Lax Monoidal Fibrations

Lax Monoidal Fibrations Marek Za w ado ws ki Inst ytut Matemat yki, Uniw e rsytet W arsza wsk i ul. S.Banac ha 2, 00-913 W arsza w a, Poland za wado@mim u w.edu.pl Decem b er 22, 2009 De dic ate d to Mihaly Makkai on the o c c asion of his 70th birth day. Abstract W e in tro duce t he notion of a lax monoidal ﬁbratio n and we show how it can b e conv eniently used to deal with v arious algebraic structures that play an impo rtan t role in some deﬁnitions, cf. [BD], [HMP], [SZ], [S] of the op etopic sets. W e pres en t the ’standard’ suc h structures, the exp onen tial ﬁbrations o f basic ﬁbra tions and three areas of applications. The ﬁrst area is related to the T -categor ies of A. B urroni. The monoids in the Burroni lax monoidal ﬁbrations form the ﬁbration of T -categ ories. The construction of the relative Burroni ﬁbrations and free T -categ ories in this c on text, allow us to extend the deﬁnition o f the set of o petop es given in [Le] to the categor y of op etopic se ts (internally to any Gro thendiec k top os, if needed). W e also show that the ﬁbration o f (1-level) m ultica tegories considered in [HMP] is eq uiv alent to the ﬁbr ation of (ﬁnitary , car tesian) po lynomial monads . This equiv a lence is induced by the equiv alence of la x monoida l ﬁbrations of amalga mated signatures, p olynomial diagrams, and polyno mial (ﬁnitary , endo) functors. Finally , we dev elop a simila r theory for symmetric sig natures, analytic diag rams (a no tion introduce d here), and (ﬁnitary , m ultiv ariable) analytic (endo)functor s, cf. [J2]. Among other things w e show that the ﬁbra tions of symmetric multicategories is equiv alent to the ﬁbra tion o f analytic monads. W e also give a characterization (Co rollary 7.6) of s uc h a ﬁbration of analytic monads. An ob ject of this ﬁbratio n is a weakly cartesia n monad on a slice of S et whose functor par t is a ﬁnitary functor weakly preser ving wide pullbacks. A morphism of this ﬁbration is a weakly ca rtesian morphism of mona ds whose functor part is a pullback functor. MS Classiﬁcation 18D10, 18D30, 18D50, 18C15 (AMS 2010) . 1 Con ten ts 1 In t roduction 3 2 Lax monoidal ﬁbrations 7 2.1 Preliminaries, ﬁbrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The basic d eﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Monoids in a lax monoidal ﬁb ratio n . . . . . . . . . . . . . . . . . . . . . . 1 0 2.4 The 2-cate gory of lax monoidal ﬁb ratio n s . . . . . . . . . . . . . . . . . . . 1 0 2.5 Simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Actions of lax monoidal ﬁbrations 13 3.1 The basic d eﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Actions of monoids along an action of a lax monoidal ﬁbr at ion . . . . . . . 14 3.3 The 2-cate gory of actions of lax monoidal ﬁbr at ions . . . . . . . . . . . . . 15 3.4 Simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 The exp onen tia l ﬁbrations 18 4.1 The exp onen tial biﬁbr ati ons in Cat / B . . . . . . . . . . . . . . . . . . . . 18 4.2 The exp onen tial ﬁbr at ions in F ib/ B . . . . . . . . . . . . . . . . . . . . . . 24 5 The Burroni ﬁbrations and op etopic set s 25 5.1 The Burroni ﬁ brations and T -categories . . . . . . . . . . . . . . . . . . . . 25 5.2 T autologo us actio n s of Burroni ﬁbrations . . . . . . . . . . . . . . . . . . . 26 5.3 Multisorted signatures vs monotone p olynomial d iag rams . . . . . . . . . . 29 5.4 Morphisms of monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.5 Relativ e Burroni ﬁb ratio ns and relativ e T -categ ories . . . . . . . . . . . . . 32 5.6 F ree r el ativ e T -catego ries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.7 A tow er of ﬁ brations f or op etopic sets . . . . . . . . . . . . . . . . . . . . . 36 5.8 A tow er of ﬁ brations f or n -categories . . . . . . . . . . . . . . . . . . . . . . 38 6 Amalgamated signatures vs p olyno mial functors 39 6.1 The amalgamat ed signatur es ﬁb ratio n p a : S ig a → S et . . . . . . . . . . . . 39 6.2 The action of p a on th e basic ﬁbration . . . . . . . . . . . . . . . . . . . . . 42 6.3 P olynomial diagrams and p olynomial functors . . . . . . . . . . . . . . . . . 43 6.4 Some p rop ertie s of the repr esen tation r ep a . . . . . . . . . . . . . . . . . . . 46 6.5 The 2-lev el amalgamated signatures ﬁbration p 2 a : S ig 2 a → S et → . . . . . . 52 6.6 Single tensor in the ﬁbration p a . . . . . . . . . . . . . . . . . . . . . . . . . 56 7 Symmetric signatures vs analytic functors 59 7.1 The symmetric s ig nature ﬁbr ati on p s : S ig s − → S et . . . . . . . . . . . . . . 59 7.2 The action of p s on th e basic ﬁbration and analytic functors . . . . . . . . . 63 7.3 A charact erization of the ﬁb ration of an al ytic fun ct ors . . . . . . . . . . . . 68 7.4 The analytic diagrams vs analytic f unctors . . . . . . . . . . . . . . . . . . . 75 7.5 Comparing the p olynomial and the analytic approac h es . . . . . . . . . . . 79 8 App endix 83 2 1 In tro duction The notion of a lax m onoidal ﬁb ration stud ie d here, is designed to h elp un derstand connections b et we en v arious deﬁnitions of op etopic sets. More sp eciﬁcally , v arious al- gebraic/ca tegorical mec h anisms used to deﬁne them. Th e primary goa l was to understand the r ela tion b et wee n the n ot ion of op etopic set of Baez-Do lan, cf. [BD] and multitopic set of Hermida-Makk ai-P ow er, cf. [HMP]. Ho wev er, it turn ed out that many other notions connected with the dev elopmen t of higher cate gories can b e successfully organized into a lax monoidal ﬁ bration. This is not a mere enco ding for its sak e but in the con text of lax monoidal ﬁbration man y notions can b e conv enien tly compared, charac terized, and deve l- op ed b ey ond wh at w as previously k no wn. This pap er pro vides man y examples of suc h applications of lax monoidal ﬁ brations but the comparison of the mentio ned deﬁnitions of op etopic sets, as w ell as a y et another deﬁnition of op etopic sets(!), will b e present ed in the forthcoming pap er [SZ]. The lax monoidal ﬁbrations pro vide a con v enient to ol to d ea l with many-lev el s truc- tures, lik e categories that ha ve ob jects and morphisms, m u lti catego ries 1 of v arious kin ds (that h a v e ob jects-t yp es and m ultiarrows-function symb ols) or T -categories of Burroni, cf. [B]. Th ese are examples of s tructures that h a v e just t wo lev els but by build ing a to w er of ﬁbrations, see Subs ec tions 5.7 and 5.8, or iterating a construction insid e a single ﬁbra- tion, cf. [S Z], [S], we can d eal with man y-lev el structur es lik e op etopic s et s, p olygraphs, n -catego ries, ω -cate gories and others. T o deﬁne monoids of interest in this setting w e are not doing it in ’one big step’ but w e divide it in to thr ee smaller steps. First we d eﬁne a ﬁbration, then we deﬁne the monoidal structure in this ﬁ bration and ﬁnally we deﬁne a ﬁbration of monoids ov er th e same b ase as th e ﬁb ration we started w ith. In that wa y if w e wan t, as w e d o in S ect ion 7, to compare multica tegories with non-standard amalgama- tion with symmetric m u ltic ategorie s we can compare the ﬁ brations of amalgamate d and symmetric signatures ﬁrst, then compare the tensors (there is m ore th an one p ossibilit y) and ﬁnally w e get a comparison of su ita ble multicate gories. A lax monoidal ﬁbr at ion is a ﬁb ratio n p : E → B equip ped with t wo functors ⊗ : E × B E → E and I : B → E comm uting ov er th e base bu t not required to b e morphisms of ﬁbr ati ons 2 . W e call such morphism lax morphism of ﬁ brations (or ﬁ bred morphisms), as the fact that a morp hism of ﬁbr a tions comm u te o ve r the base already forces some lax p reserv ation of pron e morp hisms. There are also coherence morphisms α , λ , ρ satisfying the usu al conditions but they are not required to b e isomorphisms, as in man y examples they are not. The direction of these morphism are so c h osen to cov er all our examples. The ﬁbres of suc h ﬁb ratio n are monoidal categories and reindexing fun ct ors are mon oi dal fun ct ors. It is in fact often th e case, that the ﬁbr es are str ong monoidal cat- egories b ut r eindexing functors are almost nev er strong eve n in the lax monoidal ﬁbration whose monoids are small categories! This m ak es the whole con text una voidably lax. The morphism of lax monoidal ﬁbr at ions are lax morp hisms of ﬁbrations that are monoidal in the only r easo nable sense. The 2-cells are also deﬁned in th e only reasonable w a y . T hen in the analogy with the n on-ﬁbred situation, cf. [BD ], a lax m onoidal ﬁbration ma y act on arbitrary ﬁ brations. So we hav e a 2-category of actions of lax monoidal ﬁbrations, as w ell. It is qu ite surprisin g ho w m an y th ings can b e explained in terms of actions and their exp onen tial adj oi n ts. This will b e carefully explained in Section 4 an d used man y times in the follo wing sections. 1 W e follo w mostly the terminology from [Le ], in particular for us (v arious) multicategories are th e same things as (v arious) colored set op erads. 2 This means that we d o not require that ⊗ or I send pron e (formerly cartesian) morph isms to prone morphisms. 3 The pap er is organized as follo ws. In sections 2 and 3, we introdu ce th e main n oti ons of the pap er of a lax monoidal ﬁbr a tion, an action of a lax monoidal ﬁbr at ions and 2- catego ries of these structures. T he examples presented there are v ery basic. In S ect ion 4, w e discus s the lax monoidal ﬁ brations E E → B that arises as exp onen tial ﬁbrations of biﬁb ratio ns. It turns out that the exp onen tiation in C at / B , the slice of C at o ve r the base, is muc h more in teresting than th e exp onen tiation in F ib ( B ), the category of ﬁbrations o v er B . Among suc h ﬁb ratio ns there are ev en more sp ecial o nes, the exp onen tial ﬁbrations of the b asic ﬁbrations cod : B → − → B . If B h as p ullbac ks suc h a ﬁ bration, denoted E xp ( B ) → B , alw a y s exists and if a lax monoidal ﬁbration E → B acts on the basic ﬁbr at ions cod : B → − → B we h a v e a representa tion morph ism of lax monoidal ﬁbrations E E xp ( B ) ✲ B ❅ ❅ ❅ ❘    ✠ that compares an arbitrary ﬁbration with a standard one. In Section 5, w e sho w ho w one can split the deﬁnition of a T -category of Bur roni, cf. [B] p. 225- 227, into three p arts. The ﬁbration of T -graph s, denoted p T : Gph ( T ) → C , the m onoidal part and ﬁnally th e monoids in su c h a lax monoidal ﬁbr at ions. W e call su c h ﬁbrations Burroni ﬁbrations to honor A.Burroni who w as the ﬁrst to consider them, cf. [B] p. 262. Th e monoids in suc h a ﬁbrations are exactly the T -catego ries of Burroni. Since it is not necessary to ha ve a cartesian monad 3 T to build s uc h a ﬁbration w e can reco v er that w a y all th e T -categories th at w ere considered in [B]. The ﬁbr es of such a ﬁbration p T are not n ece ssarily strong monoidal unless T is cartesian. Ho w ev er, as w e already mentioned, the reindexing fu nctors are almost nev er strong monoidal f unctors. The Burroni ﬁbrations alw a ys acts on basic ﬁbr at ions an d hence they ha ve r epresen tation morphisms of lax monoidal ﬁbration int o the stand ard ones Gph ( T ) E xp ( C ) ✲ r ep T C p T ❅ ❅ ❅ ❘    ✠ If T is cartesian then this morphism is a morphism of b iﬁbrations, Pr oposition 5.2. Th e construction of T -categories can b e made relativ e with resp ect to a ﬁ bration if the monad T is already ﬁbred. Moreo ver, in this relativ e con text the construction du e to M. Kelly , cf. [Ke] p.69, see also [BJT], toge ther w ith the characte r iza tion of T. Leinster, cf. [Le] p. 334, giv es a charact erization of those ﬁb red cartesian monads for which the fr ee relativ e T -catego ries exists. This allo ws us to extend the d eﬁnition of the set of op et op es given b y T. Leinster, cf. [Le] p. p .17 9, to the deﬁnition of the wh ole category of op etopic sets, and this category can b e bu ild int ernally in an y Grothend ie c k top os not only in S et . W e simply iterate ω times the construction of r el ativ e T -graph ﬁbration starting from the iden tity monad. In Section 6, w e sho w th at tw o seemingly diﬀerent languages used to d eﬁne op etop es and op eto pic sets, cf. [HMP] and [K o] , are in fact equiv alen t. W e sho w that the lax monoidal ﬁb ratio ns of amalgamated signatures p a : S ig a → S et and of p olynomial dia- grams p pd : P ol y D iag → S et are equiv alen t. The d iﬀerence is rather in st yle th at can b e easily explained in the con text of lax monoidal ﬁ brations. The amalgamat ed signatures are ’more concrete’ and come n at urally equipp ed with an action on the basic ﬁbration 3 The only requirement is that the category hav e pullbacks. 4 cod : S et → − → S et w hereas p olynomial diagrams come equipp ed with a representa tion in to th e exp onen tial ﬁbr at ion E xp ( S et ) → S et . Th is representa tion is the exp onen tial adjoin t of the action and its essen tial image is the lax monoidal ﬁb ratio n of (ﬁnitary) p olynomial (endo)functors p poly : P ol y → S et . As these th ree ﬁ brations are equiv alen t as lax m onoi dal ﬁbr ati ons w e obtain in particular that the ﬁb ratio n of (1-lev el) multi- catego ries with non-standard amalgamations is equiv alen t to the ﬁbr at ion of p olynomial monads (i.e. cartesian monads on slices of S et w hose functor p arts are p olynomial func- tors) and as morph isms cartesian morp hism of monads wh ose functor parts are pu llbac k functors (coun ted as morph ism in the op posite direction), see Corollary 6.13. It is p ossi- ble to giv e a natural deﬁn itio n of op eto pic sets in this conte x t, see [SZ], [S]. W e end this section by showing ho w to d ea l with the so called single tensor and 2-lev el ob jects, the original setting for the deﬁnition in [HMP]. In S ec tion 7, we dev elop a parallel theory to the one from the p revious section bu t this time we start with th e lax monoidal ﬁbration of symmetric signatures p s : S ig s → S et instead of amalgamated signatures, whose monoids f orm exactly the ﬁbr at ion symmetric m u lti catego ries. T his ﬁb ratio n is also naturally equipp ed with an action on the basic ﬁbration and taking an adjoin t w e get again repr esen tation morphism S ig s E xp ( S et ) ✲ r ep s S et p s ❅ ❅ ❅ ❘    ✠ As in the p revious case, this m orphism is f a ithful and full on isomorphisms. Its essen tial image, denoted b y p an : A n → S et , is the lax monoidal ﬁbration of m u ltiv ariable ana- lytic (endo)functors, cf. [J2 ], and analytic natural transformations b et ween them. As a consequence, the ﬁbration of symm et ric m ulticategories in S et is equiv alen t to th e ﬁbr a- tion of analytic monads. W e also pr o vide an abstract charact erization of the ﬁbr ati on of analytic fun cto r s extending the one from [J2]. W e sh o w, see T heorem 7.5, that this ﬁbration of (multiv ariable) analytic (endo)fun cto r s, whic h is a lax monoidal subﬁ bration of the exp onen tial ﬁbration E xp ( S et ) → S et , consists of ﬁn ita ry functors on slices of S et that preserve w eakly wide pu llbac ks a nd has as morphisms w eakly cartesian natural trans- formations. The pro of of this c h arac terizatio n is based on ideas fr om [J2 ] and [A V]. As a consequ ence , we obtain Corollary 7.6 sa ying that the notions of a symmetric multic at- egory and of an analytic monad are equiv alen t. So analytic functors is y et another to ol that could b e u sed to d eﬁne the category of op etopic sets. I n S ubsection 7.4 we introd uce an in termediate notion of an analytic diagrams that is r ela ted to symmetric signatures and analytic fu nctors as p olynomial d iag r ams are related to amalgamated signatures and p olynomial fu nctors. These diagrams are p olynomial d iag rams of a sp ecial kind in th e catego ry of symmetric sets σ S et , i.e. the category of pr eshea v es on the copro duct (in C at ) of ﬁn ite symmetric groups. How ev er the represen tation is given via a comp osition of ﬁve functors not three as in the case of usual p olynomial diagrams. In the last Sub sec tion of the pap er we compare the notions studied in Sections 6 and 7. W e describ e the follo wing diagram of lax monoidal ﬁbrations and their morphism s 5 S ig a S ig s ✲ K sig ❄ ι a ❄ ι s P oly D iag A n D iag ✲ K diag ❄ r ep pd ❄ r ep and P oly A n ✲ K f u ❄ ❄ C ar t ( S et ) wC ar t ( S et ) ✲ E xp ( S et ) ❅ ❅ ❅ ❘    ✠ Φ ⇒ Ψ ⇒ All the arro ws are str ong morphism s of lax monoidal ﬁbrations. W e sh o w among other things that the horizon tal m orphisms are full and faithful. The three named horizon tal arro ws are morphisms comparin g signatures, diagrams, and fun cto rs, resp ectiv ely . The four named v ertical arro ws are equiv alences of lax monoidal ﬁbr ati ons. The ﬁve unn amed arro ws are inclusions. Man y more interesti ng connections b et we en these lax monoidal ﬁbrations w ill b e explained in [S Z ]. W e ﬁnish with an observ ation th at a w eakly cartesian natural transformations b et wee n p olynomial functors are cartesian. There are t wo p ossible notions of an analytic f unctor on S et /O . The sp ecies and the analytic functors of one v ariable S et → S et and of many v ariables S et /O → S et , for a ﬁ nite s et O , were introdu ce d b y A. Joy al in [J1] and [J2] to stud y enumerativ e com binatorics. Clearly , an O -tuple of multiv ariable analytic f unctors tak en together form an en dofunctor S et /O → S et /O , that should b e considered as analytic, as wel l. Th e concept of analytic fu nctor w as stud ied in the categ ory S et , in category of ve ctor spaces V ect , cf. [J2] but also in an arbitrary monoidal categ ory , cf. [A V]. In that wa y , we h a v e t wo kin ds of analytic fun ct ors on slices of S et (and p o wers of other monoidal categories). An analytic fu ncto r from S et /O to S et /O can b e deﬁned as the left Kan extension of a functor f : B → S et /O , where B is the category of ﬁnite sets and b ijec tions, cf. [A V], or as an O -tuple of m ultiv ariable analytic fun ct ors S et /O → S et , cf. [J2 ]. The ﬁrst notion do es n ot allo w functors that are not copro ducts of fu ncto rs b et ween ﬁbres. In this pap er w e consid er only the second notion. The idea of equipping ﬁb ratio ns with some kind of a monoidal structure go es bac k to N.S. Riv ano [Sa] and M.F. Gouzou-R. Grunig, cf. [GG]. It w as take n up later b y M. Shulman in [S h] , p . 698. These n oti ons ‘ B − ⊗− cat´ egories ﬁbre´ e’ in [Sa], ‘cat ´ egorie ﬁbr´ ee sur B multiplicat iv e’ in [GG], and ‘monoidal ﬁb ration’ in [Sh] are diﬀeren t than the notion of a lax monoidal ﬁ brations presen ted h ere. Also the motiv ations in eac h case are d iﬀeren t than ours. The total category of a lax monoidal ﬁb ratio n is n ot a monoidal category and in this sense the n ot ion is closer to the notions considered in [Sa] and [GG]. On the other hand, we do not require our tensor or unit to b e morphisms of ﬁbrations (i.e. p reserv e the prone morphisms) as it wo uld eliminate most of our examples. This causes that our reindexing morphisms are n ot necessarily strong monoidal f unctors. In our applicatio ns the actions of lax monoidal ﬁbrations pla y an imp ortan t role. This does not h a v e an analog in the other approac h es. I w ould lik e to thank George Janelidze and T homas Streic h er for the con v ersations related to the matters con tained in this pap er, And re Jo yal for explaining to me some asp ects of his theory of analytic functors. Sp ecial th a nks are due to Kr zyszto f Kapulkin , Magdalena K¸ edziorek, K arol Szumi lo and Stanis la w S za w iel , the m em b ers of an inf ormal 6 Category Seminar held in Spr ing 2008 at W arsa w Universit y , f o r giving me an opp ortunit y to present the essent ial notions in tro duced and studied in this p aper. I w ould lik e also to thank th e anon ym o us referee for the v ery th orough rep ort that help ed considerably to impro v e the presen tation of the pap er. L ast bu t not lea st I w ould li k e to thank, o ur jubilee, Mihaly Makk ai for introd ucing me to the su b ject of h igher-dimensional categ ories many y ears ago and to h im and Victor Harnik for coun tless discussions of the related matters. The diagrams for this pap er we re prepared with the help of c atmac1 of Michae l Barr. 2 Lax monoidal ﬁb rati ons 2.1 Preliminaries, ﬁbrations Our standard reference no ﬁb ratio n s (opﬁbrations and biﬁ brations) is [St]. Ho wev er the terminology u sed h ere follo ws more the one used b y P . T a y or and P .T. Johnston. W e call pr one and supine m orphisms w hat [St] w ould call c artesian and c o c artesian . The ﬁbre of a (b i) ﬁbration p : E → B ov er B ∈ B will b e denoted E B . If p is a ﬁ bration, u : B → B ′ is a morph ism in B then w e ha ve (usin g axiom of c h oi ce for classes) a r eindexing functor u ∗ : E B ′ → E B deﬁned w ith the help of prone m orphisms; if p is an opﬁbration, we hav e a c or e indexing f unctor u ! : E B → E B ′ deﬁned with the help of su pine morphisms. In a b iﬁbration b oth fun cto rs exist and are adj oi n t u ! ⊣ u ∗ . The unit and counit of this adjunction will b e denoted by η u and ε u , r espectiv ely . W e call a biﬁbr at ion P : E → B c artesian if the ﬁbres of p hav e pullbac ks, u ! preserve s them a n d both η u and ε u are cartesia n natural tr ansformatio ns, for all morph isms u in B . Note that this notation suppr esses the fac t that these f unctors and n at ural transformations are related to a speciﬁc (bi,op)ﬁbration. The ﬁbration w e hav e in mind should b e alw ays read from the context . A ﬁb ratio n has ﬁ bred (co)limits of typ e K if and only if eac h ﬁ bre has (co)limits of t yp e K and reindexing functors preserv e them. In the pap er, w e consider (bi)ﬁbrations that are equipp ed additionally with a m o noidal structure. It is n ot alw ays the case that the morphisms we w ant to consider b et ween (bi)ﬁb ratio n s p reserv es all the stru ct ure inv olv ed (prone m orphisms, su pine morph isms, and /o r tensor). Th erefore as the basic morph isms b et we en ﬁ bration w e consider lax morph isms that only make the s quare b elo w commute. A lax morphism from a ﬁ bration p to p ′ is a p air of functors ( G, G ′ ) making th e square B B ′ ✲ G ′ E E ′ ✲ G ❄ p ❄ p ′ comm ute. If if p and p ′ are (op)ﬁbrations and G p reserv es prone (supine) morph isms, ( G, G ′ ) will b e called a morphism of ﬁbr ations (opﬁbr ations) . If p and p ′ are biﬁbrations and G preserv es b oth pr one and supine morp hisms, ( G, G ′ ) will b e called a morphism of biﬁbr ations . The ﬁbr e d natur al tr ansformation ( τ , τ ′ ) : ( G, G ′ ) → ( H , H ′ ) is a pair of natural transformations τ : G → H and τ ′ : G ′ → H ′ suc h that p ′ ( τ ) = τ ′ p . If g = G ′ = id B and τ ′ = id id B then a ﬁ b ered natural tr ansformation is natural transformation τ : G → H whose comp onen ts are vertic al m orphisms. A ﬁbr e d left adjoint to ( G, G ′ ) is a ﬁ bred morphism ( F , F ′ ) suc h that F ⊣ G and F ′ ⊣ G ′ are adjunctions with units and counits ( η , ε ) and ( η ′ , ε ′ ), resp ectiv ely , so that p ′ ( η ) = η ′ and p ′ ( ε ) = ε ′ . 2.2 The basic deﬁnition A lax monoidal ﬁbr ation ( p : E → B , I , ⊗ , α, λ,  ) is 7 1. a ﬁbr ati on p : E → B , 2. equ ipp ed with t wo lax morphism s of ﬁbrations ⊗ and I E × B E E ✲ ⊗ ❅ ❅ ❅ ❅ ❘ B ❄ p B ✛ I B 1 B     ✠ 3. th ree ﬁbred natural transformations α , λ ,  E × B E E ✲ ⊗ E × B ( E × B E ) ∼ = ( E × B E ) × B E E × B E ✲ ⊗ × B E ❄ E × B ⊗ ❄ ⊗ ⇒ α (where th e unnamed iso ∼ = is the canonical one b et w een pu llbac ks) i.e. there are morphisms A ⊗ ( B ⊗ C ) ( A ⊗ B ) ⊗ C ✲ α A,B ,C for any O ∈ B and any A, B , C ∈ E O so that p ( α A,B ,C ) = 1 O and these morphism s are natur al in A , B , and C , in the ob vious sense. Moreo ver B × B E E × B E ✲ I × 1 E π 2 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ E ❄ ⊗ E × B B ✛ 1 E × I E π 1        ✠ ⇒ λ ⇒  i.e. there are morp hisms A ⊗ I O A ✲ ρ A A I O ⊗ A ✲ λ A for an y O ∈ B and any A ∈ E O so that p ( ρ A ) = 1 O = p ( λ A ) and these morph isms are natur al in A . 