜 𝑇 is. In t his semisimple setting, t he e xtension 𝜄 : 𝒜 𝑇 − → 𝒜 𝑇 of the canonical embedding 𝜄 : 𝒜 𝑇 − → 𝒜 𝑇 from Equa- tion ( 2.2.2 ) to t he Cauchy completion is an equivalence of categories. 184 8.1. Extending module structures Proof. Since 𝒜 is Cauch y complete, t he adjunction 𝐹 𝑇 : 𝒜 ⇄ 𝒜 𝑇 : 𝑈 𝑇 extends to an adjunction 𝐹 𝑇 : 𝒜 ⇄ 𝒜 𝑇 : 𝑈 𝑇 , such that 𝑈 𝑇 𝐹 𝑇 = 𝑇 . It is easy to see t hat the resulting comparison functor 𝐾 : 𝒜 𝑇 − → 𝒜 𝑇 is naturall y isomor phic to t he extension 𝜄 of 𝜄 . F urt her , since 𝜄 is fully fait hful, so is 𝜄 , hence it reﬂects zero objects and so do 𝑈 𝑇 = 𝑈 𝑇 ◦ 𝜄 and 𝑈 𝑇 . Assume that 𝒜 𝑇 is semisimple. The monomor phisms and epimor phisms in 𝒜 𝑇 are split b y Proposition 2.85 , and hence they are reﬂected under t he fully faithful functor 𝜄 . Thus, 𝒜 𝑇 is an abelian categor y and 𝜄 is an exact functor . In particular 𝑈 𝑇  𝑈 𝑇 ◦ 𝜄 is exact and faithful, so it reﬂects zero objects. By Theorem 2.92 , 𝐾 is an equiv alence — in particular , 𝒜 𝑇 is semisimple. Assume no w t hat 𝒜 𝑇 is semisimple. Then 𝑈 𝑇 is exact b y Proposition 2.85 , and so it again satisﬁes the assum ptions of Theorem 2.92 . Thus, 𝐾 : 𝒜 𝑇 ∼ − → 𝒜 𝑇 is an equiv alence; in particular , 𝒜 𝑇 is semisimple. □ Deﬁnition 8.15. W e sa y t hat a right exact monad 𝑇 : 𝒜 − → 𝒜 on an abelian category 𝒜 is semisim ple if it satisﬁes t he equivalent conditions of Proposi- tion 8.14 . Since the opposite of an abelian categor y is abelian and t he opposite of a semisimple category is semisimple, an analogous result to Proposition 8.14 holds for comonads, and w e deﬁne semisimple comonads analogously . Proposition 8.16. Let 𝑇 : ℳ − → ℳ be a semisim ple lax 𝒞 -module monad on an abelian 𝒞 -module category . Then 𝑇 is extendable. Proof. By Proposition 8.14 , the functor 𝜄 : ℳ 𝑇 ∼ − → ℳ 𝑇 is an equivalence. Further , follo wing Proposition 2.129 , ℳ 𝑇 has a canonical structure of a 𝒞 - module category , such t hat t he inclusion ℳ 𝑇 ↩ − → ℳ 𝑇 is a strong 𝒞 -module functor . T ransporting the 𝒞 -module structure along 𝜄 endo ws 𝜄 with t he structure of a strong 𝒞 -module functor , pro ving the claim. □ Semisimplicity also “transf ers” to the lef t adjoint of a monad. Lemma 8.17. Let 𝑇 : 𝒜 − → 𝒜 be a right exact monad on an abelian category , and 𝑆 : 𝒜 − → 𝒜 a left adjoint to 𝑇 . Then 𝑇 is semisimple if and only if 𝑆 is semisimple. In t hat case, ther e is an equiv alence 𝒜 𝑇 ≃ 𝒜 𝑆 . Proof. By Proposition 8.14 , 𝑇 is semisimple if and only if so is 𝒜 𝑇 . Moreov er , the equivalence 𝒜 𝑇 ≃ 𝒜 𝑆 of Proposition 2.26 extends to one on t he Cauch y completions: 𝒜 𝑇 ≃ 𝒜 𝑆 . Thus, 𝒜 𝑇 is semisimple if and onl y if 𝒜 𝑆 is so, which is the case if and only if 𝑆 is semisimple. This establishes the ﬁrst claim. 185 8. Infi nite and n on -rig id re c on s tructi on The latter claim follo ws from t he equivalences 𝒜 𝑇 ≃ 𝒜 𝑇 ≃ 𝒜 𝑆 ≃ 𝒜 𝑆 . □ Proposition 8.18. Let 𝑇 : ℳ − → ℳ be a semisimple lax 𝒞 -module monad on an abelian 𝒞 -module category ℳ , and let 𝑆 : ℳ − → ℳ be a left adjoint to 𝑇 . The equiv alence ℳ 𝑇 ≃ ℳ 𝑇 of Lemma 8.17 is a 𝒞 -module equiv alence, wher e ℳ 𝑇 and ℳ 𝑆 ar e endowed with the ext ended 𝒞 -module structur es of ℳ 𝑇 and ℳ 𝑆 , r espectivel y. Proof. F ollowing the proof of Proposition 8.16 , the functor 𝜄 : ℳ 𝑇 ∼ − → ℳ 𝑇 is a 𝒞 -module equiv alence, and similar ly for ℳ 𝑆 and ℳ 𝑆 . Further , from Proposition 2.129 w e obtain a 𝒞 -module equiv alence ℳ 𝑇 ≃ ℳ 𝑆 from the 𝒞 -module equiv alence ℳ 𝑇 ≃ ℳ 𝑆 of Proposition 5.37 . The result follo ws by t he follo wing chain of 𝒞 -module equivalences: ℳ 𝑇 ≃ ℳ 𝑇 ≃ ℳ 𝑆 ≃ ℳ 𝑆 . □ 8.1.3 Lint on coequaliser s via multiactegories The re ex ists a q uite gene ral cond itio n for exte nd ab ilit y , in which one can ev en explicitly write down t he resulting 𝒞 -module structure on the Eilen- berg– Moore categor y . Our aim no w is to establish Theorems 8.25 and 8.26 . Thus, for t he rest of t his section, w e make t he follo wing assump tions. Hypothesis 8.19. • Let 𝒞 be an abelian monoidal and ℳ an abelian 𝒞 -module category . • The action functor ⊲ : 𝒞 ⊗ k ℳ − → ℳ is right exact in both variables. • Let 𝑇 : ℳ − → ℳ be a right exact lax 𝒞 -module monad. N ote that, due to for example Proposition 8.2 , many of these assump tions are already necessar y for the theory we w ould lik e to dev elop in t his chapter . In our presentation, w e closely follo w [ AHLF18 ], where an analogous result for lax monoidal monads — r at her than lax module monads — can be found. As such, w e shall of ten onl y describe t he modiﬁcations necessary to make t hese results w ork in our case. Deﬁnition 8.20. The Linton coequaliser 𝑥 ▶ 𝑚 of 𝑥 ∈ 𝒞 and ( 𝑚 , ∇ 𝑚 ) ∈ ℳ 𝑇 is: 𝑥 ▶ 𝑚 . . = coeq  𝑇 ( 𝑥 ⊲ 𝑇 𝑚 ) 𝑇 ( 𝑥 ⊲ 𝑚 ) 𝑇 ( 𝑥 ⊲ ∇ 𝑚 ) 𝜇 𝑥 ⊲ 𝑚 ◦ 𝑇𝑇 a ; 𝑥 ,𝑚  . 186 8.1. Extending module structures As ℳ 𝑇 is abelian due to Proposition 8.2 , by functoriality of colimits w e obtain a functor − ▶ = : 𝒞 ⊗ k ℳ 𝑇 − → ℳ 𝑇 . Lemma 8.21. F or 𝑥 ∈ 𝒞 , the functor 𝑥 ▶ − : ℳ 𝑇 − → ℳ 𝑇 is right exact. Proof. This immediatel y follo ws from the right exactness of 𝑇 ( 𝑥 ⊲ 𝑇 (−)) and 𝑇 ( 𝑥 ⊲ −) , as w ell as t he deﬁnition of the Linton coequaliser . □ Lemma 8.22. Ther e is a natural isomorphism 1 ▶ −  Id ℳ 𝑇 . Proof. Recall t hat for all 𝑇 -alg ebras 𝑚 ∈ ℳ 𝑇 , 𝑇 2 𝑚 𝑇 𝑚 𝑚 𝜇 𝑚 𝑇 ∇ 𝑚 ∇ 𝑚 is a coequaliser in ℳ 𝑇 , created by t he underl ying split coequaliser in ℳ . N o w , 𝑇 ( 1 ⊲ 𝑇 𝑚 ) 𝑇 ( 1 ⊲ 𝑚 ) 𝑇 2 𝑚 𝑇 𝑚 𝑇 ( 1 ⊲ ∇ 𝑚 ) 𝜇 1 ⊲ 𝑚 ◦ 𝑇𝑇 a ;1 , 𝑚 𝑇 𝜆  𝑇 𝜆  𝑇 ∇ 𝑚 𝜇 𝑚 deﬁnes an isomor phism in t he categor y of parallel pairs of morphisms in ℳ 𝑇 — t he upper square commutes due to t he naturality of t he left unitor 𝜆 of ℳ , and the lo w er by coherence for − ⊲ = . This isomor phism induces an isomorphism on t he lev el of coequalisers, 𝑚  1 ▶ 𝑚 . Further , t his construc- tion of mor phisms of parallel pairs is clearl y natural in 𝑚 , establishing the naturality of t he isomorphism of coequalisers. □ Lemma 8.23. F or any 𝑚 ∈ ℳ , the coequaliser of 𝑇 ( 𝑥 ⊲ 𝑇 2 𝑚 ) 𝑇 ( 𝑥 ⊲ 𝑇 𝑚 ) 𝑇 ( 𝑥 ⊲𝜇 𝑚 ) 𝜇 𝑥 ⊲ 𝑇 𝑚 ◦ 𝑇𝑇 a ; 𝑥 ,𝑇 𝑚 in ℳ 𝑇 is natur al in 𝑚 , split, and isomorphic to 𝑇 ( 𝑥 ⊲ 𝑚 ) . In ot her wor ds t he Linton coequaliser on a free 𝑇 -module is given by 𝑥 ▶ 𝑇 𝑚  𝑇 ( 𝑥 ⊲ 𝑚 ) . Proof. The splitting data is given b y 𝑇 ( 𝑥 ⊲ 𝑇 2 𝑚 ) 𝑇 ( 𝑥 ⊲ 𝑇 𝑚 ) 𝑇 ( 𝑥 ⊲ 𝑚 ) . 𝑇 ( 𝑥 ⊲𝜇 𝑚 ) 𝜇 𝑥 ⊲ 𝑇 𝑚 ◦ 𝑇𝑇 a ; 𝑥 ,𝑇 𝑚 𝑇 ( 𝑥 ⊲ 𝑇 𝜂 𝑚 ) 𝜇 𝑥 ⊲ 𝑚 ◦ 𝑇𝑇 a ; 𝑥 ,𝑚 𝑇 ( 𝑥 ⊲ 𝜂 𝑚 ) □ 187 8. Infi nite and n on -rig id re c on s tructi on Lemma 8.24. F or 𝑥 , 𝑦 ∈ 𝒞 , ther e is a natural isomor phism ( 𝑥 ⊗ 𝑦 ) ▶ −  𝑥 ▶ ( 𝑦 ▶ −) . Proof. W e calculate 𝑥 ▶ ( 𝑦 ▶ 𝑚 ) = 𝑥 ▶ coeq ( 𝑇 ( 𝑦 ⊲ ∇ 𝑚 ) , 𝜇 𝑥 ⊲ 𝑚 ◦ 𝑇 𝑇 a ; 𝑦 , 𝑚 )  coeq ( 𝑥 ▶ 𝑇 ( 𝑦 ⊲ ∇ 𝑚 ) , 𝑥 ▶ 𝜇 𝑥 ⊲ 𝑚 ◦ 𝑇 𝑇 a ; 𝑦 , 𝑚 )  coeq ( 𝑇 ( 𝑥 ⊲ ( 𝑦 ⊲ ∇ 𝑚 )) , 𝜇 𝑦 ⊲ 𝑥 ⊲ 𝑚 ◦ 𝑇 ( 𝑇 a ; 𝑥 , 𝑦 ⊲ 𝑚 ◦ 𝑦 ⊲ 𝑇 a ; 𝑥 , 𝑚 ))  coeq ( 𝑇 (( 𝑥 ⊗ 𝑦 ) ⊲ ∇ 𝑚 ) , 𝜇 ( 𝑥 ⊗ 𝑦 ) ⊲ 𝑇 𝑚 ◦ 𝑇 𝑇 a ; 𝑥 ⊗ 𝑦 ,𝑇 𝑚 ) = ( 𝑥 ⊗ 𝑦 ) ▶ 𝑚 , where t he ﬁrst isomorphism follo ws from Lemma 8.21 , t he second one is due to Lemma 8.23 , and t he t hir d one follow s by coherence of − ⊲ = . □ In view of Lemmas 8.22 and 8.24 , all that remains in order to establish t hat − ▶ = deﬁnes a 𝒞 -module category structure on ℳ 𝑇 is the v eriﬁcation of coher - ence axioms. While t his can be accomplished b y diag r am chasing — see [ Sea13 , Theorem 2.6.4] — w e choose t o giv e a more formal argument, mimicking the multicategorical approach used by [ AHLF18 ] in t he analogous case of mono- idal monads. More speciﬁcally , w e shall prov e t he follo wing tw o results in the remainder of t his section. Theorem 8.25. The functor − ▶ = : 𝒞 ⊗ k ℳ 𝑇 − → ℳ 𝑇 deﬁnes a lef t 𝒞 -module category structur e on ℳ 𝑇 . Assuming Theorem 8.25 , w e ha v e 𝑥 ▶ 𝑇 𝑚 = 𝑇 ( 𝑥 ⊲ 𝑚 ) , and so t he image of the inclusion 𝜄 : ℳ 𝑇 − → ℳ 𝑇 of Equation ( 2.2.2 ) is a 𝒞 -module subcategor y , where ℳ 𝑇 is endo w ed wit h t he action giv en in the proof of Corollar y 5.33 . Theorem 8.26. The unique embedding 𝜄 : ( ℳ 𝑇 , ⊲ ℳ 𝑇 ) − → ( ℳ 𝑇 , ▶ ) can be equipped wit h the structur e of a str ong 𝒞 -module functor . Let us no w recall the t heory of multicategories, see [ Her00 ; Lei04 ]. Deﬁnition 8.27. A (locall y small) multicategory C consists of • a class Ob C of objects of C , where we write 𝑥 ∈ C f or 𝑥 ∈ Ob C ; • for an y ﬁnite sequence ( 𝑥 1 , . . . , 𝑥 𝑛 , 𝑦 ) of objects in C , a set of multimor ph- isms C ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑦 ) from ( 𝑥 1 , . . . , 𝑥 𝑛 ) to 𝑦 ; • for ev er y 𝑥 ∈ C , an identity multimor phism id 𝑥 ∈ C ( 𝑥 ; 𝑥 ) ; and 188 8.1. Extending module structures • for an y 𝑦 ∈ Ob ( C ) , ( 𝑥 𝑖 ) 𝑛 𝑖 = 1 ∈ Ob ( C ) 𝑛 , as w ell as  ( 𝑢 1 1 , . . . , 𝑢 𝑘 1 1 ) , . . . , ( 𝑢 1 𝑛 , . . . , 𝑢 𝑘 𝑛 𝑛 )  ∈ Ob ( C ) 𝑘 1 +··· + 𝑘 𝑛 , a composition operation C ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑦 ) × C ( 𝑢 1 1 , . . . , 𝑢 𝑘 1 1 ; 𝑥 1 ) × · · · × C ( 𝑢 1 𝑛 , . . . , 𝑢 𝑘 𝑛 𝑛 ; 𝑥 𝑛 ) − → C ( 𝑢 1 1 , . . . , 𝑢 𝑘 𝑛 𝑛 ; 𝑦 ) , subject to natural associativity and unitality conditions; see for example [ Lei04 , Section 2.1] or [ Her00 , Deﬁnition 2.1]. Since w e want our multicategories and multiactegories to correspond to k -linear monoidal and module categories, w e should replace sets of mul- timorphisms with v ector spaces, and the Cartesian products with tensor products. How ev er , since our aim is to show Theorem 8.25 , which can be v eri- ﬁed on the lev el of the underl ying (ordinary) categories, this is not essential. The follo wing deﬁnition is a non-skew special case of [ AM24 , Deﬁnition 6.9]. Deﬁnition 8.28. Let C be a (locally small) multicategor y . A (locally small) left multiactegory M ov er C has as data: • a class Ob M of objects of M , where we write 𝑚 ∈ M for 𝑚 ∈ Ob M ; • for an y ﬁnite (possibly emp ty) sequence ( 𝑥 1 , . . . , 𝑥 𝑛 ) of objects in C and a pair of objects ( 𝑚 , 𝑚 ′ ) of M , a set M ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ; 𝑚 ′ ) of multimorphisms from ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ) to 𝑚 ′ ; • for ev er y 𝑚 ∈ M , an identity multimorphism id 𝑚 ∈ M ( 𝑚 ; 𝑚 ) ; and • for an y 𝑥 ∈ C , 𝑚 , 𝑚 ′ , ℓ ∈ M , ( 𝑥 𝑖 ) 𝑛 𝑖 = 1 ∈ Ob ( C ) 𝑛 , and  ( 𝑢 1 1 , . . . , 𝑢 𝑘 1 1 ) , . . . , ( 𝑢 1 𝑛 , . . . , 𝑢 𝑘 𝑛 𝑛 )  ∈ Ob ( C ) 𝑘 1 +··· + 𝑘 𝑛 , a composition operation M ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ; 𝑚 ′ ) × C ( 𝑢 1 1 , . . . , 𝑢 𝑘 1 1 ; 𝑥 1 ) × · · · × C ( 𝑢 1 𝑛 , . . . , 𝑢 𝑘 𝑛 𝑛 ; 𝑥 𝑛 ) × M ( 𝑢 1 𝑛 + 1 , . . . , 𝑢 𝑘 𝑛 + 1 𝑛 + 1 ; ℓ ; 𝑚 ) − → M ( 𝑢 1 1 , . . . , 𝑢 𝑘 𝑛 + 1 𝑛 + 1 ; ℓ ; 𝑚 ′ ) . (8.1.4) This data has to satisfy t he (non-mar ked) associativity and unitality axioms of [ AM24 , Deﬁnitions 4.4 and 6.9]. 189 8. Infi nite and n on -rig id re c on s tructi on Notation 8.29. From now on, we will of ten abbreviate sequences of objects using t he v ector notation similar to [ Her00 ]; e.g., writing ® 𝑥 for ( 𝑥 1 , . . . , 𝑥 𝑛 ) and ( ® 𝑥 1 , ® 𝑥 2 ) for ( 𝑥 1 1 , . . . , 𝑥 𝑛 1 1 , 𝑥 1 2 , . . . , 𝑥 𝑛 2 2 ) . W e will use t his notation for clarity and brevity , when keeping track of t he length of the tuples is not required. W e also ﬁx t he notation for the remainder of this section, letting C be a multicategor y and letting M be a C -multiactegory . Recall from [ Her00 , Chapter 8] t hat a pr e-universal arrow for a tuple ® 𝑥 of objects of C consists of an object ⊗ ( ® 𝑥 ) of C tog ether wit h a multimor phism 𝜋 ∈ C ( ® 𝑥 ; ⊗ ( ® 𝑥 )) , inducing isomor phisms C (⊗ ( ® 𝑥 ) ; 𝑦 ) ∼ − → C ( ® 𝑥 ; 𝑦 ) . If 𝜋 further induces isomorphisms C ( ® 𝑧 , ⊗ ( ® 𝑥 ) , ® 𝑧 ′ ; 𝑦 ) ∼ − → C ( ® 𝑧 , ® 𝑥 , ® 𝑧 ′ ; 𝑦 ) , w e sa y that 𝜋 is univer sal . If an y sequence of objects of C is t he domain of a univ ersal arro w , w e sa y t hat C is repr esentable . A repr esentation of C is a choice of a univ ersal arrow from ev er y sequence of objects in C . Deﬁnition 8.30. Let M be a C -multiactegory . A pre-univ ersal arrow for a tuple ® 𝑥 of objects of C and 𝑚 ∈ M consists of an object ⊲ ( ® 𝑥 ; 𝑚 ) , tog ether with a multimorphism 𝜋 ∈ M ( ® 𝑥 ; 𝑚 ; ⊲ ( ® 𝑥 ; 𝑚 )) , inducing isomorphisms M ( ⊲ ( ® 𝑥 ; 𝑚 ) ; 𝑚 ′ ) ∼ − → M ( ® 𝑥 ; 𝑚 ; 𝑚 ′ ) . If 𝜋 furt her induces isomor phisms M ( ® 𝑦 , ⊲ ( ® 𝑥 ; 𝑚 ) ; 𝑚 ′ ) ∼ − → M ( ® 𝑦 , ® 𝑥 ; 𝑚 ; 𝑚 ′ ) , then 𝜋 is said to be univer sal . Deﬁnition 8.31. W e sa y that a multimor phism 𝜌 ∈ C ( ® 𝑥 ; 𝑦 ) is pr e-universal in M if it induces isomor phisms M ( 𝑦 ; 𝑚 ; 𝑛 ) ∼ − → M ( ® 𝑥 ; 𝑚 ; 𝑛 ) . It is universal in M , if it also induces isomorphisms M ( ® 𝑧 , ® 𝑥 , ® 𝑤 ; 𝑚 ; 𝑛 ) ∼ − → M ( ® 𝑧 , 𝑦 , ® 𝑤 ; 𝑚 ; 𝑛 ) . Deﬁnition 8.32. F or C representable and a representation R of C , w e sa y t hat M is a repr esentable wit h respect to R , if for ev er y sequence ® 𝑥 of objects of C and ev ery object 𝑚 ∈ M , there is a univ ersal arro w 𝜋 ( ® 𝑥 ; 𝑚 ) ∈ M ( ® 𝑥 ; 𝑚 ; ⊲ ( ® 𝑥 , 𝑚 )) , and all morphisms of R are univ ersal in M . Recall from [ Her00 , Proposition 8.5] that universal arro ws in a multicate- gory C are closed under composition. Further , if ev er y sequence of objects in C is t he domain of a pre-univ ersal arro w and pre-univ ersal arro ws are closed under composition, t hen a pre-univ ersal arrow in C is universal. A similar result holds for multiactegories. 190 8.1. Extending module structures Lemma 8.33. 1. Com position of universal arr ows of C in M and universal arr ows of M yields univer sal arrow s in M . 