Regular pairings of functors and weak (co)monads

REGULAR P AIRINGS OF FUNCTORS AND W EAK (CO)MONADS ROBER T WISBA UER Abstract. F or functors L : A → B and R : B → A b et wee n any c ategories A and B , a p airing is defined by maps, natural in A ∈ A and B ∈ B , Mor B ( L ( A ) , B ) α / / Mor A ( A, R ( B )) β o o . ( L, R ) is an ad joint p air pro vided α (or β ) is a bijection. In this case t he composition RL defines a monad on the category A , LR defines a comonad on the category B , and there is a well-kno wn correspondence b et we en monads (or comonads) and adjoi n t pairs of f unctors. F or v arious applications it w as observ ed that the conditions for a unit o f a mon ad was too restrictive and wea ken ing it still allow ed for a useful generalised notion of a monad. This led to the introduction of we ak monads and we ak c omonads and the definitions needed we re made without referri ng to this kind of adjunction. The motiv ation for the presen t paper is to show that these notions can b e naturally derived fr om pairings of functors ( L, R, α, β ) with α = α · β · α and β = β · α · β . F ollowing closely the constructions kno wn for monads (and unital modules) and comonads (and counital como dules), we sho w that any wea k (co)mon ad on A gives r ise to a regular pairing b et wee n A and the category of c omp atible (c o)mo dules . MSC: 18A40, 18C20, 16T15. Keywor ds: pairing of f unctors; adjoint functors; weak (co) monads; r -unital monads; r -counital comonads; lifting of functors; distr i butiv e l a ws. Contents 1. Int ro duction 1 2. Pairings of functors 3 3. Monads and mo dules 5 4. Comonads and como dules 8 5. Ent wining monads and c o monads 11 6. Lifting of endofunctors to mo dules a nd como dules 13 7. Mixed en twinings and liftings 15 References 18 1. Introduction Similar to the unit of an algebr a, the existence of a unit of a mona d is essential for (most of ) the interesting prop erties of the related str uc tur es. Y et, there are numerous applications for which the req ue s t for a unit o f a mo nad is to o res trictive. Dropping the unit completely makes the theory fairly p o or a nd the question was how to weak en the conditions on a unit s uch that still an effective theory can be developped. The interest in these questions was re vived, for example, by the study of we ak Hopf algebr as by G. B¨ ohm et al. in [6] and we ak entwining stru ct ur es b y S. Caenep eel et al. in [9 ] (see a lso [1], [8]). T o ha ndle this s ituation the theory of weak monads and co monads was developped and we refer to [3] for a r ecent account on this theo ry . On any categor y , mona ds a re induced by a pair of adjoint functors and, o n the other hand, any monad ( F, µ, η ) induces an adjo int pair of functors, the free functor φ F : A → A F 1 2 R OBER T W I S BA UER and the forgetful functor U F : A F → A , where A F denotes the ca tgeory o f unital F - mo dules. This is all shown in Eilenberg-Mo o re [10]. In this corres po ndence the unitalit y of the monad is substantial and the purp ose of the present pa per is to ex hibit a simila r relationship b etw een weak (co )monads and g eneralised forms of adjunctions. T o this end, for functors L : A → B and R : B → A betw een categories A and B , we consider maps Mor B ( L ( A ) , B ) α / / Mor A ( A, R ( B )) β o o , required to b e natural in A ∈ A and B ∈ B . W e call this a p airing of functors , or a ful l p airing if we w ant to stre ss that we have maps in b o th