4. T he d iag rams A ⊗ ( B ⊗ ( C ⊗ D )) ✑ ✑ ✑ ✑ ✑ ✰ 1 A ⊗ α B ,C,D ◗ ◗ ◗ ◗ ◗ s α A,B ,C ⊗ D A ⊗ (( B ⊗ C ) ⊗ D ( A ⊗ B ) ⊗ ( C ⊗ D ) ❆ ❆ ❆ ❆ ❯ α A,B ⊗ C,D ✁ ✁ ✁ ✁ ☛ α A ⊗ B ,C,D ( A ⊗ ( B ⊗ C )) ⊗ D (( A ⊗ B ) ⊗ C ) ⊗ D ✲ α A,B ,C ⊗ 1 D 8 and A ⊗ O ( I O ⊗ B ) ( A ⊗ O I O ) ⊗ B ✲ α A,I O ,B A ⊗ O B A ⊗ O B ✲ 1 A ⊗ B ❄ 1 A ⊗ λ B ✻ ρ A ⊗ 1 B comm ute, and ﬁn ally ρ I O and λ I O are isomorphisms and ρ I O = λ − 1 I O : I O ⊗ I O − → I O where O ∈ B and A, B , C , D ∈ E O . End of the d eﬁnition of a lax monoidal ﬁbration. Remarks 1. T he tensor op eration can b e app lied to ob jects in the same ﬁ bre of p : E → B only , and to morphism s that lie o ver the same map in the base. Sometimes we emp hasize this by writing a ⊗ O b and f ⊗ u g to indicate that the tensor is in th e ﬁbre o v er O or o v er a m orphism u . So the ﬁb res are (lax) monoidal categories and reindexing ’functors’ are lax monoidal. But the total category E is not monoidal. Both facts are imp ortant for the examples we ha ve in m ind. 2. F or any u : O → Q ∈ B and A, B ∈ E Q w e ha ve (unique) morphism s ψ 0 u : I O → u ∗ ( I Q ) and ψ 2 u,A,B : u ∗ ( A ) ⊗ O u ∗ ( B ) → u ∗ ( A ⊗ Q B ) so that the triangles u ∗ ( A ) ⊗ u ∗ ( B ) A ⊗ B ✲ pr u,A ⊗ pr u,B u ∗ ( A ⊗ B ) ψ 2 u,A,B       ✒ pr u,A ⊗ B ❅ ❅ ❅ ❅ ❅ ❅ ❘ and I O I Q ✲ I u u ∗ ( I Q ) ψ 0 u     ✒ pr u,I Q ❅ ❅ ❅ ❅ ❘ comm ute, wh ere pr u,A is a p rone morphism o ver u with co domain A . Due to the fact that w e deal with ﬁ brations, lax morp hisms pr ese rv e p rone morph isms in the lax sense. Thus ev en if we d o not requ ire the tensor and the unit to b e morphisms of ﬁbrations, w e still ha ve that the ’reindexing’ functors are lax monoidal, i.e. they ’resp ect’ the monoidal structure (to some exten t). There are many m ore diagrams in v olving ψ ’s, α ’s, λ ’s and  ’s that commute. 3. T he dir ec tions of the natur al trans formatio n s α ’s, λ ’s and  in the deﬁn itio n of a lax monoidal ﬁb ratio n are so c h ose n to co ver all the examples we hav e in mind. But it is sometimes con venien t to consider natural transformations λ or  that go in the other d irect ion. 9 2.3 Monoids in a lax monoidal ﬁbration A monoid in a ﬁ bre o v er O is a triple ( M , m : M ⊗ M → M , e : I O → M ) wher e M is an ob ject in E O , m , e are morphisms in E O making the diagrams M ⊗ M M ✲ m M ⊗ ( M ⊗ M ) α − → ( M ⊗ M ) ⊗ M M ⊗ M ✲ m ⊗ 1 M ❄ 1 M ⊗ m ❄ m and M M ✲ 1 M I O ⊗ M M ⊗ M ✲ e ⊗ 1 M ✻ λ M ❄ m M ✛ 1 M M ⊗ I O ✛ 1 M ⊗ e ❄ ρ M comm ute. A morphism of monoids f : ( M , m , e ) − → ( M ′ , m ′ , e ′ ) o v er u : O → Q ∈ B is a morphism f : M − → M ′ in E o v er u such that the squares M ⊗ O M M ′ ⊗ Q M ′ ✲ f ⊗ u f ❄ m ❄ m ′ I O I Q ✲ I u M M ′ ✲ f ✻ e ✻ e ′ comm ute. Then the category of monoids is again ﬁb red o ver B and th e forgetful fu ncto r is a morphism of ﬁbrations E M on ( E , ⊗ , I ) ✛ U S et p ❅ ❅ ❅ ❅ ❘ q     ✠ In the inte r esting cases it should ha ve a (ﬁbred) left adjoint wh ic h is not lik ely to b e a morphism of ﬁbrations. Remark As w e will s ee it is n ot alwa ys true that the category of all monoids is of real in terest. If the coherence transformations are indeed not isomorphisms it may happ en that w e m a y wan t to consider only those monoids that satisfy s o me additional conditions, see 6.6. 2.4 The 2-category of lax monoidal ﬁbrations A morphism of lax monoidal ﬁbr ations ( F , K, ϕ 0 , ϕ 2 ) : ( p : E → B , ⊗ , I , α, λ,  ) − → ( p ′ : E ′ → B ′ , ⊗ ′ , I ′ , α ′ , λ ′ ,  ′ ) is data 1-3 s ub ject to conditions 4-6 b elo w ( O ∈ B , A, B , C ∈ E O ): 10 1. ( F , K ) : ( E , p ) → ( E ′ , p ′ ) a lax morp hism of ﬁbrations, 2. ϕ 0 : I K − → F ◦ I a ﬁb red natural transformation (i.e. for an y O ∈ B w e hav e a morphisms ( ϕ 0 ) O : I ′ K ( O ) − → F ( I O ) in E ′ O whic h is natur a l in O ), 3. ϕ 2 : F ( − ) ⊗ ′ F (=) − → F (( − ) ⊗ (=)) a ﬁ bred natural transformation (i.e. for A, B ∈ E O w e hav e a morphism ( ϕ 2 ) A,B : F ( A ) ⊗ ′ F ( B ) − → F ( A ⊗ B )), in E ′ O whic h is n atural in A and B ) 4. th e square F ( A ) F ( I O ⊗ A ) ✲ F ( λ A ) I ′ K ( O ) ⊗ F ( A ) F ( I O ) ⊗ F ( A ) ✲ ϕ 0 ⊗ 1 F ( A ) ✻ λ ′ F ( A ) ❄ ϕ 2 comm utes, wh ere O ∈ B an d A ∈ E O ; 5. th e square F ( A ) F ( A ⊗ I O ) ✛ F (  A ) F ( A ) ⊗ I ′ K ( O ) F ( A ) ⊗ F ( I O ) ✲ 1 F ( A ) ⊗ ϕ 0 ❄  ′ F ( A ) ❄ ϕ 2 comm utes, wh ere O ∈ B an d A ∈ E O ; 6. th e diagram F ( A ) ⊗ ′ F ( B ⊗ C ) F ( A ⊗ B ) ⊗ ′ F ( C ) F ( A ) ⊗ ′ ( F ( B ) ⊗ ′ F ( C )) ( F ( A ) ⊗ ′ F ( B )) ⊗ ′ F ( C ) ✲ α ′ F ( A ) ,F ( B ) ,F ( C ) ❄ 1 F ( A ) ⊗ ′ ϕ 2 ❄ ϕ 2 ⊗ ′ 1 F ( C ) F ( A ⊗ ( B ⊗ C )) F (( A ⊗ B ) ⊗ C ) ✲ F ( α A,B ,C ) ❄ ϕ 2 ❄ ϕ 2 comm utes, wh ere O ∈ B an d A, B , C ∈ E O . End of the deﬁnition of a morphism of lax monoidal ﬁ brations. A m orphism of lax monoidal ﬁb ratio ns is called str ong if th e transition morphism s ϕ 0 , ϕ 2 are isomorph isms and ( F , K ) is a morph ism of ﬁ brations. A tr ansformation b et ween t wo m orphisms of lax monoidal ﬁbrations is a pair of natural transformations ( τ , σ ) : ( F , K, ϕ 0 , ϕ 2 ) − → ( F ′ , K ′ , ϕ ′ 0 , ϕ ′ 2 ) suc h th a t 1. σ : K − → K ′ and τ : F − → F ′ are natural transformations, 11 2. p ′ ( τ ) = σ p , i.e. τ is ﬁbred o ve r σ , 3. th e diagrams F ( I O ) F ′ ( I O ) ✲ τ I O I ′ K ( O ) I ′ K ′ ( O ) ✲ I ′ σ O ❄ ( ϕ 0 ) O ❄ ( ϕ ′ 0 ) O and F ( A ⊗ O B ) F ′ ( A ⊗ O B ) ✲ τ A ⊗ O B F ( A ) ⊗ K ( O ) F ( B ) F ′ ( A ) ⊗ K ( O ) F ′ ( B ) ✲ τ A ⊗ K ( O ) τ B ❄ ( ϕ 2 ) A,B ❄ ( ϕ ′ 2 ) A,B comm ute, for O ∈ B . End of the d eﬁnition of a transformation b et ween t wo m orphisms of lax monoidal ﬁbr a- tions. Prop osition 2.1 The morphisms of lax monoidal ﬁbr ations induc e morphisms b etwe en the ﬁb r ations of monoids . The tr ansformations b etwe en morphisms of lax monoidal ﬁbr a- tions induc e natur al tr ansformatio ns b e twe en the induc e d fu ncto rs. Pr o of. E xercise . ✷ 2.5 Simple examples 1. Cate gories. Pr obably the simplest non-trivial example of a lax monoidal ﬁ bration (in the ab o ve sense) is the ﬁb rati on of graphs o ve r sets, sa y p : Gph − → S et , where p sends the parallel pair of arro ws to their common co domain. The tensor ( A, d A , c A : A → O ) ⊗ O ( B , d B , c B : B → O ) = ( A × O B , c A ◦ π 1 , d B ◦ π 2 : A × O B − → O ) where A × O B d enote s the pullbac k of the follo wing p ai r of morphisms A O ✲ d A A × O B B ✲ π 2 ❄ π 1 ❄ c B The unit for the tensor in the ﬁbre o v er O is a pair of iden tities on O , ( O , 1 O , 1 O : O → O ). The total cate gory of the ﬁb ratio n of monoids q : M on ( Gph ) → S et in this ﬁb ration is the category of s mall categ ories and fu nctors. Th e monoids in a ﬁb re M on ( Gph ) O are catego ries with th e set of ob jects O . 2. L amb ek’s multic ate gories. Let (( − ) ∗ , η , µ ) b e the monad f or monoids on the category S et . Then, w e can deﬁn e a ﬁb ratio n of m ultisorted sig natures p m : S ig m − → S et as follo ws. An ob ject of S ig m in the ﬁbr e o ver the set O is a triple ( A, ∂ , O ) suc h that A is a set and ∂ : A − → O × O ∗ function. ( f , u ) : ( A, ∂ A , O ) − → ( A ′ , ∂ A ′ , O ′ ) is a morphism in S ig m o ve r a f unction u : O → O ′ if f : A → A ′ is a f unction making the square 12 O × O ∗ O ′ × O ′∗ ✲ u × u ∗ A A ′ ✲ f ❄ ∂ A ❄ ∂ A ′ comm ute. T he tensor ( A, ∂ A , O ) ⊗ ( B , ∂ B , O ) = ( A × O ∗ B ∗ , ∂ , O ) of t wo ob ject in the same ﬁbre is giv en b y the pu llbac k and multiplica tion µ in the m onad ( − ) ∗ and the u nit for this tensor is ( I O , ∂ , O ) = ( O , h 1 O , η i , O ). The category of monoids in this ﬁ bration is the catego ry of Lamb ek’s multica tegories. As th is construction will b e describ ed in Section 5 in a more general case of arb itrary monad o ve r a category with pullbac ks w e don’t go into the details here. 3 Actions of lax monoidal ﬁbrations 3.1 The basic deﬁnition An action ( ⋆, ψ 2 , ψ 0 ) of a lax monoidal ﬁ bration ( E , p, I , ⊗ , α, λ,  ) on a ﬁb ratio n π : X → B is 1. a lax morph ism of ﬁbrations E × B X X ✲ ⋆ B p ❅ ❅ ❅ ❅ ❘ π     ✠ 2. a ﬁbr ed natural transformation ψ 0 X ∼ = B × B X E × B X ✲ I × B 1 X X 1 X ❅ ❅ ❅ ❅ ❘ ⋆     ✠ ψ 0 ⇒ i.e. for X ∈ X O , w e ha ve a morph ism ( ψ 0 ) X : X → I O ⋆ X in the ﬁ bre X O whic h is natural in O ; 3. a ﬁbr ed natural transformation ψ 2 E × B X X ✲ ⋆ E × B ( E × B X ) ∼ = ( E × B E ) × B X E × B X ✲ ⊗ × B X ❄ E × B ⋆ ❄ ⋆ ⇒ ψ 2 i.e. for O ∈ B , A, B ∈ E O and X ∈ X O w e h a v e a m orphism ( ψ 2 ) A,B ,X : A ⋆ ( B ⋆ X ) − → ( A ⊗ B ) ⋆ X in the ﬁbr e X O , w hic h is natural in A , B and X ; the unnamed iso ∼ = is th e canonical one b et w een pullbac ks; 13 4. m aking the p en tagon A ⋆ ( B ⋆ ( C ⋆ X )) ✑ ✑ ✑ ✑ ✑ ✰ ( ψ 2 ) A,B ,C ⋆X ◗ ◗ ◗ ◗ ◗ s 1 A ⋆ ( ψ 2 ) B ,C,X ( A ⊗ B ) ⋆ ( C ⋆ X ) A ⋆ (( B ⊗ C ) ⋆ X ) ❆ ❆ ❆ ❆ ❯ ( ψ 2 ) A ⊗ B ,C,X ✁ ✁ ✁ ✁ ☛ ( ψ 2 ) A,B ⊗ C,X (( A ⊗ B ) ⊗ C ) ⋆ X ( A ⊗ ( B ⊗ C )) ⋆ X ✛ α A,B ,C ⋆ 1 X 5. an d tw o squ ares A ⋆ X ( I ⊗ A ) ⋆ X ✲ λ A ⋆ 1 X A ⋆ X I ⋆ ( A ⋆ X ) ✲ ( ψ 0 ) A⋆X ❄ 1 A⋆X ❄ ( ψ 2 ) I , A ,X A ⋆ X ( A ⊗ I ) ⋆ X ✛  A ⋆ 1 X A ⋆ X A ⋆ ( I ⋆ X ) ✲ 1 A ⋆ ( ψ 0 ) X ❄ 1 A⋆X ❄ ( ψ 2 ) A,I ,X comm ute, wh ere O ∈ B , A, B , C ∈ E O and X ∈ X O . End of the d eﬁnition of an action of a lax monoidal ﬁbration on a ﬁ bration. An action of a lax m onoidal ﬁbration is calle d str ong if the trans iti on morphisms ψ 0 , ψ 2 are isomorphisms and ⋆ is a m o rphism of ﬁbr at ions. 3.2 Actions of monoids along an action of a lax monoidal ﬁbration Let ( ⋆, ψ 2 , ψ 0 ) b e an action of a lax m onoidal ﬁbr at ion ( p, I , ⊗ , α, λ,  ) on a ﬁ bration π : X → B , O an ob ject of B , ( M , m, e ) a m onoid in M on ( E ) O , X an ob ject in X O and ν : M ⋆ X → X a morp hism in X O . The pair ( X, ν ) is an action of ( M , m, e ) on X along the action ( ⋆, ψ 2 , ψ 0 ) (or ju st ⋆ , for short) if the follo wing d iag r ams M ⋆ X X ✲ ν M ⋆ ( M ⋆ X ) M ⋆ X ❄ 1 M ⋆ ν ❄ ν ( M ⊗ M ) ⋆ X ✲ ( ψ 2 ) M ,M ,X ✲ m ⋆ 1 X and X X ✲ 1 X I ⋆ X M ⋆ X ✲ e ⋆ 1 X ✻ ( ψ 0 ) X ❄ ν comm ute. A morphism of actions ( f , g , u ) : ( M , X , ν ) − → ( M ′ , X ′ , µ ′ ) is a triple of morphisms u : O → O ′ in B , and f : M → M ′ in M on ( E ), g : X → X ′ in X b oth ov er u so that the s quare in X X X ′ ✲ g M ⋆ X M ′ ⋆ X ′ ✲ f ⋆ g ❄ ν ❄ ν ′ 14 comm utes. The categ ory of actions Act ( E , X , ⋆ ) is ﬁbred o ve r M on ( E ). It migh t happ en that monoids in the ﬁbr e o ver O can b e interpreted as algebras for a single monoid M O in a ﬁbre o ver K ( O ). If this asso ciation is fun ct orial we h a v e commuting squares B M on ( E ) ✲ M on ( E ) Act ( E , X , ⋆ ) ✲ R ❄ q ❄ π µ B B ✲ K ✲ M ❄ 1 B ❄ q The functor R is the represen ting functor t hat in terpr ets monoids as alge bras. If the upp er square is a pullbac k then we sa y that M is the functor of metamonoid 4 and the triple ( R, M , K ) str ongly r e pr esents monoids in M on ( E ) as algebras. This means in particular that the catego r y of O -monoids is equ iv alen t to the category of M O -algebras. If the functor R is an em b edding (faithfu l an d r eﬂec ts isomorphism s) on ﬁb res then we s a y that the triple ( R, M , K ) we akly r epr e sents m onoids in M on ( E ). 3.3 The 2-category of actions of lax monoidal ﬁbrations W e d eﬁne b elo w the morp hisms of actions of lax monoidal ﬁ brations and transformations of s uc h morphism s. In th at wa y we shall deﬁne the 2-cate gory AC T I O N of actions of lax monoidal ﬁbrations on ﬁbrations. A morphism of actions of lax monoidal ﬁbr ations ( F , H , K , ϕ 0 , ϕ 2 , τ ) : ( E , p, X , π , ⋆, ψ 0 , ψ 2 ) − → ( E ′ , p ′ , X ′ , π ′ , ⋆ ′ , ψ ′ 0 , ψ ′ 2 ) consists of the data 1-4 sub ject to the conditions 5-6 b elo w: 1. f unctors F : E − → E ′ , H : X − → X ′ , K : B − → B ′ 2. a morph ism of lax m onoi dal ﬁbrations ( F , K, ϕ 0 , ϕ 2 ) : ( E , p, ⊗ , α, λ,  ) − → ( E ′ , p ′ , ⊗ ′ , α ′ , λ ′ ,  ′ ) 3. a lax morph ism of ﬁbrations ( H , K ) : ( X , π ) − → ( X ′ , π ′ ) 4. a natural transf ormatio n τ : ⋆ ′ ◦ ( F × K H ) − → H ◦ ⋆ i.e. we ha v e a morp hism τ A,X : F ( A ) ⋆ ′ H ( X ) − → H ( A ⋆ X ) whic h is natur a l in A ∈ E O , X ∈ X O and O ∈ B . So we h a v e a diagram 4 This is what seems to b e th e intension of the notion of operad for (colored) op erads introduced by J.Baez and J.Dolan in [BD]. 15 E ′ × B X ′ X ′ ✲ ⋆ ′ E × B X X ✲ ⋆ ❄ F × K H ❄ H B B ′ ❄ K ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✶ π ′ P P P P P P P ✐ π ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ P P P P P P P q in wh ic h th e triangles and int ernal squares comm u te , b ut external square comm utes up to the n atural transformation τ . 5. T he s quare H ( I O ⋆ X ) F ( I O ) ⋆ ′ H ( X ) ✛ τ I O ,X H ( X ) I ′ K ( O ) ⋆ ′ H ( X ) ✲ ( ψ ′ 0 ) H ( X ) ❄ H (( ψ 0 ) X ) ❄ ( ϕ 0 ) O ⋆ ′ 1 H ( X ) comm utes, for X ∈ X O , and O ∈ B . 6. T he h exag on F ( A ) ⋆ ′ ( F ( B ) ⋆ ′ H ( X )) ✑ ✑ ✑ ✑ ✑ ✰ ( ψ ′ 2 ) F ( A ) ,F ( B ) ,H ( X ) ◗ ◗ ◗ ◗ ◗ s 1 F ( A ) ⋆ ′ τ B ,X ( F ( A ) ⊗ ′ F ( B )) ⋆ ′ H ( X ) F ( A ) ⋆ ′ H ( B ⋆ X ) ❄ ( ϕ 2 ) A,B ⋆ ′ 1 H ( X ) ❄ τ A,B ⋆X F ( A ⊗ B ) ⋆ ′ H ( X ) τ A ⊗ B ,X H ( A ⋆ ( B ⋆ X )) H (( A ⊗ B ) ⋆ X ) ✑ ✑ ✑ ✑ ✑ ✰ ◗ ◗ ◗ ◗ ◗ s H (( ψ 2 ) A,B ,X ) comm utes, for A, B ∈ E O , X ∈ X O , and O ∈ B . End of the deﬁnition of a morphism of actions of lax m o noidal ﬁbrations. Let ( F , H , K , ϕ 0 , ϕ 2 , τ ) , ( F ′ , H ′ , K ′ , ϕ ′ 0 , ϕ ′ 2 , τ ) : ( E , p, X , π , ⋆, ψ 0 , ψ 2 ) − → ( E ′ , p ′ , X ′ , π ′ , ⋆ ′ , ψ ′ 0 , ψ ′ 2 ) b e tw o morphism s of actions of lax monoidal ﬁbrations. A tr ansform ation of morphisms of actions of lax monoidal ﬁbr ations ( ζ 0 , ζ 1 , ζ 2 ) : ( F , H , K , ϕ 0 , ϕ 2 , τ ) − → ( F ′ , H ′ , K ′ , ϕ ′ 0 , ϕ ′ 2 , τ ) consists of data 1-3 s ub ject to the condition 4 b elo w: 1. n atural transformations ζ 2 : F − → F ′ , ζ 1 : H − → H ′ and ζ 0 : K − → K ′ ; 16 2. a transformation of lax mon oi dal ﬁbrations ( ζ 2 , ζ 0 ) : ( F , K, ϕ 0 , ϕ 2 ) − → ( F ′ , K ′ , ϕ ′ 0 , ϕ ′ 2 ); 3. a ﬁbr ed natural transformation of morphisms of ﬁbrations ( ζ 1 , ζ 0 ) : ( H , K ) − → ( H ′ , K ′ ) b et we en ﬁbr a tions ( X , π ) and ( X ′ , π ′ ); 4. s o that the square F ′ ( A ) ⋆ ′ H ′ ( X ) H ′ ( A ⋆ X ) ✲ τ ′ A⋆X F ( A ) ⋆ ′ H ( X ) H ( A ⋆ X ) ✲ τ A,X ❄ ( ζ 2 ⋆ ζ 1 ) A,X ❄ ( ζ 1 ) A⋆X comm utes, for A ∈ E O and X ∈ X O . Prop osition 3.1 The morphisms of actions of lax monoidal ﬁbr ations induc e morphisms b etwe e n ﬁbr ations of actions of monoids along actions of monoidal ﬁbr ations. The tr ans- formations b etwe en morphisms of actions of lax monoidal ﬁbr ations induc e natur al tr ans- formations b etwe en the induc e d functors. Pr o of. E xercise . ✷ 3.4 Simple examples 1. T he lax monoidal ﬁbration of graphs p : Gph → S et acts naturally on the basic ﬁbration cod : S et → − → S et . The action of a graph ( d, c : A → O ) on a function F : X → O is deﬁned as the comp osition of the horizonta l arro ws on the top of the follo wing diagram O X ∗ ✛ ξ A A ⋆ X ✛ ❄ c ❄ O ✛ d in whic h the square is a pullbac k. T hen the action of monoids along this action are all p reshea v es on all s mall categories. 2. I f we replace in the previous example S et by any category C with p ullbac ks we get all internal p reshea v es on all internal categories in C . 3. T he ﬁb ratio n of multisorte d signatures also acts on the basic ﬁbration. But this example will b e describ ed in Section 5. 17 4 The exp onen tial ﬁbrations Most of the material of this section b elongs to folklore. W e p resen t it here as we need it later in this form. Cat is the category of large categories, so S et and C at are ob jects of Cat . If X is an ob ject of a cartesian closed category C then X X carries a n at ural structure of a monoid. If p X : X → B is a ﬁbr ati on then w e can form an exp onen tial ﬁb ration p : [ X ⇒ X ] → B in F ib ( B ), the cate gory of ﬁbrations o ve r B , whic h also carries a natural structur e of a lax m onoidal ﬁbration. Then an y strong action of a lax monoidal ﬁbration p E : E → B on p X : X → B give s r ise to a represen tation of p E : E → B in the lax monoidal ﬁb ration of the in ternal endomorphism s of p X : X → B , i.e. a strong morphism of lax monoidal ﬁbrations from p E : E → B to p : [ X ⇒ X ] → B in F ib ( B ). Ho w ev er, the examples of a ctions w e ha v e in mind, are al most nev er strong. But ev en in this c ase w e can still ﬁnd reasonable r epresen tations if we will consid er the exp onen tiation of p X : X → B in Cat / B instead of F ib ( B ). T o distinguish these t wo kinds of exp onen tiation w e denote the exp onen tial ob ject p X : X → B to p Y : Y → B in Cat / B as p : Y X → B . It is well kno wn , cf. [G], for p : Y X → B to b e a w ell deﬁn ed ob ject of Cat / B it is necessary and suﬃcien t for X to b e a so called Cond uc h ´ e ﬁ bration. But as w e w ant p : Y X → B to b e a ﬁbration w e shall assume that p X is a b iﬁbration, i.e. b oth ﬁbration and opﬁbration. In fact, as w e are mainly interested in the case w here X = Y , in order to get a b etter description p : Y X → B , it won’t b e a b ig restriction wh en we sh a ll assume that b oth X and Y are biﬁb ratio ns. 4.1 The exp onen tial biﬁbrations in Cat / B F or any biﬁbration p X : X → B the exp onen tial ﬁ bration p exp : X X → B in Cat / B is a lax monoidal ﬁbration with tensor b eing the comp osition of fu nctors in ﬁbres. The monoids in a ﬁ bre X X o ve r B are monads on X B , and a morphism of monoids o ve r u is a u sual morphism of monads wh ose fu nctor p art is u ∗ . The counit of the exp onent ial adjunction, the e v aluation ev X : X X × B X − → X , is an actio n of the lax monoidal ﬁbration p exp : X X → B on p X : X → B . Finally , the algebras f or this actio n are E ile n b erg-Mo ore algebras for all the monads tak en together. As we sh all n ee d it later, w e shall describ e all of this b elo w in d et ail. The case of real in terest in this pap er is when p X : X − → B is a b asic b iﬁbration cod : C → − → C of the category C with pullbac ks and very lik ely b eing just S et . Let 1 , 2 , 3 b e the obvious categories generated b y th e graphs {•} , {• → •} , {• → • → •} , resp ectiv ely . F or an ob ject B ∈ B , r B : 1 → B is the f unctor pic king the ob ject B . Similarly , r u : 2 → B is a functor pic king the morphism u : B ′ → B in B , and r u,v : 3 → B is a morph ism p ic king a comp osable p air u ◦ v in B . Let p X : X → B and p Y : Y → B b e t wo b iﬁbration. W e can form p ullbac ks 1 B ✲ r B X B X ✲ ❄ ❄ p X 2 B ✲ r u X u X ✲ ❄ ❄ p X 3 B ✲ r u,v X u,v X ✲ ❄ ❄ p X in the categ ory Cat , i.e. p rodu cts in Cat / B . If the exp onen tial ob ject p exp : Y X − → B exists in Cat / B then the ob jects of Y X B corresp ond to morp hisms from r B to p exp in Cat / B and w e h a v e a sequence of corresp ondences 18 1 Y X ✲ ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✙ B r B p exp X B Y ✲ ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✙ B p Y X B Y B ✲ sho w ing that we can (and w e will) ident ify ob jects of Y X B with fun cto rs from X B to Y B . Similarly , usin g r u and r u,v w e see that morphisms in Y X o ve r a morphism u are fun ct ors from X u to Y u comm uting o v er 2 , and the comp osable pairs are morphism s from X u,v to Y u,v comm uting o ve r 3 . The Condu c h ´ e condition is saying in elemen tary terms 5 that for any comp osable pair of morphisms v : B ′′ → B ′ , u : B ′ → B in B the squ are of th e ob vious em b eddings X v X u,v ✲ κ v X B ′ X u ✲ d u ❄ c v ❄ κ u is a p ushout in Cat . Then the comp osition of morphisms F : X u → Y o ve r u and G : X v → Y o ver v suc h that F ◦ d u = G ◦ c v is th e uniqu e fu nctor [ F , G ] : X u,v → Y such that [ F , G ] ◦ κ u = F and [ F , G ] ◦ κ v = G comp osed with the embed ding X u ◦ v → X u,v . Recall that for u : B ′ → B ∈ B w e hav e th e reindexing fu nctor u ∗ : X B ′ → X B and the coreindexing functor u ! : X B → X B ′ deﬁned with the use of prone and s upine morphisms in X . W e denote suc h functors in diﬀerent ﬁ brations b y the same sym b ols. T he f o llo wing Lemma describ es morph isms in Y X more con venien tly in ﬁve diﬀerent wa ys. Lemma 4.1 L et p X : X → B and p Y : Y → B b e two biﬁbr ations. L et u : B ′ → B b e a morphism in B , Q an obje ct in Y X B ′ i.e. a functor fr om X B ′ to Y B ′ , P an obje ct in Y X B i.e. a functor f r om X B to Y B . Ther e is a natur al c orr esp ondenc e b etwe en 1. functors fr om F : X u − → Y u over 2 such that F ◦ d u = Q and F ◦ c u = P ; 2. natur al tr ansform ations τ : Qu ∗ − → u ∗ P in Cat ( X B , Y B ′ ) ; 3. natur al tr ansform ations σ : u ! Q − → P u ! in Cat ( X B ′ , Y B ) . 4. natur al tr ansform ations τ : u ! Qu ∗ − → P in Cat ( X ( B ) , Y ( B )) ; 5. natur al tr ansform ations σ : Q − → u ∗ P u ! in Cat ( X ( B ′ ) , Y ( B ′ )) . Mor e over, if b oth p X and p Y ar e c artesian biﬁbr ations and b oth P and Q (we akly) pr eserve pul lb acks i n ﬁbr es, then under the ab ove c orr esp ondenc es the (we akly) c artesian natur al tr ansformations c orr esp ond to the (we akly) c artesian natur al tr ansformations . Note that in the ab o v e Lemma, the t w o o ccurrences of the sym b ols u ∗ in 2., and u ! in 3., d o NOT denote the same fun cto r s! In eac h of the conditions 2. to 5. one of the functors u ∗ , u ! refers to the biﬁbration p X : X → B an d one of the f unctors u ∗ , u ! to the other biﬁbr at ion p Y : Y → B . 5 This condition will n ev er b e used in the explicit form but for the interested reader we recall it h ere, cf. [G]. The functor p X : X → B is called a C onduch´ e ﬁbr ation , if for any morphism f in X and a pair of morphisms u , v in B such that p X ( f ) = u ◦ v , th ere are morphisms g and h in X such that f = g ◦ h , p X ( g ) = u and p X ( h ) = v . Moreov er, such a factorization of f is u nique up to a zigzag of morphisms in X that b elong to th e ﬁbre o ver the domain of u . 19 Pr o of. As the conditions 4. and 5. are easily seen to b e equiv alen t to 2. and 3., resp ectiv ely , we sh al l concentrat e on equiv alence 1., 2., 3. Fix u : B ′ → B in B , a functor X u Y u ✲ F ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✙ B as in 1., an d t wo natural transformations τ : Q u ∗ − → u ∗ P , σ : u ! Q − → P u ! from the diagram X B Y B ✲ P X B ′ Y B ′ ✲ Q ✻ u ∗ ❄ u ! ✻ u ∗ ❄ u ! as in 2. and 3. Let f : X ′ → X b e a morp hism in X ov er u : B ′ → B with a factorization via prone pr u,X and supine su u,X ′ morphisms in X as follo w s u ∗ ( X ) X ✲ pr u,X X ′ u ! ( X ′ ) ✲ su u,X ′ ❄ ˆ f ❄ ˇ f ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ f The mutual corresp ondence b et w een the functor F and transformations τ and σ can b e read oﬀ f rom the follo win g d iag ram F ( u ∗ ( X )) F ( X ) ✲ F ( pr u,X ) F ( X ′ ) F ( u ! ( X ′ )) ✲ F ( su u,X ′ ) ❄ F ( ˆ f ) ❄ F ( ˇ f ) P P P P P P P P P P P q F ( f ) u ! ( F ( X ′ )) ❄ σ X ′ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✶ su u,F ( X ′ ) u ∗ ( F ( X )) ❄ τ X ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✶ pr u,F ( X ) whose part is just F applied to the pr evio us diagram, having in mind that F restricted to the ﬁbre X B ′ is Q and to the ﬁ bre X B is P . T o see the remaining part of the Lemma, w e recall the d irect relation b et we en τ and σ . F rom τ we get σ as follo ws u ! Q u ! Qu ∗ u ! ✲ u ! Q ( η u ) u ! u ∗ P u ! ✲ u ! ( τ u ! ) P u ! ✲ ε u P u ! and w e get back τ from σ as follo ws 20 Qu ∗ u ∗ u ! Qu ∗ ✲ η Qu ∗ u ∗ P u ! u ∗ ✲ u ∗ ( σ u ∗ ) u ∗ P ✲ u ∗ P ( ε u ) where, as usual, η u and ε u are the u nit and the counit of the adjunction u ! ⊣ u ∗ . F rom this description it is easy to see that with the assumptions of the Lemma, τ is (weakly) cartesian if and only if σ is. ✷ The s ec ond of the ab o v e ﬁ v e ab o ve d escriptions of morp hisms in Y X seems to b e the most con v enient for us, and from no w on we shall assume that the morph isms in Y X are giv en in that f orm. The comp ositio n in Y X is deﬁned as follo ws. F or m orphisms σ : Q → P and τ : R → Q in Y X o ve r u : B ′ → B and v : B ′′ → B ′ , resp ectiv ely , we hav e ✏ ✏ ✏ ✏ ✏ ✏ ✶ τ u ∗ P P P P P P q v ∗ ( σ ) v ∗ Qu ∗ Rv ∗ u ∗ v ∗ u ∗ P ✻ ∼ = ❄ ∼ = R ( u ◦ v ) ∗ ( u ◦ v ) ∗ P ✲ σ ◦ τ where unnamed isomorphisms come from canonical isomo rphisms b et w een functors ( u ◦ v ) ∗ and u ∗ ◦ v ∗ . Th e prone m orphism o ver u : B ′ → B with the co domain P in Y X B pr u,P : u ∗ P u ! − → P is the natural transformation in Cat ( X B , Y B ′ ) d eﬁned with the help of the counit ε u u ∗ P ( ε u ) : u ∗ P u ! u ∗ − → u ∗ P Then, for an y morp hism v : B ′′ → B ′ in B and any morp hisms τ : Q → P in Y X o ve r u ◦ v w e ha ve a (unique!) m orphism ˆ τ : Q → u ∗ P u ! in Y X o ve r v d eﬁned as a natural transformation Qv ∗ ✲ Qv ∗ ( η u ) Qv ∗ u ∗ u ! v ∗ u ∗ P u ! ✲ τ u ! so that τ = pr u,P ◦ ˆ τ in Y X , i.e. the triangle of natural transformations Qv ∗ u ∗ ∼ = Q ( uv ) ∗ v ∗ u ∗ P ∼ = ( uv ) ∗ P P P P P P P P P P P P P q v ∗ u ∗ P u ! u ∗ ✲ v ∗ ( pr u,P ) ❇ ❇ ❇ ❇ ◆ ˆ τ u ∗ τ comm utes. Similarly , the supine morph ism ov er u : B ′ → B with the domain Q in Y X B ′ su u,Q : Q − → u ! Qu ∗ is the natural transformation in Cat ( X B , Y B ′ ) d eﬁned with the help of the unit η u η u Qu ∗ : Qu ∗ − → u ∗ u ! Qu ∗ Then, for an y w : B ′ → B ′′ in B and an y m orphism σ : Q → P o ver w ◦ u we ha ve a (unique) morphism ˇ σ : u ! Qu ∗ − → P in Y X o ve r w deﬁned as a natural transformation u ! Qu ∗ w ∗ ✲ u ! ( σ ) u ! u ∗ w ∗ P w ∗ P ✲ ε u w ∗ P so that the tr ia ngle 21 Q ( v u ) ∗ ∼ = Qu ∗ w ∗ u ∗ u ! Qu ∗ w ∗ ✲ ( su u,Q ) w ∗ P P P P P P P P P P P P P q u ∗ ( ˇ σ ) ❇ ❇ ❇ ❇ ◆ ( v u ) ∗ P ∼ = u ∗ w ∗ P σ comm utes in Y X . Prop osition 4.2 F or any biﬁbr ation p X : X → B the exp onential obje ct p exp : X X → B in C at / B is a biﬁbr ation and it has the structur e of