2. If composition of pre-univ ersal arrow s of C in M and pre-univer sal arrow s of M yields pre-univ ersal arrow s in M , then a pre-univ ersal morphism in M is univer sal. 3. If pr e-univer sal arrow s in C act pre-univ ersall y in M , and pre-univ ersal ar - row s of C and M are closed under composition in M , t hen M is r epresent able. Proof. F or t he ﬁrst part, let ( 𝜋 0 , 𝜌 1 , . . . , 𝜌 𝑛 , 𝜋 𝑛 + 1 ) be a sequence composable in M , with 𝜋 0 , 𝜋 𝑛 + 1 in M and 𝜌 1 , . . . , 𝜌 𝑛 ∈ C , as indicated in Equation ( 8.1.4 ), all of whose elements are univ ersal in M . The composite 𝜋 0 ◦ ( 𝜌 1 , . . . , 𝜌 𝑛 , 𝜋 𝑛 + 1 ) giv es rise to the sequence of isomor phisms M ( ® 𝑥 , ⊲ (⊗ ( ® 𝑦 1 ) , . . . , ⊗ ( ® 𝑦 𝑛 ) ; ⊲ ( ® 𝑦 𝑛 + 1 , 𝑚 )) , −)  M ( ® 𝑥 , ⊗ ( ® 𝑦 1 ) , . . . , ⊗ ( ® 𝑦 𝑛 ) ; ⊲ ( ® 𝑦 𝑛 + 1 , 𝑚 ) ; −)  M ( ® 𝑥 , . . . , ® 𝑦 1 , . . . , ® 𝑦 𝑛 + 1 ; 𝑚 ; −) , pro ving the universality of the composite. F or the second part, consider t he follo wing chain of maps: M ( ® 𝑥 , ® 𝑦 , ® 𝑧 ; 𝑚 ; −) → M ( ® 𝑥 , ⊗ ( ® 𝑦 ) , ® 𝑧 ; 𝑚 ; −) → M (⊗ ( ® 𝑥 ) , ⊗ ( ® 𝑦 ) , ⊗ ( ® 𝑧 ) ; 𝑚 ; −) → M ( ⊲ (⊗( ® 𝑥 ) , ⊗ ( ® 𝑦 ) , ⊗ ( ® 𝑧 ) ; 𝑚 ) ; −) , where t he ﬁrst morphism is induced by the pre-univ ersal arrow 𝜌 ® 𝑥 of C , the second one by the pre-univ ersal arro ws 𝜌 ® 𝑦 and 𝜌 ® 𝑧 of C , and t he t hir d is induced b y t he univ ersal arro w 𝜋 ( ® 𝑦 , ® 𝑥 , ® 𝑧 ; 𝑚 ) . The composite of all t hree maps is an isomorphism — being induced b y a composite of pre-univ ersal arro ws — and hence a pre-universal arro w . The same holds for t he composite of t he latter tw o maps. These tw o obser v ations es tablish the inv ertibility of the ﬁrst map. The t hir d part is established b y appl ying the proof of t he second part abo v e to t he case of pre-univ ersal arrow s of C . □ Assume C to be representable, let R be a representation of C , and suppose that M is representable with respect to R . Giv en a sequence 𝑥 1 , . . . , 𝑥 𝑛 of objects in C and an object 𝑚 in M , the set of wa ys in which w e ma y com- pose t he univ ersal arrows in t he giv en representations to a univ ersal arrow with domain ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ) is canonicall y in bĳection with t he set of par- enthesisations of the w ord 𝑥 1 · · · 𝑥 𝑛 𝑚 into subw ords of length at leas t tw o. Let ⋄ and ⋄ ′ be tw o such parent hesisations. W e denote t he codomains of 191 8. Infi nite and n on -rig id re c on s tructi on the corresponding univ ersal arro ws by ⋄ ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ) and ⋄ ′ ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ) , and the univ ersal arro ws themselv es b y 𝜋 ⋄ ( 𝑥 1 , .. . , 𝑥 𝑛 ; 𝑚 ) and 𝜋 ⋄ ′ ( 𝑥 1 , .. . , 𝑥 𝑛 ; 𝑚 ) . The univ ersality of 𝜋 ⋄ ( 𝑥 1 , .. . , 𝑥 𝑛 ; 𝑚 ) entails the existence of a unique arro w 𝛼 ⋄ , ⋄ ′ ® 𝑥 , 𝑚 : ⋄ ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ) − → ⋄ ′ ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ) , satisfying 𝛼 ⋄ , ⋄ ′ ® 𝑥 , 𝑚 ◦ 𝜋 ⋄ ( ® 𝑥 , 𝑚 ) = 𝜋 ⋄ ′ ( ® 𝑥 , 𝑚 ) , which is in v ertible wit h in verse 𝛼 ⋄ ′ , ⋄ ® 𝑥 , 𝑚 . Further , 𝛼 ⋄ , ⋄ ′′ ® 𝑥 , 𝑚 = 𝛼 ⋄ ′ , ⋄ ′′ ® 𝑥 , 𝑚 ◦ 𝛼 ⋄ , ⋄ ′ ® 𝑥 , 𝑚 . Thus, for any sequence 𝑥 1 , . . . , 𝑥 𝑛 , 𝑚 w e ob- tain an indiscrete categor y 19 whose objects are parent hesisations of 𝑥 1 · · · 𝑥 𝑛 𝑚 19 A categor y is called indiscrete if there exists a unique morphism from ev er y object to ev ery other object. and Hom (⋄ , ⋄ ′ ) = { 𝛼 ⋄ , ⋄ ′ ® 𝑥 , 𝑚 } . The arro w 𝛼 ⋄ , ⋄ ′ ® 𝑥 , 𝑚 is natural in 𝑚 and 𝑥 𝑖 for all 𝑖 . Remark 8.34. R ecall t hat giv en a representable multicategor y C with a ﬁx ed representation, there is a monoidal categor y 𝒞 deﬁned b y Ob 𝒞 . . = Ob C , 𝒞 ( 𝑥 , 𝑦 ) . . = C ( 𝑥 ; 𝑦 ) , and 𝑥 ⊗ 𝑦 . . = ⊗( 𝑥 , 𝑦 ) , the codomain of t he univ ersal arrow 𝜋 ⊗ ( 𝑥 , 𝑦 ) from ( 𝑥 , 𝑦 ) in the representation of C . The associator of 𝒞 is deﬁned b y the 𝛼 ( 𝑤 𝑥 ) 𝑦 ,𝑤 ( 𝑥 𝑦 ) 𝑤 , 𝑥 , 𝑦 described abo v e; one obtains similar arrow s for unitality . The indiscreteness of t he categor y of mor phisms 𝛼 ⋄ , ⋄ ′ associated to the parenthesisations of w ords in C implies the commutativity of t he pentagon and triangle diag r ams, es tablishing the coherence and w ell-deﬁnedness of 𝒞 . The indiscrete categor y described abov e establishes t he w ell-deﬁnedness of the 𝒞 -module categor y ℳ of t he follo wing deﬁnition. Deﬁnition 8.35. Let M be a representable multiactegor y ov er a representable multicategory C , with ﬁx ed representations for both. Let 𝒞 be the monoidal category associated to t he representation of C . The 𝒞 -module category ℳ associated to t he ﬁxed repr esentation of M is deﬁned by setting Ob M . . = Ob ℳ , ℳ ( 𝑚 , 𝑛 ) . . = M ( 𝑚 ; 𝑛 ) , and 𝑥 ⊲ 𝑚 . . = ⊲ ( 𝑥 , 𝑚 ) , the codomain of the univ ersal arro w 𝜋 ⊲ ( 𝑥 , 𝑚 ) . The associator ( 𝑥 ⊗ 𝑦 ) ⊲ 𝑚 ∼ − → 𝑥 ⊲ ( 𝑦 ⊲ 𝑚 ) is giv en b y t he arrow 𝛼 ( 𝑥 𝑦 ) 𝑚 , 𝑥 ( 𝑦 𝑚 ) 𝑥 , 𝑦 , 𝑚 , and similar l y for t he unitors. Ha ving es tablished suﬃcient analogues of [ Her00 ; Lei04 ], w e continue with analogues of [ AHLF18 , Section 3]. For 𝑥 𝑖 ∈ 𝒞 and 𝑚 ∈ ℳ , write 𝜑 : 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑇 𝑚 − → 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 ) for t he morphism obtained by repeated application of t he lax structure of 𝑇 , where 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 is to be read as 𝑥 1 ⊲ (· · · ⊲ ( 𝑥 𝑛 ⊲ 𝑚 )) . 192 8.1. Extending module structures W e will also consider t he coequaliser diag r am 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑇 𝑚 ) 𝑇 2 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 ) 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 ) 𝑇 𝜑 𝑇 ( 𝑥 1 ⊲ ··· ⊲ 𝑥 𝑛 ⊲ ∇ 𝑚 ) 𝜇 (8.1.5) which is reﬂexiv e, wit h 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 ) 𝑇 ( 𝑥 1 ⊲ ··· ⊲ 𝑥 𝑛 ⊲𝜂 𝑚 ) − − − − − − − − − − − → 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑇 𝑚 ) as a common section. Deﬁnition 8.36. A mor phism 𝑓 : 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 − → 𝑚 ′ is called 𝑛 -multilinear if the follo wing diagram commutes: 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑇 𝑚 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 ) 𝑇 𝑚 ′ 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 𝑚 ′ 𝜑 𝑥 1 ⊲ ··· ⊲ 𝑥 𝑛 ⊲ ∇ 𝑚 𝑇 𝑓 ∇ 𝑚 ′ 𝑓 Remark 8.37. F or a monoidal category 𝒞 , t here exists a representable mul- ticategory C , with C ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑦 ) . . = 𝒞 ( 𝑥 1 ⊗ (· · · ( 𝑥 𝑛 − 1 ⊗ 𝑥 𝑛 ) . . . ) , 𝑦 ) . Then, a multimorphism 𝜋 ∈ C ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑦 ) is univ ersal if and onl y if it is an iso- morphism 𝜋 : 𝑥 1 ⊗ · · · ⊗ 𝑥 𝑛 ∼ − → 𝑦 . Similar ly , given a 𝒞 -module category , one obtains a representable multiac- tegory M o v er C satisfying M ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ; 𝑛 ) = ℳ ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 , 𝑚 ′ ) . A uni- v ersal arro w 𝜋 ∈ M ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ; 𝑚 ′ ) is an isomorphism 𝜋 : 𝑥 1 ⊲ · · · ⊲ 𝑚 ∼ − → 𝑚 ′ . Lemma 8.38. F or a 𝒞 -module category ℳ , t he collection of multilinear mor phisms of M is stable under multicomposition with mor phisms of C , forming a 𝒞 -multiac- tegory that we denot e by M 𝜑 . Proof. F or 𝑖 = 1 , . . . , 𝑛 , consider sequences ® 𝑧 𝑖 ; morphisms ⊗ ( 𝑧 𝑖 ) 𝑔 𝑖 − → 𝑥 𝑖 , where ⊗ ( 𝑧 𝑖 ) again is giv en t he rightmost parent hesisation; and a multilinear mor ph- ism 𝑓 : 𝑥 1 ⊲ . . . ⊲ 𝑥 𝑛 ⊲ 𝑚 − → 𝑚 ′ of M . The claim follo ws immediatel y from t he 193 8. Infi nite and n on -rig id re c on s tructi on commutativity of ⊗ ( ® 𝑧 1 ) ⊲ · · · ⊲ ⊗ ( ® 𝑧 𝑛 ) ⊲ 𝑇 𝑚 𝑇 (⊗( ® 𝑧 1 ) ⊲ · · · ⊲ ⊗ ( ® 𝑧 𝑛 ) ⊲ 𝑚 ) 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑇 𝑚 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 ) 𝑇 𝑚 ′ 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 𝑚 ′ 𝜑 𝑔 1 ⊲ ··· ⊲ 𝑔 𝑛 ⊲ 𝑇 𝑚 𝑇 ( 𝑔 1 ⊲ ··· ⊲ 𝑔 𝑛 ⊲ 𝑚 ) 𝜑 𝑥 1 ⊲ ··· ⊲ 𝑥 𝑛 ⊲ ∇ 𝑚 𝑇 𝑓 ∇ 𝑚 ′ 𝑓 where the top face commutes by naturality of t he action of 𝒞 in ℳ . □ N ote t hat M 𝜑 being representable w ould pro v e Theorem 8.25 , since t hen the 𝒞 -module categor y can be equipped with coherent associators; see Re- mar k 8.34 and Deﬁnition 8.35 . Lemma 8.39. A map 𝑓 is 𝑛 -multilinear if and onl y if the map 𝑓 : 𝑇 ( 𝑥 1 ⊲ . . . ⊲ 𝑥 𝑛 ⊲ 𝑚 ) 𝑇 𝑓 − − − → 𝑇 𝑚 ′ ∇ 𝑚 ′ − − − → 𝑚 ′ coequalises Diagram ( 8.1.5 ), and 𝑓 is pre-univ ersal in M 𝜑 if and onl y if 𝑓 is the (univer sal) coequaliser of ibid. Proof. Consider t he follo wing diag r am: 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑇 𝑚 ) 𝑇 2 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 ) 𝑇 2 𝑚 ′ 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑥 𝑛 ⊲ 𝑚 ) 𝑇 𝑚 ′ 𝑇 𝑚 ′ 𝑚 ′ 𝑇 𝜑 𝑇 ( 𝑥 1 ⊲ ··· ⊲ ∇ 𝑚 ) 𝑇 2 𝑓 𝜇 𝜇 𝑚 ′ 𝑇 (∇ 𝑚 ′ ) 𝑇 𝑓 𝑓 ∇ 𝑚 ′ ∇ 𝑚 ′ ( 1 ) ( 2 ) ( 3 ) Coequalising t he pair ( 8.1.5 ) is precisely coequalising face ( 1 ) in t he abov e diagram. Thus, since face ( 2 ) commutes, it is equiv alent to ∇ 𝑚 ′ coequalising the square formed jointly b y faces ( 1 ) and ( 2 ) . 194 8.1. Extending module structures Observe that, using commutativity of f ace ( 3 ) , this square postcomposed b y ∇ 𝑚 ′ giv es precisely t he pair of morphisms in ℳ 𝑇 ( 𝑇 ( 𝑥 1 ⊲ · · · ⊲ 𝑇 𝑚 ) , 𝑚 ′ ) corresponding to t he pair of mor phisms in ℳ ( 𝑥 1 ⊲ · · · ⊲ 𝑇 ( 𝑚 ) , 𝑚 ′ ) , deﬁning multilinearity of 𝑓 , under ℳ 𝑇 ( 𝑇 (−) , = )  ℳ (− , = ) . Thus, both conditions in the ﬁrst statement are equiv alent to t he commutativity of t he diag r am. The second statement is shown analogously to [ AHLF18 , Lemma 3.5] □ Finall y , in order to establish Theorems 8.25 and 8.26 , w e need an analogue of [ AHLF18 , Proposition 3.13]. Proposition 8.40. The multicategory M 𝜑 is r epresent able. Proof. Let ( 𝑥 1 , . . . , 𝑥 𝑛 , 𝑚 ′ ) be a sequence of objects, wit h 𝑥 𝑖 ∈ 𝒞 and 𝑚 ′ ∈ ℳ . The coequalizer of Diag r am ( 8.1.5 ) deﬁnes a pre-univ ersal map in M 𝜑 with domain ( 𝑥 1 , . . . , 𝑥 𝑛 ; 𝑚 ′ ) , b y Lemma 8.39 . It remains to show t hat a composition of pre-univ ersal maps is pre-univ ersal. Thus, consider sequences ® 𝑧 1 , . . . , ® 𝑧 𝑛 and ® 𝑦 , univ ersal multimorphisms ℎ 𝑖 : ® 𝑧 𝑖 − → 𝑥 𝑖 of C , and univ ersal multimorphisms 𝑓 : ( ® 𝑦 , 𝑚 ) − → 𝑚 ′ and 𝑔 : ( ® 𝑥 , 𝑚 ′ ) − → 𝑚 ′ ′ . Behold the follo wing diagram: ® 𝑧 1 ⊲ · · · ® 𝑧 𝑛 ⊲ 𝑇 2 ( ® 𝑦 ⊲ 𝑚 ) ® 𝑧 1 ⊲ · · · ® 𝑧 𝑛 ⊲ 𝑇 ( ® 𝑦 ⊲ 𝑇 ( 𝑚 )) ® 𝑧 1 ⊲ · · · ® 𝑧 𝑛 ⊲ 𝑇 ( ® 𝑦 ⊲ 𝑚 ) 𝑇 2 ( ® 𝑧 1 ⊲ · · · ® 𝑧 𝑛 ⊲ ® 𝑦 ⊲ 𝑚 ) 𝑇 ( ® 𝑧 1 ⊲ · · · ® 𝑧 𝑛 ⊲ ® 𝑦 ⊲ 𝑇 ( 𝑚 )) 𝑇 ( ® 𝑧 1 ⊲ · · · ® 𝑧 𝑛 ⊲ ® 𝑦 ⊲ 𝑚 ) 𝑧 1 ⊲ ··· 𝑧 𝑛 ⊲𝜇 𝑦 ⊲ 𝑚 ® 𝑧 1 ⊲ ··· ® 𝑧 𝑛 ⊲ 𝑇 ( 𝜑 ) ® 𝑧 1 ⊲ ··· ® 𝑧 𝑛 ⊲ 𝑇 ( ® 𝑦 ⊲ ∇ 𝑚 ) 𝜑 𝜑 𝜇 𝑧 1 ⊲ ··· ⊲ 𝑚 𝑇 𝜑 𝑇 ( ® 𝑧 1 ⊲ ··· ® 𝑧 𝑛 ⊲ ® 𝑦 ⊲ ∇ 𝑚 ) Similar ly to t he proof of [ AHLF18 , Proposition 3.13], t his is a morphism from the top triangle-shaped diagram to t he bottom one, where the top is the action of ( ® 𝑧 1 , . . . , ® 𝑧 𝑛 ) on t he Linton pair f or ( ® 𝑦 , 𝑚 ) , while the bo ttom is the Linton pair for ( ® 𝑧 1 , . . . , ® 𝑧 𝑛 , ® 𝑦 ; 𝑚 ) . N o w , let 𝑋 ∈ M 𝜑 ( ® 𝑧 1 , . . . , ® 𝑧 𝑛 , ® 𝑦 ; 𝑚 ; ℓ ) . Consider the diag r am ® 𝑧 1 ⊲ · · · ® 𝑧 𝑛 ⊲ 𝑇 ( ® 𝑦 ⊲ 𝑚 ) ® 𝑧 1 ⊲ · · · ® 𝑧 𝑛 ⊲ 𝑚 ′ 𝑥 1 ⊲ · · · 𝑥 𝑛 ⊲ 𝑚 ′ 𝑇 ( ® 𝑧 1 ⊲ · · · ® 𝑧 𝑛 ⊲ ® 𝑦 ⊲ 𝑚 ) ℓ 𝑚 ′ ′ 𝑧 1 ⊲ ··· ⊲ 𝑓 𝜑 ℎ 1 ⊲ ··· ℎ 𝑛 ⊲ 𝑚 ′ ≃ ∃ !  𝑋 𝑔 𝑋 ∃ !  𝑋 195 8. Infi nite and n on -rig id re c on s tructi on where  𝑋 exists since, b y Lemma 8.39 , 𝑋 coequalises the bottom Linton pair in the penultimate diagram abov e, and ℎ 1 ⊲ · · · ⊲ 𝑓 is the coequaliser of t he top Linton pair in the same diag r am. Similar l y to [ AHLF18 , Lemma 3.11] one sho ws t hat  𝑋 is multilinear , and thus so is ℎ − 1 1 ⊲ · · · ⊲ ℎ − 1 𝑛 ⊲ 𝑚 ′ , since ℎ 𝑖 is in v ertible for all 𝑖 , being a pre-univ ersal, and hence univ ersal, arrow of C . This prov es t he existence of  𝑋 . Thus, t he multilinear mor phism 𝑋 factors t hrough 𝑔 ◦ ( ℎ 1 , . . . , ℎ 𝑛 , 𝑓 ) , and this factorisation is unique by uniqueness of  𝑋 and of  𝑋 . This establishes t he pre-univ ersality of t he composition 𝑔 ◦ ( ℎ 1 , . . . , ℎ 𝑛 , 𝑓 ) . □ 8 . 2 i n t e r n a l p r o j e c t i v e a n d i n j e c t i v e o b j e c t s Deﬁnition 8.41. Let 𝒞 be an abelian monoidal categor y and let ℳ be an abelian 𝒞 -module category . An object 𝑀 ∈ ℳ is said to be 𝒞 -projectiv e if, f or an y projectiv e object 𝑃 ∈ 𝒞 , the object 𝑃 ⊲ 𝑀 ∈ ℳ is projectiv e. Similarl y , 𝑀 is 𝒞 -injective if, for an y injectiv e object 𝐼 ∈ 𝒞 , the object 𝐼 ⊲ 𝑀 ∈ ℳ is injectiv e. The next result connects t he formal deﬁnition of a 𝒞 -projectiv e wit h the ordinary notion of projectiv e object in an abelian categor y . Proposition 8.42. Assume that 𝒞 and ℳ hav e enough projectiv es. Let 𝑀 be a closed object of ℳ . Then ⌊ 𝑀 , −⌋ is right exact if and only if 𝑀 is 𝒞 -projectiv e. Proof. If ⌊ 𝑀 , −⌋ is right exact and 𝑃 is a projectiv e object of 𝒞 . Then 𝑃 ⊲ 𝑀 is projectiv e because ℳ ( 𝑃 ⊲ 𝑀 , −)  𝒞 ( 𝑃 , ⌊ 𝑀 , −⌋) , which is a com posite of right exact functors, hence itself a right exact functor . Assume t hat 𝑀 is 𝒞 -projectiv e. Let colim 𝑗 𝑁 𝑗 be a ﬁnite colimit in ℳ , and 𝑋 ∈ 𝒞 . Since 𝒞 has enough projectiv es, one has 𝑋  colim 𝑖 𝑄 𝑖 , realising 𝑋 as the colimit of a ﬁnite diag r am of projectiv es. W e then hav e 𝒞 ( 𝑋 , ⌊ 𝑀 , colim 𝑗 𝑁 𝑗 ⌋)  𝒞 ( colim 𝑖 𝑄 𝑖 , ⌊ 𝑀 , colim 𝑗 𝑁 𝑗 ⌋)  lim 𝑖 𝒞 ( 𝑄 𝑖 , ⌊ 𝑀 , colim 𝑗 𝑁 𝑗 ⌋)  lim 𝑖 ℳ ( 𝑄 𝑖 ⊲ 𝑀 , colim 𝑗 𝑁 𝑗 )  lim 𝑖 colim 𝑗 ℳ ( 𝑄 𝑖 ⊲ 𝑀 , 𝑁 𝑗 )  lim 𝑖 colim 𝑗 𝒞 ( 𝑄 𝑖 , ⌊ 𝑀 , 𝑁 𝑗 ⌋)  lim 𝑖 𝒞 ( 𝑄 𝑖 , colim 𝑗 ⌊ 𝑀 , 𝑁 𝑗 ⌋)  𝒞 ( colim 𝑖 𝑄 𝑖 , colim 𝑗 ⌊ 𝑀 , 𝑁 𝑗 ⌋)  𝒞 ( 𝑋 , colim 𝑗 ⌊ 𝑀 , 𝑁 𝑗 ⌋) . □ 196 8.2. Internal projectiv e and injectiv e objects By oppositising Proposition 8.42 , w e obtain t he follo wing result. Proposition 8.43. Assume t hat 𝒞 and ℳ hav e enough injectives. Let 𝑀 be a coclosed object of ℳ . Then ⌈ 𝑀 , −⌉ is lef t exact if and only if 𝑀 is 𝒞 -injective. Deﬁnition 8.44. Let 𝒞 be an abelian monoidal, and ℳ an abelian 𝒞 -module category , wit h both having enough projectiv es. A 𝒞 -projectiv e object 𝑀 is called a 𝒞 -projectiv e 𝒞 -g ener ator if for an y projectiv e object 𝑄 in ℳ there exists a projectiv e object 𝑃 in 𝒞 , such t hat 𝑄 is a direct summand of 𝑃 ⊲ 𝑀 . Analogousl y , if 𝒞 and ℳ instead ha v e enough injectiv es, a 𝒞 -injectiv e object 𝑀 is called a 𝒞 -injective 𝒞 -cog enerat or if any injectiv e object 𝐽 of ℳ is a direct summand of an object of the form 𝐼 ⊲ 𝑀 , for an injectiv e object 𝐼 of 𝒞 . Proposition 8.45. Assume t hat 𝒞 and ℳ ha ve enough projectiv es. Let 𝑀 ∈ ℳ be a closed 𝒞 -projectiv e object. If 𝑀 is a 𝒞 -g enerat or , t hen ⌊ 𝑀 , −⌋ reﬂects zero objects. If 𝒞 -proj is a Krull– Sc hmidt category and every indecom posable projectiv e ob- ject of 𝒞 is t he projective cover of a simple object, then we hav e that ⌊ 𝑀 , −⌋ reﬂects zero objects if and onl y if 𝑀 is a 𝒞 -gener ator . Proof. Assume t hat 𝑀 is a 𝒞 -gener ator . Let 𝑁 be a non-zero object of ℳ and let 𝑄 − ↠ 𝑁 be an epimorphism from a projectiv e object in ℳ . Let 𝑃 be a projectiv e object of 𝒞 such t hat 𝑄 is a direct summand of 𝑃 ⊲ 𝑀 . Then ℳ ( 𝑃 ⊲ 𝑀 , 𝑁 ) is non-zero, since ℳ ( 𝑄 , 𝑁 ) is a direct summand thereof. Thus 0 ≠ ℳ ( 𝑃 ⊲ 𝑀 , 𝑁 )  𝒞 ( 𝑃 , ⌊ 𝑀 , 𝑁 ⌋) , sho wing that ⌊ 𝑀 , 𝑁 ⌋ is not zero. F or the latter statement, let 𝑄 ′ be an indecomposable projectiv e object of ℳ , and let 𝑆 be its simple top. Since ⌊ 𝑀 , 𝑆 ⌋ ≠ 0 , there is some 𝑉 ∈ ℳ such that 𝒞 ( 𝑉 , ⌊ 𝑀 , 𝑆 ⌋) ≠ 0 . Let 𝑃 ′ − ↠ 𝑉 be an epimor phism from a projectiv e object in 𝒞 . Then 0 ≠ 𝒞 ( 𝑉 , ⌊ 𝑀 , 𝑆 ⌋) ≃ 𝒞 ( 𝑃 ′ , ⌊ 𝑀 , 𝑆 ⌋) ≃ 𝒞 ( 𝑃 ′ ⊲ 𝑀 , 𝑆 ) . Since 𝑀 is 𝒞 -projectiv e, the object 𝑃 ′ ⊲ 𝑀 is projectiv e. Thus, 𝒞 ( 𝑃 ′ ⊲ 𝑀 , 𝑆 ) being non-zero implies that 𝑄 ′ is a direct summand of 𝑃 ′ ⊲ 𝑀 . □ More closel y follo wing t he classical case, one alter nativ ely deﬁnes a closed 𝑀 ∈ ℳ as a 𝒞 -gener ator if ⌊ 𝑀 , −⌋ is faithful, see [ DSPS19 , Deﬁnition 2.21]. Oppositisation of Proposition 8.45 yields a similar variant in terms of coclosed and 𝒞 -injectiv e objects. 