directions. Suc h a pairing is s aid to b e r e gu lar pr ovided α and β ar e reg ular maps, more precisely , α = α · β · α and β = β · α · β . In Se c tion 2, reg ular pairings of functors are defined and so me of their gener al pr op erties are describ ed. Motiv ated by substructures showing up in pairings of funcoter, in Section 3.1, q -u nital monads ( F, µ, η ) on A a re defined as endofunctors F : A → A with natural tra nsformations µ : F F → F and η : I A → F ( q uasi-unit ) a nd the sole condition that µ is as so ciative. (Non-unital) F -mo dules are defined by morphisms  : F ( A ) → A s a tisfying  ◦ µ =  ◦ F  , and the ca tegory of all F -mo dules is denoted by A − → F . F or these da ta the free and forgetful functors, φ F : A → A − → F and U F : A − → F → A . give ris e to a full pa iring. F ro m this we define r e gularity of η a nd c omp atibility for the F -modules . The q -unital monad ( F, µ, η ) is said to b e r -u nital (short fo r r e gular-u nital ) provided η is regular a nd µ is compatible as an F -mo dule. Now the free functor φ F : A → A F with the forgetful functor U F : A F → A form a re gular pairing, where A F denotes the (sub)category of compa tible F -mo dules. The dual notio ns for (non-co unita l) comonads are outlined in Section 4 and at the end of the sectio n the compar ison functors for a r egular pairing ( L, R, α, β ) are cons idered (see 4.10). In Section 5 we study the lifting of functor s b etw een c a tegories to the corr esp onding categorie s of co mpatible mo dules or co mpa tible como dules, resp ectively . This is describ ed by gener alising Beck’s distribut ive laws (see [2]), also called entwinings , a nd it turns out that most o f the diag rams a re the same as for the lifting to unital mo dules (e.g . [22]) but to comp ensate the missing unitalit y extra conditions are imp osed on the ent wining natural tr ansformatio n (e.g. Prop os ition 5.2). In this co nt ext we obtain a generalis ation of Applegate’s lifting theor em for (co)monads to weak (co)mo nads (Theorem 5.4, 5.8). Lifting a n endo functor T of A to an endofunctor T of A F leads to the q uestion when T is a weak mona d ( T F allows for the structure of a weak monad) and in Section 6 we provide conditio ns to make this happ en. The final Section 7 is co ncerned w ith weak monads ( F , µ, η ) and weak comonads ( G, δ, ε ) on a ny categor y A and the interpla y b etw een the resp ective lifting pr op erties. Her eby prop erties of the lifting G to A F and the lifting b F to A G are inv estigated (see Theorems 7.9 and 7.1 0) which g eneralise obser v atio ns known for weak bi-alg ebras (and weak Hopf algebras ). In o ur setting, notions like pr e-u nits , pr e-monads , we ak monads , demi-monads , pr e- A - c orings , we ak c orings , we ak Hopf algebr as from the litera ture (e.g. [1], [3], [7], [4], [2 1]) find their natura l environment. In the framework of 2-catego ries weak structures ar e inv estigated by B¨ o hm et al. in [3], [4] and an extensive lis t of examples o f weak structures is given ther e. REGULAR P AIRINGS 3 2. P a irings of functors Throughout A and B will denote arbitrary categ ories. By I A , A or just by I , we denote the iden tity mor phism of an ob ject A ∈ A , I F or F stands for the identit y natural transformatio n on the functor F , a nd I A means the identit y functor of a category A . W e write F − , − for the natur al transforma tio n of bifunctors deter mined by the ma ps F A,A ′ : Mor A ( A, A ′ ) → Mor