a lax monoidal ﬁbr ation, whose ﬁbr es ar e strict monoidal c ate gories. Pr o of. The fact that p exp is a biﬁbr at ion we ha ve already s ee n. W e d escribe the monoidal structure in p exp : X X → B , and lea v e the rea der to v erify the axioms. W e ha ve an ob vious isomorphism of ﬁbrations B × B X − → X whose exp onen tial adjoint in Cat / B I : B − → X X is the u nit for the tensor. Thus, for B ∈ B , the unit I B in ﬁbre X X B is the id en tit y f unctor on the ﬁbre X B . Th e tensor functor X X × B X X X X ✲ ⊗ is the exp onen tial adjoint in Cat / B to the morph ism X × B X X × B X X X × B X X ✲ ev × 1 X X X ✲ ev The tensor on ob jects is the comp osition of functors. As we will u se it later, w e describ e explicitly the action of the tensor on morphisms. Let σ : P 1 → P 0 and τ : Q 1 → Q 0 b e t wo morphisms in p exp : X X → B o ver a morph ism u : B 1 − → B 0 , i.e. they are natur a l transformations σ : P 1 u ∗ → u ∗ P 0 and τ : Q 1 u ∗ → u ∗ Q 0 in Cat ( X ( B 0 ) , X ( B 1 )). Th en their tensor σ ⊗ u τ : P 1 ⊗ B 1 Q 1 = P 1 ◦ Q 1 − → P 0 ◦ Q 0 = P 0 ⊗ B 0 Q 0 is deﬁned from the comm utativ e d iag ram b elo w P 1 Q 1 u ∗ u ∗ P 0 Q 0 ✲ σ ⊗ u τ P 1 u ∗ u ! Q 1 u ∗ u ∗ P 0 u ! u ∗ Q 0 ✲ σ u ! ∗ τ = σ ∗ u ! ( τ ) ✻ P 1 ( η u Q 1 u ∗ ) ❄ u ∗ P 0 ( ε u Q 0 ) ✷ The monoids in ( p exp : X X → B , ⊗ , I ) are monads o ve r ﬁ bres of p . A morphism of monoids ( f , u ) : ( M ′ , m ′ , e ′ ) − → ( M , m, e ) ov er u : B ′ → B is a morphism of monads from ( M , m, e ) to ( M ′ , m ′ , e ′ ) wh o se functor p art is u ∗ : X B − → X B ′ and f : M ′ u ∗ − → u ∗ M is a natural transformation satisfying the u sual conditions (see Section 5). The ev aluation morphism ev : X X × X − → X in Cat / B is the action of the lax monoidal ﬁbration p exp on the ﬁbration p X . The algebras for this action are all algebras for all monads in M on ( p exp : X X → B , ⊗ , I ) 22 tak en together, i.e. organized in to a single category ﬁ bred o ver B as well as o ver M on ( p exp : X X → B , ⊗ , I ). W e will apply this construction mainly to the basic ﬁ bration cod : C → − → C of a cat- egory C with pullbac ks (usually C = S et ). Such a ﬁbr at ion is alwa ys a cartesian biﬁ- bration. W e write p exp : E xp ( C ) − → C (or p exp, C if we wan t to indicate the category C ) for p exp : C → C → − → C . T he monoids in the exp onen tial ﬁbration p exp : E xp ( C ) − → C are monads in slices of C . The m o rphism b et w een tw o m o noids o v er u : c → c ′ ∈ C is a mor- phism of monads (in the opp osite dir ec tion!) whose fun ct or part is the pu llbac k functor u ∗ : C /c ′ − → C /c . Let C ar t ( C ) denote the sub catego r y of E xp ( C ) wh ose ob jects are p ullbac k preserving functors and cartesian natural transformations b et ween them. Moreo v er, let w C ar t ( C ) denote the su b ca tegory of E xp ( C ) whose ob jects are fun cto rs weakly pr ese rving p ullbac ks and w eakly cartesian natural transf ormati ons b etw een th em. Restricting p exp to C ar t ( C ) and w C ar t ( C ) we get fun cto r s p ca, C : C ar t ( C ) − → C and p w ca, C : w C ar t ( C ) − → C , resp ec- tiv ely . W e hav e Prop osition 4.3 The functors p ca, C and p w ca, C describ e d ab ove ar e lax monoidal b iﬁ- br ations with al l that structur e inherite d fr om p exp : E xp ( C ) → C . In p articular, the emb e ddings C ar t ( C ) w C ar t ( C ) ✲ E xp ( C ) ✲ P P P P P P P P q ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ ❄ C p ca, C p w ca, C p exp ar e morphisms of lax monoidal ﬁbr ations and of biﬁbr ations, that ar e faithful and ful l on isomorph isms. The monoids in p ca, C ( p w ca, C ) ar e (we akly) c artesian monads on slic es of C . Pr o of. T o see that p ca, C is a ﬁbration one has to notice that the pr one morph ism o ve r u : c → c ′ with the co domain P : C /c ′ → C /c ′ , b eing a pullback preserving functor, is a cartesian natural trans formatio n . Moreo v er to see that a factorizatio n via a p rone morphism of a morphism τ in E xp ( C ), b eing a cartesian natural trans formatio n b etw een pullbac k p reserving functors, is also a cartesia n natural transformation. All this follo ws directly fr om the explicit f orm ulas giv en ab o v e for the prone morphism s and the factor- ization via a prone morph ism in p exp : Y X − → B and the fact that b oth indexin g and reindexing functors in p exp preserve pullbac ks. All the ab o v e remains true if we r eplac e pullbac ks by w eak pullbac ks. Thus b oth p ca, C and p w ca, C are ﬁb ratio n s and subﬁb ratio ns of p exp . The argument that these fun cto r s are sub opﬁbrations of p exp is similar. Clearly , the comp ositio n of functors (we akly) p reserving p ullbac ks (w eakly) preserv es pullbac ks . F rom this it is easy to see that the whole lax m onoidal stru ct ure of p ca, C and p w ca, C is in herited from p exp . The remaining part of the prop osition is obvious. ✷ Remark There are man y more interesting subﬁbr at ions of p exp : E xp ( C ) → C . Th e ﬁbrations p ca, C and p w ca, C ha ve also their ’wide pu llbac k versions’. If slices of C are s uf- ﬁcien tly co complete (e.g if C is S et ) then ﬁnitary or ev en accessible f unctors form fu ll subﬁb ratio ns of p exp . Ho wev er, the functors p reserving ﬁnite limits (or just the termin al ob ject) d o n ot constitute a su bﬁbration of p exp , as the fu nctor u ! : C /c − → C /c ′ do es not preserve the terminal ob ject, in general. The follo wing result is the main reason we consider exp onen tial ﬁb ratio ns. It will b e used later man y times. 23 Prop osition 4.4 L et ( p E : E → B , I , ⊗ , α, λ,  ) b e a lax monoida l ﬁbr ation and p X : X → B b e a biﬁbr ation. Then the bije ctive c orr esp ondenc e given by the exp onential adjunction in Cat / B b etwe en morphisms E × B X X ✲ ⋆ B ❅ ❅ ❅ ❘ p X    ✠ and morph isms E X X ✲ ˇ ⋆ B p E ❅ ❅ ❅ ❘ p exp    ✠ induc es a bije ctive c orr esp ondenc e b etwe en actions ( ⋆, ψ 0 , ψ 2 ) of p E on p X and morphisms of lax monoida l ﬁbr ations ( ˇ ⋆, ϕ 0 , ϕ 2 ) fr om p E to p exp . The c orr esp ondenc e r elates the c oher enc e natur al tr ansformations as fol lows. The tr ansformat ion ϕ 0 : I − → ˇ ⋆ ◦ I for O ∈ B , is a natur al tr ansformat ion of functors ( ϕ 0 ) O : id X O − → I O ⋆ ( − ) , i.e. for X ∈ X O we have (( ϕ 0 ) O ) X = ( ψ 0 ) O ,X : X − → I O ⋆ X Mor e over, for A, B ∈ E O , we have A ⋆ ( B ( − )) ( A ⊗ B ) ⋆ ( − ) ✲ ( ψ 2 ) A,B , − ˇ ⋆ ( A ) ◦ ˇ ⋆ ( B ) ˇ ⋆ ( A ⊗ B ) ✲ ( ϕ 2 ) A,B k k i.e. X ∈ X O , (( ϕ 2 ) A,B ) X = ( ψ 2 ) A,B ,X : A ⋆ ( B ⋆ X ) − → ( A ⊗ B ) ⋆ X This c orr esp ondenc e is natur al in E . Pr o of. E xercise . ✷ F rom the ab o ve Prop ositio n follo ws that if we ha ve an act ion of the lax monoidal ﬁ bra- tion p E : E → B on a biﬁb ration p X : X → B , we get a morphism from the lax monoidal ﬁbration p E in to a lax m onoidal ﬁbr at ion w hose ﬁbres are str ic t m onoidal categories. If X is suﬃcientl y concrete (lik e S et → ) and this morphism is an em b edding we can view this kind of p henomena as repr esen tation theorems. W e r epresen t ob jects of E as endofu nctors of ﬁbres of p X : X → B , and monoids in ( p E : E → B , ⊗ , I ) as m onads o ver ﬁbres of p X : X → B . Similar things can b e said ab out morp hisms. W e will see many examples of suc h r epresen tations later. 4.2 The exp onen tial ﬁbrations in F ib/ B F ib/ B is a cartesian closed category . F or any ﬁbr ati on p X : X → B the exp onen tial ﬁbration p f iexp : [ X ⇒ X ] → B is lax monoidal w ith tensor b eing (again) the internal comp ositio n. Monoids in p f exp are compatible families of monads and cartesia n morphisms b et we en them. Ha ving a strong action ⋆ : E × B X − → X we could also represent E in [ X ⇒ X ]. But strong actions are less common, and suc h (n o n-trivial) represen tations are more diﬃcult to ac hiev e in practice. This is why we are n ot going consider this kind of exp onen tial ﬁbrations in the follo wing. 24 5 The Burr oni ﬁbr ations and op etopic sets 5.1 The Burroni ﬁbrations and T -categories Let C b e a category with p ullbac ks, h T , η , µ i a monad on C . T he catego ry Gph ( T ) is the catego ry of T -graphs. An ob ject h A, O , γ , δ i of Gph ( T ) is a sp an O T ( O ) A γ    ✠ δ ❅ ❅ ❅ ❘ in C . The morphisms γ and δ are ca lled c o doma ins and d omains of the T -graph h A, O , γ , δ i , resp ectiv ely . Sometimes we write A instead of h A, O , γ , δ i , for short, when it d oes n ot lea d to a confusion. A morphism of T -graphs h f , u i : h A, O , γ , δ i − → h A ′ , O ′ , γ ′ , δ ′ i is a pair of morp hisms f : A → A ′ and u : O → O ′ in C making the squ ares O O ′ ✲ u A A ′ ✲ f ❄ γ ❄ γ ′ T ( O ) T ( O ′ ) ✲ T ( u ) A A ′ ✲ f ❄ δ ❄ δ ′ comm ute. Let Gph ( T ) denotes the category of T -graphs and T -graph morphisms. W e ha ve a pro jection fun cto r p T : Gph ( T ) − → C sending the m orphism h f , u i : h A, O , γ , δ i − → h A ′ , O ′ , γ ′ , δ ′ i to the morphism u : O → O ′ whic h is easily s ee n to b e a ﬁbration, cf. [B] p . 235. The lax monoidal structur e in p T is deﬁned as follo ws. Let h A, O , γ A , δ A i and h B , O , γ B , δ B i b e tw o ob jects in the ﬁbre o ve r O , i.e. in Gph ( T ) O . Th en the tensor h A, O , γ A , δ A i ⊗ O h B , O , γ B , δ B i = h A ⊗ B , O , γ ⊗ , δ ⊗ i is deﬁned from the follo wing d iag ram O T ( O ) A γ A    ✠ ❅ ❅ ❅ ❘ δ A T 2 ( O ) T ( B )    ✠ T ( δ B ) ❅ ❅ ❅ ❘ T ( γ B ) T ( O ) µ O ❅ ❅ ❅ ❘ A ⊗ B π 1    ✠ π 2 ❅ ❅ ❅ ❘ in whic h the s quare is a pullbac k and γ ⊗ = γ A ◦ π 1 , δ ⊗ = µ O ◦ T ( δ B ) ◦ π 2 . The unit in the ﬁbre o ve r O is 25 O T ( O ) O 1 O    ✠ η O ❅ ❅ ❅ ❘ The coherence morphisms are deﬁ ned using the u niv ersal prop erties of pullbacks. F or an ob j ec t h A, O , γ A , δ A i on the ﬁb re o ver O the left unit morph ism is λ h A,O ,γ A ,δ A i = h 1 O , η A ) , 1 O i : h A, O , γ A , δ A i − → h O ⊗ A, O , γ ⊗ , δ ⊗ i the righ t unit morphism is  h A,O ,γ A ,δ A i = h 1 A , 1 O i : h A ⊗ O , O , γ ⊗ , δ ⊗ i − → h A, O , γ A , δ A i (the righ t unit morphism is alwa ys an isomorph ism and in fact we can assume that it is an ident it y as A ⊗ O is a pu llbac k of δ A along th e identit y). Th e asso ciat ivit y morph ism α A,B ,C : A ⊗ ( B ⊗ C ) − → ( A ⊗ B ) ⊗ C is also deﬁned similarly , using un iv ersal pr operties of pu llbac ks. W e leav e the details to the reader. Prop osition 5.1 L et C b e a c ate gory with pul lb acks. The functor p T : Gph ( T ) → C i s a biﬁbr ation and to gether with the monoidal structur e ( ⊗ , I , α, λ,  ) describ e d ab ove is a lax monoidal ﬁbr ation. The tota l c ate gory of the ﬁbr ation q T : M on ( T ) − → C of monoids in ( Gph ( T ) , p T , ⊗ , I , α, λ,  ) is e quivalent to the c ate gory of T -c ate gories of Burr oni. If mor e over, the monad ( T , η , µ ) is c artesian then the ﬁbr es of p T ar e (str ong) monoidal c ate gories, i.e. the c oher e nc e morph isms, λ and α , ar e isomorphisms. Pr o of. A simple tedious chec k. ✷ Remark If the m onad ( T , η , µ ) is cartesian then the ﬁbr es of p T : Gph ( T ) → C are strong monoidal categories but the reind exing fun ct ors are still only lax monoidal. This is already so f or th e iden tity monad (1 C , 1 1 C , 1 1 C ) on C . The category M on (1 C ) is the catego ry of internal categories in C . 5.2 T au tologous actions of Burroni ﬁbrations If ( T , η , µ ) is a monad on a category C with pullbacks th en the lax monoidal ﬁbration p T : Gph ( T ) − → C has a natural action on th e basic ﬁbration cod : C → − → C . Th e functor part Gph ( T ) × C C → C → ✲ ⋆ T C ❅ ❅ ❅ ❘ cod    ✠ is deﬁned on ob jects by O T ( O ) A γ    ✠ δ ❅ ❅ ❅ ❘ X O ❄ d A ⋆ T X O ❄ ✲ where the righ t v ertical arro w i n the ab o v e diag r am is the comp osite of the u pp er horizon tal arro ws in th e follo wing diagram 26 T ( O ) T ( X ) ✛ T ( d ) A A ⋆ T X ✛ ❄ δ ❄ O ✛ γ in whic h the square is a pullback. By adjunction, w e get a morphism of lax m onoidal ﬁbrations Gph ( T ) E xp ( C ) ✲ r ep T C p T ❅ ❅ ❅ ❘ p exp    ✠ that represents T -graphs as endofu nctors on slices of C . Und er this representat ion the T -catego ries corresp ond to (some) monads on slices of C . Example The actions ⋆ T of the lax monoidal ﬁbration deﬁned ab o ve do not preserve prone morp hisms, in general. Even if T is th e free monoid monad on S et , the action ⋆ T is only a lax morphism of ﬁbrations. T o s ee this, consider a morphism u : [1] → [0], from t wo elemen t set [1] = { 0 , 1 } to one elemen t set [0] = { 0 } . L et ( A, γ , δ ) b e an ob ject in Gph ( T ) o ver [1], s uc h that A = { a } , and ∂ ( a ) = 00, the wo rd of length 2 of zero’s, γ ( a ) = 0. T he iden tit y 1 [0] on [0] is a morp hism in S et / [0] . T hen the ob ject ( A, γ , δ ) ⋆ ([0] , 1 [0] ) (in the basic ﬁ bration o ve r S et ) h as one element in the domain, op eratio n a w ith inputs and outputs in [0], i.e. h a ; 0 , 00 i . The domain of the prone morphism u ∗ (( A, γ , δ ) ⋆ ([0] , 1 [0] )) − → ( A, γ , δ ) ⋆ ([0] , 1 [0] ) ov er u h as t w o elemen ts, in the domain, th e op eration a with the output either 0 or 1 and inpu ts as b efore, i.e. {h a ; 0 , 00 i , h a ; 1 , 00 i} . On the other hand, the image under ⋆ T of the pr one morph isms ov er u , whose co domains are ( A, γ , δ ) and 1 [0] , r espective ly , pr u,A ⋆ T pr u, 1 [0] : u ∗ ( A ) ⋆ T u ∗ ([0]) − → A ⋆ T [0] h as in th e domain of its domain eight elemen ts, i.e. the op eratio n a with b oth inp uts and outputs either 0 or 1, i.e. {h a ; 0 , 00 i , h a ; 1 , 00 i , . . . , h a ; 1 , 10 ih a ; 1 , 11 i} . Thus the d o mains of those morphisms are not isomorphic and hence the prone morphism s are not p reserv ed. As the morphism r ep T is the exp onen tial transp ose of ⋆ T in C AT / C (not in F ib ( C )) one of these morphisms can b e a morph ism of ﬁb ratio ns ev en if the other one is not. W e ha ve Prop osition 5.2 L et ( T , η , µ ) b e a c artesian monad on a c ate gory with pul lb acks C . Then the functor r ep T deﬁne d ab ove is a str ong morphism of lax monoidal ﬁbr ations and of biﬁbr ations. The image of r ep T is in p ca : C ar t ( C ) → C . Pr o of. First, we describ e the functor r ep T in details. F or an ob ject A = ( A, γ , δ ) in Gph ( T ) O , w e hav e a fun cto r r ep T ( A ) = A ⋆ T ( − ) : C /O − → C /O In the f ol lo wing, we omit the s up erscript T . F or u : O → Q in C and ( h, u ) : A → B , a morphism in Gph ( T ) o ver u , we h a v e a natur a l transformation in Cat ( C /Q , C /O ) r ep T ( h, u ) : A ⋆ u ∗ ( − ) − → u ∗ ( B ⋆ ( − )) so that for d Y : Y → Q in C , th e v alue r ep T ( h, u ) Y is d eﬁned from the follo wing diagram 27 O Q ✲ A B ✲ ❄ ❄ A B ✲ A ⋆ u ∗ ( Y ) B ⋆ Y ✲ h ⋆ u u Y ❄ T ( O ) T ( Q ) T ( u ∗ ( Y )) T ( Y ) ❄ u ∗ ( B ⋆ Y ) ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✿ ◗ ◗ ◗ ◗ ◗ s ◗ ◗ ◗ ◗ ◗ s ◗ ◗ ◗ ◗ ◗ s ◗ ◗ ◗ ◗ ◗ s    ✠ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ❄ ❄ ✲ ✲ h u T ( u ) T ( u Y ) T ( u ∗ ( d Y )) T ( d Y ) γ δ γ δ r ep T ( h, u ) Y pr u,B ⋆Y where O Q ✲ u u ∗ ( Y ) Y ✲ u Y ❄ u ∗ ( d Y ) ❄ d Y is a pullbac k. As three sides of the cu be are pullbacks ( T preserves pullb ac k s), so is the fron t square. In p artic u lar, if h is an iso, so is h ⋆ u u Y , for any d Y : Y → Q in C . Note that the morphism s pr u,B ⋆Y : u ∗ ( B ⋆ Y ) → ( B ⋆ Y and u Y , u ) : u ∗ ( Y ) → Y , in the ab o ve tw o diagrams, are prone morphisms (o ver u ) in the basic ﬁbration o ver C . In the f ol lo wing, we will deal w ith prone and sup ine morph isms in tw o other ﬁbrations p T : Gph ( T ) → C and p exp : E xp ( C ) → C . Thus in total , w e ha v e th ree diﬀerent sorts of prone morphisms. The codomain of a s upine morphism ov er u : O → Q whose domain is A = ( A, γ , δ ) is u ! ( A ) = ( A, u ◦ γ , T ( u ) ◦ δ ). The su pine morphism is su u,A = (1 A , u ) : A → u ! ( A ). W e ha ve a diagram in Gph ( T ) ov er u : A ⋆ u ∗ ( Y ) u ! ( A ) ⋆ Y ✲ 1 A ⋆ u u Y u ∗ u ! ( A ⋆ u ∗ ( Y )) u ! ( A ⋆ u ∗ ( Y )) ✲ ✻ ❄ ξ u A,Y u ∗ ( u ! ( A ) ⋆ Y ) ✛ r ep T (1 A , u ) Y ( su u,r ep T ( A ) ) Y = η u A⋆u ∗ ( Y ) ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✯ su u = 1 A⋆u ∗ ( Y ) ❄ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ u ∗ ( ξ u A,Y ) ✻ pr u,u ! ( A ) ⋆Y The m orphism ξ u A,Y is the second part of th e factorizatio n of 1 A ⋆ u u Y via a supine m or- phism. Note that the morphisms 1 A ⋆ u u Y and ξ u A,Y considered as morph isms in C are equal bu t the ﬁrs t i s a p art of a morp hism in Gph ( T ) ov er u , and th e second is in the ﬁ bre o ve r Q . By a remark b elo w the previous diagram, 1 A ⋆ u u Y is an isomorp hism an d h ence, so are ξ u A,Y and u ∗ ( ξ u A,Y ), as we ll. On e can v erify that the left hand triangle comm utes, as it is a triangle in the ﬁ bre ov er O in the basic ﬁbr at ion o ver C and commutes when comp osed with th e prone morphism pr u,u ! ( A ) ⋆Y . Th us r ep T preserve s th e sup ine morp hisms. One can v erify that the p rone m orphism in Gph ( T ) ov er u : O → Q with co domain B is ( u B , u ) : u ∗ ( B ) → B wh ere 28 O T ( O ) u ∗ ( B ) γ    ✠ δ ❅ ❅ ❅ ❘ Q T ( Q ) B γ    ✠ δ ❅ ❅ ❅ ❘ ✲ u Y ✻ u ✻ T ( u ) is a limiting cone. F or d Y : Y → Q in C with u ∗ ( Y ) Y ✲ u Y u ! u ∗ ( Y ) ❄ ε u Y ✟ ✟ ✟ ✟ ✟ ✟ ✯ pr u,Y = 1 (i.e. u Y = ε u Y ) w e can f orm a d iag r am u ∗ ( B ⋆ Y ) B ⋆ Y ✲ pr u,B ⋆Y u ∗ ( B ) ⋆ u ∗ ( Y ) B ⋆ u ! u ∗ ( Y ) ✲ u B ⋆ 1 u ∗ ( Y ) ❄ r ep T ( u B , u ) Y ❄ B ⋆ ε u Y u ∗ ( B ⋆ u ! u ∗ ( Y )) ✛ r ep T (1 u Y , u ) Y ( pr u,r ep T ( B ) ) Y P P P P P P P P P P P P P P P P q ❄ pr u,B ⋆u ! u ∗ ( Y ) (( pr u,r ep T ( B ) ) Y = u ∗ ( B ⋆ ε u Y )) in wh ic h one can verify , using p roperties of pullbac ks, that u B ⋆ u 1 u ∗ ( Y ) is p rone in the b asic ﬁbration o ver C . Th us ζ u B ,Y , th e ﬁ rst part of the factorizat ion of u B ⋆ u 1 u ∗ ( Y ) via a p rone morph ism, is an iso. As the left triangle comm utes, r ep T preserve s prone morphisms, as w ell. ✷ Remark F rom th e ab o v e pro of, it follo ws that for an y monad T on a category C with pullbac ks , we hav e morph isms u ! ( A ⋆ u ∗ ( Y )) u ! ( A ) ⋆ Y ✲ ξ u A,Y u ∗ ( B ) ⋆ u ∗ ( Y ) u ∗ ( B ⋆ u ! u ∗ ( Y )) ✲ ζ u B ,Y natural in A , B and Y , that are isomorph isms if the monad T is cartesian. Note that these isomorph isms e xpress a kind of Bec k-Ch ev alley condition for actions of lax m onoidal ﬁbrations. Example. If T is the identit y mon a d on a category with pullbac ks then r ep T sends the in ternal category C = ( C 1 , C 0 , m, i, d, c ) in C to a monad rep T ( C ) on the slic e category C /C 0 whose algebras are internal p reshea v es on C . 5.3 Multisorted signatures vs monotone p olynom ial diagrams In this section we sh all examine the consid erat ions f rom the pr evio us section on a s peciﬁc example of the fr ee monoid monad T on the category S et . No te that Gph ( T ) can b e though t of as a categ ory of m u ltiso rted signatures. An ob j ec t h A, O , γ , δ i of Gph ( T ) can b e seen as a set of op erat ions A a set of t yp es O , functions γ and δ asso ciating to op erations in A their typ es of co domains in O and their lists of t yp es of their domains in T ( O ). T o emphasize this, we shall denote the ﬁbr at ion p T : Gph ( T ) → S et for this p artic ular monad 29 T as p m : S ig m → S et . As w e already ment ioned, cf. [B], the catego ry of monoids in p m is equiv alen t to the category of Lambek’s multica tegories. The action of p m : S ig m → S et on cod : S et → − → S et is as d eﬁned ab o ve. Thus, by adjunction, we ha ve a representat ion morphism S ig m E xp ( S et ) ✲ r ep m S et p m ❅ ❅ ❅ ❘ p exp    ✠ W e shall describ e the image of this represent ation in a diﬀeren t w ay . A monotone p olyno- mial diagr am 6 o ve r the set O is a diagram of th e follo win g f orm O E ✛ s B ✲ p O ✲ t of sets a nd functions, moreo v er the ﬁbres of the morphism p are ﬁnite and linea rly ordered. W e wr ite ( t, p, s ) to denote suc h a diagram. A morphism of monotone diagr ams ( f , g , u ) : ( t, p, s ) − → ( t ′ , p ′ , s ′ ) o v er a fu nction u : O → O ′ is a trip le of functions with g : E → E ′ and f : B → B ′ so that the d iag ram O ′ E ′ ✛ s ′ O E ✛ s ❄ u ❄ g B ′ ✲ p ′ B ✲ p ❄ f O ′ ✲ t ′ O ✲ t ❄ u comm utes, and the mid dle squ a re is a pullback in th e category of p osets (i.e. g ⌈ p − 1 ( b ) : p − 1 ( b ) → p ′− 1 ( f ( b )) is an order isomorph ism, for b ∈ B ). W e comp ose morphism s of monotone p olynomial diagrams in the ob vious wa y , b y p lac ing one on top of the other. In this wa y w e deﬁn ed th e catego ry MP ol y D iag of monotone p olynomial d ia grams. T he catego ry MP oly D iag is ﬁb red o ver S et , where the p ro jection fun ct or p mpd : MP ol y D iag − → S et is giv en by ( f , g , u ) : ( t, p, s ) − → ( t ′ , p ′ , s ′ ) 7→ u : O − → O ′ This is a lax monoidal ﬁbr at ion 7 whic h also acts on the b asic ﬁbration cod : S et → − → S et . The action MP oly D iag × S et S et → S et → ✲ ⋆ S et ❅ ❅ ❅ ❘ cod    ✠ is giv en by the w ell kn o wn formula deﬁning p olynomial functors (see Section 6), i.e. f or ( t, p, s ) in MP ol y D iag O , and d X : X → O a fu nctio n, w e h a v e ( t, p, s ) ⋆ d X = t ! p ∗ s ∗ ( d X ) Th us, b y ad join tness, w e ha ve a morp hism of lax monoidal ﬁbrations 6 The name is so chosen to ind icated the obvious relation with the notion of a p olynomial d ia gram th at will b e considered in th e next section. 7 The deﬁnition of th e lax monoidal structure is left to b e deﬁned by t h e reader. It is close to th e structure on the ﬁbration of p olynomial functors deﬁned in Section 6. 30 MP oly D iag E xp ( S et ) ✲ r ep mpd S et p mpd ❅ ❅ ❅ ❘ p exp    ✠ The class of fun cto rs in the image r ep mpd coincides with the class of ﬁ nitary p olyno- mial endofunctors. How ev er the linear structure in the ﬁbres of monotone p olynomial diagrams r estric ts th e class of natural tran sformatio ns b et w een them. F or ev ery p oly- nomial transformation τ : P → Q b et w een p olynomial end ofunctors on S et I , there is a monotone morph isms b etw een monotone diagrams ( f , g , u ) : ( t, p, s ) − → ( t ′ , p ′ , s ′ ) so that r ep mpd ( f , g , u ) is isomorph ic to τ (just order ﬁbres in the p olynomial diagrams deﬁnin g P and Q in a compatible wa y). This observ ation sa ys that the essent ial image of r ep mpd consists of p olynomial functors and p olynomial natural tran sformatio ns, see Section 6. Ho we v er this is n ot saying that th e monotone p olynomial monads on p olynomial functors are the same as p olynomial monads . F or a m onad ( T , η , µ ) on a p olynomial fu nctor to b e linear means 8 , that we can ﬁn d o ne ordering of the ﬁ bres of the p olynomial diagram deﬁn- ing T so th at b oth m o rphisms η : 1 C → T and µ : T 2 → T are deﬁned by the morphisms of d ia grams resp ecting these orderings (the order of the ﬁb res of the diagram d eﬁning T 2 is determined by the order of the diagram deﬁn ing T ). As w e shall see later, this migh t b e not p ossible. W e n ote for the r eco r d Prop osition 5.3 The r epr esentations r ep m and r ep mpd ar e faithful and they ar e e qu iva- lent as morph i sms of lax monoida l ﬁbr ations into E xp ( S et ) − → S et . As a c onse quenc e, r ep mpd is a morphism of biﬁbr ations and the c ate gory of L amb ek ’s multic ate gories is e quiv- alent to the c ate gory of mono ids in MP ol y D iag . Mor e over, the monads in the image of the morphism of ﬁbr ations of monoids M on ( MP ol y D iag ) M on ( E xp ( S et )) ✲ S et ❅ ❅ ❅ ❅ ❘     ✠ induc e d by r ep mpd ar e exactly monotone monads on p olynomial functors. ✷ Remark A bad thing ab out the representa tions r ep m and r ep mpd is th a t they are not full, ev en on isomorp hism. As a consequence the monotone monads do not d et er- mine the monotone diagrams deﬁn ing them uniqu ely (up to isomorphism). The lac k of fullness on isomorp hisms is du e to th e fact that ﬁbr es in monotone diagrams are linearly ordered. As we shall see in the n ext t w o sections, similar r epresen tations of b oth (ﬁnitary) p olynomial (end o)functors an d (ﬁnitary multiv ariable) analytic (endo)functors are fu ll on isomorphisms. 5.4 Morphisms of monads Morphisms of monads indu ce morphisms of Burr oni ﬁbr ati ons and morphisms of tautol- ogous actions of Burr oni ﬁb ratio ns. In d eta ils, it lo oks as f oll o ws. Let ( S, η S , µ S ) b e a monad on C and ( T , η T , µ T ) b e a monad on D , F : C → D a fu nctor p reserving p ullbac ks, and ξ : T F → F S b e a natural transformation so that ( F , ξ ) : ( S, η S , µ S ) − → ( T , η T , µ T ) is a m onad morphism, i.e. the d iag ram 8 Here by a monotone monad w e mean a monad that is an image of a monoid in p mpd : MP oly D iag − → S et . 