197 8. Infi nite and n on -rig id re c on s tructi on Proposition 8.46. Let 𝒞 and ℳ have enough injectives, and let 𝑀 ∈ ℳ be a coclosed 𝒞 -injective object. If 𝑀 is a 𝒞 -cogener ator , then ⌈ 𝑀 , −⌉ r eﬂects zero objects. If 𝒞 -inj is a Krull – Sc hmidt cat egor y and ev ery indecom posable injective object of 𝒞 is the injective hull of a simple object, t hen ⌈ 𝑀 , −⌉ r eﬂects zero objects if and onl y if 𝑀 is a 𝒞 -cogener ator . Finall y , w e giv e a brief account of internally projectiv e and injectiv e objects in a semisimple module category . Proposition 8.47. Let 𝒞 be a monoidal category with enough projectives, and let ℳ be a semisim ple 𝒞 -module cat egor y . Since ℳ is semisimple, an y object of ℳ is 𝒞 -projectiv e and 𝒞 -injectiv e. An object 𝑀 ∈ ℳ is a 𝒞 -g enerat or if and onl y if any simple object 𝑆 ∈ ℳ is a direct summand of an object of the form 𝑃 ⊲ 𝑀 , for some 𝑃 ∈ 𝒞 -proj . Proof. Recall t hat, in a semisim ple abelian categor y , e v er y object is projectiv e. In particular , 𝑃 ⊲ 𝑀 is alw a ys projectiv e, for 𝑀 ∈ ℳ and 𝑃 ∈ 𝒞 -proj , hence ev ery object in ℳ is 𝒞 -projectiv e. N ow , let 𝑀 ∈ ℳ be a 𝒞 -gener ator . Since a simple object 𝑆 ∈ ℳ is projectiv e, it immediately follow s from 𝑀 being a 𝒞 -gener ator t hat 𝑆 is a direct summand of 𝑃 ⊲ 𝑀 , for some 𝑃 ∈ 𝒞 -proj. Lastl y , assume t hat ev er y simple is a direct summand of 𝑃 ⊲ 𝑀 , for some 𝑃 ∈ 𝒞 , and let 𝑄 ∈ ℳ -proj . As ℳ is semisim ple, w e ha v e 𝑄  ⊕ 𝑖 𝑆 𝑖 for simples 𝑆 𝑖 in ℳ . Since 𝑆 𝑖 is a direct summand of 𝑃 𝑖 ⊲ 𝑀 for some 𝑃 𝑖 ∈ 𝒞 , one calculates 𝑄  ⊕ 𝑖 𝑆 𝑖 ⊆ ⊕ ⊕ 𝑖 ( 𝑃 𝑖 ⊲ 𝑀 )  (⊕ 𝑖 𝑃 𝑖 ) ⊲ 𝑀 def = 𝑃 ⊲ 𝑀 . □ 8 . 3 r e co n s t ru c t i o n f o r l a x m o d u l e e n d o f u nc to r s The following i s our most general module categorical reconstruction result. Theorem 8.48. Let 𝒞 be a monoidal abelian category with enough projectiv es, ℳ an abelian 𝒞 -module category with enough projectives, and assume that ℓ ∈ ℳ is a closed 𝒞 -pr ojective 𝒞 -gener ator . Then ther e is an equivalence of 𝒞 -module categories ℳ ≃ 𝒞 ⌊ ℓ , − ⊲ ℓ ⌋ , wher e 𝒞 ⌊ ℓ , − ⊲ ℓ ⌋ is endowed with the ext ended 𝒞 -module structur e by means of Linton coequaliser s, see Theorem 8.25 . 198 8.3. Recons tr uction for lax module endofunctors Proof. F ollowing Example 2.50 , the functor − ⊲ ℓ is a strong 𝒞 -module func- tor . By P orism 5.29 , its right adjoint ⌊ ℓ , −⌋ is a lax 𝒞 -module functor . Thus, the resulting monad ⌊ ℓ , − ⊲ ℓ ⌋ is a right exact lax 𝒞 -module monad. Pro- position 5.35 yields that t he comparison functor 𝒞 ⌊ ℓ , − ⊲ ℓ ⌋ − → ℳ is a strong 𝒞 -module functor . Further more, due to Propositions 8.42 and 8.45 , ⌊ ℓ , −⌋ is right exact and reﬂects zero objects, so w e are able to apply Theorem 2.92 , whence the com parison functor ℳ − → 𝒞 ⌊ ℓ , − ⊲ ℓ ⌋ is an equiv alence. T ransport- ing t he 𝒞 -module structure along this equivalence, w e obtain an extended 𝒞 -module structure on 𝒞 ⌊ ℓ , − ⊲ ℓ ⌋ , which, by Theorem 8.9 , is necessarily t hat of Theorem 8.25 ; this prov es the result. □ Using Theorem 8.25 , w e can also formulate a con v erse. Theorem 8.49. Let 𝒞 be a monoidal abelian category with enough projectiv es, and let 𝑇 be a right exact lax 𝒞 -module monad on 𝒞 . Then 𝑇 1 is a closed 𝒞 -pr ojective 𝒞 -g enerat or in 𝒞 𝑇 , wit h t he 𝒞 -module category structur e on the latter given by Theor em 8.25 . There is a bĳection { ( ℳ , ℓ ) as in Theorem 8.48 } ⧸ ℳ ≃ 𝒩  ← →  Right exact lax 𝒞 -module monads on 𝒞  / 𝒞 𝑇 ≃ 𝒞 𝑆 ( ℳ , ℓ ) ↦− → ⌊ ℓ , − ⊲ ℓ ⌋ ( 𝒞 𝑇 , 𝑇 1 ) ← − [ 𝑇 Proof. The fact that 𝑇 1 is closed follo ws immediately from t he isomor phisms 𝒞 𝑇 (− ⊲ 𝑇 1 , = ) ( 𝑖 )  𝒞 𝑇 ( 𝑇 (− ⊗ 1 ) , = ) ≃ 𝒞 (− ⊗ 1 , 𝑈 𝑇 ( = )) ≃ 𝒞 (− , 𝑈 𝑇 ( = )) sho wing that ⌊ 𝑇 1 , −⌋  𝑈 𝑇 , where ( 𝑖 ) follow s from Lemma 8.23 . F or 𝑃 ∈ 𝒞 -proj projectiv e, 𝑃 ⊲ 𝑇 ( 1 )  𝑇 𝑃 is projectiv e by Proposition 8.5 . By the same result, 𝑇 1 is a 𝒞 -projectiv e generat or , and 𝒞 𝑇 has enough projectiv es. F or the latter claim, notice that ℳ ≃ 𝒞 ⌊ 𝑇 1 , − ⊲ 𝑇 1 ⌋ b y Theorem 8.48 . Lastl y , 𝒞 𝑇 ≃ 𝒞 ⌊ 𝑇 1 , − ⊲ 𝑇 1 ⌋ holds, since from ⌊ 𝑇 1 , −⌋  𝑈 𝑇 and, b y uniqueness of adjoints, 𝐹 𝑇  − ⊲ 𝑇 1 w e deduce t hat 𝑇  ⌊ 𝑇 1 , − ⊲ 𝑇 1 ⌋ . □ Using Propositions 8.43 and 8.46 in place of Propositions 8.42 and 8.45 , w e ﬁnd dual statements. 199 8. Infi nite and n on -rig id re c on s tructi on Theorem 8.50. Let 𝒞 be a monoidal abelian category wit h enough injectives, ℳ an abelian 𝒞 -module category wit h enough injectives, and assume t hat ℓ ∈ ℳ be a coclosed 𝒞 -injective 𝒞 -cogener ator . Then t here is an equiv alence of 𝒞 -module categories ℳ ≃ 𝒞 ⌈ ℓ , − ⊲ ℓ ⌉ , wher e 𝒞 ⌈ ℓ , − ⊲ ℓ ⌉ is endowed with the ext ended 𝒞 -module structur e of Theorem 8.25 . Theorem 8.51. Let 𝒞 be a monoidal abelian category with enough injectives, and 𝑆 a left exact oplax 𝒞 -module comonad on 𝒞 . Then 𝒞 𝑆 , endowed with the 𝒞 -module category s tructure of Theorem 8.25 , is a 𝒞 -module category , and 𝑆 1 is a coclosed 𝒞 -injective 𝒞 -gener ator . Ther e is a bĳection { ( ℳ , ℓ ) as in Theorem 8.50 } ⧸ ℳ ≃ 𝒩 ↔  Left exact oplax 𝒞 -module comonads on 𝒞  / 𝒞 𝑆 ≃ 𝒞 𝐺 ( ℳ , ℓ ) ↦− → ⌈ ℓ , − ⊲ ℓ ⌉  𝒞 𝑆 , 𝑆 1  ← − [ 𝑆 Observe t hat Theorems 8.48 and 8.50 do no t make an y ﬁniteness assump- tions on ℳ — such conditions are often im posed on it by the exis tence of a closed 𝒞 -projectiv e 𝒞 -gener ator , or a coclosed 𝒞 -injectiv e 𝒞 -cogenerat or . Proposition 8.52. Let 𝒞 be a ﬁnite abelian monoidal category and suppose ℳ is an abelian 𝒞 -module category , such t hat ther e exists a closed 𝒞 -projective 𝒞 -g ener ator ℓ ∈ ℳ . Then ℳ is ﬁnite abelian. Proof. By Theorem 8.48 , w e hav e a 𝒞 -module equivalence ℳ ≃ 𝒞 ⌊ ℓ , − ⊲ ℓ ⌋ . Since − ⊲ ℓ admits a right adjoint it is right exact. Since ℓ is 𝒞 -projectiv e, t he functor ⌊ ℓ , −⌋ is also right exact. Thus ⌊ ℓ , − ⊲ ℓ ⌋ is a right exact monad on the ﬁnite abelian category 𝒞 . By Proposition 8.5 , 𝒞 ⌊ ℓ , − ⊲ ℓ ⌋ is ﬁnite abelian. □ Using Proposition 8.3 in place of Proposition 8.5 , one obtains the following. Proposition 8.53. Let 𝒞 be a locall y ﬁnite abelian monoidal category , and let ℳ be an abelian 𝒞 -module category wit h enough injectives, such that ther e exis ts a coclosed Ind ( 𝒞 ) -injective Ind ( 𝒞 ) -gener ator ℓ ∈ ℳ . Then the full subcategory of compact objects in ℳ is locally ﬁnite abelian. In the presence of suitable ﬁniteness conditions, t he characterisations of adjoint functors betw een ﬁnite and locally ﬁnite abelian categories giv e suﬃ- cient conditions for an object in a module categor y to be closed or coclosed. 200 8.3. Recons tr uction for lax module endofunctors Proposition 8.54. Let 𝒞 be a ﬁnite abelian monoidal category and let ℳ be a ﬁnite abelian 𝒞 -module category . Then an object 𝑚 ∈ ℳ is closed if and only if t he funct or − ⊲ 𝑚 is right exact. Proof. This is an immediate consequence of Proposition 2.138 . □ Similar l y , using Proposition 2.137 , one show s t he follo wing. Proposition 8.55. Let 𝒞 be a locally ﬁnite abelian monoidal category and ℳ a locally ﬁnite abelian 𝒞 -module category . An object ℓ ∈ Ind ( ℳ ) is coclosed wit h r espect t o t he induced Ind ( 𝒞 ) -module s tructure on Ind ( ℳ ) if and only if − ⊲ ℓ : 𝒞 − → Ind ( ℳ ) is left exact and t he induced functor − ⊲ ℓ : Ind ( 𝒞 ) − → Ind ( ℳ ) is quasi-ﬁnit e. 8.3.1 Rigid monoidal and (ﬁnite) tensor categories We ob t ain an alg ebrai c re c o nstr uc tion re sul t in case 𝒞 is rigid monoidal. Theorem 8.56. Let 𝒞 be a rigid monoidal abelian category wit h enough projectiv es, ℳ an abelian 𝒞 -module cat egor y wit h enough projectiv es, and assume that ℓ ∈ ℳ is a closed 𝒞 -pr ojective 𝒞 -g enerat or . Then t here is an algebr a object 𝐴 ∈ 𝒞 such t hat t here is an equiv alence of 𝒞 -module categories mod 𝒞 𝐴 ≃ ℳ . Proof. By Theorem 8.48 , w e hav e 𝒞 ⌊ ℓ , − ⊲ ℓ ⌋ ≃ ℳ as 𝒞 -module categories. By Proposition 2.73 , t he monad ⌊ ℓ , − ⊲ ℓ ⌋ : 𝒞 − → 𝒞 is a strong 𝒞 -module functor , as it is lax and 𝒞 is rigid. The claim follo ws by Proposition 5.23 . □ Theorem 8.57. Let 𝒞 be a rigid monoidal abelian cat egor y with enough injectives, ℳ an abelian 𝒞 -module category , and assume that ℓ ∈ ℳ is a coclosed 𝒞 -injectiv e 𝒞 - cog enerat or . Then t her e is a coalg ebra object 𝐶 ∈ 𝒞 such t hat ther e is an equivalence of 𝒞 -module categories comod 𝒞 𝐶 ≃ ℳ . Recall the follo wing result in case t hat 𝒞 is a multitensor category . Theorem 8.58 ([ EGN O15 , Theorem 7.10.1]) . Let 𝒞 be a ﬁnite multitensor category and let ℳ be an abelian 𝒞 -module category , such that • t he functor − ⊲ = is exact in the ﬁrst variable (as 𝒞 is rigid, it is alway s exact in t he second variable); • t here exis ts a 𝒞 -projectiv e 𝒞 -gener ator . 201 8. Infi nite and n on -rig id re c on s tructi on Then t here is an alg ebra object 𝐴 in 𝒞 such t hat ℳ ≃ mod 𝒞 𝐴 . W e gener alise t his statement to the locally ﬁnite setting. Theorem 8.59. Let 𝒞 be a multitensor category , and let ℳ be an abelian 𝒞 -module category such that the Ind ( 𝒞 ) -module category Ind ( ℳ ) admits a coclosed Ind ( 𝒞 ) - injective Ind ( 𝒞 ) -cogener ator ℓ . Then ther e is a coalgebr a object 𝐶 in Ind ( 𝒞 ) such t hat Ind ( ℳ ) ≃ Comod Ind ( 𝒞 ) 𝐶 . Thus, ℳ is t he category of compact 𝐶 -comodule objects. Proof. By Theorem 8.50 , w e ha v e Ind ( ℳ ) ≃ Ind ( 𝒞 ) ⌈ ℓ , − ⊲ ℓ ⌉ as Ind ( 𝒞 ) -module categories. Obser v e t hat the Ind ( 𝒞 ) -module categor y s tructure on both Ind ( 𝒞 ) and on Ind ( ℳ ) is the ﬁnitar y extension of t he respectiv e 𝒞 -module categor y structures, follo wing Proposition 2.129 . Similarl y , − ⊲ ℓ : Ind ( 𝒞 ) − → Ind ( ℳ ) is the extension of − ⊲ ℓ : 𝒞 − → Ind ( ℳ ) . On t he ind-completion, t he lef t adjoint ⌈ ℓ , −⌉ : Ind ( ℳ ) − → Ind ( 𝒞 ) , being ﬁnitar y and preser ving compact objects by Lemma 2.139 , restricts to an oplax 𝒞 -module functor ⌈ ℓ , −⌉ : ℳ − → 𝒞 . Since 𝒞 is rigid, the restricted 𝒞 -module functor ⌈ ℓ , −⌉ : ℳ − → 𝒞 is in fact a strong 𝒞 -module functor . Its ﬁnitar y extension ⌈ ℓ , −⌉ : Ind ( ℳ ) − → Ind ( 𝒞 ) is a strong Ind ( 𝒞 ) -module functor , and t he comonad ⌈ ℓ , − ⊲ ℓ ⌉ is a strong Ind ( 𝒞 ) -module monad. By Proposition 5.23 , it is of the form − ⊗ 𝐶 for a coalgebr a object in Ind ( 𝒞 ) , and hence Ind ( ℳ ) ≃ Ind ( 𝒞 ) ⌈ ℓ , − ⊲ ℓ ⌉ ≃ Comod Ind ( 𝒞 ) 𝐶 . By Proposition 8.3 , ℳ consists of compact 𝐶 -comodule objects of 𝒞 . □ Corollar y 8.60. Let 𝐻 be a Hopf algebr a and let 𝒞 = 𝐻 vect ; in particular , we hav e a monoidal equiv alence Ind ( 𝒞 ) ≃ 𝐻 V ect . Let ℳ be an abelian 𝒞 -module category such t hat the Ind ( 𝒞 ) -module category Ind ( ℳ ) admits a coclosed Ind ( 𝒞 ) -injective Ind ( 𝒞 ) -cog enerat or . Then ther e is an 𝐻 -comodule coalgebr a 𝐶 such that ther e is an equiv alence Ind ( ℳ ) ≃ Comod 𝐻 𝐶 of Ind ( 𝒞 ) -module categories, r estricting t o a 𝒞 -module equiv alence ℳ ≃ comod 𝐻 𝐶 . 8 . 4 a n e i l e n b e rg – w at t s t h e o r e m f o r l a x m o d u l e m o na d s The f or mulation o f th e b ijec tion of Theorem 8.51 indicates a Morita aspect to t he reconstruction theor y for module categories, as the equiv alence relation imposed on monads strongl y resembles — and can be specialised to — Morita equiv alence of algebr as. 202 8.4. An Eilenberg– W atts theorem for lax module monads In Theorem 8.72 , w e characterise t he right exact lax 𝒞 -module functors betw een Eilenberg –Moore categories for lax 𝒞 -module monads in ter ms of suitable bimodule objects in t he categor y of endofunctors of 𝒞 — t his giv es a precise meaning to the notion of Morita equiv alence of monads, hence Theorem 8.75 yields a bĳection { ( ℳ , ℓ ) as in Theorem 8.50 } ⧸ ℳ ≃ 𝒩 ≃ ← →  Left exact oplax 𝒞 -mod- ule comonads on 𝒞   ≃ Morita ( ℳ , ℓ ) ↦− → ⌈ ℓ , − ⊲ ℓ ⌉ ( ℳ 𝑇 , 𝑇 1 ) ← − [ 𝑇 Let us ﬁrst introduce some v ocabular y . Deﬁnition 8.61. Let 𝒜 be an abelian category and suppose t hat 𝑇 and 𝑆 are right exact monads on 𝒜 . Viewing t hem as algebra objects in Re x ( 𝒜 , 𝒜 ) , a 𝑇 - 𝑆 -biact functor is an 𝑇 - 𝑆 -bimodule object in Re x ( 𝒜 , 𝒜 ) . In other w ords, a biact functor 𝐹 : 𝒜 − → 𝒜 is a triple ( 𝐹 , 𝐹 la , 𝐹 ra ) , consisting of a right exact endofunctor on 𝒜 tog et her with transf ormations 𝐹 la : 𝑇 𝐹 = ⇒ 𝐹 and 𝐹 ra : 𝐹 𝑆 = ⇒ 𝐹 , satisfying t he natural unitality and associativity axioms, and commuting with each other in t he sense t hat 𝑇 𝐹 𝑆 𝑇 𝐹 𝐹 𝑆 𝐹 𝑇 𝐹 ra 𝐹 la 𝑆 𝐹 la 𝐹 ra (8.4.1) commutes. The categor y of 𝑇 - 𝑆 -biact functors shall be deno ted b y 𝑇 - Biact - 𝑆 . Hypothesis 8.62. F or the rest of t his section, let us ﬁx tw o right exact monads 𝑇 and 𝑆 on an abelian category 𝒜 . Remark 8.63. For a 𝑇 - 𝑆 -biact functor 𝐹 , t he lef t 𝑇 -action 𝐹 la endo ws an y object of t he form 𝐹 𝑎 , for 𝑎 ∈ 𝒜 , wit h t he structure of a 𝑇 -module by deﬁning ∇ 𝐹 𝑎 . . = 𝐹 la ; 𝑎 . An y morphism 𝑓 ∈ 𝒜 ( 𝑎 , 𝑏 ) lifts to a map 𝐹 𝑓 : 𝐹 𝑎 − → 𝐹 𝑏 of 𝑇 -modules by naturality of 𝐹 la . In particular , t he functor 𝐹 factors through the canonical forg etful functor 𝑈 𝑇 : 𝒞 𝑇 − → 𝒞 . W e write 𝐹 = 𝑈 𝑇 ◦ ˜ 𝐹 . 