B ( F ( A ) , F ( A ′ )) for A, A ′ ∈ A . Before consider ing re gularity for natural transfor mations we recall basic prop er ties of 2.1. Reg ular morphis m s. Let A, A ′ be any o b jects in a categ ory A . Then a mor phism f : A → A ′ is ca lled r e gular provided there is a morphism g : A ′ → A with f g f = f . Clearly , in this case g f : A → A and f g : A ′ → A ′ are idempotent endomor phisms. Such a mor phis m g is not necessarily uniq ue . In particular , for g f g w e als o have f ( g f g ) f = f g f = f , a nd the identit y ( g f g ) f ( g f g ) = g f g shows that g f g is ag ain a regular morphism. If idempo tent s split in A , then every idemp otent mor phism e : A → A determines a sub o b ject of A , we denote it b y eA . If f is regular with f g f = f , then the restriction of f g is the identit y morphis m on f g A ′ and g f is the identit y on g f A . Examples for regular morphisms are retractions, coretrac tions, and isomo rphisms. F or mo dules M , N over any r ing, a mo r phism f : M → N is regula r if and only if the image and the kernel of f ar e direct summands in N and M , resp ectively . This notio n of regula rity is de r ived from von Neumann re g ularity of rings. F or mo dules (and in pr eadditive categ ories) it was considered b y Nicholson, Kasch, Mader and o thers (see [14]). W e use the termino logy als o for natur a l tra nsformations a nd functors w ith obvious interpretations. 2.2. P airing of functors. (e.g . [19, 2 .1]) Let L : A → B a nd R : B → A b e c ov a riant functors. Assume ther e a re morphisms, natural in A ∈ A and B ∈ B , α : Mor B ( L ( A ) , B ) → Mor A ( A, R ( B )) , β : Mor A ( A, R ( B )) → Mo r B ( L ( A ) , B ) . These maps corr esp ond to natural tra nsformations b etw een functors A op × B → Set . The quadruple ( L, R, α, β ) is called a (ful l) p airing (of fun ctors) . Given such a pa iring, the morphisms, for A ∈ A , B ∈ B , η A := α A,L ( A ) ( I ) : A → RL ( A ) and ε B := β R ( B ) ,B ( I ) : LR ( B ) → B corres p o nd to na tural transforma tions η : I A → RL , ε : L R → I B , which we ca ll quasi-unit and quasi-c ounit of ( L, R , α, β ), resp ectively . F ro m these the transformatio ns α a nd β ar e obtained by α A,B : L ( A ) f − → B 7− → A η A − → RL ( A ) R ( f ) − → R ( B ) , β A,B : A g − → R ( B ) 7− → L ( A ) L ( g ) − → LR ( B ) ε B − → B . Thu s the pa iring ( L, R , α, β ) is also describ ed by the quadruple ( L, R, η , ε ). Naturality of ε and η induces a n ass o ciative product and a quasi-unit for the endofunctor RL : A → A , RεL : RLR L → R L, η : I A → RL , and a coas so ciative copro duct and a quasi- counit for the endofunctor LR : B → B , Lη R : LR → LRL R, ε : LR → I B . By the Y o neda Lemma we ca n describ e co mp os itions o f α a nd β by the imag es o f the ident ity tr a nsformations of the resp ective functor s. 4 R OBER T W I S BA UER 2.3. Comp os ing α and β . Let ( L , R , α, β ) b e a pairing with quasi-unit η and quasi-counit ε . The descr iptions of α and β in 2.2 yield, for the identit y trans fo rmations I L : L → L , I R : R → R , α ( I L ) = I A η − → RL, β · α ( I L ) = L Lη − → LRL εL − → L, α · β · α ( I L ) = I A η − → RL RLη − → RL RL RεL − → RL , β ( I R ) = LR ε − → I B , α · β ( I R ) = R ηR − → RL R Rε − → R, β · α · β ( I R ) = LR Lη R − → LR LR LRε − → LR ε − → I B . The following morphisms will play a sp ecial role in wha t follows. 