31 F S F S 2 ✛ F ( µ S ) T F T 2 F ✛ µ T F ❄ ξ T F S ❄ T ( ξ ) ❄ ξ S F ✑ ✑ ✑ ✸ η T F ◗ ◗ ◗ s F ( η S ) comm utes. Then w e can deﬁne the functor from S -graph s to T -graphs C D ✲ F Gph ( S ) G p h ( T ) ✲ Gph ( F, ξ ) ❄ p S ❄ p T as O S ( O ) A γ    ✠ δ ❅ ❅ ❅ ❘ F ( O ) F S ( O ) F ( A ) F ( γ )    ✠ F ( δ ) ❅ ❅ ❅ ❘ T F ( O ) ✛ ξ O Gph ( F, ξ )( A ) ✛ ❅ ❅ ❅ ❘ ✲ where the sq uare on the righ t in a pullbac k. This functor has an obvio us structure ( ϕ 0 and ϕ 2 ) of a m orphism of lax monoidal ﬁbrations. In particular, as an y monad ( T , η , µ ) on a category C with p ullbac ks has a monad morphism to the iden tit y monad 1 C , any Burr oni ﬁbration on C has a morphism into the 1 C -ﬁbration. T his is another w ay o f sa ying that the ca tegory of T -categ ories has a forgetful functors in to the catego ry of in ternal categories in C . No w a r outine veriﬁcat ion will show that such a morphism of lax monoidal ﬁ brations of graphs together with a ﬁbred morphism of basic ﬁ brations C D ✲ F C → D → ✲ F → ❄ cod ❄ cod giv es rise to a morph ism of tautologous actions. 5.5 Relativ e Burroni ﬁbrations and relativ e T -categories The co n struction of a lax m onoi dal ﬁbration of T -graphs can b e p erformed e v en on a ﬁb red monad on a ﬁbration. Supp ose p : E → B is a ﬁbration su c h that th e ﬁ bres of p ha v e pullbac ks . Moreo ve r ( T , η , µ ) is a monad on the catego ry E so that T is a lax morp hism of ﬁbrations E E ✲ T B p ❅ ❅ ❘ p   ✠ and η , µ are ﬁb red natural transformations (i.e. their comp onen ts lie in the ﬁbres of p ). Ha ving such data we can rep eat the construction of the category of T -graphs bu t restricting the ob jects to such spans O T ( O ) A γ   ✠ δ ❅ ❅ ❘ 32 that are in ﬁbres of p (i.e. p ( γ ) = p ( δ ) = 1 p ( O ) ). The morphisms are deﬁned as b efore. In this w a y , we get a relativ e Bu rroni ﬁ bration p T : Gph ( T , p ) → E of T -graphs o v er p . Clearly , p T is a lax m o noidal ﬁbration with th e tensor structure deﬁned as b efore. Th us w e h a v e a ﬁ bration of monoids with a forgetful to Gph ( T , p ) as in the diagram Gph ( T , p ) M on ( T , p ) ✛ U T ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ p T E B ❄ p q T of fu nctors and categories. As for an y category C , the functor ! : C − → 1 int o the terminal catego ry 1 is a ﬁbr ati on, this construction is a generalizat ion of the p revious one. Remark W e can also deﬁne a basic ﬁ bration cod : E → ,p → E relativ e to a ﬁbration p : E → B , so that the ob jects of E → ,p are morphisms of E in ﬁbres of p and morphisms are comm uting squares. Th en, as p reviously for th e Burroni ﬁbrations, we ha ve a tautologous action the lax monoidal ﬁbr at ion p T : Gph ( T , p ) → E on a ﬁ bration cod : E → ,p → E Gph ( T , p ) × E E → ,p E → ,p ✲ ⋆ ( T , p ) E ❅ ❅ ❅ ❅ ❘ cod     ✠ If we tak e the exp onen tial adjoin t of this m o rphism, as in 5.2, we obtain a (relativ e) represent ation of relativ e T -graphs and r el ativ e T -catego ries. 5.6 F ree relative T -categories The full charac terization of th o se monads T for whic h the forgetful fu nctor U T deﬁned ab o v e h as a left adjoin t seem to b e u nkno wn. Ho wev er there are v arious reasonable suﬃcien t conditions, cf. [B], [Ke ], [BJT], [Le] in case the monad ( T , η , µ ) is cartesian. Recall that a monad ( T , η , µ ) on a categ ory C with ﬁnite pro ducts is cartesian if T preserve s pullbac ks and b oth η , µ are cartesian natural transformations. A. Burroni in [B] (p p. 267- 269) pr o vided one su c h c haracterizatio n and h e notice d that if s uc h an adj o in t exists U T is automatica lly monadic [B] (p. 304). He also noticed that in certain cases one can iterate the T -category construction [B] (p. 269). How ev er the condition for the iteration in [B] is to o strong 9 to b e u sed for our construction b elo w. T. Leinster in [Le] used a w eak er condition f or iteration b ut he w as in terested in iteration in p artic u lar ﬁb res rather than of the w hole ﬁbr at ion. With the h elp of th is kind of iteration he deﬁned the set of op etop es [Le] (p. 179 and App endix D). The construction of the free monoids describ ed in [Le] is the same as the earlier and more detailed, y et compact, constru ct ion describ ed in [BJT] in App endix B. The inductiv e formula deﬁning the free monoids giv en in b oth [BJT] and [Le] seem to app ear ﬁr st in [A] (p. 591) to d escrib e free algebras for a f unctor and then in a long compreh ensiv e stu dy [Ke] (p. 69) that extend s and u niﬁes some earlier deve lopmen ts of this and r ela ted su b jects. The p rerequisites for th e constru ctio n of the free monoids as w ell as the ﬁnal goals diﬀer in [BJT] and [Le ]. In [BJT] the p rerequisites are giv en d irect ly 9 One of t h e requirement is th at the monad T commutes with copro ducts. 33 in terms of the prop erties of the category and the tensor inv olv ed to get a left adjoin t to the forgetful functor from the monoids to the monoidal category . In [Le ] the prerequisites are giv en also in terms of th e prop erties of the catego ry h o w eve r the pr operty o f the tensor is n ot sp eciﬁed dir ec tly b ut through the prop ert y of the monad the tensor is coming from. Moreo v er, in [Le] the aim is not only to get a left adjoint b ut also to mak e su re that a monad (and a category it is deﬁned on) ded uced from the new adjunction s at isﬁes th e same prop erties, so that one can iterate the constru ct ion, as in [B]. Belo w we gi v e a c haracterization of those ﬁ brations p and ﬁbred monads T on them for whic h one can iterate the pro cess of taking T -graph s ov er a ﬁ bration p . In the exp osition w e use ideas from all the men tioned p apers. Th e notions of a s uitable ﬁ brations and a ﬁbrewise suitable m onad are v ery muc h insp ired b y the notions of a su ita b le category and a su ita b le monad, resp ectiv ely , cf. [Le] App endix D. The m ai n d iﬀe rence of our approac h with resp ect to [Le ] is th at w e iterate whole ﬁbrations ov er ﬁbrations an d get as a ﬁnal result the category of op etopic sets, wh erea s in [Le] the construction is d one ﬁb re by ﬁ bre and giv es the set of op etopes as a result. F rom the p ersp ectiv e of our construction th is set of op etopes is the set of cells in the terminal op etopic set. W e sa y th at a ﬁbr a tion p : E → B is suitable if and only if 1. p h as ﬁbred pullbac ks, ﬁ nite copro ducts, and ﬁltered colimits, 2. ﬁ nite copro ducts and ﬁ ltered colimits are universal in ﬁbres of p , 3. ﬁ ltered colimits comm u tes with pu llbac ks in ﬁbres of p . Let p : E → B b e a ﬁ bration with ﬁbr ed pullbac ks . A mon ad ( T , η , µ ) on E is c artesian r elative to p if a nd o n ly if ( T , η , µ ) is a ﬁbred monad o v er p (i.e. p ◦ T = p , p ( η ) = 1 p = p ( µ )) and the restriction of the monad ( T , η , µ ) to every ﬁbre of p is a cartesian monad on this ﬁbre. Let p : E → B b e a s uitable ﬁbr at ion. W e sa y that a monad ( T , η , µ ) on E is suitable r elative to p if and only if ( T , η , µ ) is cartesian relati v e to p and T p reserv es ﬁltered colimits in the ﬁbres of p . The follo wing theorem is the k ey to the deﬁnition of the to w er of ﬁbrations that deﬁnes the category of op etopic sets. Theorem 5.4 L et ( T , η , µ ) b e a su itable monad r elative to a suitable ﬁbr ation p : E → B . Then 1. the ﬁbr ation p T over p is again suitable; 2. the for getful functor U T is monadic; 3. the monad ( e T , e η , e µ ) induc e d by the adjunction F T ⊣ U T is suitable r elative to p T . Gph ( T , p ) M on ( T , p ) ✛ U T ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ p T E B ❄ p q T ✄ ✂ ✲ T ✄ ✂ ✲ e T In the pro of of this theorem we shall use the follo wing easy lemma. 34 Lemma 5.5 Supp ose p and q ar e ﬁbr ations and we have two lax morphisms of ﬁbr ations U and F , as in the diagr am ✛ U E M ✲ F ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ p B q If U is a morphism of ﬁbr ations and F is a left adjoint to U when r estricte d to e ach ﬁbr e then F is a left adjoint to U . Remark Th is Lemma could b e compared with Lemma 1 .8.9 of [Ja]. Ho wev er we don’t require the Bec k-Ch ev alley condition as we don’t exp ect F to b e a morphism of ﬁbr ati ons, as in our application it won’t b e. Pr o of of The or em 5.4. The functor Gph ( T , p ) − → E sending T -graph ( A, γ , δ ) to A creates pu llbac ks, ﬁnite copro ducts and ﬁ ltered colimits. Thus those limits and colimits ha ve the same exactness prop erties in the ﬁ bres of p T as they had in ﬁbres of p . The fact that they are ﬁb red in p T follo ws from th e fact that they are ﬁbred in p and that ﬁnite copro ducts and ﬁ lte r ed colimits are un iv ersal. Thus, that p T is a su itable ﬁbr at ion. Recall the construction of the free monoid from [Ke], [BJT] 10 , [Le]. F or an ob ject ( A, γ , δ ) in a ﬁ bre E O w e constru ct a ﬁltered diagram. W e write A for ( A, γ , δ ) and O for the unit of the tensor ( O , 1 O , η O ), for short. A 0 = O ❄ e 0 A 1 = O + A ⊗ O ❄ e 1 = 1 + (1 ⊗ e 0 ) A 2 = O + A ⊗ ( O + A ⊗ O ) ❄ e 2 = 1 + (1 ⊗ e 1 ) A 3 = O + A ⊗ A 2 . . . ❄ e 3 = 1 + (1 ⊗ e 2 ) with th e h elp of binary copro ducts and tensors. T he colimit of this diagram in E O is the unive rse of F T ( A, γ , δ ). T o see the d eﬁnition of m u lti plication for the monoid F T ( A, γ , δ ) and unit see [BJT]. As all the op erations inv olv ed are functorial in the whole ﬁbr at ion, F T is fun ct orial, as w ell. Th us, by Lemma 5.5, to show th at F T is a left adjoint to U T w e need to ve rify that they are adjoint w hen restricted to eac h ﬁbr e. But this is clear f rom [BJT], [Le]. As T preserve s ﬁltered colimits in ﬁbres of p so do es U T in th e ﬁ bres of p T and hence e T preserves them, as w ell. 10 The assumpt io ns that we h a ve on the monad ( T , η , µ ) obviously suﬃcient to for this construction to w ork . 35 The monadicit y of U T follo ws f rom Lemme 1 p ag e 304 of [B] or can b e pr o v ed directly using the ab o v e explicit construction of th e free fu nctor F T . If ( M , m, e ) is a monoid in Gph ( T , p ) − → E then u sing m and e w e can construct inductiv ely an algebra ( M , α : e T ( M ) → M ) and ha ving a e T -algebra ( M , α ) w e deﬁne a monoid by putting e equal to I O → e T ( M ) α − → M and m equal to M ⊗ M − → I + M ⊗ ( I + M ⊗ I ) → e T ( M ) α − → M . The remaining details are left for the readers. The fact that the induced monad ( e T , e η , e µ ) is cartesian relativ e to p T is also easy . ✷ R emark. The monadicit y of U T w as already n ot iced in [B] Prop osition I I.1.19 for a monad T satisfying sligh tly stronger conditions. 5.7 A tow er of ﬁbrations for op etopic sets Using the ab o v e T heorem 5.4, and starting with any ﬁbr ewise su ita ble monad T 0 on a ﬁbrewise suitable ﬁb ratio n p : E 0 → B , we can build a to wer of (ﬁbrewise suitable) lax monoidal ﬁbrations and ﬁbrewise suitable monads as in the diagram b elo w: . . . . . . E 3 = Gph ( T 2 , p T 1 ) M on ( T 2 , p T 1 ) ✛ U T 2 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ p T 2 q T 2 ✄ ✂ ✲ T 3 = f T 2 E 2 = Gph ( T 1 , p T 0 ) M on ( T 1 , p T 0 ) ✛ U T 1 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ p T 1 q T 1 ✄ ✂ ✲ T 2 = f T 1 E 1 = Gph ( T 0 , p ) M on ( T 0 , p ) ✛ U T 0 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ p T 0 E 0 B ❄ p q T 0 ✄ ✂ ✲ T 0 ✄ ✂ ✲ T 1 = f T 0 So as p an d T 0 are ﬁbrewise suitable, w e ha ve the monad T 1 = e T on p T 0 . By Theorem 5.4 p T 0 and T 1 are again suitable and hence we can rep eat the construction again. Th e iden tity monad 1 S et on S et is of course a ﬁb rewise suitable on th e ﬁb rewise suitable ﬁbration ! : S et → 1 , where 1 is the terminal category . Th u s w e can build a to w er of ﬁbrations, as ab o v e, starting form this ﬁbr at ion. W e obtain 36 . . . . . . O 3 = Gph ( T 2 , p T 1 ) M on ( T 2 , p T 1 ) ✛ U T 2 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ p T 2 q T 2 ✄ ✂ ✲ T 3 = f T 2 O 2 = Gph ( T 1 , p T 0 ) M on ( T 1 , p T 0 ) ✛ U T 1 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ p T 1 q T 1 ✄ ✂ ✲ T 2 = f T 1 O 1 = Gph ( T 0 , !) M on ( T 0 , !) ✛ U T 0 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ p T 0 O 0 = S et 1 ❄ ! q T 0 ✄ ✂ ✲ T 0 = 1 S et ✄ ✂ ✲ T 1 = f T 0 An op etopic set is an inﬁn ite sequence of ob jects { A n } n ∈ ω suc h th a t 1. A n is an ob ject in O n , 2. A n +1 lies in the ﬁbre o v er A n , i.e. p T n ( A n +1 ) = A n , for n ∈ ω . A morphism of op etopic sets { f n } n ∈ ω : { A n } n ∈ ω − → { B n } n ∈ ω is a f a mily of morphisms suc h that 1. f n : A n − → B n is a m orphism in O n 2. f n +1 lies in the ﬁb re o ver f n , i.e. p T n ( f n +1 ) = f n , for n ∈ ω . Unra veling this d eﬁnition, w e see that an opetopic set (in the ab o v e sense) is an ∞ -span as the diagram b elo w: . . . . . . A 3 T 3 ( A 3 ) ❄ γ 2 ❄ δ 2 ❅ ❅ ❅ ❅ ❅ ❅ ❘   ✠    γ 2 δ 2 A 2 T 2 ( A 2 ) ❄ γ 1 ❄ δ 1 ❅ ❅ ❅ ❅ ❅ ❅ ❘   ✠    γ 1 δ 1 A 0 T 0 ( A 0 ) A 1 T 1 ( A 1 ) ❄ γ 0 ❄ δ 0 ❅ ❅ ❅ ❅ ❅ ❅ ❘   ✠    γ 0 δ 0 37 with γ n ◦ γ n +1 = γ n ◦ δ n +1 , δ n ◦ γ n +1 = δ n ◦ δ n +1 γ n ◦ γ n +1 = γ n ◦ δ n +1 , δ n ◦ γ n +1 = δ n ◦ δ n +1 for n ∈ ω . T o describ e the terminal op eto pic set A , we need to s ta rt w it h A 0 = 1 the terminal ob j ec t in S et . And then c ho ose A n +1 as the terminal ob j ec t in the ﬁb re of p T n o ve r A n . Thus A 1 is 1 and A n +1 for n > 0 can b e tak en as the limit in the follo wing diagram: A n +1 ❄ γ n ❅ ❅ ❅ ❅ ❅ ❅ ❘ δ n A n − 1 T n − 1 ( A n − 1 ) A n T n ( A n ) ❄ γ n − 1 ❄ δ n − 1 ❅ ❅ ❅ ❅ ❅ ❅ ❘   ✠    γ n − 1 δ n − 1 The disjoin t union of the sets { A n } n ∈ ω is th e set of op etopes in the sense of T. Leinster. The p roof of th e follo wing theorem is uses ord ered face str uctures, cf. [Z], and will n ot b e giv en here. Theorem 5.6 The c ate gory of op etopic sets so deﬁne d i s e quivalent to the c ate gory of multitopic sets. Remark Internal op etopic sets. Clearly the ﬁ bration ! : S et → 1 is not the only in teresting suitable on e to start the p rocess of iteration. F or example, we can start with ! : E → 1 where E is a suﬃcien tly co complete top os. Thus, w e ha ve the categ ory of in ternal op etopic sets in any Grothendieck top os, ev en in the category of op etopic sets itself ! 5.8 A tow er of ﬁbrations for n -categories If w e start with the (ﬁbred) identit y monad 1 E on a ﬁ bration p : E → B whose ﬁbres ha ve pullbac ks then the ﬁ bration of monoids o ver p , q 1 E : M on (1 E , p ) − → E again h as pullbac ks in the ﬁbr es. Thus we can iterate th is pro cess and get another to w er of ﬁb ratio ns based on monoids, this time: 38 . . . . . . E 3 = M on (1 E 2 , p 1 E 1 ) Gph (1 E 2 , p 1 E 1 ) ✲ U 1 E 2 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ q 1 E 2 p 1 E 2 ✄ ✂ ✲ 1 E 3 E 2 = M on (1 E 1 , p 1 E 0 ) Gph (1 E 1 , p 1 E 0 ) ✲ U 1 E 1 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ q 1 E 1 p 1 E 1 ✄ ✂ ✲ 1 E 2 E 1 = M on (1 E 0 , p ) Gph (1 E 0 , p ) ✲ U 1 E 0 ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❄ q 1 E 0 E 0 B ❄ p p 1 E 0 ✄ ✂ ✲ 1 E 0 ✄ ✂ ✲ 1 E 1 If w e start w ith a s uitable ﬁbr at ion p : C → 1 , then after n -th iteration w e r eco ver the D. Bourn [Bo] construction of in tern al n -categories in C . 6 Amalgamated signatures vs p olynomial fu n ctors 6.1 The amalgamated signatures ﬁbration p a : S ig a → S et This example is one of th e main reasons for considering lax monoidal ﬁbr at ions in th e con text of higher category theory at all. The monoids in this ﬁbration are pr eci sely the (1- lev el) multic ategories with n on-sta ndard amalgamatio n. They are like the multica tegories considered b y C. Hermida M.Makk ai J . P ow er in [HMP] to deﬁne the m ultitopic s et s, except that th ere the 2-lev el version is used b y . This mo diﬁcation will b e explained at the end of the section. Notation. Let [ n ] = { 0 , . . . , n } , ( n ] = { 1 , . . . , n } , for n ∈ ω . In particular [ n ] = [0 ] ∪ ( n ] and (0] = ∅ . F or a set O , w e p ut O † n = O [ n ] , O ∗ n = O ( n ] and O † = S n ∈ ω O [ n ] , O ∗ = S n ∈ ω O ( n ] . S n acts on b oth O † n and O ∗ n on the righ t by comp ositio n (i.e. w e lea v e 0 ﬁxed in the domain of the elemen ts of O † n ). If d : [ n ] → O is a fun ct ion, then its restriction to the p ositiv e num b ers is den ot ed by d + : ( n ] → O and to [0] by d − : [0] → O . Th is restrictions establish a bijection h ( − ) − , ( − ) + i : O † → O × O ∗ . Clearly ( − ) † : S et − → S et is a f unctor. The base category of our ﬁbr ati on is S et . The total category S ig a of our ﬁb ratio n has as ob jects triples, ( A, ∂ , O ) su c h that A and O are sets and ∂ : A → O † is a fun ct ion. W e write ∂ a : [ n ] → O for the eﬀect of ∂ on a ∈ A , and n in this case will b e referred to as | a | . A morp hism ( f , σ, u ) : ( A, ∂ , O ) → ( B , ∂ , Q ) in S ig a is a pair of fu nctions f : A → B and u : O → Q , and for any a ∈ A with n = | a | a p ermutation σ a : [ n ] → [ n ] ∈ S n (with σ a (0) = 0) making the square O Q ✲ u [ n ] [ n ] ✛ σ a ❄ ∂ a ❄ ∂ f ( a ) 39 comm ute. A m orphism ( f , σ, u ) is called strict if σ a is an identit y , for a ∈ A . The pro jection f unctor p : S ig a − → S et send s the morp hism ( f , σ, u ) : ( A, ∂ , O ) → ( B , ∂ , Q ) to u : O → Q . Remarks 1. T he catego ry S ig m is isomorphic to the fu ll sub category of S ig a whose morphisms are strict. 2. W e thin k of an ob ject ( A, ∂ , O ) of S ig a as a signature with O as the set of its types, A the set of its op eration s ym b ols, and ∂ the t ypin g function asso ciating arities to function sym b ols a : ∂ a (1) , . . . , ∂ a ( | a | ) − → ∂ a (0), i.e. is ∂ a (1) , . . . , ∂ a ( | a | ) are t yp es of the argu men ts (inputs) of a an d ∂ a (0) is the type of v alues (outputs) of a . The lax monoidal structure on p a W e ha ve t w o lax morphisms of ﬁbrations S et S ig a × S et S ig a S ig a ✲ ⊗ ❄ p a S et ✛ I ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ p ′ a ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✙ 1 S et Let ( A, ∂ , O ) and ( B , ∂ , O ) b e t wo ob ject in the ﬁbr e o ver O . Their tensor ( A ⊗ O B , ∂ ⊗ , O ) is deﬁn ed as f ol lo ws A ⊗ O B = {h a, b i i i ∈ ( | a | ] : a ∈ A, b i ∈ B , ∂ a ( i ) = ∂ b i (0) , for i ∈ ( | a | ] } and for h a, b i i i ∈ ( | a | ] ∈ A ⊗ O B , ∂ ⊗ h a,b i i i = [ ∂ − a , ∂ + b i ] i : [ |h a, b i i i | ] = [ | a | X i =1 | b i | ] − → O . Note th at j ust sa ying that we ha ve a copr oduct determines th e fu nctio n [ ∂ − a , ∂ + b i ] i only up t o a p erm utation. In prin ciple w e don’t n eed more th an that for as far as [ ∂ − a , ∂ + b i ] i (0) = ∂ − a (0). But to b e o n the safe side, w e will alw ays tacitly assume that the domains of ∂ + b i are placed one after the other. F or a p ai r of maps in S ig a f = ( f , σ, u ) : ( A, ∂ , O ) → ( A ′ , ∂ , Q ) , g = ( g , τ , u ) : ( B , ∂ , O ) → ( B ′ , ∂ , Q ) o ve r th e same map u : O → Q we deﬁne the map f ⊗ u g = ( f ⊗ u g , σ ⊗ u τ , u ) : ( A ⊗ O B , ∂ ⊗ , O ) − → ( A ′ ⊗ Q B ′ , ∂ ⊗ , Q ) so that, for h a, b i i i ∈ ( | a | ] ∈ A ⊗ O B , f ⊗ u g ( h a, b i i i ∈ ( | a | ] ) = h f ( a ) , g ( b σ a ( j ) ) i j ∈ ( | f ( a ) | ] Clearly , | a | = | f ( a ) | and n = |h a, b i i i ∈ ( | a | ] | = X i ∈| a | | b i | = X i ∈| f ( a ) | | g ( b i ) | = | f ⊗ u g ( h a, b i i i ∈ ( | a | ] ) | . Moreo v er, we pu t ( σ ⊗ u τ ) h a,b i i i = [ τ + b σ a ( i ) ] i making the square 40 O Q ✲ u [ n ] [ n ] ✛ ( σ ⊗ u τ ) h a,b i i i ❄ ∂ ⊗ h a,b i i i ❄ ∂ ⊗ h f ( a ) ,g ( b i ) i i comm ute. This ends the deﬁn iti on of the tensor ⊗ . The unit I O in the ﬁb re o ver O , is I O = ( O , ∂ I O , O ) su c h that for x ∈ O , ∂ I O x : [1] → O is a constant function equal to x . W e n ot e for the record Lemma 6.1 The ﬁbr ation p a : S ig a → S et with the structur e describ e d ab ove is a lax monoidal ﬁbr ation whose ﬁbr es ar e str ong monoidal c ate gories. ✷ Pulling bac k t he monoidal str ucture W e shall describ e ho w reindexin g fu nctors in teract with the monoidal structure in the ﬁbration p a . An y ob ject B in the ﬁ bre o v er Q of p a : S ig a → S et can b e p ulled b ac k a long a function u : O → Q : O † Q † ✲ u † u ∗ ( B ) B ✲ π B ❄ ∂ ❄ ∂ O Q ✲ u th u s u ∗ ( B ) = {h b, d i : b ∈ B , d : [ | b | ] → O , such that u † ( d ) = ∂ b } ( u † ( d ) = u ◦ d ) and ∂ h b,d i = d W e ha ve u ∗ ( I Q ) = {h x, x ′ i ∈ O 2 : u ( x ) = u ( x ′ ) } and ϕ 0 : I O − → u ∗ ( I Q ) x 7→ h x, x i Moreo v er, for ob jects A and B o v er Q we hav e u ∗ ( A ⊗ B ) = {hh a, b i i i ∈ ( | a | ] , d i : h a, b i i i ∈ ( | a | ] ∈ A ⊗ B , u † ( d ) = ∂ ⊗ h a,b i i i ∈ ( | a | ] } and u ∗ ( A ) ⊗ u ∗ ( B ) = {hh a, d i , h b i , d i ii i ∈ ( | a | ] : a ∈ A, b i ∈ B , u † ( d ) = ∂ a , u † ( d i ) = ∂ b i , d ( i ) = d i (0) , for i ∈ ( | a | ] } Th us w e ha ve a transform at ion ϕ 2 ,A,B : u ∗ ( A ) ⊗ u ∗ ( B ) − → u ∗ ( A ⊗ B ) suc h th a t hh a, d i , h b i , d i ii i ∈ ( | a | ] 7→ hh a, b i i i ∈ ( | a | ] , [ d − , d + i ] i ∈ ( | a | ] i All the morp hisms deﬁ ned ab o v e π B , ϕ 0 , and ϕ 2 ,A,B are strict, i.e. with amalg amations b eing identiti es. 41 Lemma 6.2 The data u ∗ , ϕ 0 , ϕ 2 ab ove, make the usual (thr e e ) diagr ams for c oher enc e of monoid al functor (not ne c essarily str ong) c ommute. Pr o of. E xercise . ✷ Moreo v er we h a v e Prop osition 6.3 The total c ate gory of the ﬁbr ation q a : M on ( S ig a ) − → S et of mono ids in p a : S ig a − → S et is e quivalent to the c ate gory of (1-level) multic ate g o ries with non- standar d amalgamations. The ﬁbr e d for g etful functor fr om the ﬁbr ation of monoids to the ﬁbr ation of amalgamate d signatur es U : M on ( S ig a ) − → S ig a S ig a M on ( S ig a ) ✲ F S et p a ❅ ❅ ❅ ❅ ❘ q a     ✠ ✛ U has a ﬁbr e d left adjoint F , the fr e e monoid functor. Pr o of. Strictly sp eaking the m u lti catego ries with non-standard amalgamations w ere deﬁned [HMP] fr om t he single tensor and add itio nal prop ert y , called comm utativit y there. But, as it is w ell kno w n, they can b e equiv alen tly deﬁn ed u sing the total tensor, i.e. the one w e deﬁned ab o ve. F or more, see also Sub sect ion 6.6. ✷ The free functor F men tioned in the Prop osition ab o v e was describ ed in [HMP]. 6.2 The action of p a on the basic ﬁbration The lax monoidal ﬁbration p a comes equipp ed naturally w ith an action on th e basic ﬁbration cod : S et → − → S et S ig a × S et S et → S et → ✲ ⋆ S et ❅ ❅ ❅ ❘ cod    ✠ F or ( A, ∂ A , O ) in S ig a and ( X , d, O ) in S et → , the set A ⋆ X is deﬁned fr o m the follo w ing diagram O ∗ X ∗ ✛ d ∗ A A ⋆ X ✛ ❄ ∂ A, + ❄ O ✛ ∂ A, − with the square b eing a pu llbac k. ( − ) ∗ is th e free monoid fun cto r , i.e. A ⋆ X = { ( a, x 1 , . . . , x | a | ) : ∂ A a ( i ) = d ( x i ) , i = 1 , . . . , | a |} and ∂ ⋆ : A ⋆ X − → O is deﬁned by ∂ ⋆ ( a, x 1 , . . . , x | a | ) = ∂ A a (0) Th us it is the comp osition of the upp er horizont al m orphism in the ab o ve diagram. On morphisms the action ⋆ is deﬁned in the ob vious w ay . Thus w e h a v e an adj oi n t morphism of lax monoidal ﬁbrations 42 S ig a E xp ( S et ) ✲ r ep a S et p a ❅ ❅ ❅ ❘    ✠ where, as usual, E xp ( S et ) is the exp onen t in Cat /S et . 6.3 P olynomial diagrams and p olynom ial functors In this su bsectio n we collect the d eﬁnitions and facts concerning p olynomial diagrams and p olynomial fu nctors from the literature, that are n eeded in the follo wing. F or (m u c h!) more, the reader should consult [Ko], [GK] and bibliograph y there. W e deal with p olyno- mial functors based on a n arbitrary lo ca lly cartesian closed categ ory but with a sp ecia l ey e on S et and th e presheaf catego ry S et S ∗ , where S ∗ is the copro duct in C at of the (ﬁnite) symmetric groups. Th e later category will b e imp ortant in S ec tion 7. In this section, unless otherwise sp eciﬁed, E is an arbitrary lo cally cartesian closed category , and b y this w e m ea n that E has th e terminal ob ject, as well. By a p olynom ial diagr am (over O ) in E , w e mean the follo win g d iag ram in E E A ✲ p O ✛ s O ✲ t The ob ject O is an ob ject of t yp es of th e p olynomial ( t, p, s ). W e say th at a p olynomial diagram ( t, p, s ) in S et is ﬁnitary if and only if the function p has ﬁn ite ﬁbres. A morphism of p olynom ial diagr ams (over u : O → Q ) in E is a triple ( f , g , u ) of morp hism making the diagram E ′ A ′ ✲ p ′ E A ✲ p ❄ g ❄ f Q ✛ s ′ O ✛ s ❄ u Q ✲ t ′ O ✲ t ❄ u comm ute, and suc h that th e square in the midd le is a p ullbac k. Morphisms of p olynomial diagram comp ose in the o bvious w a y , by putting one on to p of the other. Let P ol y D iag ( E ) denotes the category of the p olynomial d ia grams and m orphisms b et ween them. Remark If E is th e category S et , w e can think of A as the set of op erations, and E as the set of argumen ts of all op eratio ns in A . Th u s with th is interpretatio n p − 1 ( a ) is the set of argumen ts (or arity) of a . Then s ( e ), for e ∈ p − 1 ( a ), can b e interpreted as the t yp e of th e argument e of the op eration a , and t ( a ) is the t yp e of the v alues of the op eration a . W e ha ve an obvious p ro jection fu nctor p pd, E : P ol y D iag ( E ) − → E sending ( f , g , u ) to u , whic h is a lax monoidal ﬁb ration. The tensor in ﬁ bres is giv en by comp ositio n of diagrams, cf. [GK] 1.11. Let p pd : P ol y D iag → S et denote the ﬁnitary p olynomial diagram ﬁ bration, the full su bﬁbration of p pd,S et consisting of ﬁ nitary p olynomial diagrams. By p poly : P oly → S et we d enot e the image of the ﬁbration p pd in p exp,S e t . W e shall see that p poly is a lax monoidal subﬁ bration of p exp,S e t . 