203 8. Infi nite and n on -rig id re c on s tructi on Deﬁnition 8.64. Giv en a 𝑇 - 𝑆 -biact functor 𝐹 , deﬁne ˜ 𝐹 ◦ 𝑆 − : 𝒜 𝑆 − → 𝒜 𝑇 b y ˜ 𝐹 ◦ 𝑆 − . . = coeq  𝐹 𝑆 𝐹 𝐹 ∇ 𝐹 ra  , (8.4.2) where for 𝑥 ∈ 𝒜 𝑆 w e endow 𝐹 𝑥 and 𝐹 𝑆 𝑥 with t he 𝑇 -module structures described in Remar k 8.63 . The functoriality of Deﬁnition 8.64 follo ws from t hat of colimits. Remark 8.65. By naturality of 𝐹 la and t he commutativity condition of Dia- gram ( 8.4.1 ), both mor phisms in Equation ( 8.4.2 ) are morphisms in 𝒜 𝑇 ; since the coequalisers in 𝒜 𝑇 are created b y 𝑈 𝑇 , this coequaliser is one in 𝒜 𝑇 . Proposition 8.66. Let 𝑇 and 𝑆 be right exact monads on an abelian category 𝒜 . W riting 𝜀 𝑇 . . = 𝜀 𝐹 𝑇 ⊣ 𝑈 𝑇 and 𝜀 𝑆 . . = 𝜀 𝐹 𝑆 ⊣ 𝑈 𝑆 , the following assignments ext end to an equiv alence of categories: N otice t he similarity of t his assignment and Deﬁnition 6.1 , only that w e are now pushing Φ along two adjunctions. Re x ( 𝒜 𝑆 , 𝒜 𝑇 ) ← → 𝑇 - Biact - 𝑆 Φ ↦− →  𝑈 𝑇 Φ 𝐹 𝑆 , 𝑈 𝑇 𝜀 𝑇 Φ 𝐹 𝑆 , 𝑈 𝑇 Φ 𝜀 𝑆 𝐹 𝑆  ˜ 𝐹 ◦ 𝑆 − ← − [ 𝐹 . Proof. Since 𝐹 𝑆 is right exact and 𝑈 𝑇 is exact, 𝑈 𝑇 Φ 𝐹 𝑆 is right exact. It is easy to v erify that the transf ormations 𝑈 𝑆 𝜀 𝑇 Φ 𝐹 𝑇 and 𝑈 𝑇 Φ 𝜀 𝑆 𝐹 𝑆 endo w 𝑈 𝑇 Φ 𝐹 𝑆 with the structure of a biact functor , and t hat for a natural tr ansforma- tion 𝛼 : Φ = ⇒ Φ ′ , the resulting transformation 𝑈 𝑇 𝛼 𝐹 𝑆 is a morphism of biact functors. This prov es the functoriality of Re x ( 𝒜 𝑆 , 𝒜 𝑇 ) − → 𝑇 - Biact - 𝑆 . T o see that t he con v erse assignment is functorial, obser v e t hat giv en a morphism 𝑓 : 𝐹 = ⇒ 𝐺 of biact functors, w e obtain a mor phism of for k s in 𝒜 𝑇 : 𝐹 𝑆 𝑥 𝐹 𝑥 𝐺 𝑆 𝑥 𝐺 𝑥 𝐹 ∇ 𝑥 𝐹 ra; 𝑥 𝑓 𝑆 𝑥 𝑓 𝑥 𝐺 ∇ 𝑥 𝐺 ra; 𝑥 This induces a morphism of coequalisers ˜ 𝐹 ◦ 𝑆 − = ⇒ ˜ 𝐺 ◦ 𝑆 − . 204 8.4. An Eilenberg– W atts theorem for lax module monads W e now prov e t hat t hese functors deﬁne mutuall y quasi-in v erse equival- ences. First, obser v e t hat  𝑈 𝑇 Φ 𝐹 𝑆 = Φ 𝐹 𝑆 : 𝒜 − → 𝒜 𝑇 , since 𝑈 𝑇 is fait hful and injectiv e on objects, so it is a monomor phism in t he 1-categor y Cat k . By slight abuse of notation, see Remar k 8.65 , for 𝑥 ∈ 𝒜 𝑆 w e ha v e ( Φ 𝐹 𝑆 ) ◦ 𝑆 𝑥 = coeq  Φ 𝑆 2 𝑥 Φ 𝑆 𝑥 Φ 𝜇 𝑆 𝑥 Φ 𝑆 ∇ 𝑥  , which, since Φ is right exact, is isomorphic to Φ  coeq  𝑆 2 𝑥 𝑆 𝑥 𝜇 𝑆 𝑥 𝑆 ∇ 𝑥   . V ia t he action ∇ 𝑥 : 𝑇 𝑥 − → 𝑥 , the latter coequaliser is canonically isomor phic to 𝑥 , hence w e obtain the follo wing isomorphism, which is natural in Φ : Φ ˆ ∇ 𝑥 : ( Φ 𝐹 𝑆 ) ◦ 𝑆 𝑥 ∼ − → Φ 𝑥 . Con v ersely , let 𝐹 be a biact functor and 𝑥 ∈ 𝒜 . W e ha v e  𝑈 𝑇 ( ˜ 𝐹 ◦ 𝑆 −) 𝐹 𝑆  ( 𝑥 ) = coeq  𝐹 𝑆 2 𝑥 𝐹 𝑆 𝑥 𝐹 ra ; 𝑆 𝑥 𝐹 ∇ 𝑆 𝑥 = 𝐹 𝜇 𝑆 𝑥  Observe that 𝐹 , being a right 𝑆 -module object, is presented as a coequaliser in t he categor y of such objects, which additionally splits in t he underl ying monoidal category Rex ( 𝒜 , 𝒜 ) . This is indicated in the follo wing diagram: 𝐹 𝑆 2 𝐹 𝑆 coeq ( 𝐹 ra 𝑆 , 𝐹 𝜇 𝑆 ) 𝐹 𝐹 ra 𝑆 𝐹 𝜇 𝑆 𝐹 ra ∃ ! 𝐹 ra ≃ 𝐹 𝜂 𝑆 Since 𝐹 ra is a mor phism of left 𝑇 -module objects, w e ﬁnd t hat 𝐹 ra deﬁnes an isomorphism of biact functors 𝑈 𝑇 ( ˜ 𝐹 ◦ 𝑆 −) 𝐹 𝑆  𝐹 t hat is natural in 𝐹 . □ Let us no w assume 𝒞 and ℳ to be abelian, − ⊲ = to be right exact in bo t h v ariables, and 𝑆 and 𝑇 to be right exact lax 𝒞 -module monads on ℳ . 205 8. Infi nite and n on -rig id re c on s tructi on Deﬁnition 8.67. An 𝑆 - 𝑇 -bimodule object in the categor y Re xLax 𝒞 Mod ( ℳ , ℳ ) of right exact lax 𝒞 -module endofunctors of ℳ is called a lax 𝒞 -module 𝑆 - 𝑇 - biact functor . W e denote the categor y of 𝑆 - 𝑇 -biact functors b y 𝑆 - Biact 𝒞 - 𝑇 . In other wor ds, a lax 𝒞 -module biact functor is a biact functor ( 𝐹 , 𝐹 la , 𝐹 ra ) , such t hat 𝐹 : ℳ − → ℳ is a right exact lax 𝒞 -module functor and 𝐹 la and 𝐹 ra are 𝒞 -module transf ormations. Theorem 8.68. The equiv alence Re x ( ℳ 𝑇 , ℳ 𝑆 ) ≃ 𝑆 - Biact - 𝑇 of Pr oposition 8.66 r estricts to a fait hful functor 𝒞 EW : LaxRex ( ℳ 𝑇 , ℳ 𝑆 ) − → 𝑆 - Biact 𝒞 - 𝑇 , such t hat the following diagram commutes: LaxRe x ( ℳ 𝑇 , ℳ 𝑆 ) 𝑆 - Biact 𝒞 - 𝑇 Re x ( ℳ 𝑇 , ℳ 𝑆 ) 𝑆 - Biact - 𝑇 𝒞 EW ≃ by 8.66 (8.4.3) The vertical arrow s are the respectiv e f or getful functor s, and the categories ℳ 𝑇 and ℳ 𝑆 ar e endowed with the 𝒞 -module structur es of Theorem 8.25 . Proof. By Theorem 8.26 , similar ly to [ AHLF18 , Proposition 3.10], the func- tor 𝐹 𝑇 : ℳ − → ℳ 𝑇 is a strong 𝒞 -module functor , and so is 𝐹 𝑆 . Thus, by P orism 5.29 , the functor 𝑈 𝑆 is a lax 𝒞 -module functor , and hence for a lax 𝒞 -module functor Φ : ℳ 𝑇 − → ℳ 𝑆 , t he composite 𝑈 𝑆 Φ 𝐹 𝑇 is a lax 𝒞 -module functor . Further , t he 𝑆 - 𝑇 -biact structure on 𝑈 𝑆 Φ 𝐹 𝑆 giv en in Proposition 8.66 assembles to a lax 𝒞 -module biact functor , and this extends also to 𝒞 -module biact transf ormations arising from 𝒞 -module transformations. This deﬁnes a funct or 𝒞 EW , such t hat Diagram ( 8.4.3 ) commutes. It is faithful, since so are the remaining functors in t hat diag r am. □ Lemma 8.69. Let Φ ∈ Re x ( ℳ 𝑇 , ℳ 𝑆 ) and for all 𝑚 ∈ ℳ suppose that ( Φ 𝑇 ) a ; 𝑚 : 𝑥 ⊲ ℳ Φ 𝑇 𝑚 − → Φ 𝑇 ( 𝑥 ⊲ ℳ 𝑚 ) is a lax 𝒞 -module biact funct or s tructur e on the biact funct or 𝑈 𝑆 Φ 𝐹 𝑇 . Then w e ha ve ( Φ 𝑇 ) a = 𝒞 EW ( Φ a ; 𝑇 (−) ) , wher e Φ a ; 𝑇 (−) : 𝑥 ⊲ ℳ 𝑆 Φ 𝑇 (−) = ⇒ Φ ( 𝑥 ⊲ ℳ 𝑇 𝑇 (−)) = Φ ( 𝑇 ( 𝑥 ⊲ ℳ −)) 206 8.4. An Eilenberg– W atts theorem for lax module monads is t he unique natural transf ormation making t he following diagram commute: 𝑥 ⊲ ℳ 𝑆 Φ 𝑇 − 𝑆 ( 𝑥 ⊲ ℳ Φ 𝑇 −) 𝑆 Φ 𝑇 𝑥 ⊲ ℳ − Φ 𝑇 𝑥 ⊲ ℳ − ∃ ! Φ a 𝑆 ( Φ 𝑇 ) a Φ la 𝑇 𝑥 ⊲ − (8.4.4) Proof. In order to v erify that Φ a is well-deﬁned, obser v e t hat t he morphism ( Φ la 𝑇 𝑥 ⊲ ℳ −) ◦ 𝑆 ( Φ 𝑇 ) a coequalises 𝑆 𝑆 a ; 𝑥 Φ 𝑇 and 𝑆 𝑥 ⊲ ℳ Φ la 𝑇 : 𝑆 ( 𝑉 ⊲ ) 𝑆 Φ 𝑇 Φ ( 𝑉 ⊲ ) 𝑇 𝑆 ( 𝑉 ⊲ ) 𝑆 Φ 𝑇 Φ ( 𝑉 ⊲ ) 𝑇 ( 1 ) = 𝑆 ( 𝑉 ⊲ ) 𝑆 Φ 𝑇 Φ ( 𝑉 ⊲ ) 𝑇 ( 2 ) = Abo v e, ( 1 ) follo ws by ( Φ 𝑇 ) la being a left 𝑆 -act structure, and ( 2 ) b y it being a 𝒞 -module transf ormation. By the deﬁnition of t he functor 𝒞 EW , w e ha ve 𝒞 EW ( Φ a ; 𝑇 (−) ) = Φ a ; 𝑇 (−) ◦ 𝑈 𝑆 a ; 𝑥 , Φ 𝑇 (−) , where 𝑈 𝑆 a is t he lax 𝒞 -module functor structure on 𝑈 𝑆 induced by Porism 5.29 . More gener ally , for 𝑚 ∈ ℳ t he map 𝑈 𝑆 a ; 𝑥 , 𝑚 is giv en by the composite 𝑥 ⊲ ℳ 𝑚 𝜂 𝑆 𝑥 ⊲ ℳ 𝑚 − − − − → 𝑆 ( 𝑥 ⊲ ℳ 𝑚 ) − ↠ 𝑥 ⊲ ℳ 𝑆 𝑚 , with t he latter map being t he projection onto t he follo wing coequaliser: coeq  𝑇 ( 𝑥 ⊲ 𝑇 2 𝑚 ) 𝑇 ( 𝑥 ⊲ 𝑇 𝑚 )  𝑆 ( 𝑥 ⊲ 𝑚 ) coeq  𝑇 ( 𝑥 ⊲ 𝑇 2 𝑚 ) 𝑇 𝑥 ⊲ 𝑇 𝑚  𝑥 ⊲ 𝑚 𝑇 ( 𝑥 ⊲ ∇ 𝑚 ) 𝑇 ( 𝑥 ⊲ ∇ 𝑚 ) All in all, w e ﬁnd t hat 𝒞 EW ( Φ a ; 𝑇 (−) ) is giv en by the outer pat h of 𝑥 ⊲ ℳ 𝑆 Φ 𝑇 − 𝑆 ( 𝑥 ⊲ ℳ Φ 𝑇 −) 𝑆 ( 𝑥 ⊲ ℳ Φ 𝑇 −) 𝑆 Φ 𝑇 𝑥 ⊲ ℳ − Φ 𝑇 𝑥 ⊲ ℳ − ∃ ! Φ a 𝜂 𝑥 ⊲ ℳ Φ 𝑇 − 𝑆 ( Φ 𝑇 ) a Φ la 𝑇 𝑥 ⊲ − which, b y the unitality of t he 𝑆 -act structure ( Φ 𝑇 ) la , equals ( Φ 𝑇 ) a . □ 207 8. Infi nite and n on -rig id re c on s tructi on F or 𝑚 ∈ ℳ 𝑇 , let Φ a ; 𝑚 be the unique map such t hat 𝑥 ⊲ ℳ 𝑆 Φ 𝑇 𝑚 𝑥 ⊲ ℳ 𝑆 Φ 𝑚 Φ 𝑇 ( 𝑥 ⊲ 𝑚 ) Φ ( 𝑥 ⊲ ℳ 𝑇 𝑇 𝑚 ) Φ ( 𝑥 ⊲ ℳ 𝑇 𝑚 ) 𝑥 ⊲ ℳ 𝑆 Φ ∇ 𝑚 Φ a ; 𝑇 𝑚 ∃ ! Φ a ; 𝑚 = Φ ( 𝑥 ⊲ ℳ 𝑇 ∇ 𝑚 ) (8.4.5) commutes. By the uniqueness in the deﬁning property of Φ a ; 𝑚 , t his deﬁnes a natural transf ormation Φ a : − ⊲ ℳ 𝑆 Φ ( = ) = ⇒ Φ (− ⊲ ℳ 𝑇 = ) . Lemma 8.70. F or Φ ∈ Re x ( ℳ 𝑇 , ℳ 𝑆 ) , the map Φ a : − ⊲ ℳ 𝑆 Φ ( = ) = ⇒ Φ (− ⊲ ℳ 𝑆 = ) ext ended from t he mor phisms Φ a ; 𝑇 𝑚 of Diagram ( 8.4.5 ) deﬁnes a lax 𝒞 -module structur e on Φ . Proof. In order to simplify notation, w e write ⊲ for ⊲ ℳ and ▶ for ⊲ ℳ 𝑇 and ⊲ ℳ 𝑆 , since each occurrence of eit her of these symbols is unambiguous wit h respect to which is used. W e also write 𝑥 ⊲ for 𝑥 ⊲ − , and suppress t he horizontal composition symbols in C at k , replacing t hem simpl y by concatenation. Hence, ev ery occurrence of ◦ in t he diag r ams that follo w is a vertical composition of natural transf ormations. For example, 𝑆 ( 𝑦 ⊲ ) 𝑆 ( 𝑥 ⊲ ) Φ 𝑇 𝑆 ( 𝑦 ⊲ )( Φ la 𝑇 ( 𝑥 ⊲ )◦ 𝑆 ( Φ 𝑇 ) a , 𝑥 ) − − − − − − − − − − − − − − − − − − → 𝑆 ( 𝑦 ⊲ ) Φ 𝑇 ( 𝑥 ⊲ ) represents the natural transf ormation 𝑆 ( 𝑦 ⊲ 𝑆 ( 𝑥 ⊲ Φ 𝑇 (−))) 𝑆 ( 𝑦 ⊲ 𝑆 ( Φ 𝑇 ) a ; 𝑦 ,𝑥 ) − − − − − − − − − − − → 𝑆 ( 𝑦 ⊲ 𝑆 Φ 𝑇 ( 𝑥 ⊲ −)) 𝑆 ( 𝑦 ⊲ Φ la 𝑇 ( 𝑥 ⊲ −)) − − − − − − − − − − − → 𝑆 ( 𝑦 ⊲ Φ 𝑇 ( 𝑥 ⊲ −)) . Since Φ a is uniquel y determined by Φ a ; 𝑇 (−) , it suﬃces to show t hat 𝑦 ▶ Φ ( 𝑥 ▶ 𝑇 ) Φ ( 𝑦 ▶ 𝑥 ▶ 𝑇 ) 𝑦 ▶ 𝑥 ▶ Φ 𝑇 𝑦 ▶ Φ 𝑇 ( 𝑥 ⊲ ) Φ 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) 𝑦 ⊗ 𝑥 ▶ Φ 𝑇 Φ 𝑇 ( 𝑦 ⊗ 𝑥 ) ⊲ Φ (( 𝑦 ⊗ 𝑥 ) ▶ 𝑇 ) Φ a ; 𝑦 ,𝑥 ▶ 𝑇 Φ 𝛼 2 𝑦 ▶ Φ a ; 𝑥 ,𝑇 𝛼 1 𝛼 2 𝛼 3 𝛼 5 𝛼 4 208 8.4. An Eilenberg– W atts theorem for lax module monads commutes, where w e deﬁne 𝛼 1 . . = 𝑦 ▶ Φ a , 𝑥 , 𝛼 3 . . = Φ a , 𝑦 ( 𝑥 ⊲ ) , 𝛼 4 . . = Φ a , 𝑦 ⊗ 𝑥 , and 𝛼 5 . . = Φ 𝑇 ( ⊲ a , 𝑦 , 𝑥 ) . The mor phism 𝛼 2 is obtained from Figure 8.1 , in which ev ery marked epimorphism is t he coequaliser of t he pair preceding it, ev er y dashed arrow is an induced mor phism betw een coequalisers coming from a morphism of diagrams, and t he dotted arro ws come from t he univ ersal property of coequalisers. The notation for the coequaliser ( 𝑦 , 𝑥 ) ▶ Φ 𝑇 is to emphasise t he connection wit h the multiactegorical approach of Section 8.1.3 . 𝑆 ( 𝑦 ⊲ ) 𝑆 𝑆 ( 𝑥 ⊲ ) 𝑆 Φ 𝑇 𝑆 ( 𝑦 ⊲ ) 𝑆 𝑆 ( 𝑥 ⊲ ) Φ 𝑇 𝑆 ( 𝑦 ⊲ ) 𝑆 ( 𝑥 ▶ Φ 𝑇 ) 𝑆 ( 𝑦 ⊲ ) 𝑆 ( 𝑥 ⊲ ) 𝑆 Φ 𝑇 𝑆 ( 𝑦 ⊲ ) 𝑆 ( 𝑥 ⊲ ) Φ 𝑇 𝑆 ( 𝑦 ⊲ )( 𝑥 ▶ Φ 𝑇 ) 𝑦 ▶ 𝑥 ▶ Φ 𝑇 𝑆 ( 𝑦 ⊲ )( 𝑥 ⊲ ) 𝑆 Φ 𝑇 𝑆 ( 𝑦 ⊲ )( 𝑥 ⊲ ) Φ 𝑇 ( 𝑦 , 𝑥 ) ▶ Φ 𝑇 𝑆 ( 𝑦 ⊗ 𝑥 ⊲ ) 𝑆 Φ 𝑇 ( 𝑆 ( 𝑦 ⊗ 𝑥 ) ⊲ ) Φ 𝑇 ( 𝑦 ⊗ 𝑥 ) ▶ Φ 𝑇 𝑆 ( 𝑦 ⊲ ) 𝑆 ( 𝜇 ◦ 𝑆 𝑆 a , 𝑥 ) Φ 𝑇 𝑆 ( 𝑦 ⊲ ) 𝑆 𝑆 ( 𝑥 ⊲ ) Φ la 𝑇 ( 𝜇 ◦ 𝑆 a , 𝑦 ) 𝑆 ( 𝑥 ⊲ ) 𝑆 Φ 𝑇 𝑆 ( 𝑦 ⊲ ) 𝜇 ( 𝑥 ⊲ ) 𝑆 Φ 𝑇 ( 𝜇 ◦ 𝑆 a , y ) 𝑆 ( 𝑥 ⊲ ) Φ 𝑇 𝑆 ( 𝑦 ⊲ ) 𝜇 ( 𝑥 ⊲ ) Φ 𝑇 𝑆 ( 𝑦 ⊲ )( 𝜇 ◦ 𝑆 𝑆 a , x ) Φ 𝑇 𝑆 ( 𝑦 ⊲ ) 𝑆 ( 𝑥 ⊲ ) Φ la 𝑇 ( 𝜇 ◦ 𝑆 a , 𝑦 )( 𝑥 ⊲ ) 𝑆 Φ 𝑇 ( 𝜇 ◦ 𝑆 a , 𝑦 )( 𝑥 ⊲ ) Φ 𝑇 ∃ ! 𝛽 0  𝛼 2 𝑆 ( 𝜇 ◦( 𝑆 a , 𝑦 ( 𝑥 ⊲ ))◦(( 𝑦 ⊲ ) 𝑆 a , 𝑥 )) Φ 𝑇 𝑆 ( 𝑦 ⊲ )( 𝑥 ⊲ ) Φ la 𝑇 𝑆 ( ⊲ a , 𝑦 , 𝑥 ) 𝑆 Φ 𝑇 𝑆 ( ⊲ a , 𝑦 , 𝑥 ) Φ 𝑇 ∃ ! 𝛽 1  𝑆 ( 𝜇 ◦ 𝑆 a , 𝑦 ⊗ 𝑥 ) Φ 𝑇 𝑆 ( 𝑦 ⊗ 𝑥 ⊲ ) Φ la 𝑇 Figure 8.1: Deﬁnition of the morphism 𝛼 2 . W e now ha v e t he diag r am 𝑆 ( 𝑦 ⊲ ) 𝑆 ( 𝑥 ⊲ ) Φ 𝑇 𝑆 ( 𝑦 ⊲ ) Φ 𝑇 ( 𝑥 ⊲ ) 𝑆 ( 𝑦 ⊲ )( 𝑥 ▶ Φ 𝑇 ) 𝑆 ( 𝑦 ⊲ )( 𝑥 ⊲ ) Φ 𝑇 𝑦 ▶ 𝑥 ▶ Φ 𝑇 𝑦 ▶ Φ 𝑇 ( 𝑥 ⊲ ) Φ 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) ( 𝑦 ⊗ 𝑥 ) ▶ Φ 𝑇 𝑆 (( 𝑦 ⊗ 𝑥 ) ⊲ ) Φ 𝑇 Φ 𝑇 (( 𝑦 ⊗ 𝑥 ) ⊲ ) 𝑆 ( 𝑦 ⊲ )( Φ la 𝑇 ( 𝑥 ⊲ )◦ 𝑆 ( Φ 𝑇 ) a , 𝑥 ) ( 𝜇 ( 𝑦 ⊲ )◦ 𝑆 a , 𝑦 )( 𝑥 ⊲ ) Φ 𝑇 (( Φ la ( 𝑦 ⊲ ))◦( 𝑆 ( Φ 𝑇 ) a , 𝑦 ))( 𝑥 ⊲ ) 𝛼 ′ 1 𝑆 ( ⊲ a , 𝑦 , 𝑥 ) Φ 𝑇 𝛼 1 𝛼 2 𝛼 3 𝛼 5 𝛼 4 ( Φ la (( 𝑦 ⊗ 𝑥 ) ⊲ ))◦( 𝑆 ( Φ 𝑇 ) a , 𝑦 ⊗ 𝑥 ) ( 2 ) ( 1 ′ ) ( 1 ) ( 3 ) ( 4 ) whose labelled f aces all commute, and where the morphism decorated by the label of t he face it is part of is deﬁned as t hat making said face commute, via the univ ersal property of coequalisers. 209 8. Infi nite and n on -rig id re c on s tructi on Our aim is to show t he commutativity of t he inner unlabelled face. Since all of its mor phisms are deﬁned by t he remaining inner (commutativ e) faces, and w e can reach 𝑦 ▶ 𝑥 ▶ Φ 𝑇 from 𝑆 ( 𝑦 ⊲ ) 𝑆 ( 𝑥 ⊲ ) Φ 𝑇 with tw o epimor phisms, this is implied by t he commutativity of t he outer face, which follo ws by the commutativity of the inner faces of Figure 8.2 , where • faces (1), (3), and (4) commute by the interchange la w in C at k ; • face (2) commutes by Φ la : 𝑆 Φ 𝑇 = ⇒ Φ 𝑇 being a 𝒞 -module map; • face (5) commutes by the associativity of action Φ la ; • face (6) commutes by Φ 𝑇 being a lax 𝒞 -module functor ; and • face (7) commutes by naturality of Φ la . □ 𝑆 ( 𝑦 ⊲ ) 𝑆 ( 𝑥 ⊲ ) Φ 𝑇 𝑆 ( 𝑦 ⊲ ) 𝑆 Φ 𝑇 ( 𝑥 ⊲ ) 𝑆 ( 𝑦 ⊲ ) Φ 𝑇 ( 𝑥 ⊲ ) 𝑆 𝑆 ( 𝑦 ⊲ )( 𝑥 ⊲ ) Φ 𝑇 𝑆 𝑆 ( 𝑦 ⊲ ) Φ 𝑇 ( 𝑥 ⊲ ) 𝑆 𝑆 Φ 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) 𝑆 Φ 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) 𝑆 ( 𝑦 ⊲ )( 𝑥 ⊲ ) Φ 𝑇 𝑆 ( 𝑦 ⊲ ) Φ 𝑇 ( 𝑥 ⊲ ) 𝑆 Φ 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) Φ 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) 𝑆 (( 𝑦 ⊗ 𝑥 ) ⊲ ) Φ 𝑇 𝑆 Φ 𝑇 (( 𝑦 ⊗ 𝑥 ) ⊲ ) Φ 𝑇 (( 𝑦 ⊗ 𝑥 ) ⊲ ) 𝑆 ( 𝑦 ⊲ ) 𝑆 ( Φ 𝑇 ) a , 𝑥 𝑆 𝑆 a , 𝑦 ( 𝑥 ⊲ ) Φ 𝑇 𝑆 ( 𝑦 ⊲ ) Φ la 𝑇 ( 𝑥 ⊲ ) 𝑆 𝑆 a , 𝑦 Φ 𝑇 ( 𝑥 ⊲ ) 𝑆 ( Φ 𝑇 ) a , 𝑦 ( 𝑥 ⊲ ) 𝑆 𝑆 ( 𝑦 ⊲ )( Φ 𝑇 ) a , 𝑥 𝜇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) Φ 𝑇 𝑆 𝑆 ( Φ 𝑇 ) a , 𝑦 ( 𝑥 ⊲ ) 𝜇 ( 𝑦 ⊲ ) Φ 𝑇 ( 𝑥 ⊲ ) 𝑆 Φ la 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) 𝜇 Φ 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) Φ la 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) 𝑆 ( 𝑦 ⊲ )( Φ 𝑇 ) a , 𝑥 𝑆 ( ⊲ a , 𝑦 , 𝑥 ) Φ 𝑇 𝑆 ( Φ 𝑇 ) a , 𝑦 ( 𝑥 ⊲ ) Φ la 𝑇 ( 𝑦 ⊲ )( 𝑥 ⊲ ) 𝑆 Φ 𝑇 ( ⊲ a , 𝑦 , 𝑥 ) Φ 𝑇 ( ⊲ a , 𝑦 , 𝑥 ) 𝑆 ( Φ 𝑇 ) a , 𝑦 ⊗ 𝑥 Φ la 𝑇 (( 𝑦 ⊗ 𝑥 ) ⊲ ) ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) Figure 8.2: The arrow Φ a deﬁnes a lax 𝒞 -module structure on Φ . Lemma 8.71. Let Φ , Φ ′ ∈ LaxRex ( ℳ 𝑇 , ℳ 𝑆 ) and 𝜙 : 𝑈 𝑆 Φ 𝐹 𝑇 = ⇒ 𝑈 𝑆 Φ ′ 𝐹 𝑇 be a lax 𝒞 -module biact tr ansformation. Then 𝜑 . . = 𝜙 ◦ 𝑇 − : Φ = ⇒ Φ ′ is a 𝒞 -module tr ansformation. Proof. Firs t, observ e that 𝜑 is deter mined b y 𝜑 𝑇 (−) , as indicated b y the diag r am Φ 𝑇 2 𝑚 Φ 𝑇 𝑚 Φ 𝑚 Φ ′ 𝑇 2 𝑚 Φ ′ 𝑇 𝑚 Φ ′ 𝑚 Φ 𝜇 𝑚 Φ 𝑇 ∇ 𝑚 𝜑 𝑇 2 𝑚 𝜑 𝑇 𝑚 ∃ ! 𝜑 𝑚 Φ ′ 𝑇 ∇ 𝑚 Φ ′ 𝜇 𝑚 Thus, it suﬃces to show t hat for any 𝑚 ∈ ℳ and 𝑥 ∈ 𝒞 , w e ha v e t hat Φ ′ a ; 𝑥 ,𝑇 𝑚 ◦ ( 𝑥 ▶ 𝜙 𝑚 ) = 𝜙 𝑥 ⊲ 𝑚 ◦ Φ a , 𝑥 ,𝑇 ( 𝑚 ) . (8.4.6) 210 8.4. An Eilenberg– W atts theorem for lax module monads Consider the follo wing diagram: 𝑆 ( 𝑥 ⊲ Φ 𝑇 ) 𝑆 Φ 𝑇 𝑥 ⊲ − 𝑥 ▶ Φ 𝑇 Φ 𝑇 𝑥 ⊲ − 𝑆 ( 𝑥 ⊲ Φ ′ 𝑇 ) 𝑆 Φ ′ 𝑇 𝑥 ⊲ − 𝑥 ▶ Φ ′ 𝑇 Φ ′ 𝑇 𝑥 ⊲ − 𝑆 ( Φ 𝑇 ) a , 𝑥 𝑆 ( 𝑥 ⊲𝜙 ) Φ la 𝑇 𝑥 ⊲ − 𝑆 𝜙 ′ 𝑥 ⊲ − Φ a , 𝑥 ,𝑇 (−) 𝜙 𝑥 ⊲ − 𝑆 ( Φ ′ 𝑇 ) a , 𝑥 𝑥 ▶ 𝜙 − Φ ′ la 𝑇 𝑥 ⊲ − Φ ′ a , 𝑥 ,𝑇 (−) Its top and bott om faces commute b y deﬁnition of Φ a , as seen from Dia- gram ( 8.4.4 ); its left face commutes b y the deﬁnition of 𝑥 ▶ − ; and its right face by 𝜙 being a biact transf ormation. The back face commutes since 𝜙 is a 𝒞 -module transformation. The commutativity of the front face is precisely Equation ( 8.4.6 ), and since t he top-left projection map is an epimor phism, it suﬃces to v erify its commutativity after precomposing with it, which follo ws easil y from the commutativity of t he remaining faces. □ Theorem 8.72. The fait hful functor 𝒞 EW : LaxRex ( ℳ 𝑇 , ℳ 𝑆 ) − → 𝑆 - Biact 𝒞 - 𝑇 of Theor em 8.68 is an equiv alence of categories. Proof. Fullness follo ws from Lemma 8.71 , so it is left t o pro v e that 𝒞 EW is essentiall y sur jectiv e. Suppose t hat 𝐹 ∈ 𝑇 - Biact 𝒞 - 𝑆 . Since t he functor of Proposition 8.66 is essentially sur jectiv e, w e ma y assume t hat 𝐹  𝑈 𝑆 Φ 𝐹 𝑇 , for some right exact Φ : ℳ 𝑇 − → ℳ 𝑆 . By Lemma 8.70 , ( Φ , Φ a ) ∈ LaxRe x ( ℳ 𝑇 , ℳ 𝑆 ) , and 𝐹  𝒞 EW (( Φ , Φ a )) due to Lemma 8.69 , giving essential sur jectivity . □ Deﬁnition 8.73. W e sa y t hat tw o right exact lax 𝒞 -module monads 𝑇 and 𝑆 on ℳ are Morita equivalent if t here are 𝐹 ∈ 𝑇 - Biact 𝒞 - 𝑆 and 𝐺 ∈ 𝑆 - Biact 𝒞 - 𝑇 such that 𝐺 ◦ 𝑇 𝐹  Id ℳ 𝑇 and 𝐹 ◦ 𝑆 𝐺  Id ℳ 𝑆 as lax 𝒞 -module biact functors. Proposition 8.74. T wo right exact lax 𝒞 -module monads 𝑇 and 𝑆 on ℳ ar e Morita equiv alent if and onl y if t here is a 𝒞 -module equivalence ℳ 𝑇 ≃ ℳ 𝑆 , where the Eilen- ber g –Moore categories are endowed with the extended Linton coequaliser structur e of Theor em 8.25 . 