2.4. Natural endomo rphi sms. With the notions from 2.2, we define the natural tra ns- formations ϑ := R ( β α ( I L )) : RL RLη / / RLR L RεL / / RL, ϑ := αβ ( R ( I L )) : RL ηRL / / RLR L RεL / / RL, γ := L ( αβ ( I R )) : LR Lη R / / LRLR LRε / / LR, γ := β α ( L ( I R )) : LR Lη R / / LRLR εLR / / LR, which hav e the prop erties RεL · RLϑ = ϑ · R εL, R εL · ϑR L = ϑ · RεL, ϑ · ϑ = ϑ · ϑ ; LRγ · L η R = Lη R · γ , γ LR · L η R = Lη R · γ , γ · γ = γ · γ . 2.5. Definitions . Let ( L, R, α, β ) b e a pairing (see 2.2). W e call α r e gular if α · β · α = α ; α symmetric if ϑ = ϑ . β re gu lar if β · α · β = β ; β s ymmet ric if γ = γ ; ( L, R , α, β ) r e gular if α = α · β · α and β = β · α · β . The following prop er ties are easy to verify: (i) If α is r egular, then β · α ( I L ), ϑ and ϑ are idempotent and ϑ · η = η = ϑ · η ; furthermore, for β ′ := β · α · β , ( L, R , α, β ′ ) is a reg ular pair ing. (ii) If β is regular , then α · β ( I R ), γ and γ are idemp otent a nd ε · γ = ε = ε · γ ; furthermore, for α ′ := α · β · α , ( L, R, α ′ , β ) is a regular pairing. An y pair ing ( L, R , α, β ) with β · α = I o r α · β = I is regula r. The second condition defines the semiadjoint funct ors in Medvedev [16]. With ma nipulations known from r ing theory one can show how pairings w ith reg ular comp onents can b e related with adjunctions provided idemp otents split. 2.6. Re lated adjunctions . Let ( L, R, α, β ) be a pa iring (with quasi-unit η , qua si-counit ε ) and assume α to be regular. If the idemp otent h := β · α ( I L ) : L Lη − → LR L εL − → L splits, that is, there a re a functor L : A → B and natural transfor mations p : L → L , i : L → L with i · p = h a nd p · i = I L , REGULAR P AIRINGS 5 then the natural tra nsformations η : I A η / / RL Rp / / RL , ε : LR iR / / LR ε / / I B , as quasi-unit and qua si-counit, define a pair ing ( L , R , α, β ) with β · α = I . If α · β = I , then ( b L, R, α , β ) is an adjunction. In ca se the na tural tra nsformation β is r egular, similar c o nstructions apply if we assume that the idemp otent α · β ( I R ) : R ηR − → RLR Rε − → R splits. The prop erties o f the ( RL , R εRη ) and ( LR, Lη R, ε ) mentioned in 2.2 motiv ate the defi- nitions in the nex t section. 3. Monads and modules 3.1. q -uni tal monads and the i r mo dules. W e call ( F , µ ) a funct or with pr o duct (o r non-unital monad ) provided F : A → A is an endofunctor on a categor y A and µ : F F → F is a natural tra nsformation satisfying the ass o ciativity condition µ · F µ = µ · µF . F or ( F, µ ), a (non-un ital) F -mo dule is defined as an ob ject A ∈ A with a mor phis m  : F ( A ) → A in A sa tisfying  · F  =  · µ A . Morphisms b etw een F -mo dules ( A,  ), ( A ′ ,  ′ ) ar e mor phisms f : A → A ′ in A with  ′ · F ( f ) = f ·  . The s et of all these is deno ted by Mor F ( A, A ′ ). With these morphisms, (non-unital) F -mo dules form a ca tegory which we denote by A − → F . By the asso c ia tivity condition on µ , fo r every A ∈ A , ( F ( A ) , µ A ) is an F -mo dule and this leads to the free functor and the forge tful functor, φ F : A → A − → F , A 7→ ( F ( A ) , µ A ) , U F : A − → F → A , ( A,  ) 7→ A. A triple ( F, µ, η ) is s a id to b e a q -unital monad o n A provided ( F , µ ) is a functor with pro duct and η : I A → F is any natura l tra nsformation, called a quasi-un it (no additional prop erties are r equired). One alwa ys ca n de fine natural transfor mations ϑ : F F η − → F F µ − → F , ϑ : F ηF − → F F µ − → F . Note that for any A ∈ A , ϑ A is in A F and ϑ A is not necessar ily so . Given q -unital monads ( F , µ, η ), ( F ′ , µ ′ , η ′ ) on A , a natur al transfo rmation h : F → F ′ is called a morphism of q -u nital monads if µ ′ · hh = h · µ and η ′ = h · η . The existence of a quasi-unit a llows the following generalisa tion of the Eilenber g-Mo ore construction for (unital) monads. 