43 In our terminology , the connection b et w een p olynomial diagrams and p olynomial func- tors ca n b e expressed as the fact that the ﬁbration of p olynomial diagrams comes e quipp ed with a represen tation morp hism into E xp ( E ), i.e. a morphism of lax m onoi dal ﬁbr at ion, cf. Pr oposition 6.7 b elo w, P oly D iag ( E ) E xp ( E ) ✲ r ep pd, E E p pd, E ❅ ❅ ❅ ❅ ❘ p exp     ✠ whose essen tial imag e is, b y deﬁnition, th e (lax monoidal ) ﬁbration of (ﬁn ita ry) p olynomial (endo)functors and p olynomial transform at ions b et ween th em. W e shall recall this now. An y morphism u : O → Q in a lo ca lly cartesian closed category E induces three fun ct ors E /O E /Q ✛ u ∗ ✲ u ∗ ✲ u ! suc h that u ∗ is a pullbac k f unctor and u ! ⊣ u ∗ ⊣ u ∗ . The u nit and counit of the adju nction u ! ⊣ u ∗ will b e denoted b y η u and ε u , resp ect iv ely and the un it and counit of the adjunction u ∗ ⊣ u ∗ will b e denoted by ¯ η u and ¯ ε u , r espectiv ely . F or an ob ject ( t, p, s ) ov er O , we d eﬁne a f unctor r ep pd, E ( t, p, s ) = t ! p ∗ s ∗ : E /O − → E /O F or a morph ism of p olynomial diagrams ( f , g , u ) : ( t, p, s ) − → ( t ′ , p ′ , s ′ ), w e deﬁne a morphism r ep pd, E ( f , g , u ) : r ep pd, E ( t, p, s ) → r ep pd, E ( t ′ , p ′ , s ′ ) in E xp ( E ) o ver u , as follo ws. W e ha ve a diagram of categories, fu ncto rs and n at ural transformations E /Q ✲ s ′∗ E /O ✲ s ∗ ✻ u ∗ E /E ′ E / A ′ ✲ p ′ ∗ E /E E / A ✲ p ∗ ✻ g ∗ ❄ f ! E /Q ✲ t ′ ! E /O ✲ t ! ❄ u ! ✻ f ∗ ε f ⇓ ✻ u ∗ ∼ = ∼ = ∼ = where ε f : f ! f ∗ → 1 E / A ′ is the counit of the adjunction f ! ⊣ f ∗ . Thus we hav e a natural transformation t ′ ! ( ε f ) p ′ ∗ s ′∗ : t ′ ! f ! f ∗ p ′ ∗ s ′∗ − → t ′ ! p ′ ∗ s ′∗ and passin g through the natural isomorphisms indicated in the ab o ve diagram we get the corresp onding a natural trans formatio n r ep pd, E ( f , g , u ) as follo ws: u ! t ! f ∗ p ′ ∗ s ′∗ − → t ′ ! p ′ ∗ s ′∗ via righ t s quare iso, and this u ! t ! p ∗ g ∗ s ′∗ − → t ′ ! p ′ ∗ s ′∗ via middle square iso, and this u ! t ! p ∗ s ∗ u ∗ − → t ′ ! p ′ ∗ s ′∗ via left square iso. Finally , taking the ad join t ( u ! ⊣ u ∗ ) of this morphism w e get r ep pd, E ( f , g , u ) : t ! p ∗ s ∗ u ∗ − → u ∗ t ′ ! p ′ ∗ s ′∗ 44 whic h is a morphism from r ep pd, E ( t, p, s ) to r ep pd, E ( t ′ , p ′ , s ′ ) in E xp ( E ) o v er u . The essen tial image of the fu nctor r ep pd, E is, b y deﬁnition, the ﬁbr ati on of p olynomial (endo)functors and p olynomial trans formatio ns p poly , E : P oly ( E ) − → E . By taking the exp onen tial adjoin t to r ep pd, E w e obtain an action of th e ﬁ bration p pd, E on the basic ﬁbration cod : E → − → E P oly D iag ( E ) × E E → E → ✲ ⋆ poly , E E ❅ ❅ ❅ ❅ ❘ cod     ✠ ev en if this p oint of view is less customary . The case E = S et W e shall mak e h ere the ab o v e abs trac t deﬁn itio ns concrete in case E = S et . A functor P : S et /Q → S et /Q is a p olynomial functor 11 if and on ly if it is isomorph ic to one of form Π ( t,p,s ) = t ! p ∗ s ∗ : S et /Q → S et /Q for some p olynomial diagram ( t, p, s ). Thus for d : Y → Q in S et /Q w e hav e t ! p ∗ s ∗ ( Y , d ) = {h b, ~ y i : b ∈ B , ~ y : p − 1 ( b ) → Y , d ◦ ~ y = s ⌈ p − 1 ( b ) } where s ⌈ p − 1 ( b ) is the restriction of the function s to the ﬁb re of the fu nctio n p o ver the elemen t b ∈ B . It is a routine to ve rify that a p olynomial fun ct or is ﬁnitary if and only if it comes from a ﬁnitary diagram. A morphism of p olynomial d ia grams ( f , g , 1 Q ) : ( t, p, s ) → ( t ′ , p ′ , s ′ ) deﬁnes a natural transformation Π ( f ,g ) : Π ( t,p,s ) − → Π ( t ′ ,p ′ ,s ′ ) so that for d : Y → Q in S et /Q , and h b, ~ y i ∈ t ! p ∗ s ∗ ( Y , d ), we h a v e Π ( f ,g ) ( h b, ~ y i ) = h f ( b ) , ~ y ◦ ( g ⌈ p − 1 ( b ) ) − 1 i A natur al transf ormati on b etw een p olynomial fu nctors is p olynomial if and only if it is giv en b y the morp hism of p olynomial diagrams deﬁnin g them. The follo win g Theorem is due to many authors. The pr ec ise accoun t of this can b e found in [GK], 1.18 and 1.19. How ev er the pro of, based on ideas of [A V], seems to b e new. Theorem 6.4 F or any set Q , the f unctor Π Q : ( P oly D iag ) Q − → N at ( S et /Q , S et /Q ) deﬁne d ab ove is f aith f ul, ful l on isomorp hisms and its essential image c onsists of ﬁnitary functors pr e serving wide pul lb acks and c artesian natur al tr ansformations. A functor F : S et /Q − → S et /Q is thin if there is q ∈ Q suc h th a t F = i q ◦ ev q ◦ F and ev q ◦ F (1) = 1. T he functors i q and ev q are inclusion an d ev aluation functors, r espectiv ely , see Subsection 7.3. Pr o of. First note that an y functor P : S et /Q → S et /Q is a copro duct of thin f unctors and P p reserv es wide pu llbac ks and ﬁltered colimits if all the thin factors in th e coprod uct do. As Π Q preserve s coprod ucts w e can a ssume that P is thin sa y , P = i q ◦ ev q ◦ P for some 11 There is an ob vious notion of a p olynomial functor P : S et /O → S et /Q with O not assumed to b e equal to Q , but this can b e considered a sp ecial case of the ab o ve deﬁ n ition, as such functors are (some) p olynomial fun ct ors S e t /O + Q → S e t /O + Q . F or details see [GK]. 45 q ∈ Q . Thus P q = ev q ◦ P : S et /Q → S et p reserv es all limits an d is ﬁnitary . Hence, by the c haracterization of the representable fu nctors c.f. [CWM] page 130, it is represent ed b y an ob ject s : E → Q in S et /Q with E ﬁnite. Now it is a matter of a s imple chec k, that with the diagram E 1 ✲ p Q ✛ s Q ✲ t in ( P oly D iag ) Q , wh ere t ( ∗ ) = q , the fun cto r Π( t, p, s ) is isomorph ic to P . T o sho w that the functor Π Q is fu ll and faithful it is enough to consider morphisms b et we en d iag rams with one op eratio n and cartesian n at ural transformations b et w een cor- resp onding thin functors, as other cartesian n atural transformations b et wee n other p oly- nomial functors m ust come from those. So sup pose that we ha ve a cartesian n at ural transform at ion b et ween t wo suc h functors τ : P = t ! ◦ p ∗ ◦ s ∗ − → P ′ = t ′ ! ◦ p ′ ∗ ◦ s ′∗ where E ′ 1 ✲ p ′ Q ✛ s ′ Q ✲ t ′ As P and P ′ are thin we must ha ve t = t ′ , say t ( ∗ ) = q ∈ Q . Thus τ q = ev q ( τ ) : P q → P ′ q is a cartesian natural transformation b et wee n functors that preserve s the terminal ob ject. Hence, as any comp onen t of τ q is a pullbac k of ( τ q ) 1 Q whic h is a morphism from the terminal ob ject to itself, τ q is cartesian if and only if it is an isomorp hism. By the Y oneda Lemma, the natural isomorphisms τ q : P q → P ′ q in C at ( S et /Q , S et ) b et w een the fun cto rs represent ed by s : E → Q and s ′ : E ′ → Q ′ corresp ond to isomorphism s E E ′ ✛ g Q s ❅ ❅ ❘ s ′   ✠ in S et /Q . But th ose isomorphisms g are exactly the functions g making the diagram E ′ 1 ✲ p ′ E 1 ✲ p ❄ g − 1 ❄ Q ✛ s ′ Q ✛ s ❄ 1 Q Q ✲ t Q ✲ t ❄ 1 Q a morphism in ( P ol y D iag ) Q , i.e. making th e left square comm u te and th e midd le square a pullbac k. The reader m a y verify that if w e take a morph ism of diagrams corresp onding to τ q , then Π Q will send it b ac k to τ q . ✷ Remark An analog of Theorem 6.4 do es not h ol d in all lo cally cartesian closed cat- egories. Eve n if E is a presh ea f category , then the endofun ct ors on slices of E that are ﬁnitary and preserv e wide pu llbac ks do not necessarily come fr om p olynomial diagrams in E . F or E = S et → , the fun ct or send ing ( x : X 0 → X 1 ) to ( h 1 X 0 , x i : X 0 → X 0 × X 1 ) is ﬁnitary and preserve s all limits but is not p olynomial. 6.4 Some prop erties of the represen tation r ep a The main ob jectiv e of this section is to establish some prop erties of the r epresen tation r ep a and then sho w (C orol lary 6.13) that the 1-lev el m ulticategories with non-standard amalgamat ions are th e same as the cartesian monads on slices of S et whose functor part is ﬁnitary and preserv es wide pullbac k s. 46 As some statemen ts concerning p olynomial diagrams and fu nctors hold in greater gen- eralit y , in arbitrary lo cally cartesian closed categories, w e start in this more general con- text. First, w e describ e su pine and prone morph isms in p pd, E : P ol y D iag ( E ) → E . Let u : O → Q b e a morphism in E . The su pine morph ism su u, ( t,p,s ) : ( t, p, s ) → ( u ◦ t, p, u ◦ s ) o ve r u with th e d o main b eing the p olynomial d iag r am ( t, p, s ) in E o ve r O is deﬁn ed by the diagram Q E ✛ u ◦ s O E ✛ s ❄ u ❄ ✲ p ✲ p 1 E ❄ 1 B B Q ✲ u ◦ t B O ✲ t ❄ u i.e. su u, ( t,p,s ) = (1 B , 1 E , u ). Th e prone morp hism pr u, ( t,p,s ) : ( ˜ t, ˜ p, ˜ s ) → ( t, p, s ) ov er u with the domain b eing the p olynomial diagram ( t, p, s ) in E o ver Q is deﬁ ned b y th e diagram Q E ✛ s O s ∗ ( O ) ✛ ¯ s ❄ u ❄ ✲ p ¯ u ❄ ˆ u B Q ✲ t p ∗ s ∗ ( O ) O ❄ u p ∗ p ∗ S ∗ ( O ) ❍ ❍ ❍ ❍ ❍ ❥   ✠ ¯ ε p s ∗ ( O ) ¯ p ˜ t ❅ ❅ ❅ ❅ ❅ ❅ ❘ ˜ B ❄ h ˜ E ❄ ¯ h ❍ ❍ ❍ ❍ ❍ ❍ ❥ ˜ p           ✠ ˜ s i.e. pr u, ( t,p,s ) = ( ˆ u ◦ h, ¯ u ◦ ¯ ε p s ∗ ( O ) ◦ ¯ h, u ). The ab o v e diagram is constr ucte d in th e f o llo wing w ay . First we app ly the functor t ! p ∗ s ∗ to the (left most) morph ism u : O → Q (in E /Q ) to get t ◦ ˆ u : p ∗ s ∗ ( Q ) → Q . ¯ ε p is the counit of the adju nctio n p ∗ ⊣ p ∗ . Then we p ull it back along u to get ˜ t and h . Finally , pulling bac k h along ¯ p we get ˜ p and ¯ h = ¯ p ∗ ( h ), the last part of the p olynomial diagram ( ˜ t, ˜ p, ˜ s ). The morphism ˜ s is d eﬁned as the comp osition ¯ s ◦ ¯ ε p s ∗ ( O ) ◦ ¯ p ∗ ( h ), and the ob jects ˜ B and ˜ E are u ∗ t ! p ∗ s ∗ ( O ) and ¯ p ∗ ( ˜ B ), resp ectiv ely . W e note for th e record Prop osition 6.5 The pr one and supine morphisms in the biﬁbr ation p pd, E : P ol y D iag ( E ) → E ar e as de- scrib e d ab ove. Pr o of. Routine veriﬁcat ion. ✷ The follo wing Lemma collec ts some kn o wn facts that will b e used in Prop osition 6.7. Lemma 6.6 L et u , u ′ , s and p b e morphisms in a lo c al ly c artesian close d c ate gory E , such that cod ( u ′ ) = dom ( u ) , cod ( u ) = cod ( s ) = dom ( p ) . Then 1. s ∗ ( ε u ) ∼ = ( ε s ∗ ( u ) ) s ∗ ; 2. p ∗ ( ε u ) ∼ = ( ε p ∗ ( u ) ) p ∗ ; 3. ε u ◦ u ′ ∼ = ε u ◦ ( u ! ( ε u ′ ) u ∗ ) . ✷ 47 Prop osition 6.7 L et E b e a lo c al ly c artesian close d c ate gory. The morphism P oly D iag ( E ) E xp ( E ) ✲ r ep pd, E E p pd, E ❅ ❅ ❅ ❅ ❘ p exp     ✠ deﬁne d in the pr evious se ction is a morphism of biﬁbr ations. Pr o of. W e need to sho w that r ep pd, E preserve s p rone and supin e morp hisms. W e sho w preserv ation of su pine morph isms ﬁrst. Let su u, ( t,p,s ) = (1 B , 1 E , u ) : ( t, p, s ) → ( u ◦ t, p, u ◦ s ) b e a sup ine morphism in P oly D iag ( E ). The representa tion of the su pine morph ism r ep pd, E ( su u, ( t,p,s ) ) is a natu- ral transformation which is isomorphic to the adjoin t ( u ! ⊣ u ∗ ) to ( ut ) ! (1 B ) ! 1 ∗ B p ∗ ( us ) ∗ ( ut ) ! p ∗ ( us ) ∗ ✲ ( ut ) ! ( ε 1 B ) p ∗ ( us ) ∗ The ab o v e morphism is isomorphic to the iden tity n atural transform at ion u ! t ! p ∗ s ∗ u ∗ u ! t ! p ∗ s ∗ u ∗ ✲ 1 u ! t ! p ∗ s ∗ u ∗ and the s upine m orphism with the co domain r ep pd, E ( t, p, s ) o ver u is a natural tr ansfor- mation ( t ! p ∗ s ∗ ) u ∗ u ∗ u ! ( t ! p ∗ s ∗ ) u ∗ ✲ η ( t ! p ∗ s ∗ ) u ∗ adjoin t to the ab o ve iden tit y morphism. Thus r ep pd, E preserve s th e sup ine morp hisms. F or the prone m orphism we u se the nota tion in tro duced a t the b eginning o f t he sec tion. Let p r u, ( t,p,s ) = ( ˆ u ◦ h, ¯ u ◦ ¯ ε p s ∗ ( O ) ◦ ¯ p ∗ ( h ) , u ) : ( ˜ t, ˜ p, ˜ s ) → ( t, p, s ) b e a prone morp hism. W e ha ve a diagram of categories, fu nctors and natural transformations E /Q E /E ✲ s ∗ E /O E /s ∗ ( O ) ✲ ¯ s ∗ ✻ u ∗ ❄ u ! ε u ⇓ ✲ p ∗ ✻ ¯ u ∗ ❄ ¯ u ! ε ¯ u ⇓ ✻ ˆ u ∗ ❄ ˆ u ! ε ˆ u ⇓ E /B E /Q ✛ t ! E /p ∗ s ∗ ( O ) E /O ✻ u ∗ ❄ u ! ε u ⇓ E /p ∗ p ∗ S ∗ ( O ) ❍ ❍ ❍ ❍ ❍ ❥    ✒ ( ε p s ∗ ( O ) ) ∗ ¯ p ∗ ✻ h ∗ ❄ h ! ε h ⇓ ˜ t ! ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ E / ˜ B ✻ ¯ h ∗ E / ˜ E ✲ ˜ p ∗ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ❃ ˜ s ∗ In tuitiv ely sp eaking, the adjoin t ( u ! ⊣ u ∗ ) natural transformation to rep pd, E ( pr u, ( t,p,s ) ) is isomorphic to t ! ( ε ˆ uh ) p ∗ s ∗ = t ! ( ε ˆ u ◦ ( ˆ u ! ( ε h ) ˆ u ∗ )) p ∗ s ∗ i.e. it is deﬁn ed with the help of the counits ε ˆ u and ε h . The adjoint to the natural transformation pr u,r ep pd, E ( t,p,s ) , b eing the prone morphism in E xp ( S et ) ov er u with the co domain r ep pd, E ( t, p, s ), is (( ε u ) t ! p ∗ s ∗ ) ◦ ( u ! u ∗ t ! p ∗ s ∗ ( ε u )). T o sho w that these adjoin ts are isomorphic, we sh o w, usin g the ab o v e Lemma 6.6, th at the counit ε ˆ u can b e ’mo v ed left’ to the ’left’ counit ε u and the counit ε h can b e ’mo ved righ t’ to the ’righ t’ counit ε u . In the s equence of morphisms b elo w, we mark on the right side of the line how w e pass from a line to another. Numbers 1. 2. 3. refer to Lemma 6.6, MEL is the mid dle exc hange la w. W e h a v e 48 r ep pd, E ( ˜ t, ˜ p, ˜ s ) r ep pd, E ( t, p, s ) ✲ r ep pd, E ( pr u, ( t,p,s ) ) def pr ˜ t ! ˜ p ∗ ˜ s ∗ u ∗ u ∗ t ! p ∗ s ∗ ✲ r ep pd, E ( ˆ uh, ¯ uε p S ∗ ( O ) ¯ h, u ) u ! ⊣ u ∗ u ! ˜ t ! ˜ p ∗ ˜ s ∗ u ∗ t ! p ∗ s ∗ ✲ def rep t ! ( ˆ uh ) ! ( ˆ uh ) ∗ p ∗ s ∗ t ! p ∗ s ∗ ✲ t ! ( ε ˆ uh ) p ∗ s ∗ 3. t ! ˆ u ! h ! h ∗ ˆ u ∗ p ∗ s ∗ t ! p ∗ s ∗ ✲ t ! ( ε ˆ u ◦ ( ˆ u ! ( ε h ) ˆ u ∗ )) p ∗ s ∗ = t ! ˆ u ! h ! h ∗ ˆ u ∗ p ∗ s ∗ t ! p ∗ s ∗ ✲ ( t ! ( ε ˆ u ) p ∗ s ∗ ) ◦ ( t ! ˆ u ! ( ε h ) ˆ u ∗ p ∗ s ∗ ) 1. u ! u ∗ t ! ˆ u ! ˆ u ∗ p ∗ s ∗ t ! p ∗ s ∗ ✲ ( t ! ( ε ˆ u ) p ∗ s ∗ ) ◦ (( ε u ) t ! ˆ u ! ˆ u ∗ p ∗ s ∗ ) MEL u ! u ∗ t ! ˆ u ! ˆ u ∗ p ∗ s ∗ t ! p ∗ s ∗ ✲ (( ε u ) t ! p ∗ s ∗ ) ◦ ( u ! u ∗ t ! ( ε ˆ u ) p ∗ s ∗ ) 2. u ! u ∗ t ! p ∗ ¯ u ! ¯ u ∗ s ∗ t ! p ∗ s ∗ ✲ (( ε u ) t ! p ∗ s ∗ ) ◦ ( u ! u ∗ t ! p ∗ ( ε ¯ u ) s ∗ ) 1. u ! u ∗ t ! p ∗ s ∗ u ! u ∗ t ! p ∗ s ∗ ✲ (( ε u ) t ! p ∗ s ∗ ) ◦ ( u ! u ∗ t ! p ∗ s ∗ ( ε u )) u ! ⊣ u ∗ u ∗ t ! p ∗ s ∗ u ∗ u ! u ∗ t ! p ∗ s ∗ ✲ u ∗ t ! p ∗ s ∗ ( ε u ) def rep u ∗ r ep pd, E ( t, p, s ) u ! u ∗ u ∗ r ep pd, E ( t, p, s ) ✲ u ∗ r ep pd, E ( t, p, s )( ε u ) def pr u ∗ r ep pd, E ( t, p, s ) u ! r ep pd, E ( t, p, s ) ✲ pr u,r ep pd, E ( t,p,s ) Th us r ep pd, E preserve s the prone morphisms, as w ell. ✷ F rom no w on till the en d of this sub sec tion w e shall consider ﬁbrations o ver S et on ly . The follo win g ap pears in [GK]. I t is an immediate consequence of Theorem 6.4 and Prop osition 6.7. Prop osition 6.8 The r epr e senta tion morphism r ep pd,S et is a morphism of lax monoidal ﬁbr ations which is faithful and ful l on i som orphisms. ✷ Corollary 6.9 We have a se quenc e of morph isms of lax monoidal ﬁbr ations P oly D iag P oly ✲ r ep pd,S et C ar t ( S et ) ✲ E xp ( S et ) ✲ with the ﬁrst b eing an e qui val enc e of biﬁbr ations and the fol lowing two b eing inclusions ful l on isomorph isms. The c omp osition of these morphisms is (isomorp hic to) r ep pd,S et . Pr o of. This is an immediate consequence of Prop ositions 6.7 and 6.8. ✷ W e shall construct a m orphism of lax monoidal ﬁb rati ons S ig a P oly D iag ✲ ι a S et p a ❅ ❅ ❅ ❘ p pd    ✠ Let ( A, ∂ , O ) b e a signature in ( S ig a ) O . Th e functor ι a sends this signature to a polynomial functor deﬁned b y the follo wing p olynomial diagram E A A ✲ p A O ✛ s A O ✲ t A 49 where E A = a a ∈ A ( | a | ] = {h a, i i : a ∈ A, i ∈ ( | a | ] } p A is the ﬁrst pro j ec tion, s A ( a, i ) = ∂ A a ( i ), and t A ( a ) = ∂ A a (0), for a ∈ A and i ∈ ( | a | ]. Moreo v er, for the morphism s of signatures ( f , σ, u ) : ( A, ∂ A , O ) − → ( B , ∂ B , Q ) in S ig a o ve r u : O → Q w e deﬁne a comm uting d iag r am E B B ✲ p B E A A ✲ p A ❄ g ❄ f Q ✛ s B O ✛ s A ❄ u Q ✲ t B O ✲ t A ❄ u where g ( a, i ) = h f ( a ) , σ − 1 a ( i ) i , for h a, i i ∈ E A . The s quare in th e middle is easily seen to b e a pu llbac k. Thus the ab o ve diagram is a morphism of p olynomial d ia grams ( f , g , u ) : ( t A , p A , s A ) − → ( t B , p B , s B ) in P oly D iag . W e ha ve Prop osition 6.10 The morphism ι a deﬁne d ab ove is a morphism of lax monoidal ﬁbr a- tions, an e quivalenc e of biﬁbr ations, and it makes the triangle of morphisms of lax monoidal ﬁbr ations over S et S ig a P oly D iag ✲ ι a E xp ( S et ) r ep a ❅ ❅ ❅ ❘ r ep pd    ✠ c ommute, up to a ﬁbr e d natur al isomorphism. Pr o of. ι a is faithful f rom the construction. Let ( f , g , u ) : ( t A , p A , s A ) → ( t B , p B , s B ) b e a morphism of p olynomial diagrams ov er u : O → Q . Th en ( f , σ, u ) : ( A, ∂ A , O ) → ( B , ∂ B , Q ) s uc h that σ a = ( g ⌈ p − 1 ( a ) ) − 1 , f or a ∈ A , is a morph ism in S ig a o ve r u . Moreo ver ι a ( f , σ, u ) = ( f , g , u ). Thus ι a is f ull. T o see that ι a is essen tially surjectiv e as well, ﬁx a diagram E A ✲ p O ✛ s O ✲ t in P ol y D iag . F or a ∈ A , choose b iject ions τ a : ( n a ] → p − 1 ( a ), for some n a = | p − 1 ( a ) | ∈ ω . Putting ∂ A a ( i ) ( t ( a ) if i = 0 sτ a ( i ) otherwise. w e h a v e that ι ( A, ∂ A , O ) is isomorp hic to ( t, p, s ), i.e. ι a is essentiall y surjectiv e as well. The v eriﬁcation that the tr ia ngle comm utes is also easy and we lea v e it to the reader. ✷ Corollary 6.11 We have a se quenc e of morph i sms of lax monoidal ﬁbr ations S ig a P oly D iag ✲ ι a P oly ✲ r ep pd C ar t ( S et ) ✲ E xp ( S et ) ✲ 50 with the ﬁrst two b eing e quivalenc es of biﬁbr ations and the fol lowing two b eing inclusions of biﬁbr ations ful l on isomorphism s. The c omp osition of al l four morph isms is (isomor phic to) r ep a , and henc e r ep a is a morphism lax monoidal ﬁbr ations, morphism of biﬁbr ations, faithful, and fu l l on isomorph isms. Pr o of. This is an immediate consequence of Corollary 6.9 and Prop osition 6.10. ✷ Theorem 6.12 The essential image of the morphism of lax monoidal ﬁbr ations r ep a in E xp ( S et ) → S et c onsists of the ﬁnitary endofunctors pr eserving wide pul lb acks and c arte- sian natur al tr ansformations. Pr o of. By Pr oposition 6.10 the image of r ep a in p exp is p poly : P ol y → S et , th e image of r ep pd . By Prop osition 6.7, r ep pd preserve s prone morphism s. By Corollary 6.9, the image of r ep pd is con tained in th e lax monoidal su bﬁbration C ar t ( S et ) → S et . Th u s, it is enough to v erify th e statemen t on ﬁbres. But this is the conte n t of Th eo rem 6.4. ✷ By Prop osition 2.1, an y m orphism of lax monoidal ﬁbration indu ce s a morphism of the corresp onding ﬁbrations of monoids. W e ﬁn ish t his section by sp elling the most important instance of this fact, announced at the b eginnin g of this subsection, that f ol lo ws from Corollary 6.11. Corollary 6.13 The ﬁbr ation of multic ate gories with non-standar d amalga mation is e quivalent to the ﬁbr ation that has as obje cts ﬁnitary c artesian monads on slic es of S et whose functor p art pr eserves wide pul lb acks. A morph ism in that ﬁbr ation b etwe en mon- ads over S et /O and S et /Q over a function u : O → Q i s a c artesian morphism of monads whose functor p art is the pul lb ack functor u ∗ : S et /Q − → S et /O . Example Th e follo wing example sho ws that s ome p olynomial diagrams can b e equipp ed with a monoid stru cture in the ﬁ bration of p olynomial diagrams P ol y D iag but that this monoid stru cture (un ique in this case) cannot b e lifted to th e ﬁ bration of mono- tone diag rams MP oly D iag . Th is is to explain wh y there are few er monotone monads than p olynomial ones. Th e signature has three t y pes O = { x 0 , x 1 , x 2 } and sev en op eratio ns. Three op eratio n s that will serve as units in the mon oi d 1 x 0 : x 0 → x 0 , 1 x 1 : x 1 → x 1 , 1 x 2 : x 2 → x 2 and four other with t yp ing as display ed: x 1 x 1 f 0    ❅ ❅ ❅ x 0 x 2 f 1 x 1 x 1 x 2 f 2    ❅ ❅ ❅ x 0 x 2 x 2 f 3    ❅ ❅ ❅ x 0 Then, no m at ter how we order en tries in f 2 : either x 1 < x 2 or x 2 < x 1 , w e won’t b e able to deﬁne one of th e multiplica tions x 1 x 2 f 1 1 x 1 f 0    ❅ ❅ ❅ x 0 x 2 x 1 1 x 1 f 1 f 0    ❅ ❅ ❅ x 0 This pr oblem disapp ears if we can switc h entries in the r esult, whic h is p ossible in amal- gamated signatures and p olynomial d ia grams. 51 6.5 The 2-lev el amalgamated signatures ﬁbration p 2 a : S ig 2 a → S et → W e d escribe b elo w a lax monoidal ﬁbr ati on suc h that its category of monoids con tains as a full sub category th e catego r y of 2-lev el multica tegories with n o n-standard amalgamation, c.f. [HMP]. It is fairly clear that this construction can b e farther generalize by making the structure of ob jects ev en more in volv ed but righ t no w we don’t s ee an y real applications for suc h structures. The ﬁbration p 2 a : S ig 2 a → S et → The base categ ory of our ﬁb ratio n is S et → . A typica l ob ject of S et → is a fu nction ˙ ( − ) : O → ¨ O , d enote d by ~ O . O is referr ed to as the set of ob jects of ~ O , ¨ O is referr ed to as the set of types of ~ O , and ˙ ( − ) is the typing of ~ O . A morphism in S et → , d enote d ~ u = ( u, ¨ u ) : ~ O → ~ Q , is a pair of morph ism making the sq uare ¨ O ¨ Q ✲ ¨ u O Q ✲ u ❄ ˙ ( − ) ❄ ˙ ( − ) comm ute. T he n ota tion ~ O and ~ u will b e used exclusiv ely in this subsection. In spite of the fact that we think of O and ¨ O as d isjoin t sets, it is con ve nien t to ’test’ elements of those sets for equalit y in the sen se th at either they b oth b elong to one set and are equal or otherwise w e mov e one of the elemen ts to ¨ O and th ere they are equal. F ormally , we deﬁne the ‘graded equalit y’ ˙ = so that if x, y ∈ O + ¨ O , then x ˙ = y iﬀ      x = y if x, y ∈ O or x, y ∈ ¨ O x = ˙ y if y ∈ O and x ∈ ¨ O ˙ x = y if x ∈ O and y ∈ ¨ O By ~ O † w e d enote the sum S n ∈ ω ~ O † n , wh ere ~ O † n = { d : [ n ] → O + ¨ O : d (( n ]) ⊆ O } is the set of functions from [ n ] t o the disj oin t sum O + ¨ O suc h that positive in tegers are se n t to ob jects and 0 is sent either to an ob ject or a t y pe. Extending the previous con ve n tions w e will write If ˙ ( − ) : O → ¨ O is an ident it y , w e write O † for ~ O † . F or d : [ n ] → O + ¨ O we ha ve restrictions of d to d + : ( n ] + → O + ¨ O and to d − : { 0 } → O + ¨ O . The total catego r y of our ﬁbration p 2 a : S ig 2 a → S et → has as ob jects triples ( A, ∂ A , ~ O ), suc h that A is a set, ~ O is an ob ject of S et → and ∂ A : A → ~ O † is a function. A morphism ( f , σ, ~ u ) : ( A, ∂ , ~ O ) → ( B , ∂ , ~ O ) is a triple such th at f : A → B is a function, ~ u : ~ O → ~ Q is a morphism in S et → and for an y a ∈ A with | a | = n , σ ∈ S n is a p ermutation such that O + ¨ O Q + ¨ Q ✲ u + ¨ u [ n ] [ n ] ✛ σ a ❄ ∂ a ❄ ∂ f ( a ) comm utes. T he pro jection fu nctor p : S ig 2 a → S et → sends ( f , σ, ~ u ) : ( A, ∂ , ~ O ) → ( B , ∂ , ~ Q ) to ~ u : ~ O → ~ Q . F or ~ u : ~ O → ~ Q in S et → , we h a v e a p ullbac k op eratio n 52 ~ O † ~ Q † ✲ ~ u † u ∗ ( B ) B ✲ π ❄ ∂ ❄ ∂ where ~ u † ( d ) = ( u + ¨ u ) ◦ d , w here d : [ n ] → O + ¨ O . Then ~ u ∗ ( B ) = {h d, b i : d : [ | b | ] → O + ¨ O , b ∈ B , ~ u † ( d ) = ∂ b } The monoidal structure in p 2 a W e ha ve t w o lax morphisms of ﬁbrations S ig 2 a × S et → S ig 2 a S ig 2 a ✲ ⊗ S et → p ′ ❅ ❅ ❅ ❘ p 2 a    ✠ S et → S ig 2 a ✲ I S et → id ❅ ❅ ❅ ❘ p 2 a    ✠ The tensor ( A ⊗ ~ O B , ∂ ⊗ , ~ O ) of tw o ob jects ( A, ∂ , ~ O ) and ( B , ∂ , ~ O ) in a ﬁbr e ( S ig 2 a ) ~ O is deﬁned as follo ws A ⊗ ~ O B = {h a, b i i i ∈ ( | a | ] : a ∈ A, b i ∈ B , ∂ b i (0) ˙ = ∂ a ( i ) } and for h a, b i i i ∈ ( | a | ] ∈ A ⊗ O B , ∂ ⊗ ( h a, b i i i ) = [ ˙ ∂ − a , ∂ + b i ] i : [ | a | X i =1 | b i | ] = [0] + a i ( | b i | ] − → O + ¨ O where ˙ ∂ − a ( ∗ ) = ∂ ⊗ ( h a, b i i i )( ∗ ) is ‘ ∂ a ( ∗ ) as muc h as p ossible’, i.e. ˙ ∂ + a ( ∗ ) = ( ˙ ( ∂ a ( ∗ )) if ∂ a ( ∗ ) ∈ O and ∃ i ∈ ( | a | ] ∂ b i ( ∗ ) ∈ ¨ O ∂ a ( ∗ ) otherwise. Remark This deﬁnition is slightly more complicated than p ossible to mak e sure that comp ositio n with iden tities (that ha ve ob jects from O , rather than t yp es fr om ¨ O , as co domains) is n eutral. The tensor on morphisms is deﬁned as in S ig a . F or a pair of maps in S ig 2 a f = ( f , σ, ~ u ) : ( A, ∂ , ~ O ) → ( A ′ , ∂ , ~ Q ) , g = ( g , τ , ~ u ) : ( B , ∂ , ~ O ) → ( B ′ , ∂ , ~ Q ) o ve r th e same map ~ u : ~ O → ~ Q in S et → w e d eﬁne the map f ⊗ ~ u g = ( f ⊗ ~ O g , σ ⊗ ~ u τ , ~ u ) : ( A ⊗ ~ O B , ∂ ⊗ , ~ O ) − → ( A ′ ⊗ ~ Q B ′ , ∂ ⊗ , ~ Q ) so that, for h a, b i i i ∈ ( | a | ] ∈ A ⊗ ~ O B , f ⊗ ~ u g ( h a, b i i i ∈ ( | a | ] ) = ( h f ( a ) , g ( b σ a ( j ) ) i j ∈ ( | f ( a ) | ] ) Clearly , | a | = | f ( a ) | and n = |h a, b i i i ∈ ( | a | ] | = X i ∈| a | | b i | = X i ∈| f ( a ) | | g ( b i ) | = | f ⊗ ~ u g ( h a, b i i i ∈ ( | a | ] ) | . Moreo v er, we pu t ( σ ⊗ ~ u τ ) h a,b i i i = [ σ − a , τ + b σ a ( i ) ] i making the triangle 53 O Q ✲ ~ u [ n ] [ n ] ✛ ( σ ⊗ ~ u τ ) h a,b i i i ❄ ∂ ⊗ h a,b i i i ❄ ∂ ⊗ h f ( a ) ,g ( b i ) i i comm ute. This ends the deﬁn iti on of the tensor ⊗ . The unit I ~ O in the ﬁbre ( S ig 2 a ) ~ O , is ( O , ∂ I O , ~ O ) where, for x ∈ O , the f unction ∂ I ~ O x : [1] → O is constant equ a l x . Remark As w e ment ioned earlier, w e would like to p ut ∂ ~ O x ( ∗ ) = ˙ x to b e su re that the co domains are alwa ys the types of ~ O but this w ould not w ork as the compatibilit y of the iden tities on the left in 2-lev el m u ltic ategories w ith non-standard amalgamation m u st b e on the ob jects of ~ O . Lemma 6.14 The ﬁbr ation p 2 a : S ig 2 a − → S et → to ge th er with the ab ove deﬁne d tensor ⊗ and unit I is a lax monoida l ﬁbr ation whose ﬁbr es ar e str ong monoidal c ate gories. The ﬁbr e d for getful functor U 2 a S ig 2 a M on ( S ig 2 a ) ✲ F 2 a S et → p 2 a ❅ ❅ ❅ ❅ ❘ q 2 a     ✠ ✛ U 2 a has a ﬁbr e d left adjoint F 2 a , the fr e e monoid fu ncto r. Pr o of. Th e fact that p 2 a is a lax monoidal ﬁbration is a s imple chec k. The construction of a free monoid functor F 2 a is essential ly the same as the one giv en in [HMP], see also [A] [Ke], [BJT]. ✷ The full em b eddings The 2-lev el amalgamate d signatures ﬁb ratio n p 2 a : S ig 2 a − → S et → con tains as a lax monoidal su bﬁbration the amalgamated signatures ﬁbr ati on p a : S ig a − → S et , and as a consequence the category of monoids M on ( S ig a ) with resp ect to p a has a full ﬁbred em b eddin g into th e category of monoids M on ( S ig 2 a ) with resp ect to p 2 a . Moreo ver, the catego ry of 2-lev el m ulticatego ries w ith non-standard amalgamation of [HMP] is a full sub category of M on ( S ig 2 a ), as w ell. Belo w w e describ e this in detail. The ﬁrst em b edd ing S et S et → ✲ δ S ig a S ig 2 a ✲ ι ❄ p a ❄ p 2 a is giv en b y the diagonal fu ncto r δ and an inclusion ι . Th e fun cto r δ h a s b oth adjoin ts, sa y l ⊣ δ ⊣ r , asso ciating co domain and d oma in, resp ectiv ely . Th e fun cto r ι h as also b oth ﬁbred adjoints L ⊣ ι ⊣ R . Th e left adjoin t L is deﬁned b y comp ositi on. F or ξ in S ig a w e ha ve L ( ξ : Y − → ˙ O × O ∗ ) = ˙ ξ where ˙ ξ is the comp osition 54 Y ˙ O × O ∗ ✲ ξ ˙ O × ˙ O ∗ = ˙ O † ✲ 1 × ˙ ( − ) ∗ The righ t ad join t R is deﬁn ed by the p ullbac k. W e hav e R ( ξ : Y − → ˙ O × O ∗ ) = ˜ ξ where ˜ ξ is giv en by the follo wing pullbac k O † ˙ O × O ∗ ✲ ˙ ( − ) × 1 O ∗ ˜ Y Y ✲ ❄ ˜ ξ ❄ ξ The second em b edd ing is a f unctor Φ : Multicat − → M on ( S ig 2 a ) from the category Multicat of 2-lev el m u lti catego ries with n on-standard amalgamation to the category of monoids in the lax m onoi dal ﬁbr at ion p 2 a : S ig 2 a → S et → . Let C b e an ob ject of Multicat. Let A = A ( C ), O = O ( C ), ¨ O = ¨ O ( C ) denote arr o ws, ob jects and lo we r lev el ob jects (i.e. t yp es) in C , resp ectiv ely . The monoid Φ( C ) is in the ﬁ bre ov er ˙ ( − ) : O → ¨ O , i.e. ~ O . The universe of Φ ( C ) is A and the t yp ing fun cti on ∂ A : A → ~ O † is deﬁned as follo ws. F or a ∈ A we ha v e source of a , s ( a ) : ( | a | ] → O and target of a , t ( a ) ∈ ¨ O . The f unction ∂ a : [ | a | ] − → O + ¨ O is equal s ( a ) on ( | a | ] and ∂ a (0) = ( x if a = 1 x t ( a ) otherwise. i.e. it is the ob ject x if a is the iden tit y on x and it is the t yp e t ( a ) otherwise. The unit map ( e, σ, id ~ O ) : ( O , ∂ I ~ O , ~ O ) → ( A, ∂ A , ~ O ) is sending x ∈ O to 1 x ∈ A , and σ x = 1 [1] . The m u lti plication ( µ A , σ, id ~ O ) : ( A ⊗ ~ O A, ∂ ⊗ , ~ O ) − → ( A, ∂ , ~ O ) is deﬁned with the help of simulta neous comp osition op eration, see [HMP]. W e ﬁr st describ e exactly the tensor ( A ⊗ ~ O A, ∂ ⊗ , ~ O ). W e h a v e A ⊗ ~ O A = {h a, a i i i ∈ ( | a | ] : a i , a ∈ A, ∂ a i (0) ˙ = ∂ a ( i ) and for h a, a i i i ∈ ( | a | ] ∈ A ⊗ ~ O B , ∂ ⊗ , + h a,a i i i = a i ∂ + a i : ( |h a i , b i i | ] = a i ( | a i | ] − → O + ¨ O and ∂ ⊗ , + h a,a i i i (0) = ( ˙ ( ∂ a (0)) if ∂ a (0) ∈ O and ∃ i ∈ ( | a | ] ∂ a i (0) ∈ ¨ O ∂ a (0) otherewise. F or h a, a i i i ∈ ( | a | ] ∈ A ⊗ ~ O A , we h a v e the simultaneo us comp osition b = a ( a i /i : i ∈ ( | a | ]) together with amalgamati ng maps ϕ i : s ( a i ) → s ( b ) f or i ∈ ( | a | ] s uc h that the function [ ϕ i ] i : a i s ( a i ) − → s ( b ) is a b iject ion. W e put µ A ( h a, a i i i ) to b e equal b , and σ b is suc h that σ − b = [ ϕ i ] i . The remaining d eta ils, as well as the deﬁnition on m orphisms, are easy . The category S ig 2 a and the tensor ⊗ ~ O are so deﬁned that id en tities must hav e ob jects as co domains. As w e menti oned at the b eginning, this is so to mimic the b eha vior of iden tities in Mul- ticat. Ho w ever, all the other arro w s in the monoids coming fr om Multicat ha ve typ es as co domains. In fact, this c h aract erizes the monoids coming from Multicat. W e h a v e 55 Prop osition 6.15 The functor Φ : Multicat − → M on ( S ig 2 a ) is ful l and faithful and its essential image c onsists of those monoids in which al l arr ows but identities have typ es as c o domains (i.e . either ∂ a (0) ∈ ¨ O or a = 1 ∂ a (0) , for a ∈ A ). The mutic ate gories with the obje ct of obje cts e qual to ˙ ( − ) : O → ¨ O ar e sent to the monoids in the ﬁbr e over ~ O . Pr o of. S imple v eriﬁcation. ✷ 6.6 Single tensor in the ﬁbration p a In th is su bsectio n we describ e another widely used tensor in the ﬁbr a tion p a . T o d istinguish these t wo tensors, we shall call the one considered so far the total tensor and denoted it, in this subsection only , b y ⊗ t . Th e tensor w e are going to d iscuss here, the single tensor , will b e d enot ed b y ⊗ s . Both tensors hav e the same unit. Let ( A, ∂ A , O ) and ( B , ∂ B , O ) b e tw o ob ject in the ﬁ bre ( S ig a ) O . The single tensor ( A ⊗ s B , ∂ ⊗ s , O ) is d eﬁned as f oll o ws A ⊗ s B = {h a, i, b i : a ∈ A, b ∈ B , i ∈ ( | a | ] ∂ A a ( i ) = ∂ B b (0) } and for h a, i, b i ∈ A ⊗ s B , ∂ ⊗ s ( h a, i, b i ) = ∂ A a ⌈ ( | a | ] −{ i } ) + ∂ B , − b . Note th at con trary to th e case of the total tensor ⊗ t , th e coherence morphisms f or the asso cia tivit y α and the righ t unit ρ are not isomorphism s. This example toget her with the Burroni ﬁ brations we re the main motiv ation for c ho osing the d irect ions of coherence morphisms in the deﬁnition of lax monoidal ﬁbrations. There is a long debate whether s ingle or total tensor is more con ve nien t. There are argumen ts for eac h. The ﬁb ratio n p a : S ig a → S et equipp ed with single tensor also acts on the basic ﬁbration, but this action is not so muc h in us e. Th e action ⋆ s S ig a × S et S et → S et → ✲ ⋆ s S et ❅ ❅ ❅ ❘ cod    ✠ is d eﬁned as f oll o ws. F or ( A, ∂ A , O ) in S ig a and ( X, d X ) in ( S et → ) O w e put ( A, ∂ A , O ) ⋆ s ( X, d X ) = ( A ⋆ s X, d s ), where A ⋆ s X = {h a, i, x i : ∂ A a ( i ) = d X ( x ) } and d s ( a, i, x ) = ∂ A a (0). One can easily v erify that this extends to a deﬁnition of an action of the lax monoidal ﬁbr ati on ( S ig a , p a , ⊗ s , I ) on the basic ﬁ bration. So, b y adjun cti on w e get a morphism of lax m onoidal ﬁbr at ions S ig a E xp ( S et ) ✲ r ep s a S et p a ❅ ❅ ❅ ❘ p exp    ✠ 56 W e can describ e th is representati on equiv alentl y as a r epresen tation of the ﬁbr ati on of p olynomial diagrams. That is, if w e consider in the ﬁb ratio n of p olynomial d ia grams the image un der equiv alence of categories ι a of the tensor ⊗ s in p a , w e h a v e a tensor, also denoted ⊗ s , in p pd : P ol y D iag → S et . Th en p pd considered with this tensor h as a representa tion r ep s pd corresp onding to th e representat ion r ep s a , i.e. a morph ism of lax monoidal ﬁbrations P oly D iag E xp ( S et ) ✲ r ep s pd S et p pd ❅ ❅ ❅ ❘ p exp    ✠ that sends the diagram ( t, p, s ) to the fun ct or t ! p ! s ∗ . The representa tion r ep s pd is faithful but not full ev en on isomorp hisms, as the diagrams E A ✲ p O ✛ s O ✲ t E A ′ ✲ p ′ O ✛ s O ✲ t ′ ha ve isomorphic repr esen tations if and only if t ◦ p = t ′ ◦ p ′ . W e wan t to d escribe the im ag e of r ep s pd . W e sh all call a p olynomial d ia gram ( t, p, s ), a line ar diagr am 12 if and only if p is an isomorphism. W e shall denote by p ld : L in D iag → S et th e f ull subﬁb ratio n of p pd whose ob jects in the total category are linear diagrams. The image of r ep s pd is the same as r ep ld , the image of r ep s pd restricted to p ld . The linear diagrams are closed under b oth tensors ⊗ t and ⊗ s and b oth tensors agree on them. Th u s b oth r epresen tations r ep t pd and r ep s pd coincide on linear diagrams. This statemen t charac terizes the linear diagrams. This is wh y the representa tion of linear diagrams is d enote d b y r ep ld , with no sup erscript. Prop osition 6.16 The total c ate gory of the image of the morphism of lax monoidal ﬁ- br ations L in D iag E xp ( S et ) ✲ r ep ld S et p ld ❅ ❅ ❅ ❘ p exp    ✠ c onsists of e ndo f unctor s on slic es of S et that pr eserves c olimits and wide pul lb acks as obje cts and c artesian natur al tr ansforma tions as morphisms. Pr o of. F rom the c h arac terization of the image of p pd in p exp the necessit y of the conditions is obvio us. On the other hand, by Th eo rem 6.4, any endofunctor P on a slices S et /O that pr eserv es colimits and wid e pullbac ks is of the form t ! p ∗ s ∗ for some p olynomial diagram E A ✲ p O ✛ s O ✲ t Recall that p has ﬁnite ﬁbres. Sin ce P p reserv es the initial ob ject the ﬁ bres of p cannot b e empt y . Since p preserves binary copro duct the ﬁbr es of p cannot h a v e more than one elemen t. T h us p is an isomorphism , and the d iag ram r epresen ting P is linear. The c haracterization of n at ural transformations in the image of r ep ld follo ws directly from Theorem 6.4. ✷ The p olynomial d ia grams of form 12 This name and notion is t aken from [Ko]. 57 E O ✲ p O ✛ s O ✲ 1 O are also closed under the t ensor ⊗ t in p pd , and fo rm a monoi dal full subﬁbration of p pd con- sidered with th e total tensor ⊗ . Su c h diagrams corresp ond to signatures that hav e exactly one op eration of eac h (output) t yp e. The r epresen tations of suc h diagrams are en dofunc- tors P on S et /O that send a function d to a fun cti on P ( d ) that has as ﬁbres ﬁnite pro ducts of ﬁb res of d . F or th is reason, w e call suc h p olynomial diagrams monomial diagr ams and the fu ll su bﬁbration of monomial diagrams will b e den ot ed by p md : M ono D iag → S et . The comp ositio n of r ep pd with the inclus io n giv es a r epresen tation morph ism r ep md M ono D iag E xp ( S et ) ✲ r ep md S et p md ❅ ❅ ❅ ❘ r ep exp    ✠ W e ha ve Prop osition 6.17 The total c ate gory of the i ma ge of the morphism r ep md of lax monoidal ﬁbr ations M ono D iag E xp ( S et ) ✲ r ep md S et p md ❅ ❅ ❅ ❘ p exp    ✠ c onsists of ﬁnitary e nd ofunctors on slic es of S et that pr eserves limits and c artesian natur al tr ansformations. Pr o of. This follo ws fr om Theorem 6.4 and the observ ation, that f o r an y p olynomial diagram ( t, p, s ) the p olynomial fu nctor t ! p ∗ s ∗ preserve s th e terminal ob ject 1 if and only if t is an isomorph ism. ✷ Remarks 1. W e wan t to p oin t out that the ﬁbr at ions of linear diagrams p ld and monomial dia- grams p md are certain ﬁ brations of graphs. The category Gph has as ob jects parallel pairs of functions s, t : A → O . A morphism of graphs ( f , u ) : ( s, t ) → ( s ′ , t ′ ) is a pair of fu nctio ns f : A → A ′ u : O → Q making the diagram E ′ Q ✲ p ′ E O ✲ p ❄ f ❄ u Q ✛ s ′ O ✛ s ❄ u comm ute. T he category of cartesian graphs cGph is a sub catego ry con taining the same ob jects as Gph and a morph ism ( f , u ) in cGph if and only if the righ t square ab o v e is a pullbac k. Th ese cate gories are ﬁb red o v er S et and th e pro jection f unctors p G : Gph → S et , p cG : Gph → S et send a morphism ( f , u ) to u . Both ﬁ brations ha ve a lax monoidal s tructure w ith tensor giv en b y the obvious pullbacks. Moreo v er, w e ha ve equiv alences of lax monoidal ﬁbrations Gph L in D iag ✲ S et p G ❅ ❅ ❘ p ld   ✠ cGph M ono D iag ✲ S et p cG ❅ ❅ ❘ p md   ✠ 58 sending a graph ( t, s : A → O ) to a linear diagram ( t, 1 A , s ) and a cartesian grap h ( p, s : A → O ) to a monomial diagram ( t, p, 1 O ). 2. S o far w e hav en’t said an yth ing ab out monoids in p a with the sin gl e tensor ⊗ s . W e can pass from m u ltiplic ation w it h resp ect to the total tensor to multiplicat ion with resp ect to the single tensor by p utting identit ies in to all place s but one. Thus we ha ve an emb edding of the monoids with resp ect to the total tensor in to the the monoids with resp ect to the single tensor. T o c haracterize the image of this embedd ing we shall use a certain n at ural isomorphism inv olving ⊗ s and the binary copro duct in ﬁbres +. Note that for any signatures A , B , C in the same ﬁbr e o ver O of S ig a , we ha ve an isomorphism: (( A ⊗ s B ) ⊗ s C ) + ( A ⊗ s ( C ⊗ s B )) ∼ = (( A ⊗ s C ) ⊗ s B ) + ( A ⊗ s ( B ⊗ s C )) that ’repairs’ the lac k of strong asso ciativit y for the tensor ⊗ s . Intuitiv ely , b oth sides of the isomorp hism con tain the part ( A ⊗ s ( C ⊗ s B )) + ( A ⊗ s ( B ⊗ s C )) i.e. the A ’ into whic h w e plug either a B with plugged in a C or a C w ith plugged in a B . The r emai ning p art on b oth s ides of the ab o ve isomorphism is A ’s in to w hic h w e p lug dir ec tly a B and a C . Clearly , this isomorphism, υ A,B ,C is natural in A , B and C . υ A,A,A ma y lo ok trivial b ut it is not! Then one can ve rify that a monoid ( M , m, e ) in S ig a with resp ect to ⊗ s comes from a monoid in S ig a with resp ect to ⊗ t , if the follo win g diagram (( M ⊗ s M ) ⊗ s M ) + ( M ⊗ s ( M ⊗ s M )) ∼ = (( M ⊗ s M ) ⊗ s M ) + ( M ⊗ s ( M ⊗ s M )) ( M ⊗ s M ) + ( M ⊗ s M ) M ✲ [ m, m ] ❄ ( m ⊗ 1 M ) + (1 M ⊗ m ) ( M ⊗ s M ) + ( M ⊗ s M ) ✛ [ m, m ] ❄ ( m ⊗ 1 M ) + (1 M ⊗ m ) comm utes, w here the unn a med isomorph ism is υ M ,M ,M . This condition corresp onds to the comm u ta tivit y condition in the multicate gories with non-standard amalga- mations in [HMP]. 7 Symmetric signatures vs analytic functors 7.1 The symmetric signature ﬁbration p s : S ig s − → S et The category of symmetric sets The category of symmetric s et s is equiv alen t to the category of sp ecies, cf. [J1], ho w ever the presenta tion is sligh tly d iﬀeren t. A sym metric set ( A, α ) is a graded set { A n : n ∈ ω } w ith (right) actions of symmetric groups α n : A n × S n → A n , for n ∈ ω . W e write a ∈ A to mean that a ∈ ` n A n and if a ∈ A then we write | a | = n to mean that a ∈ A n . Thus, f or a ∈ A , we ha ve a ∈ A | a | . In case α ( a, σ ) is d eﬁned w e us ually w rite it as a · σ if it do es not lead to a confusion. A morphism of symmetric sets f : ( A, α ) → ( B , β ) is a family of m orphisms of actio ns f n : ( A n , α n ) → ( B n , β n ) for n ∈ ω , i.e. it is a fun cti on f : A → B comm u ting with the actions α and β , in short. W e call su c h morphisms e quivariant . Th e category of symmetric sets will b e denoted b y σ S et . σ S et is (equiv alen t to) of the presheaf catego ry 59 S et S op ∗ , wh ere S ∗ the copro duct of (ﬁnite) symmetric groups in C at . C le arly the groups S ∗ act on O † on th e right by comp osition, lea ving 0 ﬁxed. T his symmetric set on O † will b e denoted by O ‡ . Any f unction u : O → Q induces an equiv arian t map u ‡ : O ‡ → Q ‡ , so that u ‡ ( d ) = u ◦ d . Th u s we h a v e a fun ct or ( − ) ‡ : S et − → σ S et The op erad of symmetries S Recall 13 that the u niv erses of sym metric groups h S n i n ∈ ω form the u nderlying sets of an op erad, called the op erad of symmetries S . Th e comp ositions ∗ : S k × ( S n 1 × . . . × S n k ) − → S P k i =1 n i ( τ , σ 1 , . . . , σ k ) 7→ τ ∗ ( σ 1 , . . . , σ k ) where, for τ ∈ S k , σ 1 ∈ S n 1 , . . . , σ k ∈ S n k , 1 ≤ m 0 ≤ k , 1 ≤ m 1 ≤ n k m 0 are give n by τ ∗ ( σ 1 , . . . , σ k )( k 1 + . . . + k m 0 − 1 + m 1 ) = k τ − 1 (1) + . . . + k τ − 1 ( τ ( m 0 ) − 1) + σ m 0 ( m 1 ) The ﬁbration p s T aking the pullback of the basic ﬁ bration on the category of symm et ric s et s cod : σ S et → − → σ S et along the functor ( − ) ‡ S et σ S et ✲ ( − ) ‡ S ig s σ S et → ✲ ❄ p s ❄ cod w e obtain the symmetric signature ﬁbration p s . W e d escribe the category S ig s explicitly . An ob ject of S ig s o ve r the set O is a quadruple ( A, α, ∂ A , O ) s uc h that ( A, α ) is a symmetric set, ∂ A : ( A, α ) → O ‡ is an equiv arian t map called the typing (or proﬁle in [BD]) map of the signature. W e w rite ∂ A a : [ n ] → O for the eﬀect of ∂ A on a ∈ A , and n in this case can b e r eferred to as | a | . The fact that ∂ A is equiv arian t means that we ha v e ∂ A a · σ = ∂ A a ◦ σ , for a ∈ A and σ ∈ S | a | . A m orphism ( f , u ) : ( A, α, ∂ A , O ) − → ( B , β , ∂ B , Q ) in S ig s o ve r a function u : O → Q is a comm uting square of equiv arian t maps: O ‡ Q ‡ ✲ u ‡ ( A, α ) ( B , β ) ✲ f ❄ ∂ A ❄ ∂ B The monoidal structure in the ﬁbres of p s W e deﬁne t wo lax morph isms of ﬁbrations S ig s × S et S ig s S ig s ✲ ⊗ S et p ′ s ❅ ❅ ❅ ❘ p s    ✠ S et S ig s ✲ I S et id ❅ ❅ ❅ ❘ p s    ✠ 13 F or example from [Le] pp. 51-54. 60 Let ( A, α, ∂ A , O ), ( B , β , ∂ B , O ) b e tw o ob j ec ts in the ﬁb re o ve r O in the ﬁ bration p s . The tensor pr o duct ( A, α, ∂ A , O ) ⊗ O ( B , β , ∂ B , O ) = ( A ⊗ O B , α ⊗ O β , ∂ ⊗ , O ) is deﬁned as f ol lo ws ( A ⊗ O B ) n = = {h a, h b i i i ∈ ( | a | ] , σ i : X i | b i | = n, b i ∈ B , a ∈ A, ∂ b i (0) = ∂ a ( i ) , for i ∈ ( | a | ] , σ ∈ S n } / ∼ where the equiv alence relation ∼ is deﬁned as follo ws: h a · τ ; h b τ ( i ) · σ τ ( i ) i i ; σ i ∼ h a, h b i i i , τ ∗ ( σ τ (1) , . . . , σ τ ( | a | ) ) ◦ σ i where τ ∈ S | a | , σ i ∈ S | b i | , σ ∈ S P i | b i | , ∗ is the comp ositio n in the op erad of symmetries, and ◦ is the usual comp ositio n of p ermutati ons. The equiv alence class of the element h a, h b i i i ∈ ( | a | ] , σ i w ith b e d enote d b y [ h a, h b i i i ∈ ( | a | ] , σ i ] ∼ . Let κ i : ( | b i | ] → ( P i | b i | ] b e the i -th inclusion in to the copro duct, for i = 1 , . . . , | a | . Clearly , there are man y suc h inclusions that make ( P i | b i | ] into a copro duct of ( | b i | ]’s (in S et ) but we w ill alw a ys mean the simplest, that is embedd ing blo c ks ( | b i | ] one after the other in to ( P i | b i | ] (i.e. κ i ( j ) = j + P i − 1 k =1 | b k | for j ∈ ( | b i | ]). W e deﬁne ∂ ⊗ ([ h a, h b i i i ∈ ( | a | ] , σ i ] ∼ ) : [ X i | b i | ] − → O so that ∂ ⊗ ([ h a, h b i i i ∈ ( | a | ] , σ i ] ∼ )(0) = ∂ a (0) and the squares ( | b i | ] O ✲ ∂ b i ( P i | b i | ] ( P i | b i | ] ✲ σ − 1 ✻ κ i ❄ ∂ ⊗ [ h a, h b i i i ,σ i ] ∼ comm ute, for all i ∈ ( | a | ]. So the typ e of the co domain of the ’op eration’ [ h a, b i , σ i i ∈ ( | a | ] ] ∼ in A ⊗ O B is the same as the t y pe of the co domain of a in A and the t yp es of the domain of the ’op eration’ [ h a, h b i i i ∈ ( | a | ] , σ i ] ∼ in A ⊗ O B are the t yp es of the domains of b i ’s in B put one next to th e other and p erm uted b y σ . The action of S n on ( A ⊗ O B ) n is deﬁned so th at [ h a, h b i i i ∈ ( | a | ] , σ i ] ∼ · σ ′ = [ h a, h b i i i ∈ ( | a | ] , σ ◦ σ ′ i ] ∼ for σ ′ ∈ S P i | b i | . F or t wo morph isms ( f , u ) : ( A, α, ∂ , O ) − → ( B , β , ∂ , Q ) , ( f ′ , u ) : ( A ′ , α ′ , ∂ , O ) − → ( B ′ , β ′ , ∂ , Q ) o ve r u , w e deﬁne their tensor to b e ( f ⊗ u f ′ , u ) : ( A ⊗ O A ′ , α ⊗ O α ′ , ∂ ⊗ , O ) − → ( B ⊗ Q B ′ , β ⊗ Q β ′ , ∂ ⊗ , Q ) 61 in the follo wing w ay . F or h a, h a ′ i i i ∈ ( | a | ] , σ i ∈ A ⊗ O A ′ , w e put f ⊗ u f ′ ([ h a, h a ′ i i i ∈ ( | a | ] , σ i ] ∼ ) = [ h f ( a ) , h f ′ ( a ′ i ) i i ∈ ( | a | ] , σ i ] ∼ Note that | a | = | f ( a ) | and w e ha ve ∂ B f ( a ) ( i ) = u ◦ ∂ A a ( i ) = u ◦ ∂ A ′ a ′ i (0) = ∂ B ′ f ′ ( a ′ i ) (0) so [ h f ( a ) , h f ′ ( a ′ i ) i i ∈ ( | f ( a ) | ] , σ i ] ∼ b elongs to B ⊗ Q B ′ indeed. Th is end s the deﬁnition of the tensor pro duct fu nctor ⊗ in p s . The unit I O = ( O , 1 , ∂ , O ) for th e tensor ⊗ O in the ﬁ bre ( S ig s ) O is d eﬁned as f oll o ws. F or x ∈ O , ∂ x : [1] → O is a fu nction suc h that ∂ x (0) = ∂ x (1) = x . So only the group S 1 acts on O and it acts trivially . The asso cia tion O 7→ I O is clearly the ob ject part of a lax morphism of ﬁbrations, as it should b e. Lemma 7.1 The functors ⊗ and I to gether with obvious asso ciativity, lef t unit, and right unit isomorph i sms α , λ , ρ make the ﬁbr es of p s into (str ong) mono idal c ate gories. Pulling bac k t he monoidal str ucture in p s An y ob ject h B , β , ∂ , Q i in the ﬁbre ( S ig s ) Q can b e pulled bac k along a fu nctio n u : O → Q O ‡ Q ‡ ✲ u ‡ u ∗ ( B ) B ✲ π B ❄ ∂ u ∗ ( B ) ❄ ∂ B where u ‡ ( d ) = u ◦ d . W e hav e u ∗ ( B ) = {h b, d i : b ∈ B , d : [ | b | ] → O , such that u ◦ d = ∂ b } and ∂ u ∗ ( B ) h b,d i = d The action in u ∗ ( B ) applies the p ermutat ion to b oth arguments, i.e. h b, d i · σ = h b · σ , d ◦ σ i Let, for x ∈ O , d x [1] → O b e the function suc h that d x (0) = d x (1) = x . W e h a v e u ∗ ( I Q ) = {h x, d x i : x ∈ O } and ϕ 0 : I O − → f ∗ ( I O ′ ) x 7→ h u ( x ) , d x i Moreo v er, for h A, α i and h B , β i in ( S ig s ) Q , w e hav e u ∗ ( A ⊗ B ) = {hh a, h b i i i ∈ ( | a | ] , σ i , d + a + a i ∈ ( | a | ] d − b i i : u ‡ ( d + a + a i ∈ ( | a | ] d − b i ) = ∂ + a + a i ∈ ( | a | ] ∂ − b i } ( d + a : [0] : → O , d − b i : ( | b i | ] : → O , for i ∈ ( | a | ]) and u ∗ ( A ) ⊗ u ∗ ( B ) = {hh a, d i , hh b i , d i ii i ∈ ( | a | ] , σ i : u ‡ ( d ) = ∂ a , u ‡ ( d i ) = ∂ a i , for i ∈ ( | a | ] } 62 ( d : [ | a | ] : → O , d i : [ | b i | ] : → O , for i ∈ ( | a | ]). Thus w e ha ve a transformation ϕ 2 ,A,B : u ∗ ( A ) ⊗ u ∗ ( B ) − → u ∗ ( A ⊗ B ) suc h th a t hh a, d i , hh b i , d i ii i ∈ ( | a | ] , σ i 7→ hh a, h b i i i ∈ ( | a | ] , σ i , d + a + a i ∈ ( | a | ] d − b i i Lemma 7.2 The map ( π B , u ) : u ∗ ( B ) → B is a pr one arr ow over u . The data u ∗ , ϕ 0 , ϕ 2 ab ove make the usual (thr e e) diagr ams of a (lax) monoida l functor c ommute, i.e. p s e quipp e d with ⊗ , I , α , λ , ρ is a lax monoidal ﬁbr ation. ✷ Moreo v er, we hav e Prop osition 7.3 The total c ate gory of the ﬁbr ation of monoids q s : M on ( S ig s ) − → S et is e quivalent to the c ate gory of symmetric multic ate gories. The ﬁbr e d f or getful functor fr om the ﬁb r ation of monoids to the ﬁbr ation of symmetric sig na tur es U : M on ( S ig s ) − → S ig s S ig s M on ( S ig s ) ✲ F s S et p s ❅ ❅ ❅ ❅ ❘ q s     ✠ ✛ U s is a morphism of ﬁbr ations and has a left adjoint F s , the fr e e monoid fu ncto r, which is a lax morphism of ﬁbr ations. ✷ Remark T he p s is a b iﬁbration as, for an ob ject ( A, α, ∂ A , O ) and a function u : O → Q the morphism (1 A , u ) : ( A, α, ∂ A , O ) − → ( A, α, u ‡ ◦ ∂ A , Q ) is a su pine morph ism. 7.2 The action of p s on the basic ﬁbration and analytic functors The ﬁbration p s acts on the basic ﬁ bration cod : S et → − → S et as follo ws S ig s × S et S et → S et → ✲ ⋆ S et ❅ ❅ ❅ ❘ cod    ✠ In the follo wing, w e often d enote an ob ject ( A, α, ∂ A , O ) in S ig s as A and an ob ject d X : X → O in S et → as X , when it do es not lead to confusion. The ob ject d ⋆ : A ⋆ X − → O is deﬁned as the quotien t of the set { ( a, ~ x ) : a ∈ A, ~ x : ( | a | ] → X, ∂ A, + a = d X ◦ ~ x } b y an equiv alence ∼ so that ( a, ~ x ) ∼ ( a · σ, ~ x ◦ σ ) for a ∈ A , ~ x : ( | a | ] → X , and σ ∈ S | a | . Th e function d ⋆ : A ⋆ X → O is deﬁn ed as d ⋆ ([ a, ~ x ] ∼ ) = ∂ a (0) 63 The action ⋆ is deﬁned on morphisms as follo ws. F or maps ( f , u ) : ( A, α, ∂ A , O ) − → ( B , β , ∂ B , Q ) in S ig s , ( g , u ) : ( X , d X ) → ( Y , d Y ) in S et → o ve r u : O → Q and for an element [ a, ~ x ] ∼ ∈ A ⋆ X w e p ut f ⋆ u g ([ a, ~ x ] ∼ ) = [ f ( a ) , g ◦ ~ x ] ∼ so that the s quare O Q ✲ u A ⋆ X B ⋆ Y ✲ f ⋆ u g ❄ d ⋆ ❄ d ⋆ comm utes. W e ha ve an adjoint morp hism of lax monoidal ﬁ brations S ig s E xp ( S et ) ✲ r ep s S et p s ❅ ❅ ❅ ❘    ✠ F or an ob ject A in ( S ig s ) O , w e ha ve a fun cto r r ep s ( A ) = A ⋆ ( − ) : S et /O − → S et /O and for a m orphism ( f , u ) : A → B in S ig s o ve r u : O → Q we h a v e a natural transforma- tion r ep s ( f , u ) : A ⋆ u ∗ ( − ) − → u ∗ ( B ⋆ ( − )) so that for a morphism g : Y → Y ′ in ( S et → ) Q w e h a v e a commuting square A ⋆ u ∗ ( Y ′ ) u ∗ ( B ⋆ Y ′ ) ✲ r ep s ( f , u ) Y ′ A ⋆ u ∗ ( Y ) u ∗ ( B ⋆ Y ) ✲ r ep s ( f , u ) Y ❄ 1 A ⋆ u ∗ ( g ) ❄ u ∗ (1 B ⋆ g ) W e n ot e for the r ec ord, that for [ a, ~ x ] ∼ ∈ A ⋆ u ∗ ( Y ) so that a ∈ A and ~ x : ( | a | ] → u ∗ ( Y ) w e h a v e r ep s ( f , u ) Y ([ a, ~ x ] ∼ ) = h ∂ A a (0) , [ f ( a ) , u Y ◦ ~ x ] ∼ i where O Q ✲ u u ∗ ( Y ) Y ✲ u Y ❄ u ∗ ( d Y ) ❄ d Y 64 is a p ullbac k in S et . W e ha ve Prop osition 7.4 The morphism of lax monoida l ﬁbr ations r ep s deﬁne d ab ove is a mor- phism of bi ﬁbr ations and it pr eserves c opr o ducts in the ﬁbr es. Pr o of. The pro of of this prop osition can b e made more abstract b ut we p refer it to b e concrete. Preserv ation of copro duct is trivial. First, w e shall sh o w that r ep s is a morphism of ﬁbr at ions. Let B = ( B , β , ∂ B , Q ) b e a symmetric signature. T he prone morphism ov er u : O → Q in the ﬁ bration p s is deﬁned via pullbac k in th e category of symmetric sets O ‡ Q ‡ ✲ u ‡ u ∗ ( B , β ) ( B , β ) ✲ u B ❄ ∂ u ∗ ( B ) ❄ ∂ B W e write u ∗ ( B ) for ( u ∗ ( B , β ) , ∂ u ∗ ( B ) , O ). Then the pr one m orphism is pr u,B = ( u B , u ) : u ∗ ( B ) → B . W e will u se the usual r epresen tation of this pu llbac k in S et i.e. u ∗ ( B ) = {h b, d i : b ∈ B , d : [ | b | ] → O , u ◦ d = ∂ B b } and ∂ u ∗ ( B ) ( h b, d i ) = d, u B ( h b, d i ) = b for h b, d i ∈ u ∗ ( B ). W e will sho w that the f ol lo wing n at ural transformations u ∗ ( B ) ⋆ u ∗ ( − ) u ∗ ( B ⋆ ( − )) ✲ r ep s ( pr u,B ) = r ep s ( u B , u ) and u ∗ ( B ⋆ u ! u ∗ ( − )) u ∗ ( B ⋆ ( − )) ✲ pr u,r ep s ( B ) = u ∗ ( B ⋆ ε u ( − ) ) in C at ( S et /Q , S et /O ) are isomorphic as ob jects of C at ( S et /Q , S et /O ) /u ∗ ( B ⋆ ( − ) ) . T o th is end w e shall deﬁne a natur al isomorp hism u ∗ ( B ) ⋆ ( − ) u ∗ ( B ⋆ u ! u ∗ ( − )) ✲ ξ (and its in verse) so that r ep s ( u B , u ) = u ∗ ( B ⋆ ε u ( − ) ) ◦ ξ , i.e. for any d Y : Y → Q the triangle u ∗ ( B ) ⋆ ( d Y ) u ∗ ( B ⋆ u ! u ∗ ( d Y )) ❄ ξ d Y u ∗ ( B ⋆ d Y ) ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✿ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ③ r ep s ( u B , u ) d Y u ∗ ( B ⋆ ε u d Y ) (1) comm utes. W e ﬁx d Y : Y → Q ∈ S et /Q . The follo wing diagram [ | b | ] O ✲ d ( | b | ] u ∗ ( Y ) ✲ ~ x ❄ ❄ Q ✲ u Y ✲ u Y u ∗ ( d Y ) ❄ d Y ∂ B b ✻ ~ y ❄ (2) 65 where the right hand squ are is a pullbac k is to ﬁx the notation. W e do not assu me no w that other morphisms exist, b ut if they do they ha ve domains and c o domains as displa yed. Similarly , other ﬁ gures in this diagram are not assumed to commute unless w e explicitly sa y so. W e will refer often to this diagram in the rest of the p roof. W e note that, f or b ∈ B , h o, [ b, ~ y ] ∼ i ∈ u ∗ ( B ⋆ d Y ) iﬀ o ∈ O , u ( o ) = ∂ B b (0) , ∂ B , + b = d Y ◦ ~ y h o, [ b, ~ x ] ∼ i ∈ u ∗ ( B ⋆ u ! u ∗ ( d Y )) iﬀ o ∈ O , u ( o ) = ∂ B b (0) , ∂ B , + b = u ◦ u ∗ ( d Y ) ◦ ~ x [ h b, d i ~ x ] ∼ ∈ u ∗ ( B ) ⋆ u ∗ ( d Y ) iﬀ u ◦ ∂ B b = d, d + = u ∗ ( d Y ) ◦ ~ x With the ab o ve notation we sp ell out the three fu nctions o ccurring in (1): u ∗ ( B ) ⋆ u ∗ ( d Y ) u ∗ ( B ⋆ d Y ) ✲ r ep s ( u B , u ) d Y [ h b, d i , ~ x ] ∼ h d (0) , [ b, u Y ◦ ~ x ] ∼ i ✲ and u ∗ ( B ⋆ u ! u ∗ ( d Y )) u ∗ ( B ⋆ d Y ) ✲ u ∗ ( B ⋆ ε u d Y ) h o, [ b, ~ x ] ∼ i h o, [ b, u Y ◦ u Y ◦ ~ x ] ∼ i ✲ and u ∗ ( B ) ⋆ u ∗ ( d Y ) u ∗ ( B ⋆ u ! u ∗ ( d Y )) ✲ ξ d Y [ h b, d i , ~ x ] ∼ h d (0) , [ b, ~ x ] ∼ i ✲ [ h b, ¯ d i , ~ x ] ∼ h o, [ b, ~ x ] ∼ i ✛ where ¯ d : [ | b | ] → O is so d eﬁned that ¯ d (0) = o and ¯ d + = u ∗ ( d Y ) ◦ ~ x . No w a simple chec k sho w s that (1) comm u tes, i.e. r ep s preserve s p rone morph isms. No w w e shall show that r ep s preserve s sup ine morph isms, i.e. it is a morph ism of opﬁbrations. Let ( A, α, ∂ A , O ) b e a symmetric signature. T he supine morp hism su u,A in p s o ve r u : O → Q with domain A is d eﬁned from the square O ‡ Q ‡ ✲ u ‡ ( A, α ) ( A, α ) ✲ 1 A ❄ ∂ A ❄ u ‡ ◦ ∂ A w e w rite A ! for ( Aα, u ‡ ◦ ∂ A , Q ) and su u,A = (1 A , u ) : A → A ! . W e shall sho w that the natural trans formatio n s A ⋆ u ∗ ( − ) u ∗ ( A ! ⋆ ( − )) ✲ r ep s ( su u,A ) = r ep s (1 A , u ) and 66 A ⋆ u ∗ ( − ) u ∗ u ! A ⋆ u ∗ ( − ) ✲ su u,r ep s ( A ) = η u A⋆u ∗ ( − ) are isomorphic in A⋆u ∗ ( − ) \ C at ( S et /Q , S et /O ). W e shall deﬁne a n atural isomorphism u ∗ u ! ( A ⋆ u ∗ ( − )) u ∗ ( A ⋆ ( − )) ✲ ζ (and its in verse) so that r ep s (1 A , u ) = ζ ◦ η u A⋆u ∗ ( − ) , i.e. for any d Y : Y → Q the triangle u ∗ ( A ! ⋆ d Y ) u ∗ u ! ( A ⋆ u ∗ ( d Y )) ✻ ζ d Y A ⋆ u ∗ ( d Y ) ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✿ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ③ r ep s (1 A , u ) d Y η u A⋆u ∗ ( d Y ) (3) comm utes. Using the notation from diagram (1) we note that, for a ∈ A , we h a v e [ a, ~ x ] ∼ ∈ A ⋆ u ∗ ( d Y ) iﬀ ∂ A, + a = u ( d Y ) ◦ ~ x h o, [ a, ~ y ] ∼ i ∈ u ∗ ( A ! ⋆ d Y ) iﬀ o ∈ O , o = ∂ A a (0) , u ◦ ∂ A, + a = d Y ◦ ~ y h o, [ a, ~ x ] ∼ i ∈ u ∗ u ! ( A ⋆ u ∗ ( d Y )) iﬀ o ∈ O , u ( o ) = u ◦ ∂ A a (0) , u ◦ ∂ A, + a = d Y ◦ ~ x No w w e sp ell out explicitly the function o ccurring in (3). A ⋆ u ∗ ( d Y )) u ∗ ( A ! ⋆ d Y ) ✲ r ep s (1 A , u ) d Y [ a, ~ x ] h ∂ A a (0) , [ a, u Y ◦ ~ x ] ∼ i ✲ and A ⋆ u ∗ ( d Y )) u ∗ u ! ( A ⋆ u ∗ ( d Y )) ✲ η u A⋆u ∗ ( d Y ) [ a, ~ x ] h ∂ A a (0) , [ a, ~ x ] ∼ i ✲ and u ∗ u ! ( A ⋆ u ∗ ( d Y )) u ∗ ( A ! ⋆ d Y ) ✲ ζ d Y h o, [ a, ~ x ] ∼ i h o, [ a, u Y ◦ ~ x ] ∼ i ✲ [ h o, [ a, ~ x ] ∼ i h o, [ a, ~ y ] ∼ i ✛ In the last corresp ondence, ~ y 7→ ~ x : ( | a | ] → u ∗ ( Y ) is deﬁ ned using the f ac t that right square in (2) is a p ullbac k and d Y ◦ ~ y = u ◦ ∂ A, + a . Again, a simple c hec k sho ws that (3) comm utes, i.e. r ep s is a morphism o f opﬁbrations, as w ell. ✷ Later w e w ill show that r ep s is f ai thful and full on isomorphism s. 67 The ﬁbration that is the essent ial im ag e of the representat ion r ep s will b e d enote d b y p an : An → S et , we tak e it as the deﬁnition of the ﬁbration of (multi v ariable) analytic (endo)functors and analytic tran sformatio ns b et wee n them. Thus by an analytic functor on S et /O , wh ere O is a set, we unders ta nd a functor (isomorphic to one) of th e form A ⋆ ( − ) : S et /O → S et /O for a symmetric signature A = ( A, α, ∂ A , O ). Moreov er, by an analytic tr ansformation o ve r a fu nction u : O → Q b et ween t wo analytic f unctors A ⋆ ( − ) : S et /O → S et /O and B ⋆ ( − ) : S et /Q → S et /Q w e mean a n at ural transformation of the form r ep s ( f , u ) : A ⋆ u ∗ ( − ) − → u ∗ ( B ⋆ ( − )) for a morph ism symmetric signatures ( f , u ) : ( A, α, ∂ A , O ) − → ( B , β , ∂ B , Q ). Remarks 1. Note that for the one elemen t set, say [0], w e ha ve S et / [0] ∼ = S et an d the ﬁbre of p an o ve r [0] is (isomorphic to) the categ ory of usual (one-v ariable) analytic functors that was c haracterized in [J2 ], see also [A V], as the category of ﬁ nitary endofunctors on S et that w eakly preserve pullbac ks and weakly cartesian natural tr a nsformations b et we en them. W e will see in the next section ho w this c haracterizatio n extends from this ﬁbre to the wh ole ﬁ bration. 2. I n [J2] multiv ariable analytic fu nctors we re d eﬁned as certain functors S et I → S et , w ere I is a ﬁnite set. Ignoring th e ‘size problems’, this d eﬁnition can b e extended to inﬁnite sets Q b y sa ying that the class(!) of multiv ariable analytic fu nctors S et /Q → S et is a coﬁltered ’limit’ of the classes(!) of analytic fun cto rs S et I → S et wh ere I ⊆ Q and I is ﬁn ite . In th ese terms , what we call an analytic functor on S et /Q , is just a Q -tup le of m u ltiv ariable analytic fu nctors S et /Q → S et . 3. T he multiv ariable analytic f unctors S et /Q → S et can b e also d esc rib ed m ore ex- plicitly a voiding all the ‘size problems’. Let Q b e a set and q ∈ Q . W e ha ve the ev aluatio n fun cto r and th e inclusion of a ﬁbre S et /Q S et ✲ ev q S et /Q ✲ i q suc h that ev q ( X, d ) = d − 1 ( z ) and i q ( B ) : B → Q is the function deﬁned by i q ( B )( b ) = z for b ∈ B . A multivariable analytic functor fr om S et /Q to S et is a functor of the form S et /Q S et /Q ✲ B ⋆ ( − ) S et ✲ ev q where ( B , β , ∂ B , Q ) is a symmetric signature, z ∈ Q and ev z is the ev aluation fun ct or suc h that ev z ( d : X → Q ) = d − 1 ( z ). If we r estrict sym metric signatur es to those for whic h ∂ B b (0) = z then su c h fun cto rs, as we shall see, determine the signatures up to an isomorp hism. 7.3 A cha r acterization of the ﬁbration of analytic functors The follo wing extends the c haracterizatio n of analytic functors and analytic transforma- tions, c.f. [J2 ], from the ﬁbre o ve r [0] to the wh ole ﬁbr at ion of analytic fun cto rs p an . Theorem 7.5 The lax monoidal ﬁbr ation p an : An − → S et has as its obje cts ﬁnitary endofunctors on c ate gories S et /Q that we akly pr eserve wide pul lb acks and we akly c artesian natur al tr ansformations as morphisms b etwe en them. Before we pr o v e a series of lemmas needed to establish th e ab o v e theorem, we s hall immediately present the follo wing ob vious Corollary th at is ev en m ore inte resting. 68 Corollary 7.6 The ﬁb r ation of symmetric multic ate gories is e quivalent to the ﬁbr ation of we akly c artesian analytic monads and we akly c artesian morphisms of monads who se functor p arts ar e pul lb ack functors b etwe en them. Under this c orr esp ondenc e the fr e e symmetric multic ate gories c orr esp ond to the fr e e analytic monads. ✷ The follo wing deﬁnition is an extension of a notion fr om [A V]. Let Q b e a set. Th e functor F : S et /Q → S et is sup erﬁnitary if and only if there is an ob ject d : I → Q in S et /Q with I ﬁnite suc h that, for an y d X : X → Q in S et /Q F ( X, d X ) = [ f :( I , d ) → ( X,d X ) F ( f )( F ( I , d )) i.e. the element s of F ( I , d ) generates the w hole f unctor F . The follo wing tw o Lemmas and their pro ofs are ’colored’ v ersions of Th eorem 2.6 and Corollary 2.7 and their pro ofs from [A V]. Lemma 7.7 L et F : S et /Q → S et b e a sup erﬁnitary functor. Then F is a multivariable analytic functor if and only if F we akly pr eserves pul lb acks. Pr o of. By obser v ations of V. T rnko v´ a, cf. [T], any fu ncto r F : S et /Q → S et is a copro duct (indexed b y F (1 Q )) o f functors that preserv es the terminal ob ject. If F = ` i F i , then F we akly preserves pullbac ks (is analytic) if and only if all F i ’s do (are). Thus it is enough to pr o v e the lemma for a sup erﬁnitary functor F : S et /Q → S et suc h that F (1 Q ) = 1. So supp ose that F w eakly preserve s pullbacks. Fix a minim al ob ject d F : ( n ] → Q suc h th a t for an y d X : X → Q F ( X, d X ) = [ f :(( n ] ,d F ) → ( X,d X ) F ( f )( F (( n ] , d F )) By ’minimal’ w e mean that there is n o prop er sub ob ject of (( n ] , d F ) in S et /Q with this prop ert y . Thus there is an elemen t g F ∈ F ( d F ) that it is not in the image of any prop er inclusion in to d F . The pair ( g F , d F ) or just the elemen t g F if d F is u ndersto od, will b e called generic , cf. [J2], [A V]. Thus, if we hav e a morph ism X ( n ] ✲ f Q d X ❅ ❅ ❅ ❘ d F    ✠ in S et /Q suc h that g F ∈ F ( f )( X, d X ) then f is on to. Therefore, an y endomorphism of (( n ] , d F ) in S et /Q lea ving g F ﬁxed, is a bijection. W e can deﬁne a s ubgroup of S n as follo ws G F = { σ : d F → d F ∈ S et /Q : F ( σ )( g F ) = g F } ⊆ S n Let F o b e S n / G F , the set of right cosets of S n o ve r G F . The class of τ in S n / G F will b e denoted by [ τ ] ∼ F . W e ha ve a right action, sa y ϕ , of S n on F o acting by comp osition on the righ t. W e d eﬁne ∂ F o : F o → Q ‡ so that, for τ ∈ S n , ∂ F o , + [ τ ] ∼ F = d F ◦ τ : ( n ] → Q. and ∂ F o [ τ ] ∼ F (0) = z wh ere z is any elemen t of Q (if Q is empt y there is nothing to pr o v e). The fu nctor ev z ◦ ( F o , ϕ, ∂ F o , Q ) ⋆ ( − ) will b e d enote d fr om no w on s imply as F o ⋆ ( − ). F or d X : X → Q in S et /Q w e p ut 69 F o ⋆ ( d X ) F ( d X ) ✲ κ F ( d X ) [[ τ ] ∼ F , u : d F → d X ] F ( u )( g F ) ✲ W e shall sh o w that κ F : F o ⋆ ( − ) − → F is a natural isomorph ism. T o this aim it is enough to sho w ( a ) for every d X : X → Q and x ∈ F ( X , d X ) there is u : d F → d X in S et /Q suc h th at x = F ( u )( g F ); ( b ) for u, v : d F → d X w e ha ve, F ( u )( g F ) = F ( v )( g F ) if and only if there is σ ∈ G F suc h that u = v ◦ σ ; ( c ) κ F is natural. W e establish ( a ) and ( b ). Th en, as F is a functor, ( c ) will b e obvious. Ad (a). It is enough to s ho w ( a ) for elemen ts x ∈ F (( n ] , d F ). As F weakly p reserv es pullbac ks and F (1) = 1, F w eakly preserv es b inary pro ducts. W e hav e a binary pro duct ( n ] ( n ] × Q ( n ] ✛ π 1 d F ❅ ❅ ❅ ❅ ❘ Q ❄ d π ( n ] ✲ π 2 d F     ✠ of d F with itself in S et /Q and hence a weak pr odu ct F ( d F ) F ( d π ) ✛ F ( π 1 ) F ( d F ) ✲ F ( π 2 ) in S et . Hence th ere is p ∈ F ( d π ) suc h th at F ( π 1 )( p ) = g F , F ( π 2 )( p ) = x Since F is sup erﬁnitary and by assump tion on d F , there are a morphism f : d F → d π and y ∈ F ( d F ) su c h that F ( f )( y ) = p . Thus g F = F ( π 1 )( p ) = F ( π 1 ◦ f )( y ), and hence π 1 ◦ f : d F → d F is epi and th en iso. Pu tti ng u = π 2 ◦ f ◦ ( π 1 ◦ f ) − 1 : d F − → d F , we ha v e x = F ( π 2 ) = F ( π 2 ◦ f )( y ) = F ( π 2 ◦ f ◦ ( π 1 ◦ f ) − 1 )( g F ) = F ( u )( g F ) as needed. Ad (b). If for some σ ∈ G F w e h a v e u = v ◦ σ , then F ( u )( g F ) = F ( v ◦ σ )( g F ) = F ( v )( F ( σ )( g F )) = F ( v )( g F ) No w assume that F ( u )( g F ) = F ( v )( g F ). W e form a pu llbac k in S et /Q (( n ] , d F ) ( X, d X ) ✲ v ( P , d P ) (( n ] , d F ) ✲ ¯ v ❄ ¯ u ❄ u As F w eakly p reserv es p ullbac ks, there is p ∈ F ( d P ) suc h that F ( ¯ u )( p ) = g F = F ( ¯ v )( p ). Using ( a ) we get f : (( n ] , d F ) → ( P , d P ) such th at F ( f )( g F ) = p . Th u s ¯ u ◦ f , ¯ v ◦ f ∈ G F . W e ha ve u = u ◦ ( ¯ v ◦ f ) ◦ ( ¯ v ◦ f ) − 1 = v ◦ ( ¯ u ◦ f ) ◦ (¯ v ◦ f ) − 1 and ( ¯ u ◦ f ) ◦ ( ¯ v ◦ f ) − 1 ∈ G F . No w supp ose that F is a comp osition of functors 70 S et /Q S et /Q ✲ B ⋆ ( − ) S et ✲ ev z for a symmetric signature ( B , β , ∂ B , Q ) and z ∈ Q . S ince F (1 Q ) = 1, ( B , β ) has jus t one orbit. W e can assume that ∂ B b (0) = z for b ∈ B . Thus by a sligh t abuse we s hall identify B ⋆ ( − ) with F . Let ( X, d X ) ( Z, d Z ) ✲ f ( P , d P ) ( Y , d Y ) ✲ π Y ❄ π X ❄ g b e a pu llbac k in S et /Q . W e need to sho w th at B ⋆ X B ⋆ Z ✲ B ⋆ f B ⋆ P B ⋆ Y ✲ B ⋆ π Y ❄ B ⋆ π X ❄ B ⋆ g is a weak pullbac k. Fix b ∈ B and let n = | b | . S upp ose f or ( n ] X ✲ u Q ∂ B , + b ❅ ❅ ❅ ❘ d X    ✠ ( n ] Y ✲ v Q ∂ B , + b ❅ ❅ ❅ ❘ d Y    ✠ w e h a v e [ b, f ◦ u ] ∼ = ( B ⋆ f )([ b, u ] ∼ ) = ( B ⋆ g )([ b, v ] ∼ ) = [ b, g ◦ v ] ∼ i.e. th ere is σ ∈ S n suc h th a t b · σ = b, f ◦ u ◦ σ = g ◦ v Using the prop ert y of the ab o ve pu llbac k, w e get w : ( n ] → P su c h th at π X ◦ w = u ◦ σ, π Y ◦ w = v Then B ⋆ π Y ([ b, w ] ∼ ) = [ b, π Y ◦ w ] ∼ = [ b, v ] ∼ and B ⋆ π X ([ b, w ] ∼ ) = [ b, π X ◦ w ] ∼ = [ b · σ, u ◦ σ ] ∼ = [ b, u ] ∼ ✷ Recall that the fun ct or F : S et /Q − → S et /Q is thin if there is z ∈ Q su c h that F = i q ◦ ev q ◦ F and ev q ◦ F (1) = 1. Clearly , eve r y functor F : S et /Q − → S et /Q is a copro duct of thin f unctors indexed b y the domain of F (1 Q ) and ev ery natural transform at ion b et ween suc h fu nctors is a copro duct of transf ormatio n s b et ween thin fu nctors. W e h a v e Lemma 7.8 L et F : S et /Q → S et /Q b e a ﬁnitary functor. The f ol lowing ar e e quivalent 1. F is a multivariable analytic fu nctor ; 2. F we akly pr eserves wide pul lb acks of p ower ≤ ℵ 0 + | Q | ; 3. F we akly pr eserves wide pul lb acks. 71 Pr o of. As ev ery fun ct or S et /Q → S et /Q is a c opro duct of thin functors, w e can assume that F is thin. Th u s w e consider F as a functor S et /Q → S et suc h th a t F (1) = 1. 3 . ⇒ 2 . is obvious and 1 . ⇒ 3 . can b e prov ed as in Lemm a 7.7. In order to sho w 2 . ⇒ 1 . we shall sho w th at F is sup erﬁn ita r y and u se Lemma 7.7. Supp ose on the con trary that for eac h n ∈ ω and f : ( n ] → Q there is d f : X f → Q and x f ∈ F ( d f ) su c h th at x f 6∈ S h : f → d f F ( h )( f ). S ince F w eakly preserve s p ullbac ks of p o wer ≤ ℵ 0 + | Q | , there is p ∈ F ( Q n ∈ ω , f :( n ] → Q ( X f , d f )) suc h that F ( π f )( p ) = x f for n ∈ ω and f : ( n ] → Q . Since F is ﬁn ita r y , there are ( m ] Q n ∈ ω , f :( n ] → Q ( X f , d f ) ✲ g Q f 0 ❅ ❅ ❅ ❘    ✠ and y ∈ F (( m ] , f 0 ) suc h th at F ( g )( y ) = p . Th en π f 0 ◦ g : (( m ] , f 0 ) → ( X f 0 , d f 0 ) and F ( π f 0 ◦ g )( y ) = F ( π f 0 ( p )) = x f 0 con trary to the assu mption. ✷ The follo w ing fact is not n eeded for the p roof of Th eo rem 7.5 b ut it follo ws easily from the pro ofs o f the ab o v e Lemm a s and puts some ligh t on the corresp ondence b et w een orb its of ( B , β ) and elements of B ⋆ (1 Q ). Sc holium 7.9 L et F : S et /Q → S et /Q b e a ﬁnitary functor that pr eserves wide pul lb acks. Then F is i som orphic to a functor ( B , β , ∂ B , Q ) ⋆ ( − ) : S et /Q → S et /Q for a symmetric signatur e ( B , β , ∂ B , Q ) such that ( B , β ) c ontains as many orbits as the c ar dinality of the domain F (1 Q ) orbits. In p articular, if F is thin iﬀ ( B , β ) has exactly one orbit. ✷ Remark In the pr oof of the ab o v e Lemma 7.7 we in tro duced the notions of a m inimal ob ject, a generic element and a generic p a ir. The n o tion of a generic element is a v ariant of a notion introd uced in [J2 ], see also [A V]. F or the late r use, w e describ e b elo w s uc h generic pairs for the functors of form B ⋆ ( − ), wh ere ( B , β ) h as one orbit. Lemma 7.10 L et ( B , β , ∂ B , Q ) b e a symmetric signatur e su ch that ( B , β ) has a single orbit, b ∈ B and let n = | b | . Then the p ositive typing for b , i.e. ∂ B , + b : ( n ] → Q is the minimal obje ct for functor B ⋆ ( − ) : S et /Q → S et /Q and [ b, 1 ( n ] ] ∼ ∈ B ⋆ (( n ] , ∂ B , + b ) is a generic element for B ⋆ ( − ) . Mor e gener al ly, let d Y : Y → Q , ~ y : Y → Q and b ∈ B . Then h b, ~ y i r epr esents a generic element [ b, ~ y ] ∼ ∈ B ⋆ d Y for B ⋆ ( − ) if and only if ~ y is a bije ction and ∂ B , + b = d Y ◦ ~ y . Pr o of. E xercise . ✷ Lemma 7.11 L et ( f , u ) : ( A, α, ∂ A , O ) − → ( B , β , ∂ B , Q ) b e a morphism of symmetric signatur e s i n S ig s over a function u : O → Q . Then the natur al tr ansformatio n in C at ( S et /Q , S et /O ) r epr esenting ( f , u ) r ep s ( f , u ) : r ep s ( A ) ◦ u ∗ − → u ∗ ◦ r ep s ( B ) is we akly c artesian. 72 Pr o of. T h us w e h a v e a comm uting square O ‡ Q ‡ ✲ u ‡ A B ✲ f ❄ ∂ A ❄ ∂ B in the catego ry of s ymmetric sets. Let Y Y ′ ✲ g Q d Y ❅ ❅ ❅ ❘ d Y ′    ✠ b e a morph ism in S et → /Q i.e. in S et /Q . Th en we pullbac k this morphism along u , and we get a diagram u ∗ ( Y ′ ) Y ′ ✲ u ∗ ( Y ) Y ✲ u Y ❄ u ∗ ( g ) ❄ g O Q ✲ u u Y ′ ❄ u ∗ ( d Y ′ ) ❄ d Y ′ u ∗ ( d Y ) ✲ d Y ✛ in whic h thr ee squares are pullb ac k s. W e n ee d to sho w that th e square A ⋆ u ∗ ( Y ′ ) u ∗ ( B ⋆ Y ′ ) ✲ r ep s ( f , u ) d Y ′ A ⋆ u ∗ ( Y ) u ∗ ( B ⋆ Y ) ✲ r ep s ( f , u ) d Y ❄ 1 A ⋆ u ∗ ( g ) ❄ u ∗ (1 B ⋆ g ) is a weak pullbac k. So let [ a, ~ x ] ∼ ∈ A ⋆ u ∗ ( Y ′ ) i.e. a ∈ A , ~ x : ( | a | ] → u ∗ ( Y ), so th at ∂ A, + a = u ∗ ( d Y ) ◦ ~ x and let h o, [ b, ~ y ] ∼ i ∈ u ∗ ( B ⋆ Y ) i.e. o ∈ O , b ∈ B and ~ y : ( | b | ] → Q , so that u ( o ) = ∂ B b (0) and ∂ B , + b = d Y ◦ ~ y . Moreo v er, assume that r ep s ( f , u ) d Y ′ ([ a, ~ x ] ∼ ) = h ∂ A a (0) , [ f ( a ) , u Y ′ ◦ ~ x ] ∼ i = = h o, [ b, g ◦ ~ y ] ∼ i = u ∗ (1 B ⋆ y ′ )( h o, [ b, ~ y ] ∼ i ) i.e. ∂ A a (0) = o , and there is σ ∈ S | a | suc h that f ( a ) = b · σ and u Y ′ ◦ ~ x = g ◦ ~ y ◦ σ . Thus, using the upp er pu llbac k ab o v e, w e get a fu nctio n ~ z : ( | a | ] → u ∗ ( Y ) suc h that u Y ◦ ~ z = ~ y ◦ σ , u ∗ ( g ) ◦ ~ z = ~ x . Then 1 A ⋆ u ∗ ( g )([ a, ~ z ] ∼ ) = [ a, u ∗ ( g ) ◦ ~ z ] ∼ = [ a, ~ x ] ∼ . Moreo v er o = ∂ A a (0) , and h f ( a ) , u Y ◦ ~ z i = h b · σ, ~ y ◦ σ i ∼ h b, ~ y i i.e. r ep s ( f , u ) d Y ([ a, ~ z ] ∼ ) = h o, [ b, ~ y ] ∼ i . Thus the ab o ve square is a wea k p ullbac k, as required. ✷ 73 Lemma 7.12 L et ( A, α, ∂ A , O ) and ( B , β , ∂ B , O ) b e two symmetric signatur es in ( S ig s ) O . If ξ : A ⋆ ( − ) − → B ⋆ ( − ) is a we akly c artesian natur al tr ansformation then ther e is a uni que morphism of symmetric signatur es ( f , 1 O ) : ( A, α, ∂ A , O ) − → ( B , β , ∂ B , O ) in ( S ig s ) O such that r ep s ( f , 1 O ) = ξ . Pr o of. Let ( A, α, ∂ A , O ), ( B , β , ∂ B , O ) b e tw o symmetric signatures in ( S ig s ) O and let ξ : A ⋆ ( − ) − → B ⋆ ( − ) b e a w eakly cartesian natural transformation. By remark after Sc holium 7.9, we can assu me that b oth ( A, α ) and ( B , β ) h a v e one orb it . Th en the existence of ξ as ab o ve implies th at for some z ∈ O w e ha ve ∂ A a (0) = ∂ B b (0) for all a ∈ A and b ∈ B . Hence w e sh all not consider these v alues an y more. Fix d A : ( n A ] → O and x A ∈ A ⋆ d A so that ( d A , x A ) is a generic p air for the fu nctor A ⋆ ( − ), and d B : ( n B ] → O and x B ∈ B ⋆ d B so that ( d B , x B ) is a generic pair for the f unctor B ⋆ ( − ). Thus there is a morphism ( n B ] ( n A ] ✲ u O d B ❅ ❅ ❅ ❘ d A    ✠ in S et /O , suc h that B ⋆ u ( x B ) = ξ d A ( x A ). Sin ce ξ is w eakly cartesian, the square A ⋆ d A B ⋆ d A ✲ ξ d A A ⋆ d B B ⋆ d B ✲ ξ d B ❄ A ⋆ u ❄ B ⋆ u is a weak pullbac k and there is x ∈ A ⋆ d B , suc h that ξ d B ( x ) = x B , and A ⋆ u ( x ) = x A . Since x A is generic for A ⋆ ( − ) there is a morph ism ( n A ] ( n B ] ✲ v O d A ❅ ❅ ❅ ❘ d B    ✠ in S et /O , such that A ⋆ v ( x A ) = x . Th u s A ⋆ ( u ◦ v )( x A ) = x A and u ◦ v is iso. By naturalit y of ξ B ⋆ v ( ξ d A ( x A )) = x B and B ⋆ ( v ◦ u )( x B ) = x B and v ◦ u is iso. Th erefore, b oth u and v are bijections, and n A = n B = n . Thus we can assume that u is an identit y , d A = d B = d and x B = ξ d ( x A ). Moreo v er, b y Lemma 7.10, w e can assume that x A = [ a, 1 ( n ] ] ∼ for some a ∈ A , n = | a | and d = ∂ A, + a . F urthermore, again b y Lemma 7.10, we can assume that ξ d ( x A ) = x B = [ b, ~ x ] ∼ for some b ∈ B and a bijection ~ x : ( n ] → ( n ] suc h that ∂ B , + b = ∂ A, + a ◦ ~ x . Th us ∂ B , + b · ( ~ x ) − 1 = ∂ B , + b ◦ ( ~ x ) − 1 = ∂ A, + a . Hence the asso ciatio n a 7→ b · ( ~ x ) − 1 extends to a morphism of symmetric signatures ( f , 1 O ) : ( A, α, ∂ A , O ) → ( B , β , ∂ B , O ) suc h that f ( a · σ ) = ( b · ( ~ x ) − 1 ) · σ . W e s hall show that r ep s ( f , 1 O ) = ξ . First note that r ep s ( f , 1 O ) d ( x A ) = r ep s ( f , 1 O )([ a, 1 ( n ] ] ∼ ) = [ f ( a ) , 1 ( n ] ] ∼ = = [ b · ( ~ x ) − 1 , 1 ( n ] ] ∼ = [ b, ~ x ] ∼ = x B = ξ d ( x A ) i.e. r ep s ( f , 1 O ) and ξ agree on x A . No w let d X : X → O and x ∈ A ⋆ d X b e arbitrary . Since x A is generic we h a v e a morphism w : d → d X suc h that A ⋆ w ( x A ) = x . No w using the naturalit y of ξ and r ep s ( f , 1 O ) on w , i.e. serial comm u tat ivit y of the diagram 74 A ⋆ d X B ⋆ d X A ⋆ d B ⋆ d ❄ A ⋆ w ❄ B ⋆ w ✲ ξ d X ✲ r ep s ( f , 1 O ) d X ✲ ξ d ✲ r ep s ( f , 1 O ) d w e get that r ep s ( f , 1 O ) d X ( x ) = ξ d X ( x ) and hence r ep s ( f , 1 O ) = ξ . If ( g , 1 O ) : ( A, α, ∂ A , O ) → ( B , β , ∂ B , O ) is another morphism of symmetric signatures suc h th a t r ep s ( g , 1 O ) = ξ , then in particular [ g ( a ) , 1 ( n ] ] ∼ = r ep s ( g , 1 O ) d ( x A ) = ξ d ( x A ) = r ep s ( f , 1 O ) d ( x A ) = [ f ( a ) , 1 ( n ] ] ∼ This im plies that g ( a ) = f ( a ) and, since ( A, α ) has one orbit, ( g, 1 O ) = ( f , 1 O ), a s required. ✷ Pr o of of The or em 7.5. F rom Lemm a 7.8 w e kno w that the ob jects in the essentia l image of the represen tations r ep s are ﬁnitary functors t hat w eakly preserv e wide pullbac ks. F rom Lemma 7.11 w e kno w that morph isms in the essen tial image are w eakly cartesian natural transformations. Let ξ : A ⋆ ( − ) − → B ⋆ ( − ) b e a morp hism in E xp ( S et ) ov er u : O → Q which is a weakly cartesian natural transformation. By Pr oposition 7.4 r ep s is a prone morphism s of ﬁbrations. Hence ξ can b e factored in E x p ( S et ), in an essentiall y unique wa y , via a prone morphism r ep s ( pr u,B ) : u ∗ ( B ) ⋆ ( − ) → B ⋆ ( − ) in p exp and v er ti cal morphisms ξ ′ : A ⋆ ( − ) → u ∗ ( B ) ⋆ ( − ) in the ﬁbre o v er O , so that ξ = r ep s ( pr u,B ) ◦ ξ ′ . F rom Prop osition 4.3 b oth morp hisms pr u,B ⋆ ( − ) , ξ ′ are wea kly cartesian. As again b y Lemma 7.12 r ep s is faithful and full on w eakly cartesian arro ws in ﬁbres, we obtain that r ep s is faithful and full on w eakly cartesian arro w s in the whole ﬁbration. ✷ 7.4 The analytic diagrams vs analytic functors In section 6, w e ha v e sho wn that the concepts of amalgamated signature, p olynomial d ia- gram and p olynomial fu nctor are equiv alen t wh en organized int o lax monoidal ﬁbrations. The signatures are