211 8. Infi nite and n on -rig id re c on s tructi on Proof. This is a direct consequence of Theorem 8.72 , using t hat an equivalence of categories is right exact. □ Combining Proposition 8.74 with Theorem 8.51 , w e ﬁnd the following. Theorem 8.75. Let 𝒞 be a monoidal abelian category wit h enough injectives, and let 𝑇 be a left exact lax 𝒞 -module comonad on 𝒞 . There is a bĳection { ( ℳ , ℓ ) as in Theorem 8.50 } ⧸ ℳ ≃ 𝒩 ≃ ← →  Left exact oplax 𝒞 -module comonads on 𝒞   ≃ Morita ( ℳ , ℓ ) ↦− → ⌈ ℓ , − ⊲ ℓ ⌉ ( ℳ 𝑇 , 𝑇 1 ) ← − [ 𝑇 212 I am big! It’s the pictures t hat got small. Norma De smond ; Sunset Boulev ard H O P F T R I M O D U L E S 9 The orem s 8 .48 to 8 .50 giv e ge nera l re c on s truct ion res ul ts in t he non- rigid setting, which has recentl y seen increased interes t in multiple areas, see for example [ DSPS19 ; ALSW21 ] or [ Str24 b , Corollar y 10.11]. How ev er , lax 𝒞 -module monads on 𝒞 are less accessible t han mere algebra objects of 𝒞 , and ma y indeed appear not v ery accessible in gener al. T o show that t his need not be t he case, w e again turn to the Hopf-algebraic setting. Recall that a ring category — the non-rigid counterpart of a tensor categor y , see [ EGN O15 , Sections 4.2 and 5.4] — which admits a ﬁbre functor to vect is monoidall y equiv alent to t he categor y of ﬁnite-dimensional lef t 𝐵 -comodules o v er a (not necessaril y Hopf) bialgebra 𝐵 . In this chapter , w e show t hat the categor y of left exact ﬁnitar y lax 𝐵 V ect -module endofunctors of 𝐵 V ect is monoidall y equiv alent to t he categor y 𝐵 𝐵 V ect 𝐵 of Hopf trimodules 20 : 𝐵 - 𝐵 -bico- 20 The “trimodule” terminology is based on [ Sho09 ] and was suggested b y Ulrich Krähmer , in lieu of more un wieldy notations like “ ( 2 , 1 ) -Hopf module”. modules with an additional left 𝐵 -action that is a 𝐵 - 𝐵 -bicomodule morphism. Theorem 9.2 . F or a bialg ebra 𝐵 and 𝒱 . . = 𝐵 V ect , ther e is a monoidal equiv alence 𝐵 𝐵 V ect 𝐵 ≃ Le xfLax 𝒱 Mod ( 𝒱 , 𝒱 ) between the category of Hopf trimodules, and t he cat egor y of left exact ﬁnitary lax 𝒱 -module endofunctors on 𝒱 . Such structures feature prominentl y in the quasi-bialgebraic gener alisa- tion of t he fundamental t heorem of Hopf modules, see [ HN99 ; Sar17 ]. This equiv alence matches a lax 𝐵 V ect -module monad wit h a Hopf trimodule al- gebr a: a Hopf trimodule 𝐴 tog ether with maps 𝐴 □ 𝐴 − → 𝐴 and 𝐵 − → 𝐴 satisfying the usual associativity and unitality axioms for an algebr a object. If 𝐵 is inﬁnite-dimensional, Theorem 8.50 should yield reconstruction in terms of an oplax 𝐵 V ect -module comonad ⌈ 𝑋 , − ⊲ 𝑋 ⌉ , using t he existence of injectiv e objects in 𝐵 V ect . Since our result giv es an algebraic realisation onl y for the lax 𝐵 V ect -module functors, w e restrict ourselv es to t he case where 213 9. Hopf trim odul es − ⊲ 𝑋 admits both a left and a right adjoint, b y assuming − ⊲ 𝑋 to be exact. In that case, w e obtain an adjunction ⌈ 𝑋 , − ⊲ 𝑋 ⌉ ⊣ ⌊ 𝑋 , − ⊲ 𝑋 ⌋ . As t he monad is right adjoint, t he Eilenberg –Moore categor y of ⌈ 𝑋 , − ⊲ 𝑋 ⌉ is not equiv alent to t he category of modules o v er t he associated Hopf trimod- ule algebr a ⌊ 𝑋 , 𝑋 ⌋ , but rather to t he categor y of its contramodules : structures extensiv ely studied in the setting of so-called semi-inﬁnite homological al- gebr a, [ P os10 ]. On the other hand, t he Kleisli categories are equivalent, and once again control t he 𝐵 V ect -module structure, which can t hus be read oﬀ directl y from the Hopf trimodule algebra. W e obtain t he follo wing result. Theorem 9.23 . Let ℳ be a locally ﬁnite abelian 𝐵 vect -module category satisfying t he assumptions of Theorem 8.50 . There exis ts a Hopf trimodule alg ebra 𝐴 ∈ 𝐵 𝐵 vect 𝐵 , such t hat t here is an equivalence Ind ( ℳ ) ≃ 𝐴 -Contramod of 𝐵 vect -module categor - ies, where the 𝐵 vect -module structur e on t he right-hand side is extended from the category of free 𝐴 -module. This equiv alence res tricts to ℳ ≃ 𝐴 -contr amod . Since t his algebr aic realisation of our reconstruction results ma y at ﬁrst glance seem diﬃcult to apply in calculations, w e giv e tw o explicit examples in which w e deter mine a trimodule algebr a for a giv en module categor y . The ﬁrst is Section 9.4 , where 𝐵 is t he semig roup algebr a for t he unique tw o-element monoid which is not a g roup. This example is intended to be v er y similar t o [ DSPS19 , Example 2.20], in that a sim ple object of 𝒞 acts as zero on an indecomposable object. W e also show t hat applying t he ordinary , “rigid” recons truction procedure on this exam ple yields the same alg ebra object in 𝐵 vect as t hat corresponding to t he regular action of 𝐵 vect on itself, and thus fails to classify module categories. Our second exam ple is giv en in Section 9.5 . Supposing 𝐵 to be arbitrary , w e determine the Hopf trimodule algebr a under lying t he module functor giv en by the ﬁbre functor 𝐵 vect − → vect . Finall y , t he Morita-t heoretic results of Theorem 8.75 giv e us a precise notion of Morita equiv alence for Hopf trimodule algebr as wit h respect to their contramodules in Deﬁnition 9.26 . This results in a Morita Theorem for 𝐵 vect -module categories, Theorem 9.28 . As another application, w e use trimodule reconstruction to giv e a categor - ical interpretation of a variant of t he fundamental theorem of Hopf modules. Proposition 9.38 and Corollar y 9.45 . The functor 𝐵 V ect − → 𝐵 𝐵 V ect 𝐵 corr esponds to t he inclusion of str ong 𝐵 V ect -module endofunct ors in t he cat egory of lax 𝐵 V ect - module endofunctors, under t he Y oneda lemma and Theor em 9.2 . The latter — and 214 9.1. Bicomodules and their g r aphical calculus hence also t he former — functor is an equiv alence if and only if 𝐵 vect is left rigid, which is t he case if and only if 𝐵 has a twist ed antipode. In Proposition 9.49 , w e show t hat t he fusion operators of a Hopf monad 𝑇 on a monoidal category 𝒲 can be interpreted as coherence cells for the canonical oplax module structure on 𝑇 . This giv es a strong con v erse to the fact that (op)lax module functors o v er a rigid categor y are automaticall y strong, b y pro viding a distinguished module functor which is strong if and onl y if the category is rigid. This result, interpreting additional structure of a monad on a monoidal categor y as mor phisms for an (op)lax module functor structure on said monad, is similar to the results of [ FLP24 , Theorems 3.17 and 3.18], which studies Frobenius monoidal functors rather than Hopf monads. 9 . 1 b i co m o d u l e s a n d t h e i r g r a p h i c a l c a l c u lu s Befo re in tr od ucing H opf t rimo dule s , let us ﬁrst talk about bicomodules o v er 𝐵 in t he sense of Section 2.6 and their string diag r ammatic representation. Hypothesis 9.1. F or this chap ter , ﬁx a ﬁeld k and a bialgebr a 𝐵 o v er it. W rite 𝒱 . . = 𝐵 V ect for t he categor y of left 𝐵 -comodules. After introducing all of t he relev ant concepts and notation, our ﬁrst goal is to prov e the follo wing theorem. Theorem 9.2. Ther e is a monoidal equiv alence 𝐵 𝐵 V ect 𝐵 − → Le xfLax 𝒱 Mod ( 𝒱 , 𝒱 ) 𝑋 ↦− → ( 𝑋 □ − , 𝜒 ) (9.1.1) between the category of Hopf trimodules, and the category of left exact ﬁnitary lax 𝒱 - module endofunctor s on 𝒱 , where 𝜒 is the inter chang e mor phism of Deﬁnition 9.6 , see Equation ( 9.2.4 ) The quasi-in v erse of Equation ( 9.1.1 ) ev aluates a lef t exact ﬁnitar y lax 𝒱 - module functor at the injectiv e g enerat or 𝐵 . W e defer the proof of Theorem 9.2 until Section 9.2 . Notation 9.3. A bicomodule ov er 𝐵 consists of a v ector space 𝑋 , a left coaction 𝜆 : 𝑋 − → 𝐵 ⊗ k 𝑋 , and a right coaction 𝜌 : 𝑋 − → 𝑋 ⊗ k 𝐵 , such t hat ( 𝐵 ⊗ k 𝜌 ) ◦ 𝜆 = ( 𝜆 ⊗ k 𝐵 ) ◦ 𝜌 . 215 9. Hopf trim odul es In graphical notation, w e write = 𝐵 𝑋 𝐵 𝐵 𝐵 𝑋 𝑋 𝑋 (9.1.2) Exam ple 9.4. Analogousl y to Example 2.108 , w e can form t he cotensor product of tw o bicomodules ( 𝑋 , 𝜆 𝑋 , 𝜌 𝑋 ) and ( 𝑌 , 𝜆 𝑌 , 𝜌 𝑌 ) o v er 𝐵 . This is t he equaliser 𝑋 □ 𝑌 𝑋 ⊗ k 𝑌 𝑋 ⊗ k 𝐵 ⊗ k 𝑌 . 𝜌 𝑋 ⊗ k 𝑌 𝑋 ⊗ k 𝜆 𝑌 Our graphical calculus has to diﬀerentiate betw een the tensor product of tw o bicomodules ov er k and their cotensor product. In particular , w e hav e to indicate which additional transformations are possible with t he latter: t he equalised actions in the cotensor product will be annotated in g re y . 𝑋 □ 𝑌 𝑋 𝑋 𝑌 𝑌 1 1 𝑋 ⊗ k 𝑌 𝑋 𝑋 𝑌 𝑌 The fact t hat w e ma y interchange t he appropriate lef t and right actions will be indicated thusly : 1 1 𝑋 𝑌 𝑋 𝑌 𝐵 1 1 𝑋 𝑌 𝑋 𝑌 𝐵 1 = 𝑋 □ 𝜆 𝑌 = 𝜌 𝑋 □ 𝑌 9 . 2 f ro m h o p f t r i m o d u l e s t o l a x m o d u l e f u nc t o r s In this chap ter w e g ive a proof of Theorem 9.2 . Deﬁnition 9.5. A Hopf trimodule ov er 𝐵 consists of a bicomodule 𝑋 tog ether with a lef t 𝐵 -action 𝛼 : 𝐵 ⊗ k 𝑋 − → 𝑋 , such t hat 𝛼 is a left and right 𝐵 -comodule morphism. W e write 𝑋 ∈ 𝐵 𝐵 V ect 𝐵 . 216 9.2. From Hopf trimodules to lax module functors Alternativ el y , Deﬁnition 9.5 could impose t he conditions that 𝜆 and 𝜌 are left and right 𝐵 -module morphisms, respectiv el y . This equivalence is easily seen in a string diag r ammatic reformulation: = = = = 𝐵 𝑋 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 = = = = 𝐵 𝑋 𝐵 𝑋 𝐵 𝐵 𝐵 𝑋 𝐵 𝑋 𝑋 𝑋 𝐵 𝐵 𝑋 𝑋 𝐵 𝑋 𝐵 𝑋 (9.2.1) Deﬁnition 9.6. Let 𝑋 ∈ 𝐵 𝐵 V ect 𝐵 . For all 𝑀 , 𝑁 ∈ 𝐵 V ect , deﬁne t he inter chang e morphism 𝜒 𝑀 ,𝑁 : 𝑀 ⊗ k ( 𝑋 □ 𝑁 ) − → 𝑋 □ ( 𝑀 ⊗ k 𝑁 ) b y N otice t he similarity of t he interchange morphism with the Y etter– Drinf eld braiding of Example 2.54 . 2 2 1 1 𝑀 𝑋 𝑁 𝐵 𝑋 𝑀 𝑁 Observe in particular that 1 1 = because t he image of t he coaction 𝑀 − → 𝐵 ⊗ k 𝑀 is contained in the cotensor product 𝐵 □ 𝑀 . Lemma 9.7. The inter chang e mor phism is a well-deﬁned lef t 𝐵 -comodule mor phism. Proof. T o sho w w ell-deﬁnedness, w e need to sho w that t he image of 𝜒 𝑀 ,𝑁 is contained in 𝑋 □ ( 𝑀 ⊗ k 𝑁 ) , for all 𝑀 , 𝑁 ∈ 𝐵 V ect . This follo ws by t he calculation in Figure 9.1 . 217 9. Hopf trim odul es 2 2 1 1 𝑀 𝑋 𝑁 𝐵 𝑋 𝐵 𝑁 𝑀 2 2 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 1 = 2 2 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 2 = 2 2 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 = 2 2 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 ( 9.2.1 ) = 1 1 1 1 1 1 1 1 Figure 9.1: The image of 𝜒 𝑀 , 𝑁 is contained in 𝑋 □ ( 𝑀 ⊗ k 𝑁 ) . Graphicall y , t he fact that 𝜒 is a left 𝐵 -comodule mor phism means t hat 2 2 1 1 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 2 2 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 1 1 = Using this, t he calculation in Figure 9.2 ﬁnishes t he proof. □ 2 2 1 1 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 2 2 1 1 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 2 2 1 1 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 2 2 1 1 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 2 2 𝑀 𝑋 𝑁 𝑋 𝐵 𝑁 𝑀 1 1 = coass = = ( 9.2.1 ) = Figure 9.2: The arrow 𝜒 is a morphism of lef t 𝐵 -comodules. 218 9.2. From Hopf trimodules to lax module functors Lemma 9.8. The inter chang e mor phism is a braiding; i.e., it is natur al in bot h variables and the following diagrams commute for all 𝑀 , 𝑁 , 𝑃 ∈ 𝐵 V ect : 𝑀 ⊗ k ( 𝑁 ⊗ k ( 𝑋 □ 𝑃 )) 𝑀 ⊗ k ( 𝑋 □ ( 𝑁 ⊗ k 𝑃 )) ( 𝑀 ⊗ k 𝑁 ) ⊗ k ( 𝑋 □ 𝑃 ) 𝑋 □ (( 𝑀 ⊗ k 𝑁 ) ⊗ k 𝑃 ) 𝑋 □ ( 𝑀 ⊗ k ( 𝑁 ⊗ k 𝑃 )) 𝑀 ⊗ k 𝜒 𝑁 ,𝑃 𝜒 𝑀 , 𝑁 ⊗ k 𝑃 𝛼 𝜒 𝑀 ⊗ k 𝑁 ,𝑃 𝛼 (9.2.2) k ⊗ k ( 𝑋 □ 𝑀 ) 𝑋 □ ( k ⊗ k 𝑀 ) 𝑋 □ 𝑁 𝜒 k , 𝑁 𝑋 □ 𝜆 𝑁 𝜆 𝑋 □ 𝑀 (9.2.3) Proof. The interchange morphism is alw a ys natural in its second v ariable. T o prov e naturality in t he ﬁrst variable, let 𝑓 : 𝑀 − → 𝑀 ′ be a lef t comodule morphism. Then 𝑓 2 2 1 1 𝑓 2 2 1 1 2 2 1 1 𝑓 = = 𝑀 𝑋 𝑁 𝑀 𝑋 𝑁 𝑀 𝑋 𝑁 𝑋 𝑀 ′ 𝑁 𝑋 𝑀 ′ 𝑁 𝑋 𝑀 ′ 𝑁 where t he ﬁrst equality follo ws b y 𝑓 being a lef t comodule mor phism, and the second one is t he naturality of t he braiding. Diagram ( 9.2.2 ) commuting is equivalent to t he follo wing diagram, which is seen to be tr ue b y associativity of the action on 𝑋 . 𝑀 𝑁 𝑋 𝑃 𝑀 𝑁 𝑋 𝑃 𝑀 𝑀 𝑁 𝑁 𝑃 𝑃 𝑋 𝑋 = 1 1 2 2 3 3 2 2 1 1 3 3 Diagram ( 9.2.3 ) follo ws by t he unitality of t he action on 𝑋 . □ Lemmas 9.7 and 9.8 taken together say that t he w ell-deﬁned arrow 𝜒 : − ⊗ k ( 𝑋 □ = ) = ⇒ 𝑋 □ (− ⊗ k = ) satisﬁes Diag r ams ( 9.2.2 ) and ( 9.2.3 ). This, in turn, yields the follo wing result. 219 9. Hopf trim odul es Proposition 9.9. The pair ( 𝑋 □ − , 𝜒 ) deﬁnes a lax 𝐵 V ect -module functor . W e can extend t his correspondence to morphisms. Lemma 9.10. Let 𝑓 ∈ 𝐵 𝐵 V ect 𝐵 ( 𝑋 , 𝑌 ) be a morphism of Hopf trimodules. Then 𝑓 □ − : 𝑋 □ − = ⇒ 𝑌 □ − is a 𝐵 V ect -module transf ormation. Proof. W e hav e to prov e t hat  𝑓 □ ( 𝑀 ⊗ k 𝑁 )  ◦ 𝜒 𝑋 𝑀 ,𝑁 = 𝜒 𝑌 𝑀 ,𝑁 ◦  𝑀 ⊗ k ( 𝑓 □ 𝑁 )  . In our graphical language, this means t hat 2 2 1 1 𝑀 𝑋 𝑁 𝑋 𝑀 𝑁 𝑓 2 2 1 1 𝑀 𝑋 𝑁 𝑋 𝑀 𝑁 𝑓 = which follo ws immediately from 𝑓 being a module mor phism. □ Recall t he notation 𝒱 . . = 𝐵 V ect . As a result of the previous considerations, there exists a w ell-deﬁned functor Σ : 𝐵 𝐵 V ect 𝐵 − → Le xfLax 𝒱 Mod ( 𝒱 , 𝒱 ) , 𝑋 ↦− → ( 𝑋 □ − , 𝜒 ) , (9.2.4) T o ﬁnish the proof of Theorem 9.2 , w e ha v e to show that Σ is monoidal, as w ell as an equivalence of categories. Lemma 9.11. Let 𝑋 , 𝑌 ∈ 𝐵 𝐵 V ect 𝐵 . Their cot ensor product 𝑋 □ 𝑌 is a Hopf trimod- ule, wher e the left action is given diagonally by 𝑏 ( 𝑥 ⊗ 𝑦 ) . . = 𝑏 ( 1 ) 𝑥 ⊗ 𝑏 ( 2 ) 𝑦 . Proof. Firs t, w e sho w that t he diagonal action of 𝐵 is well-deﬁned as a map 𝐵 ⊗ k ( 𝑋 □ 𝑌 ) − → 𝑋 □ 𝑌 ; i.e., t hat 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝑋 𝐵 𝑌 𝑋 𝐵 𝑌 = 1 1 1 1 220 9.2. From Hopf trimodules to lax module functors This follo ws by t he calculation in Figure 9.3 . It is left to show that the action is a left and right 𝐵 -comodule morphism. The former case follo ws by Figure 9.4 , and in t he latter case one calculates: 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 ( 9.2.1 ) = nat. 