3.2. q -unital monads and pairings. F or a q-unital mona d ( F, µ, η ) we o bta in a pairing ( φ F , U F , α F , β F ) with the maps, for A ∈ A , ( B ,  ) ∈ A − → F , α F : Mor F ( φ F ( A ) , B ) → Mor A ( A, U F ( B )) , f 7→ f · η A , β F : Mor A ( A, U F ( B )) → Mor F ( φ F ( A ) , B ) , g 7→  · F ( g ) . The quasi-unit η is ca lled r e gular if α F is regula r, that is, I A η − → F = I A η − → F F η − → F F µ − → F, and we say η is symmetric if α F is so, that is, ϑ = ϑ . An F - mo dule  : F ( A ) → A in A − → F is said to b e c omp atible if β F α F (  ) =  , that is F ( A )  − → A = F ( A ) F η A − → F F ( A ) µ A − → F ( A )  − → A. In particular , the natur al transforma tio n µ : F F → F is compatible if F F µ − → F = F F F η F − → F F F µF − → F F µ − → F . 6 R OBER T W I S BA UER It is eas y to se e that this implies F F ϑϑ − → F F µ − → F = F F µ − → F . Let A F denote the full sub categor y of A − → F made up by the compa tible F -mo dules . If µ is compatible, the imag e of the fr ee functor φ F lies in A F and (by restriction or co r estriction) we get the functor pa ir (keeping the notatio n fo r the functors ) φ F : A → A F , U F : A F → A , and a pairing ( φ F , U F , α F , β F ) b etw een A a nd A F . Since for ( A,  ) in A − → F , β F ( I U F ( A ) ) =  , the compatibility condition o n  implies that β · α · β (  ) = β (  ), i.e., β is regular in ( φ F , U F , α F , β F ) when restr icted to A F . 3.3. Definition. A q -unital monad ( F, η , µ ) is called r -u nital if η is reg ular and µ is compatible; we ak m onad if ( F , η , µ ) is r -unital and η is symmetric. Summarising the obser v atio ns fro m 3 .2 we hav e: 3.4. Prop ositi on. L et ( F , µ, η ) b e a q -unital monad. (1) The fol lowing ar e e qu ivalent: (a) ( F , µ, η ) is an r -unital monad; (b) ( φ F , U F , α F , β F ) is a r e gu lar p airing of fun ctors b etwe en A and A F . (2) The fol lowing ar e e qu ivalent: (a) ( F , µ, η ) is we ak m onad; (b) ( φ F , U F , α F , β F ) is a r e gu lar p airing b etwe en A and A F with α F symmetric. A quasi-unit η tha t is reg ular a nd sy mmetric is na med pr e-unit in the litera ture (e.g. [11, Definition 2.3 ]); for the notion of a weak monad (also called demimonad ) see e.g . [3], [4]. In c ase η is a unit, q -unital monads, r - unital mona ds a nd weak monads all are (unital) monads. In (non- unital) alg ebras over commutativ e r ings, r -unital monads ar e o btained from idemp otents while weak monads corr esp ond to central idemp otents (see 3.7). 3.5. Prop erties of w eak monads. L et ( F , µ, η ) b e a we ak monad. (i) ϑ : F → F is a morphism of q -unital monads; (ii) for any ( A, ϕ ) ∈ A F , F ( A ) ϕ − → A = F ( A ) ϕ − → A η A − → F ( A ) ϕ − → A and A η A − → F ( A ) ϕ − → A is an idemp otent F -m orphism. In a q - unital monad ( F , µ, η ), if η is regular , a compatible multiplication for F can be found. Mor e precisely o ne can easily show: 3.6. Prop ositi on. L et ( F , µ, η ) b e a q -unital monad. (1) If η is r e gular, then, for e µ := µ · F µ · µF η F : F F → F , ( F , e µ, η ) is an r - unital monad. (2) If µ is c omp atible, then, for e η := µ · F η · η : I A → F , ( F, µ, e η ) is an r -unital monad. (3) If ( F , µ, η ) is an r -unital monad, then for b µ : F F ηF F η − → F F F F µF F − → F F F µF − → F F µ − → F , ( F, b µ, η ) is a we ak monad. As a sp ecial case, we consider q -unital monads on the catego ry R M of mo dules ov er a commutativ e r ing R with unit. In the terminology used here this c o mes o ut as follows. REGULAR P AIRINGS 7 3.7. No n-unital al g ebras. A q -u nital R -algebr a ( A, m, u ) is a non-unital R -algebr a ( A, m ) with some R -linea r ma p u : R → A . Put e := u (1 R ) ∈ A . The n: (1) u is r egular if and only if e is an idemp otent in A . (2) u is reg ular and symmetric if and only if e is a central idemp otent (then Ae is a unital R -subalg ebra of A ). (3) µ is compa tible if and only if ab = a eb for all a, b ∈ A . (4) If u is reg ular, then e m ( a ⊗ b ) := aeb , for a, b ∈ A , defines an r -unital alg ebra ( A, e m, u ) ( e m and u are reg ular). (5) If u is r egular, then b m ( a ⊗ b ) := eaebe , for a, b ∈ A , defines an r -unital algebra ( A, b m, u ) with u symmetric . Clearly , the q -unital algebra s ( A, m, u ) ov er R co rresp ond to the q -unital monads given by ( A ⊗ R − , m ⊗ − , u ⊗ − ) on R M . F or a n A -mo dule  : A ⊗ M → M , writing as usual  ( a ⊗ m ) = am , the co mpatibiliy condition comes o ut as am = aem for a ll a ∈ A , m ∈ M . 3.8. Mo nads acting on functors. Let T : A → B b e a functor and ( G, µ ′ , η ′ ) a q -unital monad on B . W e call T a left G -mo dule if there exists a natural transformatio n  : GT → T such that GGT G − → GT  − → T = GGT µ ′ T − → GT  − → T , and we call it a c omp atible G -mo dule if in addition GT  − → T = GT Gη ′ − → GGT µ ′ T − → GT  − → T . 3.9. Prop o s ition. Le t T : A → B b e a funct or and ( G, µ ′ , η ′ ) a we ak m onad on B . Then the fol lowing ar e e quivale nt: (a) t her e is a functor T : A → B G with T = U G T ; (b) T is a c omp atible G -mo dule. Pro of. (b) ⇒ (a) Given T as a co mpatible G -mo dule with  : GT → T , a functor with the required pro pe r ties is T : A → B G , A 7→ ( T ( A ) ,  A : GT ( A ) → T ( A )) . (a) ⇒ (b) F or any A ∈ A , there a re morphisms ρ A : GT ( A ) → T ( A ) and we claim that these define a natural transfor mation ρ : GT → T . F or this w e hav e to show that, for a n y morphism f : A → b A , the middle recta ng le is commut ative in the diag r am GGT ( A ) µ ′ T ( A ) / / GT ( A ) GT ( f )   ρ A z z ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ GT ( A ) Gη ′ T A O O GT ( f )   ρ A / / T ( A ) T ( f )   GT ( b A ) Gη ′ T b A   ρ b A / / T ( b A ) GGT ( b A ) µ ′ T ( b A ) / / GT ( b A ) . ρ b A c c ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ The top and bo ttom diag rams are commutativ e by compa tibility of the G -mo dules, the right tr ap ezium is co mmu tative since T ( f ) is a G -morphism, and the outer pa ths commute by symmetry o f η ′ . Thus the inner dia gram is commutativ e showing na turality of ρ . ⊔ ⊓ 8 R OBER T W I S BA UER 4. Comonads an d comodul es In this section we sketch the transfer o f the constructions for monads to co mo nads. 4.1. q -counital comonads and their com o dules . A functor with c opr o duct (or non- c oun ital c omonad ) is a pa ir ( G, δ ) where G : A → A is an endo functor and δ : G → GG is a natura l trans fo rmation sub ject to the coass o ciativity condition Gδ · δ = δ G · δ . F or ( G, δ ), a (non-c ounital) G -c omo dule is defined as a n ob ject A ∈ A with a morphism υ : A → G ( A ) in A such that Gυ · υ = δ A · υ . Morphisms