the m ost exp lic it and the functors are the most abstr ac t among these concepts. The diagrams constitute a useful and imp ortant link b etw een them. In section 7.2, we ha ve describ ed the direct connection b et w een lax monoidal ﬁ brations of symmetric signatures and of analytic f unctors. W e pro vid e h ere the missin g link in this app roac h, the analytic diagrams. T hey corresp ond to analytic functors in m uc h the same wa y as p olynomial diagrams corresp ond to p olynomial fun cto rs. In fact, these represent ing d i- agrams will constitute a fu ll sub category of the category of p olynomial d iag rams in the catego ry of symmetric sets σ S et . Ho w ev er the mon oi dal structure is not inherited from p poly ,σS et : P ol y D iag ( σ S et ) → σ S et . In th e remaining part of the pap er we wan t to indi- cate the r ele v ant deﬁn itio ns and the ob vious facts lea ving a more comprehensive study of the analytic diagrams to another pap er. Recall that the diagonal functor δ : S et → σ S et indu ced b y the unique functor S ∗ → 1, has b oth adjoin ts or b ⊣ δ ⊣ f ix . This adjunction can b e also sliced. F or details concerning all such functors that w e sh all consider in the follo wing see the Ap pen dix. By a pseudo-analytic diagr am (over a set O ) in S et w e mean a diagram in σ S et ( E , ε ) ( A, α ) ✲ p δ ( O ) ✛ s δ ( O ) ✲ t suc h th a t the ﬁbr es of p are ﬁnite. The ob ject O is an ob ject of t yp es of the p olynomial ( t, p, s ). A morphism of pseudo- analytic diagr ams (over a function u : O → Q ) is a triple ( f , g , u ), w ith f and g morphisms in σ S et , making the diagram 75 ( E ′ , ε ′ ) ( A ′ , α ′ ) ✲ p ′ ( E , ε ) ( A, α ) ✲ p ❄ g ❄ f δ ( Q ) ✛ s ′ δ ( O ) ✛ s ❄ δ ( u ) δ ( Q ) ✲ t ′ δ ( O ) ✲ t ❄ δ ( u ) comm ute and the square in the midd le is a pullb ac k . Morphisms of pseudo-analytic dia- grams comp ose in the obvio us wa y , b y pu tting one on top of the other. Let P A n D iag denotes the categ ory of pseud o-a nalytic diagrams and morphisms b e- t wee n them. The categ ory of analytic diagrams A n D iag is th e slice P A n D iag o ve r the pseudo-analytic diagram R δ (1) ( R, ρ ) ✛ δ (1) ✲ δ (1) ✲ where R n = { n } × ( n ] with action ρ n ( h n, i i , τ ) = h n, τ − 1 ( i ) i for τ ∈ S n . As δ (1) is the terminal sym metric set all the morphism s in the ab o ve diagram are uniqu ely d ete rmined. Th us an analytic diagram is a pseudo-analytic d iag ram ( R, ρ ) δ (1) ✲ ( E , ε ) ( A, α ) ✲ p ❄ ❄ δ ( O ) ✛ δ ( O ) ✲ so that p is a pullback of ( R, ρ ) − → δ (1) along ( A, α ) − → δ (1). Thus the ﬁ bre of p o ve r a ∈ A can and will b e identiﬁed as { a } × ( | a | ] with the acti on of S | a | so that h a, i i · τ = h a · τ , τ − 1 ( i ) i , for τ ∈ S | a | . With suc h an id en tiﬁcation if ( f , g , u ) : ( t, p , s ) → ( t ′ , p ′ , s ′ ) is a morphism of analytic diagrams then g ( a, i ) = h f ( a ) , i i , f or h a, i i ∈ E . Hence we shall not sp ecify g in the morp hism of an a lytic diagrams an ymore and we s hall denote it when necessary as ¯ f . W e ha ve an obvious p ro jection fu nctor p and : A n D iag − → S et, sending ( f , u ) to u , wh ic h is a lax m onoi dal ﬁ bration. Ho wev er the tensor in ﬁbres is not the one indu ced by the tensor of those diagrams as if th ey were p olynomial d ia grams in P oly D iag ( σ S et ) → σ S et . Its will b e d escribed b elo w in an indir ec t w ay . As in the case of th e p olynomial diagram ﬁbration, the ﬁbr at ion of analytic dia- grams comes equipp ed with a rep resen tation morph ism int o the exp onen tial ﬁbration E xp ( S et ) → S et . The r epresen tation fun cto r A n D iag E xp ( S et ) ✲ r ep and S et p and ❅ ❅ ❅ ❅ ❘     ✠ is d eﬁned as follo ws. F or an analytic d iag ram ( t, p, s ) o ver O as disp la y ed ab o v e, w e deﬁne a functor r ep and ( t, p, s ) from S et /O to S et /O as the comp osition of ﬁve fun cto rs S et /O ✲ δ /O σ S et /δ ( O ) ✲ s ∗ σ S et / ( E ,ε ) ✲ p ∗ σ S et / ( A,α ) ✲ t ! σ S et /δ ( O ) S et /O ✲ or b /O i.e. we tak e a d iag onal f unctor δ /O to mo ve from the ’set con text’ to th e ’symm etric set con text’, then w e app ly the usu al p olynomial fun cto r o v er the category of symmetric sets 76 and w e come back again to the set con text via or b /O , by taking orb its of wh at ev er we collect ed on th e wa y . W e describ e it with more details in three steps. F or an ob ject ( X , d X : X → O ) in S et /O the d omai n of p ∗ s ∗ ( δ /O )( X, d X ) in σ S et / ( A,α ) is th e set {h a, ~ x i : a ∈ A, ~ x : p − 1 ( a ) → X , d X ◦ ~ x = π O ◦ s ⌈ p − 1 ( a ) } where π O : δ ( O ) = ω × O → O is the obvious p ro jection, equ ipp ed with an action ξ acting b y conjugation, i.e. for σ ∈ S | a | ξ ( h a, ~ x i , σ ) = h a, ~ x i · σ = h a · σ, ~ x (( − ) · σ − 1 ) · σ i The t yping fu nctio n send s h a, ~ x i to a ∈ A . Th e functor t ! c hanges only th e t yping, i.e. t ! p ∗ s ∗ ( δ /O )( X, d X ) in σ S et /δ ( O ) has t ypin g sendin g h a, ~ x i to h| a | , t ( a ) i ∈ δ ( O ). Finally , or b /O asso cia tes the orbits to w hat we’v e got so far, i.e. the domain of ( or b /O ) t ! p ∗ s ∗ ( δ /O )( X, d X ) in S et /O is the set of equ iv alence classes [ a, ~ x ] ∼ of pairs h a, ~ x i as ab o v e, d ivided by the action ξ . The v alue of this morphism on the class [ a, ~ x ] ∼ is sen t to π O t ( a ) ∈ O . F or a morphism of analytic diagrams ( f , u ) : ( t, p, s ) − → ( t ′ , p ′ , s ′ ) o ver u as deﬁned ab o v e, w e d eﬁne a morphism in E xp ( S et ) o ver u , i.e. a natural transformation r ep and ( f , u ) : ( or b /O ) t ! p ∗ s ∗ ( δ /O ) u ∗ − → u ∗ ( or b /Q ) t ′ ! p ′ ∗ s ′∗ ( δ /Q ) using the diagram S et /O ✲ δ /O σ S et /δ ( O ) ✲ s ∗ σ S et / ( E ,ε ) ✲ p ∗ σ S et / ( A,α ) ✲ t ! σ S et /δ ( O ) S et /O ✲ or b /O ✻ u ∗ ✻ δ ( u ) ∗ ✻ ¯ f ∗ ✻ f ∗ ❄ f ! ❄ δ ( u ) ! ❄ u ! ✻ u ∗ S et /Q ✲ δ /Q σ S et /δ ( Q ) ✲ s ′∗ σ S et / ( E ′ ,ε ′ ) ✲ p ′ ∗ σ S et / ( A ′ ,α ′ ) ✲ t ′ ! σ S et /δ ( Q ) S et /Q ✲ or b /Q ε f ⇓ as follo ws. By adjunction u ! ⊣ u ∗ it is enough to deﬁne a natural trans formatio n b et we en these functors u ! ( or b /O ) t ! p ∗ s ∗ ( δ /O ) u ∗ − → ( or b /Q ) t ′ ! p ′ ∗ s ′∗ ( δ /Q ) and using the comm utativit y of some squ ares (including Bec k-Chev alley condition) we deﬁne a natural transformation b et wee n fun cto rs isomorphic to th ose ab o ve ( or b /Q ) t ′ ! ( ε f ) p ′ ∗ s ′∗ ( δ /Q ) : ( or b /Q ) t ′ ! f ! f ∗ p ′ ∗ s ′∗ ( δ /Q ) − → ( or b /Q ) t ′ ! p ′ ∗ s ′∗ ( δ /Q ) T racing this d eﬁnition bac k through the adj unctions we ﬁnd that the so deﬁned natural transformation r ep and ( f , g , u ) app lie d to an ob ject ( Y , d Y : Y → Q ) in S et /Q sends the elemen t [ a, ~ y : p − 1 ( a ) → u ∗ ( Y )] ∼ in the domain of ( or b /O ) t ! p ∗ s ∗ ( δ /O ) u ∗ ( Y , d Y ) to the element h t ( a ) , [ f ( a ) , u Y ◦ ~ y ] ∼ i in the domain of u ∗ ( or b /Q ) t ′ ! p ′ ∗ s ′∗ ( δ /Q )( Y , d Y ), where the notation is as in the follo wing diagram 77 Q O ✛ u Y u ∗ ( Y ) ✛ u Y ❄ d Y ❄ u ∗ ( d Y ) ω × O ✛ π O p − 1 ( a ) ⊆ E ✛ s ⌈ p − 1 ( a ) ❍ ❍ ❍ ~ y ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ② W e lea v e for the reader the ve riﬁcation that the so deﬁned r ep and is a lax morph ism of ﬁbrations. In ord er to sho w that the essential image of r ep and is the ﬁ bration of analytic fu nctors, w e s hall deﬁn e ﬁ rst a morph ism (in fact an equiv alence) of ﬁbrations S ig s A n D iag ✲ ι s S et p s ❅ ❅ ❅ ❅ ❘ p and     ✠ T o an ob ject ( A, α, ∂ A , O ) in S ig s , ι s assigns an analytic diagram as f ol lo ws δ ( O ) ✛ s A ( E A , α ) ( A, α ) ✲ p A δ ( O ) ✲ t A where E A = {h a, i i : a ∈ A, i ∈ ( | a | ] } and α ( h a, i i , σ ) = h a, i i · σ = h a · σ , · σ − 1 ( i ) i Moreo v er s A ( a, i ) = h| a | , ∂ A a ( i ) i , p A ( a, i ) = a, t A ( a ) = h| a | , ∂ A a (0) i for h a, i i ∈ E A , a ∈ A . If ( f , u ) : ( A, α, ∂ A , O ) − → ( A ′ , α ′ , ∂ A ′ , Q ) is a morph ism in S ig s o ve r u : O → Q , then ι s assigns to it the f ol lo wing morphism of d ia grams ( E ′ A ′ , α ′ ) ( A ′ , α ′ ) ✲ p ′ A ′ ( E A , α ) ( A, α ) ✲ p A ❄ ¯ f ❄ f δ ( Q ) ✛ s ′ A ′ δ ( O ) ✛ s A ❄ δ ( u ) δ ( Q ) ✲ t ′ A ′ δ ( O ) ✲ t A ❄ δ ( u ) so that ¯ f ( a, i ) = h f ( a ) , i i f or h f ( a ) , i i ∈ E A , as b efore. Prop osition 7.13 The asso ciation ι s is an e quiv al enc e of ﬁbr ations. Pr o of. ι s is in fact an isomorphism if w e restrict only to those analytic diagrams that ( E , ε ) ( A, α ) ✲ p δ ( O ) ✛ s δ ( O ) ✲ t for whic h ( E , ε ) is id en tiﬁed with ( A, α ) × δ ( O ) ( R, ρ ). T h us it is an equiv alence in deed. ✷ Prop osition 7.14 The fol lowing triangle of morphisms of ﬁbr ations S ig s A n D iag ✲ ι s E xp ( S et ) r ep s ❅ ❅ ❅ ❘ r ep and    ✠ 78 c ommutes up to an isomorph ism. Pr o of. All the necessary items were d eﬁned. W e shall chec k that the v alues of b oth functors on ob jects (that are fun cto r s on slices of S et ) agree on ob jects. The remaining details are left for the reader. Let A = ( A, α, ∂ A , O ) b e a symmetric signature and X = ( X , d X ) an ob j ec t in S et /O . Both v alues r ep s ( A )( X ) and r ep and ◦ ι s ( A )( X ) are fun cti ons in to the set O whose d omains are (= can b e identi ﬁed) with the set {h a, ~ x i : a ∈ A, ~ x : ( | a | ] → X , d X ◦ ~ x = ∂ A, + a } divided by an equiv alence relation. In the f o rmer case th e relatio n identiﬁes the pair h a, ~ x i with h a · σ, ~ x ◦ σ i for σ ∈ S | a | . In the second case the relation identiﬁes the element s of the same orbit of the action s uc h that for σ ∈ S | a | h a, ~ x i · σ = h a · σ, ~ x (( − ) · σ − 1 ) · σ i F or i ∈ ( | a | ], we ha v e ( ~ x (( − ) · σ − 1 ) · σ )( i ) = ~ x ( i · σ − 1 ) · σ = ~ x ( σ ( i )) · σ = ~ x ◦ σ ( i ) The last equ ali t y follo ws from the fact that the actio n in X (in fact δ ( X )) is constan t. Th us these equiv alence r el ations are the same and hence the whole m orphisms in to O sending the equiv alence class of [ a, ~ x ] ∼ to ∂ A, + a (0) are the same. ✷ As ι s is an equiv alence of ﬁbr at ions b y Pr oposition 7.13 and the essentia l image of r ep s is (by deﬁnition) the ﬁbration of analytic fu nctors, w e get f rom the ab o v e Pr oposition 7.14 Corollary 7.15 The essential image of the r epr esentation functor A n D iag E xp ( S et ) ✲ r ep and S et p and ❅ ❅ ❅ ❘    ✠ is the ﬁbr ation of analytic functors. 7.5 Comparing the p olynom ial and the analytic approac hes In Section 6 w e ha ve sho w n that the lax monoidal ﬁbrations of amalgamated signatures, p olynomial diagrams and p olynomial fun cto r s are equiv alen t. In the pr evio us Su bsectio n 7.4 we ha ve intro duced the n oti on of an analytic diagram an d we h a v e sh o wn that th e lax monoidal ﬁb rations of symmetric signatures, analytic diagrams, and analytic f unctors are equiv alen t. Thus in eac h case, we ha ve three diﬀerent w a ys of p resen ting essen tially the same notion. Belo w we compare these notion at all three leve ls, i.e. we s hall deﬁne the missing fun ct ors K sig , K diag and natural transformations Φ, Ψ in the follo wing diagram 79 S ig a S ig s ✲ K sig ❄ ι a ❄ ι s P oly D iag A n D iag ✲ K diag ❄ r ep pd ❄ r ep and P oly A n ✲ K f u ❄ ❄ C ar t ( S et ) wC ar t ( S et ) ✲ E xp ( S et ) ❅ ❅ ❅ ❘    ✠ Ψ ⇒ Φ ⇒ All the arro ws are morphisms of lax m onoidal ﬁbr at ions and of biﬁbrations o ver S et . The four named vertic al arro ws are equiv alence of lax m onoi dal ﬁb rations. Th e ﬁve un named arro ws are inclusions. The thr ee named horizon tal arro ws are morphisms comparing sig- natures, diagrams, and functors, resp ectiv ely . So w e b egin by describin g the fu nctor S ig a S ig s ✲ K sig S et p a ❅ ❅ ❅ ❘ p s    ✠ The morph ism ( f , σ, u ) : ( A, ∂ A : A → O † , Q ) − → ( B , ∂ B , Q ) in S ig a o ve r u is sent to a morphism ( s ( f , σ ) , u ) : ( s ( A ) , α, ∂ s ( A ) : s ( A ) → O ‡ , O ) − → ( s ( B ) , β , ∂ s ( B ) , Q ) so that s ( A ) = {h a, τ i : n ∈ ω , a ∈ A n , τ ∈ S n } and, for h a, τ i ∈ s ( A ) ∂ s ( A ) ( a, τ ) = ∂ A a ◦ τ : [ | a | ] → O , s ( f , σ )( a, τ ) = h f ( a ) , σ − 1 a ◦ τ i W e ha ve, for h a, τ i ∈ s ( A ) , ∂ s ( B ) ◦ s ( f , σ )( a, τ ) = ∂ s ( B ) ( f ( a ) , σ − 1 a ◦ τ ) = = ∂ B f ( b ) ◦ σ − 1 a ◦ τ = u ◦ ∂ A a ◦ τ = u ‡ ◦ ∂ A ( a, τ ) i.e. th e square O ‡ Q ‡ ✲ u ‡ ( s ( A ) , α ) ( s ( B ) , β ) ✲ s ( f , σ ) ❄ ∂ s ( A ) ❄ ∂ s ( B ) comm utes and K sig is a well deﬁ ned functor. W e note for the record Prop osition 7.16 The func to r K sig is fu l l, faithful, and its essential image c onsists of those symmetric signatur es that have fr e e actions. 80 Pr o of. S imple c h ec k. ✷ Next we deﬁne the functor P oly D iag A n D iag ✲ K diag S et p pd ❅ ❅ ❅ ❘ p and    ✠ T o a p olynomial diagram O E ✛ s A ✲ p O ✲ t K diag asso cia tes an analytic diagram ( ˜ E , ˜ ε ) ( ˜ A, ˜ α ) ✲ ˜ p δ ( O ) ✛ ˜ s δ ( O ) ✲ ˜ t so that ˜ A = {h a, h i : a ∈ A, h : ( | a | ] ∼ = − → p − 1 ( a ) } ˜ E = {h a, h, i i : a ∈ A, h : ( | a | ] ∼ = − → p − 1 ( a ) , i ∈ ( | a | ] } where | a | is th e num b er of elements of p − 1 ( a ). F or h a, h i ∈ ˜ A w e hav e ˜ α ( h a, h i , τ ) = h a, h ◦ τ i , ˜ t ( a, h ) = h| a | , t ( a ) i and for h a, h, i i ∈ ˜ E , ˜ ε ( h a, h, i i , τ ) = h a, h ◦ τ , τ − 1 ( i ) i , ˜ p ( a, h, i ) = h a, h i , ˜ s ( a, h, i ) = h| a | , s ◦ h ( i ) i . T o a morphism of p olynomial diagrams E ′ A ′ ✲ p ′ E A ✲ p ❄ g ❄ f Q ✛ s ′ O ✛ s ❄ u Q ✲ t ′ O ✲ t ❄ u K diag asso cia tes a morph ism of analytic diagrams δ ( O ) ✛ ˜ s ′ δ ( O ) ✛ ˜ s ❄ δ ( u ) ( ˜ E ′ , ˜ ε ′ ) ( ˜ A ′ , ˜ α ′ ) ✲ ˜ p ′ ( ˜ E , ˜ ε ) ( ˜ A, ˜ α ) ✲ ˜ p ❄ ˜ g ❄ ˜ f δ ( Q ) ✲ ˜ t ′ δ ( O ) ✲ ˜ t ❄ δ ( u ) so that, for h a, h i ∈ ˜ A and h a, h, i i ∈ ˜ E ˜ f ( a, h ) = h f ( a ) , g ⌈ p − 1 ( a ) ◦ h i , ˜ g ( a, h, i ) = h f ( a ) , g ⌈ p − 1 ( a ) ◦ h, i i . This ends the deﬁnition of K diag . No w w e shall d eﬁne the natural isomorph ism Φ. Fix ( A, ∂ A , O ) in S ig a . W e need to deﬁne a morphism Φ ( A,∂ A ,O ) : K diag ◦ ι a ( A, ∂ A , O ) − → ι s ◦ K sig ( A, ∂ A , O ) in A n D iag in the ﬁ bre o ve r O , i.e. a m orphism of analytic diagrams 81 δ ( O ) ✛ s s ( A ) δ ( O ) ✛ f s A ❄ δ (1 O ) ( E s ( A ) , α ) ( s ( A ) , α ) ✲ p s ( A ) ( g E A , e ε ) ( ˜ A, ˜ α ) ✲ f p A ❄ Φ 1 ❄ Φ 0 δ ( O ) ✲ t s ( A ) δ ( O ) ✲ f t A ❄ δ (1 O ) K diag ◦ ι a ( A, ∂ A , O ) = ι s ◦ K sig ( A, ∂ A , O ) = ❄ Φ ( A,∂ A ,O ) An element of e A is a pair h a, h i such that a ∈ A and h : ( | a | ] → p − 1 ( a ) = {h a, i i : i ∈ ( | a | ] } is a b iject ion. An element of ( s ( A ) is a pair h a, τ i so that a ∈ A and τ ∈ S | a | . T h us we can put Φ 0 ( a, h ) = h a, π 2 ◦ h i with π 2 ( a, i ) = i , for i ∈ ( | a | ]. Clearly , Φ 0 is a bijection. An elemen t of g E A is a triple h a, h, i i so that h a, h i ∈ e A and i ∈ ( | a | ]. An elemen t of E s ( A ) is a triple h a, τ , i i su c h that h a, τ i ∈ s ( A ) and i ∈ ( | a | ]. Clearly , we put Φ 1 ( a, h ) = h a, π 2 ◦ h, i i , and Φ 1 is a b ijec tion as w ell. W e ha ve Prop osition 7.17 The tr ansformatio n Φ : K diag ◦ ι a → ι s ◦ K sig deﬁne d ab ove is a natur al isomorph ism. Pr o of. W e ha ve already seen that the comp onents of Φ are isomorphisms. The v eriﬁcation that Φ is natural is left f o r the r ea der. ✷ Finally , w e d eﬁne the natural isomorphism Ψ. W e ﬁx a p olynomial diagram O E ✛ s A ✲ p O ✲ t in P oly D iag O . W e need to deﬁne a morphism Ψ ( t,p,s ) : K f u ◦ r ep pd ( t, p, s ) − → r ep and ◦ K diag ( t, p, s ) in A n in the ﬁ bre o ver O , i.e. a n at ural transformation Ψ ( t,p,s ) : t ! p ∗ s ∗ − → ( or b /O ) ˜ t ! ˜ p ∗ ˜ s ∗ ( δ /O ) b et we en endofun ct ors on S et /O . T o this end, we need to deﬁne its comp onen ts (Ψ ( t,p,s ) ) ( X,d X ) : t ! p ∗ s ∗ ( X, d X ) − → ( or b /O ) ˜ t ! ˜ p ∗ ˜ s ∗ ( δ /O )( X, d X ) for any ob ject ( X , d X : X → O ) in S et /O . Fix ( X , d X ) in S et /O . An elemen t of t ! p ∗ s ∗ ( X, d X ) is a pair h a, ~ x i so that a ∈ A , ~ x : ( | a | ] → X is a fun ctio n such that d X ◦ ~ x = s ⌈ p − 1 ( a ) . An elemen t [ h a, h, ~ x i ] ∼ of ( or b /O ) ˜ t ! ˜ p ∗ ˜ s ∗ ( δ /O )( X, d X ) is an equiv alence class of triples h a, h, ~ x i s o that h a, ~ x i is an elemen t of t ! p ∗ s ∗ ( X, d X ) and h : ( | a | ] → p − 1 ( a ) is a b ijec tion. The action of S | a | is deﬁned so that h a, h, ~ x i · τ = h a, h ◦ τ , ~ x i . Th u s an y tw o triples h a, h, ~ x i and h a ′ , h ′ , ~ x ′ i are identiﬁed if and only if a = a ′ and ~ x = ~ x ′ . Thus we can put (Ψ ( t,p,s ) ) ( X,d X ) ( a, ~ x ) = {h a, h, ~ x i : h : ( | a | ] ∼ = − → p − 1 ( a ) } i.e. we asso ci ate to h a, ~ x i the equiv alence class of all triples wh ose second comp onen t is a bijection of ( | a | ] and p − 1 ( a ). As these sets ha ve, by d eﬁnition, the same n u m b er of elemen ts, Ψ is w ell deﬁn ed. W e ha ve 82 Prop osition 7.18 The tr ansformation Ψ : K f u ◦ r ep pd − → r ep and ◦ K diag deﬁne d ab ove is a natur al isomorphism. Pr o of. As b efore, the veriﬁcatio n th a t Ψ is natural is left for the r ea der. F rom the considerations ab o ve it sh ould b e clear that for any p olynomial diagram ( t, p, s ) and ( X, d X ) in S et /O (Ψ ( t,p,s ) ) ( X,d X ) is a bijection. S o (Ψ ( t,p,s ) ) is a natural isomorph ism and hence Ψ is an isomorphism, as w ell. ✷ In that wa y w e ha ve completed the d escription of the diagram of categories, fun ct ors and natur al transform at ions fr om the b eginning of this sub secti on. Thus, w e kno w that the wh ole diagram comm utes (at least u p to an equiv alence), m oreo ver th e named hori- zon tal fun ct ors are equiv alences of categories. As we note, Prop osition 7.16, K sig is fu ll and faithful. Th erefore b oth K diag and K f u are full and faithful, as w ell. Hence using the c h arac terizatio ns of ﬁbrations of p olynomial and analytic functors, Prop osition 6.12, Theorem 7.5, w e ob ta in a statemen t, a bit su rprising at ﬁrst sight . Corollary 7.19 Any we akly c artesian natur al tr ansformation b etwe en p olynomial func- tors is c artesian. 8 App endix W e sp ell b elo w in detail some w ell known deﬁnitions of v arious adjoin t functors b et we en slices of S et and σ S et . First, recall that the u nique functor S ∗ → 1 indu ce s by comp ositio n the diagonal functor δ that has b oth adjoints S et σS et ✲ δ ✛ f ix ✛ or b or b ⊣ δ ⊣ f ix . The f unctor δ sends set X to ω × X , i.e. to ω copies of X , with n -th copy of X equipp ed with a trivial actio n of S n . The f unctor or b send s a s ymmetric set ( A, α ) to the set of its orb its with resp ect to all actions A /α . The functor f ix sends a symm et ric set ( A, α ) to the pro duct o v er ω of the sets of ﬁx p oin ts with resp ect to eac h action S n , i.e. f ix ( A, α ) = Y n ∈ ω f ix n ( A n , α n ) where f ix n ( A n , α n ) = { a ∈ A n : a · σ = a for σ ∈ S n } . The fu ncto r f ix is n ot us ed directly but its existence shows that δ preserves colimits. F or an y set O the ab o ve ad junction can b e sliced, i.e. w e ha ve functors S et /O σ S et /δ ( O ) ✲ δ /O ✛ f ix /O ✛ or b /O suc h th a t or b /O ⊣ δ /O ⊣ f ix /O . F or d X : X → O in S et /O δ /O ( X, d X ) = δ ( d X ) : δ ( X ) → δ ( O ) is the sliced diagonal functor. Moreo ver, for d Y : ( Y , ζ ) → δ ( O ) in σ S et /δ ( O ) , w e ha ve or b /O (( Y , ζ ) , d Y ) : or b ( Y , ζ ) − → O 83 so that or b /O (( Y , ζ ) , d Y )([ y ] ∼ ) = o if d Y ( y ) = h n, o i for some n ∈ ω . Finally , for (( Y , ζ ) , d Y ) as ab o v e f ix /O (( Y , ζ ) , d Y ) : a o ∈ O Y n ∈ ω f ix n,o (( Y , ζ ) , d Y ) − → O is the ob vious p ro jection function, w here f ix n,o (( Y , ζ ) , d Y ) = { y ∈ Y n : d Y ( y ) = o, y · σ = y for σ ∈ S n } Next w e recall the pullbac k functor an d its adjoint in the category of symmetric sets σ S et . An y m orphism p : ( E , ε ) → ( A, α ) in σ S et ind uces three fun cto r s σ S et / ( E ,ε ) σ S et / ( A,α ) ✛ p ∗ ✲ p ∗ ✲ p ! so th at p ! ⊣ p ∗ ⊣ p ∗ . p ∗ is d eﬁned b y pulling b ac k along p , p ! is d eﬁned by comp osing w ith p . T he actions are deﬁned in the obvi ous w a y . F or d X : ( X, ξ ) → ( E , ε ) the u niv erse of p ∗ (( X, ξ ) , d X ) is {h a, ~ x : E a → ( X , ξ ) i : a ∈ A, d X ◦ ~ x = i a } where E a and i a are deﬁned from the follo wing p ullbac k in σ S et E a S | a | ✲ ( E , ε ) ( A, α ) ✲ p ✻ i a ✻ ¯ a and ¯ a : S | a | − → ( A, α ) is the m orphism fr om the symmetric set S | a | (with action of S | a | on the righ t) sending iden tit y on ( | a | ] to a . The action in p ∗ (( X, ξ ) , d X ) is d eﬁned by conjugation h a, ~ x i · σ = h a · σ, ~ x (( − ) · σ − 1 ) · σ i and the t yp ing sends h a, ~ x i to a . Th us w e can d ra w a diagram of categories and f unctors S et /O S et /Q ✛ u ∗ σ S et /δ ( O ) σ S et /δ ( Q ) ✛ δ ( u ) ∗ ✻ δ /O ✻ δ /Q ❄ or b /O ❄ f ix /Q ✲ u ∗ ✲ δ ( u ) ! ❄ f ix /O ❄ or b /Q ✲ δ ( u ) ∗ ✲ u ! in whic h we hav e a natural isomorphism of fun ct ors δ ( u ) ∗ ◦ δ /Q ∼ = δ /O ◦ u ∗ and hence of their left or b /Q ◦ δ ( u ) ! ∼ = u ! ◦ or b /O and righ t adjoin ts f ix /Q ◦ δ ( u ) ! ∼ = u ! ◦ f ix /O 84 References [A] J. Adamek, F r e e algebr as and automata r e alizations in the language of c ate gories . Commen tationes Math. Univ ersitatis Carolinae, V ol. 1 5, 19 74, No. 4, pp. 589-602 (a v aiable on line: h ttp://dml.cz/dmlcz/1055 83). [A V] J. Adamek, J. V elebil, Analy tic fu nctor s and we ak pul lb acks . Th eo ry and Ap pli- cations of Categories, V ol. 21, 2008, No. 11, pp . 191-20 9. [BD] J. Baez, J. Dolan, Higher-dimensional algebr a III: n-Cate gories and the algebr a of op etop es . Ad v ances in Math. 135 1998, pp. 145-206. [BJT] H.J. Baues, M. Jibladze, A.T onks, Cohomology of monoids in monoidal cate- gories, in Op erads: pro ceedings of renaissance conferences (Hartford/Luminy , 1995) , 137-1 65, Contemp. Math. 202, Amer. Math. S oc., Pro vidence, RI, 1997. [Bo] D. Bourn, La tour de ﬁb ratio ns exactes des n -c at ´ egories, Cahiers de T op et Geom Diﬀ. Categoriques, tome 21, no 2, 1980, pp. 161-189. [B] A. Burr oni, T-catego ries (cat ´ egories dans un triple), Cahiers de T op et Geom Diﬀ. Categoriques, tome 12, no 3, 1971, pp. 215-321. [CJ] A. Carb oni, P .T. John stone, Connected limits, F amilial representabilit y and Artin glueing, Math. Structures in Comp. Sci. 5, 1995. p p. 441-459. [GK] N. Gambino, J. Ko c k, P olynomial functors and p olynomial monads, arXiv:0906 .4931v1 [math.CT], 2009. [G] J. Giraud, M ´ eto de de la descen t. M ´ emoires de la S oc. Math. d e F r ance M ´ em. 2 1964. [GG] M.F. Gouzou, R. Gru nig, Fibrations relativ e. Seminaire d e Theorie des Cate- gories d irige par J. Benab ou, No ve m b re 1976. [HMP] C. Hermida, M. Makk ai, J. P ow er, On we ak higher dimensional c ate gories I P arts 1,2,3, J. Pure an d Applied Alg. vol. 153, 2000, pp. 221-2 46, vol 157, 2001, p p. 247-2 77, v ol 166, 2002, pp. 83-104. [HMZ] V. Harnik, M. Makk ai, M. Z a w adowski, Computads and M ultitop i c Sets , arXiv:0811 .3215v1 [math.CT] [Ja] B. J ac obs, Categorical Logic and Type Theory , Studies in Logic and F ound . of Math, v ol 141, Elsevier 1999. [J1] A. Jo y al, Une theorie com binatoire des series formelles, Adv an ce s in Math. 42 1981, p p. 1-82. [J2] A. Jo y al, F oncteurs analytiques et esp´ eces d e structures, Lecture Notes Math. 1234, S pringer 1986, 126-159. [Ke] G.M. Kelly , A uniﬁed treatmen t of transﬁnite constru cti ons for free algebras, free monoids, colimits, associated shea ves, and so on, Bull. Austral. Math. So c. v ol 22 1980, pp . 1-83. [Ko] J. K oc k, P olynomial functors and trees, arXiv:0807.2 874v2 [math.CT]. 2009. 85 [Ko] J. K oc k, A.Jo yal, M.Batanin, J-F Masca ri, Polynomial functors and op etop es, Adv ances in Math., vol 224, 2010 p p. 2690-2737. [Le] T. Leinster, Higher Op er ads, Higher Cate gories . Lo ndon Math. S oc. Lecture Note Series. Cam bridge Univ ersity Press, Cam b ridge, 2004. (math.CT/030504 9) [CWM] S. MacLane, Cate gories for the working Mathematician , Graduate T ext in Math., Springer, 2nd ed, 1998. [Sa] N. Saa v edr a-Ri v ano, Categories T annakiennes, Lecture Notes Ma th. 265, Springer 1972. [Sh] M. Shulman, F ramed Bicategories and Monoidal Fibration, Theory and Appli- cations of Categories, V ol. 20, No. 18, 2008, pp . 650-738 . [St] T. Streic her , Fib er e d Cate gories ´ a la B´ enab ou , 2005. [S] S. Szawiel , Popr awno ´ s´ c deﬁnicji zbior”ow op etop owych w r ozw l”oknieniach monoidal nych (Corr e ctness of the deﬁnition of op etopic sets in monoidal ﬁbr a- tions) , Master Thesis, W arsaw Univ er sit y , 2009, p p. 1-62. (in Po lish) [SZ] S. Sza wiel, M. Zaw ado wski, Comp aring algabr aic deﬁntions of op etopic sets . In preparation. [T] V. T rnko v´ a, Some pr op erties of set fu ncto rs , Commen t. Math. Univ. Carolin”c, 10 1969, pp. 323-352. [Z] M. Za wado wski, O n or der e d fac e structur es and many-to-one c omputads , arXiv:0708 .2659v2 [math.CT] 86

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