𝜎 = coass = = □ 𝐵 𝑋 𝑌 𝐵 𝑋 𝑌 𝑋 𝐵 𝑌 𝑋 𝑌 ( 9.2.1 ) = 1 1 1 1 𝐵 𝐵 𝑋 𝑌 𝑋 𝑌 coass = 1 1 𝐵 𝐵 𝑋 𝑌 𝑋 𝑌 1 = 1 1 𝐵 𝐵 𝑋 𝑌 𝑋 𝑌 = 1 1 𝐵 𝐵 𝑋 𝑌 𝑋 𝑌 ( 9.2.1 ) = 1 1 𝐵 Figure 9.3: The diagonal action is w ell-deﬁned on the cotensor product. 𝐵 𝑋 𝑌 𝑌 𝑋 𝐵 1 1 = 𝐵 𝑋 𝑌 𝑌 𝑋 𝐵 1 1 coass = 𝐵 𝑋 𝑌 𝑌 𝑋 𝐵 1 1 = 𝐵 𝑋 𝑌 𝑌 𝑋 𝐵 1 1 ( 9.2.1 ) = 𝐵 𝑋 𝑌 𝑌 𝑋 𝐵 1 1 Figure 9.4: The action of the cotensor product is a right 𝐵 -comodule morphism. 221 9. Hopf trim odul es Proposition 9.12. Let 𝜒 𝑋 and 𝜒 𝑌 be as in Deﬁnition 9.6 . Then 𝜒 𝑋 ⋄ 𝜒 𝑌  𝜒 𝑋 □ 𝑌 , wher e ⋄ is the multiplicative cell for lax module mor phisms; i.e.,  𝑋 □ − , 𝜒 𝑋  ⋄  𝑌 □ − , 𝜒 𝑌  . . =  𝑋 □ 𝑌 □ − , 𝜒 𝑋 ⋄ 𝜒 𝑌  . In ot her wor ds, t he functor Σ from Equation ( 9.2.4 ) is str ong monoidal. Proof. Associativity follo ws from coassociativity of t he coaction and naturality of the braiding: 𝑀 𝑋 𝑌 𝑁 𝑀 𝑁 𝑋 𝑌 𝑀 𝑋 𝑌 𝑁 𝑀 𝑁 𝑋 𝑌 𝑀 𝑋 𝑌 𝑁 𝑀 𝑁 𝑋 𝑌 = = U nitality is immediate. □ T o com plete the proof of Theorem 9.2 , it is left to show that the strong monoidal functor Σ from Equation ( 9.2.4 ) is an equiv alence. W e will sho w that it is fully faithful and essentially sur jectiv e. Proposition 9.13. The functor Σ is fully fait hful. Proof. T o show t hat Σ is faithful, suppose t hat 𝑓 , 𝑔 : 𝑋 − → 𝑌 are morphisms of Hopf trimodules, such that 𝑓 □ −  𝑔 □ − . Then 𝑋 □ 𝐵 𝑌 □ 𝐵 𝑋 𝑌 𝑓 □ 𝐵 𝑔 □ 𝐵   𝑓 𝑔 commutes, so the result follo ws. T o see t hat Σ is full, suppose that 𝜑 : 𝑋 □ − = ⇒ 𝑌 □ − is a 𝒱 -module transf ormation. By Proposition 2.109 , w e to show t hat t he induced arro w ˆ 𝜑 𝐵 : 𝑋  𝑋 □ 𝐵 𝜑 𝐵 − − − → 𝑌 □ 𝐵  𝑌 222 9.2. From Hopf trimodules to lax module functors is a mor phism of Hopf trimodules. F or a Hopf trimodule 𝑀 ∈ 𝐵 𝐵 V ect 𝐵 , t he morphism 𝜑 𝑀 : 𝑋 □ 𝑀 − → 𝑌 □ 𝑀 and t he induced morphism ˆ 𝜑 𝐵 can, in string diag r ams, be depicted like t his: 𝜑 𝑀 𝑋 𝑀 𝑌 𝑀 1 1 1 1 𝜑 : 𝑋 □ − − → 𝑌 □ − 𝜑 𝑀 𝑋 𝑌 1 1 1 ˆ 𝜑 𝐵 : 𝑋 − → 𝑌 1 Figure 9.5 collects sev eral properties of 𝜒 and 𝜑 in this notation. The arrow ˆ 𝜑 𝐵 is a lef t and right comodule mor phism by the calculations in Figure 9.6 . Likewise, that 𝜑 𝐵 is a left module mor phism follo ws from: 𝜑 𝐵 𝑋 𝑌 1 1 1 1 𝐵 𝜑 𝐵 𝑌 1 1 1 1 𝑋 𝐵 ( 9.2.1 ) = 𝜑 𝐵 ⊗ 𝐵 𝑌 1 1 1 1 𝑋 𝐵 9 . 5 = 𝜑 𝐵 𝑌 1 𝑋 𝐵 9 . 5 = 1 1 1 𝜑 𝐵 𝑌 1 𝑋 𝐵 = 1 1 1 𝜑 𝐵 𝑌 1 𝑋 𝐵 1 1 1 = □ 𝜑 𝑀 𝑋 𝑀 2 2 2 2 𝑁 1 1 𝑋 𝑀 𝑁 = 𝑋 𝑀 2 2 𝑁 1 1 𝑋 𝑀 𝑁 𝜑 𝑁 ⊗ 𝑀 𝜒 𝑌 𝑁 , 𝑀 ◦ ( 𝑁 ⊗ 𝜑 𝑀 ) = 𝜑 𝑀 ⊗ 𝑁 ◦ 𝜒 𝑋 𝑁 , 𝑀 𝜑 𝑀 𝑋 𝑀 1 1 = 𝜑 𝑀 ∈ 𝐵 vect ( 𝑋 □ 𝑀 , 𝑌 □ 𝑀 ) 1 1 𝐵 𝑌 𝑀 𝜑 𝑀 𝑋 𝑀 1 1 1 1 𝐵 𝑌 𝑀 𝜑 𝑀 ′ 𝑋 𝑀 1 1 = ( 𝑌 ⊗ 𝑓 ) ◦ 𝜑 𝑀 = 𝜑 𝑀 ′ ◦ ( 𝑋 ⊗ 𝑓 ) 1 1 𝑌 𝑀 ′ 𝜑 𝑀 𝑋 𝑀 1 1 1 1 𝑌 𝑀 ′ 𝑓 𝑓 2 2 𝜑 𝑀 ⊗ 𝑉 𝑋 𝑀 = ∀ 𝑉 ∈ vect 𝜑 𝑀 ⊗ 𝑉 = 𝜑 𝑀 ⊗ 𝑉 𝑌 𝑀 𝑉 𝑉 𝜑 𝑀 𝑋 𝑀 𝑌 𝑀 𝑉 𝑉 1 1 1 1 1 1 1 1 Figure 9.5: Properties of the arrow s 𝜒 and 𝜑 . 223 9. Hopf trim odul es 𝜑 𝐵 𝑋 𝑌 1 1 1 1 𝐵 ( 9.1.2 ) = 𝜑 𝐵 𝑋 𝑌 1 1 1 1 𝐵 𝜑 𝐵 𝑋 𝑌 1 1 1 1 𝐵 9 . 5 = 𝜑 𝐵 𝑋 𝑌 1 1 1 1 𝐵 𝜑 𝐵 𝑋 𝑌 1 1 1 1 𝐵 1 = 𝜑 𝐵 𝑋 𝑌 1 1 1 1 𝐵 = 𝜑 𝐵 ⊗ 𝐵 𝑋 𝑌 1 1 1 1 9 . 5 = 𝜑 𝐵 𝑋 𝑌 1 1 1 1 9 . 5 = 𝐵 𝐵 𝜑 𝐵 𝑋 𝑌 1 1 1 1 𝐵 = Figure 9.6: V eriﬁcation that ˆ 𝜑 𝐵 is a left and right comodule morphism. Proposition 9.14. The functor Σ is essentially surjective. Proof. Suppose t hat Φ ∈ Le xfLax 𝒱 Mod ( 𝒱 , 𝒱 ) . Since Φ is left exact and ﬁnitar y , b y Proposition 2.109 there exists a 𝐵 - 𝐵 -bicomodule 𝑋 , such t hat Φ  𝑋 □ − as functors. W e will show t hat if 𝑋 □ − is a lax module morphism, t hen 𝑋 comes equipped with a lef t module structure, making it into a Hopf trimodule. Deﬁne a 𝐵 -action on 𝑋 in the following w a y: 𝜒 𝐵 ,𝐵 𝐵 𝐵 𝐵 𝐵 𝑋 𝑋 1 1 1 1 1 𝜒 𝐵 ,𝐵 : 𝐵 ⊗ ( 𝑋 □ 𝐵 ) − → 𝑋 □ ( 𝐵 ⊗ 𝐵 ) 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 1 1 1 1 1 𝛼 : 𝐵 ⊗ 𝑋 − → 𝑋 Abo v e, w e ha v e chosen to represent 𝜒 𝐵 , 𝐵 — see Deﬁnition 9.6 — b y a single bo x. The claim is that t his action tur ns 𝑋 into a Hopf trimodule. F or brevity , w e shall omit the grey action indicators in our string diagram- matic proofs, sa v e when performing the respectiv e transf ormation. T o show t hat 𝛼 is a bicomodule mor phism, it suﬃces to note t hat t he lef t comodule morphism 𝜒 𝐵 , 𝐵 is also a right comodule morphism, as 𝛼 is then composed of bicomodule morphisms. N otice t hat 𝜒 is a natural transformation of functors which are left exact ﬁnitary in each variable, from 𝐵 V ect ⊗ k 𝐵 V ect to 𝐵 V ect . Since 𝜒 𝐵 , 𝐵 is t he component of the transformation 𝜒 𝐵 , − : 𝐵 ⊗ k ( 𝑋 □ −) = ⇒ 𝑋 □ ( 𝐵 ⊗ k −) 224 9.2. From Hopf trimodules to lax module functors taken at t he injectiv e gener ator of 𝐵 V ect , it follo ws that it is a right comodule morphism; see Propositions 2.84 and 2.109 . Similar l y , 𝜒 𝐵 , 𝐵 is the com ponent of 𝜒 − ,𝐵 : − ⊗ k ( 𝑋 □ 𝐵 ) = ⇒ 𝑋 □ (− ⊗ k 𝐵 ) taken at the injectiv e g enerator , so 𝜒 𝐵 , 𝐵 also intertwines this additional comodule structure: 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝐵 𝐵 𝐵 = 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝐵 𝐵 𝐵 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝐵 𝐵 𝐵 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝐵 𝐵 𝐵 = Let us now v erify t hat 𝛼 is in fact a 𝐵 -module mor phism. For t he multi- plicativ e identity of 𝛼 , one calculates 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 = 𝜒 ( 𝐵 ⊗ 𝐵 ) , 𝐵 𝐵 𝑋 𝑋 𝐵 (i) = 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝑋 𝐵 𝐵 𝜒 𝐵 , ( 𝐵 ⊗ 𝐵 ) = 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝑋 𝐵 𝐵 𝜒 𝐵 ,𝐵 1 1 1 = 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝑋 𝜒 𝐵 ,𝐵 1 1 where (i) uses the fact that ( 𝑋 □ − , 𝜒 ) is a 𝐵 V ect -module functor . The unitality of 𝛼 follo ws from t he naturality of 𝜒 , the unital axiom of ( 𝑋 □ − , 𝜒 ) being a 𝐵 V ect -module functor , and unitality of 𝐵 and 𝑋 : 𝜒 𝐵 ,𝐵 𝑋 𝑋 𝜒 1 , 𝐵 𝑋 𝑋 𝑋 𝑋 𝑋 𝑋 = = = It is lef t to show t hat Σ ( 𝛼 ) = 𝜒 𝐵 , 𝐵 . The necessar y calculation is giv en in Figure 9.7 , where equation (i) holds because 𝜒 𝐵 , 𝐵 is a right comodule morphism in both variables, and (ii) follo ws b y naturality of 𝜒 . □ 225 9. Hopf trim odul es 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 1 1 1 1 1 ↦− → 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 (i) = 𝜒 𝐵 , 𝐵 𝐵 𝑋 𝑋 𝐵 (ii) = 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 = 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 1 = 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 1 1 𝐵 𝐵 2 2 2 = 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝐵 𝐵 2 2 = 𝜒 𝐵 ,𝐵 𝐵 𝑋 𝑋 𝐵 𝐵 𝐵 Figure 9.7: V eriﬁcation that Σ ( 𝛼 ) = 𝜒 𝐵 ,𝐵 . 9 . 3 c o n t r a m o d u l e r e co n s t ru c t i o n ov e r h o p f t r i m o d u l e a l g e b r a s We s tr ess t ha t w e are still w or king under Hypotheses 8.1 and 9.1 . Deﬁnition 9.15. A Hopf trimodule algebr a consists of an algebr a object in the monoidal categor y  𝐵 𝐵 V ect 𝐵 , □ , 𝐵  . More explicitly , it consists of a Hopf trimodule 𝐴 ∈ 𝐵 𝐵 V ect 𝐵 , tog et her wit h trimodule mor phisms 𝜇 : 𝐴 □ 𝐴 − → 𝐴 and 𝜂 : 𝐵 − → 𝐴 , satisfying associativity and unitality conditions. Deﬁnition 9.16. A lef t module ov er a Hopf trimodule algebr a is a lef t 𝐴 -mod- ule object in t he 𝐵 𝐵 V ect 𝐵 -module category 𝐵 V ect in the sense of Deﬁnition 2.98 . As such, it consists of a lef t 𝐵 -comodule 𝑀 tog ether wit h a 𝐵 -comodule ho- momorphism 𝐴 □ 𝑀 − → 𝑀 , satisfying associativity and unitality conditions with respect to 𝜇 and 𝜂 . Similar l y to Deﬁnition 9.16 w e deﬁne right modules ov er 𝐴 . A module o v er 𝐴 is free if it is of t he form ( 𝐴 □ 𝑁 , 𝜇 □ 𝑁 ) . W e can also dualise this deﬁnition: w e refer to objects of t he category 𝐵 𝐵 V ect 𝐵 as Hopf termodules . A Hopf termodule coalg ebra is a coalgebra object 𝐶 in the monoidal category  𝐵 𝐵 V ect 𝐵 , ⊗ 𝐵 , 𝐵  , and a comodule over 𝐶 is a left 𝐵 -module 𝑁 tog et her with a 𝐵 -module homomorphism 𝑁 − → 𝐶 ⊗ 𝐵 𝑁 , satisfying usual coaction axioms. Right comodules o v er 𝐶 are deﬁned analogously . Remark 9.17. U nder t he equiv alence of Equation ( 9.2.4 ) between 𝐵 𝐵 V ect 𝐵 and Le xfLax 𝒱 Mod ( 𝒱 , 𝒱 ) , the 𝐵 𝐵 V ect 𝐵 -module category 𝒱 = 𝐵 V ect corresponds to 226 9.3. Contramodule reconstruction ov er Hopf trimodule algebras the ev aluation action of Le xfLax 𝒱 Mod ( 𝒱 , 𝒱 ) on 𝒱 . Thus, t he category of modules o v er a Hopf trimodule algebra 𝐴 is precisel y t he Eilenberg– Moore category for t he monad 𝐴 □ − . The subcategor y of free modules ov er 𝐴 is the image of t he Kleisli categor y for this monad under t he canonical embedding ( 𝐵 V ect ) 𝐴 □ − 𝜄 − − → ( 𝐵 V ect ) 𝐴 □ − . Let 𝐴 ∈ 𝐵 𝐵 V ect 𝐵 be a Hopf trimodule algebr a such t hat 𝐵 𝐴 is quasi-ﬁnite, meaning t hat t he corresponding lax 𝐵 V ect -module functor 𝐴 □ − has a lef t adjoint cohom 𝐵 − ( 𝐴 , −) , which is canonically a comonad b y Proposition 2.26 . Deﬁnition 9.18. A left contramodule over 𝐴 is an object in ( 𝐵 V ect ) cohom 𝐵 − ( 𝐴 , −) . Explicitl y , t his means that a lef t contramodule is a lef t 𝐵 -comodule 𝑀 tog ether wit h a left 𝐵 -comodule homomorphism 𝑀 − → cohom 𝐵 − ( 𝐴 , 𝑀 ) , endo wing 𝑀 with t he structure of a comodule in 𝐵 V ect o v er cohom 𝐵 − ( 𝐴 , −) . A contramodule is free if it lies in the image of the canonical embedding ( 𝐵 V ect ) cohom 𝐵 − ( 𝐴 , −) 𝜄 ↩ − − → ( 𝐵 V ect ) cohom 𝐵 − ( 𝐴 , −) . In other w ords, it is of t he form cohom 𝐵 − ( 𝐴 , 𝑀 ) , for a lef t 𝐵 -comodule 𝑀 , and its coaction is of the form cohom 𝐵 − ( 𝐴 , 𝑀 ) 𝜇 ∗ − − → cohom 𝐵 − ( 𝐴 □ 𝐴 , 𝑀 )  cohom 𝐵 − ( 𝐴 , cohom 𝐵 − ( 𝐴 , 𝑀 )) . W e denote t he categor y of free contramodules b y 𝐴 -ContramodF ree. Similar l y , the categor y 𝐶 -Contramod of lef t contramodules ov er a Hopf termodule coalgebr a 𝐶 is t he Eilenberg –Moore category ( 𝐵 V ect ) Hom 𝐵 − ( 𝐶 , −) for the monad Hom 𝐵 − ( 𝐶 , −) right adjoint to the comonad 𝐶 ⊗ 𝐵 − . The free contramodules are deﬁned analogously to t he case of Hopf trimodules. Remark 9.19. Contramodules similar to those of Deﬁnition 9.18 w ere studied extensiv ely in more homological settings, see in particular [ P os10 ]. Forg etting the left 𝐵 -module structure, a Hopf trimodule algebr a yields a so-called semialg ebr a , and b y for getting t he left 𝐵 -comodule structure, a Hopf termodule coalgebr a yields a so-called bocs — bimodule object, coalgebra structure. Lemma 9.20. Let 𝐴 be a Hopf trimodule algebr a. The action 𝑉 ⊲ ( 𝐴 □ 𝑀 ) . . = 𝐴 □ ( 𝑉 ⊗ 𝑀 ) , f or all 𝑉 ∈ V ect and 𝑀 ∈ 𝐵 vect 227 9. Hopf trim odul es endows the category 𝐴 -ModFree of fr ee 𝐴 -modules with a canonical module cat- egory structur e over 𝐵 V ect . Similar ly , 𝐴 -ContramodF ree is a module category over 𝐵 V ect by means of the action 𝑉 ⊲ cohom 𝐵 − ( 𝐴 , 𝑁 ) . . = cohom 𝐵 − ( 𝐴 , 𝑉 ⊗ 𝑁 ) . These two 𝐵 V ect -module categories are canonically equiv alent. Proof. By deﬁnition, we ha v e 𝐴 -ModFree = ( 𝐵 V ect ) 𝐴 □ − . Since 𝐴 is a Hopf trimodule alg ebra, the monad 𝐴 □ − is a lax 𝐵 V ect -module monad, so the Kleisli categor y ( 𝐵 V ect ) 𝐴 □ − is a 𝐵 V ect -module categor y , as described in Corol- lary 5.33 , and it is easy to check t hat t he assignments described in the lemma correspond precisel y to t hose of ibid . Similar considerations apply to 𝐴 -Contr amodFree, using the equality 𝐴 -ContramodF ree = ( 𝐵 V ect ) cohom 𝐵 − ( 𝐴 , −) and the fact t hat, since cohom 𝐵 − ( 𝐴 , −) is the lef t adjoint of t he lax 𝐵 V ect - module monad 𝐴 □ − , it is an oplax 𝐵 V ect -module comonad, b y Porism 5.29 . The asserted equiv alence betw een t hese tw o 𝐵 V ect -module categories follo ws directly from Proposition 5.37 . □ Recall from [ T ak77 , 1.12] that Cohom 𝐵 − ( 𝐴 , −) is exact if and onl y if 𝐴 is an injectiv e lef t 𝐵 -comodule. Proposition 9.21. If 𝐵 𝐴 is injective as a left 𝐵 -module, then the 𝐵 V ect -module category structur e on 𝐴 -ContramodF ree of Lemma 9.20 extends to a 𝐵 V ect -module category structur e on 𝐴 -Contramod , in t he sense of Theorem 8.25 . Proof. F or the comonadic dual of Theorem 8.25 , it suﬃces to notice t hat the comonad Cohom 𝐵 − ( 𝐴 , −) is left exact, which is t he case if and onl y if 𝐵 𝐴 is injectiv e, by [ T ak77 , 1.12]. □ Theorem 9.22. Let 𝐵 be a bialgebr a and let ℳ be a locally ﬁnite abelian module category over 𝐵 vect , which admits a 𝐵 V ect -injective 𝐵 V ect -cogener ator 𝑋 ∈ Ind ( ℳ ) , such that − ⊲ 𝑋 : 𝐵 vect − → Ind ( ℳ ) is exact and quasi-ﬁnit e. Then t here is a Hopf trimodule algebr a ( 𝐴 , 𝜇 , 𝜂 ) , such t hat 𝐵 𝐴 is quasi-ﬁnite and injective, hence ﬁnitel y cog enerat ed injective, such t hat ther e is an equiv alence of 𝐵 V ect -module categories Ind ( ℳ ) ≃ 𝐴 -Contramod , between Ind ( ℳ ) and t he category of lef t 𝐴 -contramodules. The 𝐵 V ect -module struc- tur e on 𝐴 -Contramod is ext ended, in the sense of Deﬁnition 8.10 , fr om t he category 𝐴 -ContramodF ree of free left 𝐴 -contramodules. This equivalence res tricts to an equivalence ℳ ≃ 𝐴 -contr amod , between ℳ and t he category of ﬁnite-dimensional 𝐴 -contramodules. 