b etw een G -como dules ( A, υ ), ( A ′ , υ ′ ) ar e morphisms g : A → A ′ in A satis- fying υ ′ · g = G ( g ) · υ , a nd the set of all these is denoted by Mor G ( A, A ′ ). With these morphisms, (non-counital) G -como dules form a c a tegory which we denote by A − → G . F or this there ar e the obvious free and forgetful functors φ G : A → A − → G , U G : A − → G → A . A triple ( G, δ, ε ) is said to b e a q -c ounital c omonad provided ( G, δ ) is a functor with copro duct a nd ε : G → I A is any natural tra nsformation, called a quasi-c ounit . One can alwa ys define natural transforma tions γ : G δ − → GG Gε − → G, γ : G δ − → GG εG − → G. Morphisms of q -co unital comonads are defined in an obvious wa y (dual to 3.1). 4.2. q -couni tal comonads and pairings. F or ( G, δ, ε ), the functors φ G and U G allow for a pair ing ( U G , φ G , α G , β G ) where, for A ∈ A and ( B , υ ) ∈ A − → G , α G : Mor A ( U G ( B ) , A ) → Mor G ( B , φ G ( A )) , f 7→ G ( f ) · υ , β G : Mor G ( B , φ G ( A )) → Mor A ( U G ( B ) , A ) , g 7→ ε A · g. The quasi-c o unit ε is ca lled r e gular if β G is regula r, that is, G ε − → I A = G δ − → GG Gε − → G ε − → I A , and we say η is symmetric provided φ G is so, that is γ = γ . A (non-counital) G -como dule ( B , υ ) is said to b e c omp atible provided α G β G ( υ ) = υ , that is B υ − → G ( B ) = B υ − → G ( B ) δ B − → GG ( B ) Gε B − → G ( B ) . In particular , δ is compatible if G δ − → GG = G δ − → GG δG − → GGG GεG − → GG. This obviously implies G δ − → GG = G δ − → GG γ γ − → GG. By A G we denote the full s ubca tegory of A − → G whose ob jects are co mpa tible G - c omo dules. If δ is compatible, the imag e o f the free functor φ G lies in A G and (b y r estriction and corestrictio n) we obtain the functor pair ing (keeping the notation for the functors) φ G : A → A G , U G : A G → A , leading to a pair ing ( U G , φ G , α G , β G ) b et ween A a nd A G . Since for ( B , υ ) in A − → G , α G ( I U G ( B ) ) = υ , the co mpatibility conditio n on υ implies that α G · β G · α G ( υ ) = α G ( υ ), i.e., α is regular in ( U G , φ G , α G , β G ) when restr icted to A G . 4.3. Definition. A q -co unital comonad ( G, δ, ε ) is called r -c ounital if ε is r egular a nd δ is compa tible; we ak c omonad if it is r - c ounital and ε is sy mmetric. F ro m the co nstructions ab ov e we obtain: REGULAR P AIRINGS 9 4.4. Prop ositi on. L et ( G, δ, ε ) b e a q -c ounital c omonad. (1) The fol lowing ar e e qu ivalent: (a) ( G, δ, ε ) is an r -c ounital c omonad ; (b) ( U G , φ G , α G , β G ) is a r e gu lar p airing of functors b etwe en A and A G . (2) The fol lowing ar e e qu ivalent: (a) ( G, δ, ε ) is we ak c omonad ; (b) ( U G , φ G , α G , β G ) is a r e gular p airing of functors b etwe en A and A G with β G symmetric. Similar to the situation for modules, for any (counital) comonad ( G, δ, ε ), all non-counital G -como dules are compa tible (i.e., A − → G = A G ). 4.5. Prop erties of w eak comonads. L et ( G, δ, ε ) b e a we ak c omonad. (i) γ : G → G is an idemp otent morphism of q -c ou n ital c omonads; (ii) for any ( B , υ ) ∈ A G , B υ − → G ( B ) = B υ − → G ( B ) ε B − → B υ − → G ( B ) and B υ − → G ( B ) ε B − → B is an idemp otent G -morphism. Prop erties of pair ings c a n improv ed in the following sense. 