228 9.3. Contramodule reconstruction ov er Hopf trimodule algebras Proof. Let us again write 𝒱 . . = 𝐵 V ect . Since − ⊲ 𝑋 : 𝐵 vect − → Ind ( ℳ ) is right exact, its extension to Ind ( 𝐵 vect )  𝒱 admits a right adjoint ⌊ 𝑋 , −⌋ b y Pro- position 2.133 . Similarl y , since − ⊲ 𝑋 : 𝐵 vect − → Ind ( ℳ ) is lef t exact, so is its extension to 𝒱 , which is also quasi-ﬁnite b y assumption. Thus, by Proposi- tion 2.134 , the funct or − ⊲ 𝑋 : 𝒱 − → Ind ( ℳ ) also has a left adjoint ⌈ 𝑋 , −⌉ . W e thus obtain an oplax 𝒱 -module comonad ⌈ 𝑋 , − ⊲ 𝑋 ⌉ on 𝒱 , and a right adjoint lax 𝒱 -module monad ⌊ 𝑋 , − ⊲ 𝑋 ⌋ . By Theorem 9.2 , w e ha ve an isomorphism ⌊ 𝑋 , − ⊲ 𝑋 ⌋  ⌊ 𝑋 , 𝐵 ⊲ 𝑋 ⌋ □ − of lax 𝒱 -module monads, where ⌊ 𝑋 , 𝐵 ⊲ 𝑋 ⌋ is a Hopf trimodule algebra. Let us denote ⌊ 𝑋 , 𝐵 ⊲ 𝑋 ⌋ by 𝐴 . Since 𝐴 □ − admits a lef t adjoint, 𝐵 𝐴 is quasi-ﬁnite. By uniqueness of adjoints, an isomorphism ⌈ 𝑋 , − ⊲ 𝑋 ⌉  cohom 𝐵 − ( 𝐴 , −) follo ws. This is an isomor phism of 𝒱 -module comonads, since the 𝒱 -module comonad structure is lif ted from the isomor phic oplax 𝒱 -module monad structures on t he respectiv e right adjoints. By Theorem 8.50 , w e ha v e an equivalence Ind ( ℳ ) ≃ 𝒱 ⌈ 𝑋 , − ⊲ 𝑋 ⌉ of 𝒱 -module categories. In particular , t he functor ⌈ 𝑋 , −⌉ is comonadic and hence exact, see Theorem 2.92 . By assump tion, so is its right adjoint − ⊲ 𝑋 , whence t he comonad ⌈ 𝑋 , − ⊲ 𝑋 ⌉ is exact, which, by [ T ak77 , 1.12] and since it is isomor phic to cohom 𝐵 − ( 𝐴 , −) , implies t he injectivity of 𝐵 𝐴 . The 𝒱 -module structure on 𝒱 ⌈ 𝑋 , − ⊲ 𝑋 ⌉ is extended from 𝒱 ⌈ 𝑋 , − ⊲ 𝑋 ⌉ . By t he isomorphism ⌈ 𝑋 , − ⊲ 𝑋 ⌉  cohom 𝐵 − ( 𝐴 , −) of the inv ol v ed comonads, w e ha v e an equiv alence of 𝒱 -module categories 𝒱 ⌈ 𝑋 , − ⊲ 𝑋 ⌉ ≃ 𝒱 cohom ( 𝐴 , −) , and an equiv alence 𝒱 ⌈ 𝑋 , − ⊲ 𝑋 ⌉ ≃ 𝒱 cohom ( 𝐴 , −) . Further , b y deﬁnition 𝒱 cohom ( 𝐴 , −) = 𝐴 -Contramod and 𝒱 cohom ( 𝐴 , −) = 𝐴 -ContramodFree , which yields the desired equivalence Ind ( ℳ ) ≃ 𝐴 -Contramod. The restricted equiv alence ℳ ≃ 𝐴 -contramod follo ws from Proposition 8.3 . In particular , w e use t he obser v ation that a contramodule is compact if and onl y if its underl ying 𝐵 -comodule is such, if and only if it is ﬁnite-dimensional. □ In the ﬁnite-dimensional setting, a similar result uses projectiv e objects. Theorem 9.23. Let 𝐵 be a ﬁnite-dimensional bialg ebr a and let ℳ be a ﬁnite abelian module category over 𝐵 vect , which admits an object 𝑋 ∈ ℳ such t hat 𝑋 is a 𝐵 vect - projectiv e 𝐵 vect -g enerat or; and − ⊲ 𝑋 : 𝐵 vect − → ℳ is exact. 229 9. Hopf trim odul es Then t here is a ﬁnit e-dimensional Hopf termodule coalg ebra 𝐶 ∈ 𝐵 𝐵 vect 𝐵 , such t hat 𝐵 𝐶 is projective and ther e is an equivalence of 𝐵 vect -module categories ℳ ≃ 𝐶 -contramod between ℳ and t he category of ﬁnite-dimensional lef t 𝐶 -contr amodules. The 𝐵 vect - module structur e on 𝐶 -contramod is extended from the free left 𝐴 -contramodules. Proof. This follo ws analogousl y to Theorem 9.22 , using Theorem 8.48 instead of Theorem 8.50 and Proposition 8.5 instead of Proposition 8.3 . □ In the semisimple case, contramodules become equiv alent to modules. Deﬁnition 9.24. W e sa y t hat a Hopf trimodule algebr a 𝐴 is semisimple if t he monad 𝐴 □ − is semisimple in t he sense of Deﬁnition 8.15 . Proposition 9.25. F or a semisimple Hopf trimodule algebr a 𝐴 ∈ 𝐵 𝐵 V ect 𝐵 , t here is an equiv alence 𝐴 -Mod ≃ 𝐴 -Contramod of module categories over 𝐵 V ect . Proof. This is Proposition 8.18 applied to t he monad 𝐴 □ − . □ 9.3.1 Morit a equiv alence for contramodules Deﬁnition 9.26. W e sa y that tw o Hopf trimodule algebras 𝐴 , 𝐴 ′ ∈ 𝐵 𝐵 vect 𝐵 , such that 𝐵 𝐴 and 𝐵 𝐴 ′ are quasi-ﬁnite, are contraMorit a equiv alent if t he comon- ads cohom 𝐵 − ( 𝐴 , −) and cohom 𝐵 − ( 𝐴 ′ , −) are Morita equivalent. More explicitl y , Deﬁnition 9.26 sa ys t hat 𝐴 and 𝐴 ′ are contraMorita equi- v alent if there are 𝑀 , 𝑁 ∈ 𝐵 𝐵 V ect 𝐵 with coaction mor phisms in 𝐵 𝐵 V ect 𝐵 la 𝑀 ,𝐴 : 𝑀 − → cohom 𝐵 − ( 𝐴 , 𝑀 ) , ra 𝑁 , 𝐴 : 𝑁 − → 𝑁 □ 𝐵 cohom 𝐵 − ( 𝐴 , 𝐵 ) , la 𝑁 , 𝐴 ′ : 𝑁 − → cohom 𝐵 − ( 𝐴 ′ , 𝑁 ) , ra 𝑀 ,𝐴 ′ : 𝑀 − → 𝑀 □ 𝐵 cohom 𝐵 − ( 𝐴 ′ , 𝐵 ) , such that • there is an isomor phism in 𝐵 𝐵 V ect 𝐵 betw een t he equaliser of 𝑀 □ 𝐵 cohom 𝐵 − ( 𝐴 ′ , 𝐵 ) □ 𝐵 𝑁 𝑀 □ 𝐵 𝑁 𝑀 □ 𝐵 cohom 𝐵 − ( 𝐴 ′ , 𝑁 ) 𝑀 □ 𝐵 𝜕 𝐴 ′ ,𝐵 , 𝑁 ≃ ra 𝑀 , 𝐴 ′ □ 𝐵 𝑁 230 9.4. A semisimple example of non-rigid reconstruction and 𝐴 , intertwining the tw o coactions 𝐴 − → 𝐴 □ 𝐵 cohom 𝐵 − ( 𝐴 , 𝐵 ) and 𝐴 − → cohom 𝐵 − ( 𝐴 , 𝐴 ) coming from t he alg ebra structure on 𝐴 , wit h the coactions on t he equaliser induced b y ra 𝑁 , 𝐴 and la 𝑀 ,𝐴 . Here, 𝜕 𝐴 ′ , 𝐵 ,𝑁 is t he isomor phism of [ T ak77 , 1.13], which is inv ertible because 𝐵 𝐴 ′ is injectiv e. • a similar isomorphism in 𝐵 𝐵 V ect 𝐵 betw een a similarl y deﬁned equaliser rev ersing the roles of 𝑀 and 𝑁 , and 𝐴 ′ exists. Proposition 9.27. F or 𝐴 , 𝐴 ′ as in Deﬁnition 9.26 , t he following are equivalent: (i) 𝐴 and 𝐴 ′ ar e contraMorit a equivalent, (ii) 𝐴 -Contramod ≃ 𝐴 ′ -Contramod as 𝐵 V ect -module categories, and (iii) 𝐴 -contramod ≃ 𝐴 ′ -contramod as 𝐵 vect -module categories. Proof. The equivalence of (i) and (ii) is an instance of Proposition 8.74 , for 𝒞 = 𝐵 V ect . Since an equivalence of categories preserv es compact objects, (iii) follo ws from (ii) . Finall y , if 𝐹 is an equivalence as in (iii) , then Ind ( 𝐹 ) is an equiv alence as in (ii) . □ Combining Proposition 9.27 wit h Theorem 9.22 , w e ﬁnd t he follo wing algebr aic realisation of Theorem 8.75 : Theorem 9.28. Ther e is a bĳection { ( ℳ , 𝑋 ) as in Theorem 9.22 } ⧸ ℳ ≃ 𝒩  ← →          Left ﬁnitel y cogener ated injective Hopf trimodule alg ebras over 𝐵           ≃ cMorita . 9 . 4 a s e m i s i m p l e e x a m p l e o f n o n - r i g i d r e co n st ru c t i o n We now gi ve a concre te ap plica ti on of Theorems 8.50 and 9.22 , describing module categories for non-rigid categories where t he ordinary reconstruction procedure for rigid categories w ould not yield the correct result. Let 𝑆 = { 𝑒 , 𝑠 } be the commutativ e tw o-element monoid which is not a group, wit h identity element 𝑒 . In particular , 𝑠 2 = 𝑠 is an idempotent. The category rep ( 𝑆 ) of ﬁnite-dimensional modules o v er t he monoid algebra k [ 𝑆 ] is semisim ple, wit h tw o isoclasses of simple objects giv en b y 1 -dimensional representations k triv and k gr p . In k triv , t he element 𝑠 acts b y 1 , while in k gr p it acts by 0 . The standard comultiplication on k [ 𝑆 ] , in which 𝑒 and 𝑠 231 9. Hopf trim odul es are g roup-like, makes tur ns k [ 𝑆 ] into a bialgebr a. The categor y corep ( 𝑆 ) of ﬁnite-dimensional comodules o v er k [ 𝑆 ] is equivalent to that of 𝑆 -graded ﬁnite-dimensional spaces, vect 𝑆 . Thus, it too is semisimple of r ank tw o — w e denote t he simple objects b y 𝛿 𝑒 and 𝛿 𝑠 . The monoidal structure on rep ( 𝑆 ) satisﬁes k gr p ⊗ k gr p  k gr p . In fact, this presentation of the Grothendieck ring [ rep ( 𝑆 )] of rep ( 𝑆 ) determines rep ( 𝑆 ) uniquel y as a semisimple monoidal categor y , see [ SCZ23 , Proposition 3.5]. Thus, w e ﬁnd monoidal equivalences rep ( 𝑆 ) ≃ corep ( 𝑆 ) ≃ vect 𝑆 , since in the latter case w e ha v e 𝛿 𝑠 ⊗ 𝛿 𝑠  𝛿 𝑠 . Let us no w focus on t he coalgebraic setup of Theorem 8.50 , studying t he module categories o v er corep ( 𝑆 ) . A semisimple corep ( 𝑆 ) -module category ( ℳ , ⊲ ) is indecomposable 21 if 21 A 𝒞 -module category ℳ is called indecomposable if it does not split as a direct sum of 𝒞 -module subcategories. and onl y if t he matrix [ 𝛿 𝑠 ] ℳ describing the action of 𝛿 𝑠 on the Grothendieck ring [ ℳ ] in t he basis of simple objects for ℳ is indecomposable. But [ 𝛿 𝑠 ] is idempo tent, so it mus t be t he 1 × 1 -matrix ( 1 ) , and, as a categor y , w e must ha v e ℳ ≃ vect . W e then either ha v e 𝛿 𝑠 ⊲ −  Id ℳ , or 𝛿 𝑠 ⊲ −  0 . One can sho w t hat an y tw o module categories satisfying one of t hese conditions are equiv alent, by showing t hat the second cohomology g roup 𝐻 2 ( 𝑆 , k × ) is trivial, similar ly to [ EGN O15 , Example 7.4.10]. Indeed, 𝜓 ∈ 𝑍 2 ( 𝑆 , k × ) must satisfy the relation 𝜓 ( 𝑒 , 𝑠 ) = 𝜓 ( 𝑒 , 𝑒 ) = 𝜓 ( 𝑠 , 𝑒 ) , and thus can be written as 𝜓 ( 𝑥 , 𝑦 ) = 𝜎 ( 𝑥 ) − 𝜎 ( 𝑥 𝑦 ) + 𝜎 ( 𝑦 ) , b y setting 𝜎 ( 𝑒 ) . . = 𝜓 ( 𝑒 , 𝑒 ) and 𝜎 ( 𝑠 ) . . = 𝜓 ( 𝑠 , 𝑠 ) . The condition 𝛿 𝑠 ⊲ −  Id ℳ is satisﬁed b y the ﬁbre functor Fr : corep ( 𝑆 ) − → v ect . Assume instead t hat w e hav e an indecomposable module categor y ℳ satisfying 𝛿 𝑠 ⊲ −  0 . Let 𝑋 ∈ ℳ be a sim ple object. One can verify t hat 𝑋 satisﬁes the assump tions of Theorem 8.50 . Giv en 𝑉 ∈ corep ( 𝑆 ) , w e ha v e ⌈ 𝑋 , 𝑉 ⊲ 𝑋 ⌉ 𝑠 = corep ( 𝑆 )(⌈ 𝑋 , 𝑉 ⊲ 𝑋 ⌉ , 𝛿 𝑠 )  corep ( 𝑆 )( 𝑉 ⊲ 𝑋 , 𝛿 𝑠 ⊲ 𝑋 ) , and similar l y for 𝛿 𝑒 . Using 𝛿 𝑠 ⊲ 𝑋  0 and 𝛿 𝑒 ⊲ 𝑋  𝑋 , w e ﬁnd t hat ⌈ 𝑋 , 𝛿 𝑒 ⊲ 𝑋 ⌉  𝛿 𝑒 and ⌈ 𝑋 , 𝛿 𝑠 ⊲ 𝑋 ⌉ = 0 . T o describe t he bicomodule corresponding t o the functor ⌈ 𝑋 , − ⊲ 𝑋 ⌉ under the correspondence of Proposition 2.109 , w e obser v e that a bicomodule can be identiﬁed wit h an 𝑆 × 𝑆 -graded space, and t he cotensor product is giv en by ( 𝑉 □ 𝑊 ) 𝑥 , 𝑦 =  𝑧 ∈ 𝑆 𝑉 𝑥 , 𝑧 ⊗ 𝑊 𝑧 , 𝑦 . 232 9.5. A Hopf trimodule algebr a constructing t he ﬁbre functor Using t his, w e ﬁnd t hat ⌈ 𝑋 , − ⊲ 𝑋 ⌉  𝛿 𝑒 , 𝑒 □ − , where 𝛿 𝑒 , 𝑒 is t he one- dimensional space concentrated in deg ree ( 𝑒 , 𝑒 ) . The unit Hopf trimodule homomorphism k [ 𝑆 ] − → ⌈ 𝑋 , k [ 𝑆 ] ⊲ 𝑋 ⌉ must t hus ha v e t he submodule k { 𝑠 } as its kernel, so the reconstructing Hopf trimodule ⌈ 𝑋 , k [ 𝑆 ] ⊲ 𝑋 ⌉ is giv en b y k [ 𝑆 ]/ k { 𝑠 } . As a bicomodule, it is indeed concentrated in degree ( 𝑒 , 𝑒 ) , and as a module it is isomorphic to k gr p . The composition ⌈ 𝑋 , 𝛿 𝑒 ⊲ 𝑋 ⌉ ⊗ ⌈ 𝑋 , 𝛿 𝑒 ⊲ 𝑋 ⌉ − → ⌈ 𝑋 , 𝛿 𝑒 ⊲ 𝑋 ⌉ sending id 𝑋 ⊗ id 𝑋 to id 𝑋 corresponds to t he algebr a structure k [ 𝑆 ]/ k { 𝑠 } □ k [ 𝑆 ]/ k { 𝑠 } ≃ k [ 𝑆 ]/ k { 𝑠 } ⊗ k [ 𝑆 ]/ k { 𝑠 } − → k [ 𝑆 ]/ k { 𝑠 } ( 𝑒 + k { 𝑠 }) ⊗ ( 𝑒 + k { 𝑠 }) ↦− → ( 𝑒 + k { 𝑠 }) , and this multiplication deﬁnes a Hopf trimodule algebra. T o check that t he categor y k [ 𝑆 ]/ k { 𝑠 } -Contr amod satisﬁes the properties w e ha v e assumed of ℳ previousl y , note that t he free module k [ 𝑆 ]/ k { 𝑠 } □ 𝐵 is one-dimensional, hence semisimple, sho wing t hat the category of modules o v er k [ 𝑆 ]/ k { 𝑠 } is semisimple of rank one. Thus, by Proposition 9.25 w e ha v e k [ 𝑆 ]/ k { 𝑠 } -Contramod ≃ k [ 𝑆 ]/ k { 𝑠 } -Mod . Further , 𝛿 𝑠 ⊲ k [ 𝑆 ]/ k { 𝑠 } = k [ 𝑆 ]/ k { 𝑠 } □ 𝛿 𝑠 = 0 , which corresponds to 𝛿 𝑠 ⊲ 𝑋 = 0 . If w e instead follo w the reconstruction procedure described in [ EGN O15 , Chapter 7], t he reconstructing alg ebra object w ould be t he object 𝐴 of corep ( 𝑆 ) representing the presheaf ℳ (− ⊲ 𝑋 , 𝑋 ) . W e see that w e must ha v e 𝐴  𝛿 𝑒 , and the algebra structure on 𝛿 𝑒 is unique. Ho w ev er , 𝛿 𝑒  1 corep ( 𝑆 ) and so Mod ( 𝛿 𝑒 ) ≃ corep ( 𝑆 ) ; ℳ , sho wing that t he reconstruction procedure for module categories ov er rigid monoidal categories fails in t his non-rigid case. It also illustrates t hat t his failure can be obser v ed already in the semisimple case. 9 . 5 a h o p f t r i m o d u l e a l g e b r a c o n s t r u c t i n g t h e f i b r e f u n c t o r Thi s s ecti on di scus ses th e g ener al constructi on of a Hopf trimodule algebr a, which co v ers the remaining indecomposable semisimple module category ov er the categor y corep ( 𝑆 ) considered in Section 9.4 . 233 9. Hopf trim odul es Proposition 9.29. The k -vect or space 𝐵 ⊗ k 𝐵 forms a Hopf trimodule with coactions 𝑏 ⊗ 𝑐 ↦− → 𝑏 ( 1 ) ⊗ 𝑏 ( 2 ) ⊗ 𝑐 , 𝑐 ⊗ 𝑏 ↦− → 𝑐 ⊗ 𝑏 ( 1 ) ⊗ 𝑏 ( 2 ) , and action 𝑥 ⊗ 𝑏 ⊗ 𝑐 ↦− → 𝑥 ( 1 ) 𝑏 ⊗ 𝑥 ( 2 ) 𝑏 . W e shall denot e it by 𝐵 • 𝐵 . Proof. It is clear t hat 𝐵 • 𝐵 is a bicomodule, as both of t he coactions are giv en b y that of 𝐵 on t he respectiv e Let. Thus, w e ﬁrst v erify t hat the action is a morphism of left and right comod- ules; i.e., that 𝐵 • 𝐵 is in 𝐵 V ect 𝐵 and 𝐵 𝐵 V ect . The former follo ws b y Figure 9.8 , and the latter is due to the calculation in Figure 9.9 . □ 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 def = 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 bialg = 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 coass = 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 def = Figure 9.8: The action of 𝐵 • 𝐵 is a morphism of lef t comodules. 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 def = 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 bialg = 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 coass = 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 = 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 def = Figure 9.9: The action of 𝐵 • 𝐵 is a morphism of right comodules. 234 9.5. A Hopf trimodule algebr a constructing t he ﬁbre functor Proposition 9.30. The Hopf trimodule 𝐵 • 𝐵 from Proposition 9.29 is an algebr a object in 𝐵 𝐵 V ect 𝐵 . Proof. W e deﬁne t he multiplication and unit of 𝐵 • 𝐵 b y 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 1 1 𝜇 : ( 𝐵 • 𝐵 ) □ ( 𝐵 • 𝐵 ) − → 𝐵 • 𝐵 𝜂 = Δ : 𝐵 − → 𝐵 • 𝐵 𝐵 𝐵 𝐵 where w e ha v e denote the counit 𝜀 of 𝐵 with a small white dot, and ha v e emplo y ed the same numbering system for representing t he cotensor product as in Section 9.1 . It is clear that 𝜇 is associativ e, left unitality follo ws from 𝐵 𝐵 1 1 𝐵 𝐵 𝐵 𝐵 1 1 𝐵 𝐵 𝐵 𝐵 1 1 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 bialg = bialg = bialg = and right unitality is similar . One immediately has t hat 𝜇 is a 𝐵 V ect 𝐵 -morphism, and for 𝜂 = Δ this follo ws from coassociativity . T o ﬁnish t he proof w e need onl y show that both are mor phisms of left 𝐵 -modules, which follo ws from F igure 9.10 and t he follo wing calculation: 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 = 𝐵 𝐵 𝐵 𝐵 = 𝐵 𝐵 𝐵 𝐵 = □ Deﬁnition 9.31. Let 𝐽 : ( 𝐵 • 𝐵 ) -ModF ree − → V ect be the functor giv en b y 𝐽 (( 𝐵 • 𝐵 ) □ 𝑀 ) = 𝑀 and by local maps 𝐽 𝑀 ,𝑁 : ( 𝐵 • 𝐵 ) -ModF ree  ( 𝐵 • 𝐵 ) □ 𝑀 , ( 𝐵 • 𝐵 ) □ 𝑁  − → Hom k ( 𝑀 , 𝑁 ) 𝜎 ↦− → ( 𝜀 ⊗ 𝜀 □ id 𝑁 ) ◦ ( 𝜎 ( 1 ⊗ 1 ⊗ −)) Lemma 9.32. Deﬁnition 9.31 yields an equiv alence of 𝐵 V ect -module categories. 235 9. Hopf trim odul es 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 1 1 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 1 1 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 1 1 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 𝐵 1 1 = = = Figure 9.10: The unit 𝜂 = Δ is a morphism of lef t 𝐵 -modules. Proof. Firs t, assume t hat 𝐽 is a functor ; w e will show it is an equiv alence of ( 𝐵 V ect )-module categories. Let 𝑉 ∈ V ect . Essential sur jectivity follo ws b y 𝑉  k dim 𝑉  𝐽 (( 𝐵 • 𝐵 ) □ k dim 𝑉 triv ) . The fact that 𝐽 is fully faithful follo ws by 𝐽 𝑀 ,𝑁 being deﬁned as t he com- posite of the isomor phisms ( 𝐵 • 𝐵 ) -ModF ree  ( 𝐵 • 𝐵 ) □ 𝑀 , ( 𝐵 • 𝐵 ) □ 𝑁  −◦ 𝜂 − − → 𝐵 V ect ( 𝑀 , ( 𝐵 • 𝐵 ) □ 𝑁 ) ( 𝐵 ⊗ 𝜀 □ 𝑁 )◦− − − − − − − − − → 𝐵 V ect ( 𝑀 , 𝐵 ⊗ 𝑁 ) ( 𝜀 ⊗ 𝑁 )◦− − − − − − − → v ect ( 𝑀 , 𝑁 ) . The ﬁrst morphism is an isomorphism of the form 𝒜 𝑇 ( 𝑇 𝑋 , 𝑇 𝑌 ) ∼ − → 𝒜 ( 𝑋 , 𝑇 𝑌 ) , for t he monad 𝑇 . . = ( 𝐵 • 𝐵 ) □ − on the category 𝒜 . . = 𝐵 V ect , see Remar k 8.4 . The second isomorphism is simpl y postcom position wit h t he isomor phism ( 𝐵 • 𝐵 ) □ 𝑁 = 𝐵 ⊗ 𝐵 □ 𝑁 𝐵 ⊗ 𝜀 □ 𝑁 − − − − − → 𝐵 ⊗ 𝑁 , since 𝐵 □ 𝑁 𝜀 □ 𝑁 − − − → 𝑁 is an isomorphism. The t hir d is an isomorphism ℬ 𝐶 ( 𝐶 𝑋 , 𝐶 𝑌 ) ∼ − → ℬ ( 𝐶 𝑋 , 𝑌 ) , for the comonad 𝐾 . . = 𝐵 ⊗ − on the categor y ℬ . . = V ect . Lastl y , 𝐽 is a ( 𝐵 V ect )-module functor : for 𝑊 ∈ 𝐵 V ect , w e hav e 𝑊 ⊲ 𝐽 ( 𝐵 • 𝐵 □ 𝑀 ) = 𝑊 ⊗ 𝑀 = 𝐽 ( 𝐵 • 𝐵 □ ( 𝑊 ⊗ 𝑀 )) = 𝐽 ( 𝑊 ⊲ 𝐵 • 𝐵 □ 𝑀 ) . Thus, if 𝐽 deﬁnes a functor , it is an equiv alence of ( 𝐵 V ect )-module categories. 236 9.6. The fundamental theorem of Hopf modules It is left to v erify t he functoriality of 𝐽 . Let 𝜎 ∈ ( 𝐵 • 𝐵 ) -ModF ree (( 𝐵 • 𝐵 ) □ 𝑀 , ( 𝐵 • 𝐵 ) □ 𝑁 ) , 𝜏 ∈ ( 𝐵 • 𝐵 ) -ModF ree (( 𝐵 • 𝐵 ) □ 𝑁 , ( 𝐵 • 𝐵 ) □ 𝑊 ) . W e represent t hese mor phisms graphically b y 𝐵 𝐵 𝑀 𝐵 𝐵 𝑁 1 1 𝜎 ∈ ( 𝐵 • 𝐵 ) -ModF ree (( 𝐵 • 𝐵 ) □ 𝑀 , ( 𝐵 • 𝐵 ) □ 𝑁 ) 𝜎 𝑀 𝑁 1 1 𝐽 ( 𝜎 ) ∈ Hom k ( 𝑀 , 𝑁 ) 𝜎 1 1 1 1 Funct oriality of 𝐽 follo ws by Figure 9.11 ; (i) and (ii) hold due to 1 1 1 1 1 1 = Equality (iii) is satisﬁed b y 𝜏 being a ( 𝐵 • 𝐵 ) -module morphism, (iv) f ollows b y the follo wing calculation, using that 𝐵 is a bialgebr a: 𝜀 ( 𝑏 ( 1 ) 𝑏 ( 2 ) ) ⊗ 𝑏 ( 3 ) = 𝜀 ( 𝑏 ( 1 ) ) 𝜀 ( 𝑏 ( 2 ) ) ⊗ 𝑏 ( 3 ) = 𝜀 ( 𝑏 ( 1 ) 𝜀 ( 𝑏 ( 2 ) )) ⊗ 𝑏 ( 3 ) = 𝜀 ( 𝑏 ( 1 ) ) ⊗ 𝑏 ( 2 ) = 𝑏 , and (v) holds b y an analogous computation. □ Corollar y 9.33. The monad ( 𝐵 • 𝐵 ) □ − is semisimple, and ther e is an equiv alence of 𝐵 V ect -module categories ( 𝐵 • 𝐵 ) -Mod ≃ V ect . 9 . 6 t h e f u n da m e n ta l t h e o r e m o f h o p f m o d u l e s Deﬁnition 9.34 ([ Mon93 , Proposition 1.5.10]) . Let 𝐵 be a bialgebr a. A twist ed antipode for 𝐵 is an antipode for t he bialgebr a 𝐵 cop . 22 22 Equiv alently , a twisted antipode is an antipode for 𝐵 op . Remark 9.35. A Hopf algebr a admits a twisted antipode ˜ 𝑆 if and only if its o wn antipode 𝑆 is in v ertible, in which case w e ha v e 𝑆 − 1 = ˜ 𝑆 . N otabl y , a ﬁnite-dimensional Hopf algebr a alwa ys admits a twisted antipode. 237 9. Hopf trim odul es 𝜎 𝑀 1 1 1 1 𝜏 1 1 1 1 𝑊 𝜎 𝑀 1 1 1 1 𝜏 1 1 𝑊 (i) = 𝜎 𝑀 1 1 1 1 𝜏 1 1 𝑊 (ii) = 2 2 𝜎 𝑀 1 1 1 1 𝜏 1 1 𝑊 (iii) = 2 2 𝜎 𝑀 1 1 1 1 𝜏 1 1 𝑊 (iv) = id 𝑀 1 1 1 1 𝑀 𝑀 𝑀 (v) = Figure 9.11: The assignment 𝐽 is functorial. Lemma 9.36 below is a w ell-known result, see for example [ EGN O15 , Theorem 5.4.1.(2)]. W e emphasise that while its former part is seemingly opposite to t he claim of [ EGN O15 , Theorem 5.4.1.(2)], identifying t he presence of an antipode wit h right, rather t han left, rigidity , this is because w e consider left, r ather than right, comodules. The latter claim of Lemma 9.36 f ollo ws from the former by using the equivalence ( 𝐵 V ect ) rev ≃ 𝐵 op V ect . Lemma 9.36. Let 𝐵 be a bialgebr a. Then 𝐵 vect is right rigid if and onl y if 𝐵 admits an antipode. Similar ly , 𝐵 vect is lef t rigid if and onl y if 𝐵 admits a twist ed antipode. The next t heorem was shown in t he setting of quasi-bialgebr as in [ HN99 , Proposition 3.11] and [ Sar17 , Theorem 3.9], as a gener alisation of the structure theorem for Hopf modules of Larson and Sw eedler [ LS69 ]. Theorem 9.37. Let 𝐵 be a bialgebr a. The functor − ⊗ 𝐵 : 𝐵 V ect − → 𝐵 V ect 𝐵 𝐵 is an equiv alence if and onl y if 𝐵 admits an antipode. In t he rest of t his section w e sho w that a variant of Theorem 9.37 can be deduced from the follo wing gener al statement. Proposition 9.38. Let 𝒞 be a left closed monoidal category such that every left 𝒞 - module endofunctor of 𝒞 is str ong — in ot her wor ds,that the monoidal embedding Str 𝒞 Mod ( 𝒞 , 𝒞 ) ↩ − → Lax 𝒞 Mod ( 𝒞 , 𝒞 ) is an equiv alence. Then 𝒞 is left rigid. 238 9.6. The fundamental theorem of Hopf modules Proof. Since 𝒞 is left closed, for ev er y 𝑥 ∈ 𝒞 there is a right adjoint ⌊ 𝑥 , −⌋ to the strong 𝒞 -module endofunctor − ⊗ 𝑥 . By Porism 5.29 , the functor ⌊ 𝑥 , −⌋ is a lax 𝒞 -module endofunctor of 𝒞 , and t hus, by assump tion, it is a strong module endofunctor . Proposition 2.51 yields an object ∨ 𝑥 such t hat t here is an isomor phism of 𝒞 -module functors ⌊ 𝑥 , −⌋  − ⊗ ∨ 𝑥 . Thus w e obtain an adjunction of 𝒞 -module functors − ⊗ 𝑥 ⊣ − ⊗ ∨ 𝑥 , which b y [ Kel72 , pp. 102– 103] sho ws t hat ( ∨ 𝑥 , 𝑥 ) is a dual pair in 𝒞 . It follo ws that 𝒞 is lef t rigid. □ Remark 9.39. Giv en a rigid monoidal category 𝒞 , no tice t hat ev ery lax 𝒞 -mod- ule endofunctor on 𝒞 is automaticall y strong by Proposition 2.73 . T ogether with Proposition 9.38 , t his yields a characterisation when a (left) closed mo- noidal categor y is (left) rigid, solely based on its lax 𝒞 -module functors. This is a special property of module functors, as in Section 3.1.2 w e ha v e already seen that ordinary adjunctions do not suﬃce for such a characterisation. W e now deﬁne t he functor 𝐵 ⊗ − analogous to − ⊗ 𝐵 of Theorem 9.37 . Deﬁnition 9.40. Deﬁne t he functor 𝐵 ⊗ − : 𝐵 V ect − → 𝐵 𝐵 V ect 𝐵 b y sending a left 𝐵 -comodule 𝑀 to t he trimodule 𝐵 ⊗ 𝑀 whose lef t and right coaction, as w ell as left action, is giv en b y 𝑏 ⊗ 𝑚 ↦− → 𝑏 ( 1 ) 𝑚 (− 1 ) ⊗ 𝑏 ( 2 ) ⊗ 𝑚 ( 0 ) ; 𝑏 ⊗ 𝑚 ↦− → 𝑏 ( 1 ) ⊗ 𝑚 ⊗ 𝑏 ( 2 ) ; 𝑎 ⊗ 𝑏 ⊗ 𝑚 ↦− → 𝑎 𝑏 ⊗ 𝑚 ; where we extends to mor phisms in the natural w ay . It is easy to v erify t hat 𝐵 ⊗ 𝑀 deﬁnes a Hopf trimodule and also t hat 𝐵 ⊗ − deﬁnes a functor . Remark 9.41. Comparing Deﬁnition 9.40 wit h Theorem 9.37 , we remar k on tw o diﬀerences. Firs t, t he domain of t he functor w e s tudy is t he category of comodules ov er 𝐵 , rather t han modules. This is merely to maintain consistence with Section 9.2 , and a variant f or modules can be formulated wit hout greater diﬃculty . Second, the functor w e study endo ws a left comodule wit h an additional comodule structure on the right, and a module structure on t he left, rather than on the right. W e will see t hat it is related to t he functor V ect − → 𝐵 V ect 𝐵 , which is known to be an equivalence if 𝐵 admits a twisted antipode; see for example [ Mon93 , Section 1.9.4]. W e now giv e a categorical interpretation of t he functor of Deﬁnition 9.40 . Recall the notation 𝒱 . . = 𝐵 V ect of Section 9.2 . 239 9. Hopf trim odul es Proposition 9.42. The diagr am Str 𝒱 Mod ( 𝒱 , 𝒱 ) LexfLax 𝒱 Mod ( 𝒱 , 𝒱 ) 𝒱 rev 𝐵 𝐵 V ect 𝐵 ≃ ≃ commutes up to a monoidal natural isomorphism. Its upper horizontal arrow is the natur al inclusion functor , its lef t vertical arrow is the equiv alence of Proposition 2.51 , its right v ertical arr ows is the equivalence of Theorem 9.2 , and its lower horizontal arrow is t he functor of Deﬁnition 9.40 . Proof. Chasing a functor 𝐹 in t he diag r am, w e ﬁnd t he follo wing: 𝐹 𝐹 𝐹 k triv 𝐵 ⊗ 𝐹 k triv 𝐹 ( 𝐵 ⊗ k triv ) 𝐹 𝐵 ≃ ≃ It is easy to v erify that t he isomor phism indicated in the bottom row of t his chase is natural in 𝐹 , and also monoidal. □ Corollar y 9.43. If the functor 𝐵 ⊗ − of Deﬁnition 9.40 is an equivalence, then 𝐵 admits a twist ed antipode. Proof. By Proposition 9.42 , 𝐵 ⊗ − is an equivalence if and only if t he embedding Str 𝒱 Mod ( 𝒱 , 𝒱 ) ↩ − → Le xfLax 𝒱 Mod ( 𝒱 , 𝒱 ) is an equiv alence, so assume that it is. F or 𝑀 ∈ 𝐵 vect , t he right adjoint of − ⊗ k 𝑀 is ﬁnitar y by Proposition 2.134 , since − ⊗ k 𝑀 sends ﬁnite-dimensional comodules to ﬁnite-dimensional co- modules. The right adjoint is t hen a ﬁnitar y left exact lax 𝒱 -module functor . By assumption it is strong, and t hus of the form − ⊗ ∨ 𝑀 , for ∨ 𝑀 ∈ 𝒱 . Since the ﬁbre functor 𝑈 : 𝒱 − → V ect is strong monoidal, w e hav e ∨ 𝑀  𝑀 ∗ . This sho ws t hat 𝐵 vect is left rigid; b y Lemma 9.36 , 𝐵 admits a twisted antipode. □ Proposition 9.44. If a bialgebr a 𝐵 admits a twis ted antipode, t hen t he funct or 𝐵 ⊗ − of Deﬁnition 9.40 is an equiv alence. 240 9.7. Fusion operators for Hopf monads Proof. Since t he functor 𝒱 ↩ − → LexfLax 𝒱 Mod ( 𝒱 , 𝒱 ) is alw a ys full y f aithful, so is the functor 𝐵 ⊗ − b y Proposition 9.42 . It remains to show t hat 𝐵 ⊗ − is essentiall y sur jectiv e. F ollowing [ Mon93 , Section 1.9.4], for 𝑋 ∈ 𝐵 V ect 𝐵 the composite map Γ : 𝑋 Δ 𝑋 − − − → 𝑋 ⊗ 𝐵 t ⊗ 𝐵 − − − − → 𝑋 co 𝐵 ⊗ 𝐵 𝜎 𝑋 co 𝐵 , 𝐵 − − − − − → 𝐵 ⊗ 𝑋 co 𝐵 is an isomorphism in 𝐵 V ect 𝐵 , where 𝑋 co 𝐵 = { 𝑚 ∈ 𝑋 | Δ 𝑟 𝑋 ( 𝑚 ) = 𝑚 ⊗ 1 } and t is the projection diag r ammaticall y represented b y ˜ 𝑆 𝐵 𝐵 The in v erse of Γ is giv en by t he restriction ∇ 𝑋 | co 𝐵 : 𝐵 ⊗ 𝑋 co 𝐵 − → 𝑋 of the left 𝐵 -action on 𝑋 . F or 𝑋 ∈ 𝐵 V ect 𝐵 , the space 𝑋 co 𝐵 of coin variants is a lef t 𝐵 -subcomodule of 𝑋 , since 𝑋 is a bicomodule. It is easy to check that ∇ 𝑋 | co 𝐵 : 𝐵 ⊗ 𝑋 co 𝐵 − → 𝑋 is a left 𝐵 -comodule mor phism, and thus a morphism in 𝐵 𝐵 V ect 𝐵 , making the functor 𝐵 ⊗ − essentially sur jectiv e. □ Corollar y 9.45. A bialg ebra 𝐵 admits a twisted antipode if and onl y if the functor 𝐵 ⊗ − of Deﬁnition 9.40 is an equivalence. Remark 9.46. Here w e see t hat it is crucial that Section 9.2 identiﬁes Hopf trimodules with ﬁnitary lax module functors. Giv en an inﬁnite-dimensional comodule 𝑀 o v er 𝐵 , t he functor Hom k ( 𝑀 , −) : 𝐵 V ect − → 𝐵 V ect is endow ed with lax 𝐵 V ect -structure b y virtue of Theorem 5.28 , but is not ﬁnitar y itself. 9 . 7 f u s i o n o p e r at o r s f o r h o p f m o na d s Sim ilar l y to ou r study o f Theorem 9.37 in Section 9.6 , t he results of Bruguières, Lack, and Virelizier in [ BL V11 ] can be used to characterise ri- gidity of a monoidal category . Lemma 9.47. Let 𝐹 : 𝒞 ⇄ 𝒟 : 𝑈 be an oplax monoidal adjunction. A variant of this result using “Doi– K oppinen data ” appears in [ KKS15 , Section 5.5]. The str ong monoidal structur e of 𝑈 turns 𝒞 into a 𝒟 -module category , by deﬁning − ⊲ = . . = 𝑈 (−) ⊗ = . 241 9. Hopf trim odul es W ith r espect to t his 𝒟 -module structur e, t he bimonad 𝑇 . . = 𝑈 𝐹 on 𝒞 becomes an oplax 𝒟 -module monad. The coher ence morphism is given by 𝑇 a ; 𝑑 , 𝑐 : 𝑇 ( 𝑑 ⊲ 𝑐 ) = 𝑇 ( 𝑈 𝑑 ⊗ 𝑐 ) 𝑇 2; 𝑈 𝑑 , 𝑐 − − − − → 𝑇 𝑈 𝑑 ⊗ 𝑇 𝑐 𝑈 𝜀 𝑑 ⊗ 𝑇 𝑐 − − − − − − → 𝑈 𝑑 ⊗ 𝑇 𝑐 , (9.7.1) for all 𝑐 ∈ 𝒞 and 𝑑 ∈ 𝒟 . In particular , for 𝑥 ∈ 𝒞 , the coher ence mor phism 𝑇 a ; 𝐹 𝑥 , 𝑐 is pr ecisely t he right fusion operat or for 𝑇 at ( 𝑥 , 𝑐 ) . In ot her wor ds, 𝑇 a ; 𝐹 𝑥 , 𝑐 = 𝑇 rf ; 𝑥 , 𝑐 . Proof. The ﬁrst statement is a w ell-kno wn result on transport of structure. F or 𝑦 ∈ 𝒟 , t he 𝒟 -module structure on 𝑈 is giv en by t he mor phisms 𝑈 m : 𝑑 ⊲ 𝑈 𝑦 = 𝑈 𝑑 ⊗ 𝑈 𝑦 ∼ − → 𝑈 ( 𝑑 ⊗ 𝑦 ) = 𝑈 ( 𝑑 ⊲ 𝑦 ) . F or t he second statement, obser v e t hat b y Porism 5.29 , 𝐹 inherits an oplax 𝒟 -module functor structure from t he strong 𝒟 -module structure on 𝑈 . The oplax 𝒟 -module functor on 𝑇 is giv en by the upper part of t he follo wing diagram, where the low er part is included clarity: 𝑈 𝐹 ( 𝑈 𝑑 ⊗ 𝑐 ) 𝑈 ( 𝐹𝑈 𝑑 ⊗ 𝐹 𝑐 ) 𝑇 ( 𝑑 ⊲ 𝑐 ) 𝑈 ( 𝑑 ⊗ 𝐹 𝑐 ) 𝑈 𝑑 ⊗ 𝑈 𝐹 𝑐 𝑈 𝐹 ( 𝑑 ⊲ 𝑐 ) 𝑈 ( 𝑑 ⊲ 𝐹 𝑐 ) 𝑈 𝑑 ⊗ 𝑇 𝑐 𝑑 ⊲ 𝑈 𝐹 𝑐 𝑑 ⊲ 𝑇 𝑐 𝑈 𝐹 2; 𝑈 𝑑 , 𝑐 𝑈 ( 𝜀 𝑑 ⊗ 𝐹 𝑐 ) 𝑈 m ; 𝑑 , 𝐹 𝑐 T o see that t his mor phism equals t hat deﬁned in Equation ( 9.7.1 ), observe that t he follo wing diagram of functors commutes by naturality of 𝑈 2 : 𝑈 ( 𝐹 𝑈 ⊗ 𝐹 ) 𝑈 𝐹𝑈 ⊗ 𝑈 𝐹 𝑈 ( Id 𝒟 ⊗ 𝐹 ) 𝑈 ⊗ 𝑈 𝐹 𝑈 2; 𝐹 𝑈 , 𝐹 𝑈 ( 𝜀 ⊗ 𝐹 ) 𝑈 𝜀 ⊗ 𝑈 𝐹 𝑈 2;Id , 𝐹 Since by deﬁnition w e ha v e t hat 𝜇 = 𝑈 𝜀 𝐹 , substituting 𝑑 = 𝐹 𝑥 in Equa- tion ( 9.7.1 ) yields 𝑇 a ; 𝐹 𝑥 , 𝑐 = 𝑇 rf ; 𝑥 , 𝑐 , which pro v es t he result. □ 242 9.7. Fusion operators for Hopf monads Lemma 9.48. Let 𝒞 be a category , 𝑓 , 𝑔 : 𝑥 − → 𝑦 a pair of parallel morphisms in 𝒞 , such that the coequaliser coeq ( 𝑓 , 𝑔 ) exists. If 𝐺 , 𝐻 : 𝒞 − → 𝒟 is a pair of funct ors pr eserving t hat coequaliser , 𝛼 : 𝐺 = ⇒ 𝐻 a natur al tr ansformation, then 𝛼 𝑥 and 𝛼 𝑦 being in vertible implies that so is 𝛼 coeq ( 𝑓 , 𝑔 ) . Proof. Let 𝑐 . . = coeq ( 𝑓 , 𝑔 ) and write 𝑝 : 𝑦 − ↠ 𝑐 for the canonical projection. W e are in t he follo wing situation: 𝐺 𝑥 𝐺 𝑦 𝐺 𝑐 𝐻 𝑥 𝐻 𝑦 𝐻 𝑐 𝐺 𝑓 𝐺 𝑔 𝛼 𝑥 𝐺 𝑝 𝛼 𝑦 𝐻 𝑓 𝐻 𝑔 𝐻 𝑝 Then w e calculate 𝐻 𝑝 ◦ 𝛼 𝑦 ◦ 𝐺 𝑓 = 𝐻 𝑝 ◦ 𝐻 𝑓 ◦ 𝛼 𝑥 = 𝐻 𝑝 ◦ 𝐻 𝑔 ◦ 𝛼 𝑥 = 𝐻 𝑝 ◦ 𝐺 𝑔 ◦ 𝛼 𝑥 , which, b y the univ ersal property of 𝐺 𝑐 , im plies t he existence of a unique arro w ! : 𝐺 𝑐 − → 𝐻 𝑐 that is equal to 𝛼 𝑐 . Since 𝛼 𝑥 and 𝛼 𝑦 are in v ertible w e also obtain a unique arro w 𝐻 𝑐 − → 𝐺 𝑐 , which is easil y seen to be in v erse to 𝛼 𝑐 . □ Proposition 9.49. The right fusion operat ors of bimonad 𝑇 on a monoidal category 𝒞 are in vertible if and onl y if t he coherence morphisms for the oplax 𝒞 𝑇 -module structur e on 𝑇 of Lemma 9.47 are inv ertible. In ot her wor ds, 𝑇 is right Hopf if and only if it is a str ong 𝒞 𝑇 -module monad. Proof. Since, b y Lemma 9.47 , the right fusion operat ors f or 𝑇 are a special case of the coherence mor phisms for the 𝒞 𝑇 -module functor structure on 𝑇 , it is immediate t hat 𝑇 being a strong 𝒞 𝑇 -module monad implies in v ertibility of the right fusion operators. T o sho w t he conv erse, assume that t he right fusion oper ators are in v ertible, in particular , b y Lemma 9.47 , the mor phisms 𝑇 a ; 𝐹 𝑥 , 𝑐 are in v ertible. Recall an y 𝑇 -alg ebra ( 𝑥 , 𝛼 ) ∈ 𝒞 𝑇 is the coequaliser of t he diag r am 𝑇 2 𝑥 𝑇 𝑥 𝑥 𝜇 𝑥 𝑇 𝛼 𝛼 in 𝒞 𝑇 . In other w ords, 𝑥 is a 𝑈 -split coequaliser of mor phisms from 𝐹𝑇 𝑥 to 𝐹 𝑥 , since 𝐹 𝑥 is the free 𝑇 -module on 𝑥 . 243 9. Hopf trim odul es Consider the natural transf ormation 𝑇 a ; − , 𝑐 : 𝑇 (− ⊲ 𝑐 ) − → − ⊲ 𝑇 𝑐 . W e ha v e 𝑇 (− ⊲ 𝑐 ) = 𝑇 ( 𝑈 (−) ⊗ 𝑐 ) = (− ⊗ 𝑐 ) ◦ 𝑈 , which means t hat 𝑇 (− ⊲ 𝑐 ) preserves t he coequaliser coeq ( 𝑇 𝛼 , 𝜇 𝑥 ) b y ﬁrst sending it to a split coequaliser under 𝑈 , and t hen preser ving it, since split coequalisers are absolute. In a similar fashion, − ⊲ 𝑇 𝑐 = 𝑈 (−) ⊗ 𝑇 𝑐 = (− ⊗ 𝑇 𝑐 ) ◦ 𝑈 also preserv es this coequaliser . The in v ertibility of 𝑇 a ; 𝑥 , 𝑐 no w follo ws from t he assumed in v ertibility of 𝑇 rf ; 𝑥 , 𝑐 = 𝑇 a ; 𝐹 𝑥 , 𝑐 and 𝑇 rf ; 𝑇 𝑥 ,𝑐 = 𝑇 a ; 𝐹 𝑇 𝑥 , 𝑐 , b y Lemma 9.48 . □ 244 B I B L I O G R A P H Y [ A C12 ] Marcelo Aguiar and Stephen Chase. Generalized Hopf modules for bimon- ads. In: Theor y and Applications of Categories 27 (2012), pp. 263 –326 (cit. on pp. 2 , 5 , 117 , 122 , 123 , 129 ). [ AEBSJ01 ] Jiří A dámek, R obert El Bashir, Manuela Sobral, and V elebil Jiří. On functors which are lax epimor phisms. In: Theory Appl. Categ. 8 (2001), pp. 509 –521. iss n : 1201-561X (cit. on p. 76 ). 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Die Arbeit wurde bisher w eder im Inland noch im A usland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde v org elegt. T on y Zorman Dresden, 01. April 2025

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