4.6. Prop ositi on. L et ( G, δ, ε ) b e a q -c ounital c omonad. (1) If ε is r e gular, then, for e δ : G δ − → GG Gδ − → GGG GεG − → GG , ( G, e δ , ε ) is an r -c ounital c omonad. (2) If δ is c omp atible, then, for e ε : G δ − → GG Gε − → G ε − → I A , ( G, δ , e ε ) is an r - c ounital c omonad. (3) If ( G, δ, ε ) is a r e gular quasi-c omonad, then, for b δ : G δ − → GG Gδ − → GGG GGδ − → GGGG εGGε − → GG, ( G, b δ , ε ) is a we ak c omonad. As a sp ecial ca s e, cons ider non-counita l comona ds o n the category R M of mo dules over a commutativ e r ing R with unit. In our terminology this comes out as follows. 4.7. Non-counital coalgebras. A q - c oun ital c o algebr a ( C, ∆ , ε ) is a non-counita l R - coalgebr a ( C, ∆) with so me R -linear map ε : C → R . W riting ∆( c ) = P c 1 ⊗ c 2 for c ∈ C , we have: (1) ε is re gular if and only if for any c ∈ C , ε ( c ) = P ε ( c 1 ) ε ( c 2 ). (2) ε is symmetr ic if and only if P c 1 ε ( c 2 ) = P ε ( c 1 ) c 2 . (3) ∆ is compatible if and only if ∆( c ) = P c 1 ⊗ c 2 ε ( c 3 ). (4) If ε is regula r, then e ∆( c ) := P c 1 ⊗ ε ( c 2 ) c 3 defines an r -co unital coalgebra ( C, e ∆ , ε ). (5) If ( C, ∆ , ε ) is an r -counital comonad, then b ∆( c ) := P ε ( c 1 ) c 2 ⊗ c 3 ε ( c 4 ) defines an r -co unital coalg ebra ( C, b ∆ , ε ) with ε symmetric. Clearly , the q -co unital coalgebras ( C, ∆ , ε ) ov er R corresp ond to the q -counital comonads given by ( C ⊗ R − , ∆ ⊗ − , ε ⊗ − ) on R M . F rom this the c o mpatibility conditions fo r C - como dules are derived (see 4.2). 4.8. W eak corings and pre- A -corings . Let A b e a ring with unit 1 A and C a non-unital ( A, A )-bimo dule whic h is unital as right A -mo dule. Assume there are ( A, A )-bilinear maps ∆ : C → C ⊗ A C , ε : C → A, 10 R OBER T W I S BA UER where ∆ is coas so ciative. ( C , ∆ , ε ) is called a right unit al we ak A -c oring in [21], provided for a ll c ∈ C , ( ε ⊗ I C ) · ∆ ( c ) = 1 A · c = ( I C ⊗ ε ) · ∆( c ) , which rea ds in (o bvious) Sweedler no tation as P ε ( c 1 ) c 2 = 1 A · c = P c 1 ε ( c 2 ) . F ro m the eq uations ( I C ⊗ ε ⊗ I C ) · ( I C ⊗ ∆) · ∆( c ) = P c 1 ⊗ 1 A · c 2 = P c 1 ⊗ c 2 = ∆( c ) , ( I C ⊗ ε ⊗ I C ) · (∆ ⊗ I C ) · ∆( c ) = P 1 A · c 1 ⊗ c 2 = 1 A · ∆( c ) , it follows by co asso cia tiv ity that 1 A · ∆( c ) = ∆( c ). Summaris ing we see that, in this cas e, ( C , ∆ , ε ) induces a weak comonad on the categor y A M − → of left non-unital A -mo dules (= A M since A has a unit). ( C , ∆ , ε ) is called an A -pr e-c oring in [7, Section 6], if ( ε ⊗ I C ) · ∆ ( c ) = c, ( I C ⊗ ε ) · ∆( c ) = 1 A · c, which rea ds (in Sweedler notation) as c = P ε ( c 1 ) c 2 , 1 A · c = P c 1 ε ( c 2 ) . Similar to the co mputation ab ov e we obtain that 1 A · ∆ ( c ) = ∆( c ). Now ( C , ∆ , ε ) induces an r -counital comona d on A M − → but ε is not sy mmetric. Notice that in b oth case s cons idered a bove, restr ic tio n and corestrictio n of ∆ and ε yield an A -c oring ( A C , ∆ , ε ) (e.g. [21, Prop os ition 1.3 ]). 4.9. Comonads acting on functors. Let T : A → B b e a functor and ( G, δ, ε ) a weak comonad on B . W e call T a left (non- c ounital) G -c omo dule if there ex is ts a natural transformatio n υ : T → GT such that T υ − → GT υ G − → GGT = T υ T − → GT δ − → GGT , and we call it a c omp atible G -c omo dule if, in addition, T υ − → GT = T υ − → GT δ − → GGT Gε − → GT . Dual to Prop o s ition 3.9, g iven a w eak comona d ( G, δ, ε ) o n B , a functor T : A → B is a compatible G -como dule if a nd o nly if there is a functor T : A → B G with T = U G T . The mo tiv ation for consider ing gener alised mona ds and comonads came from structures observed while handling full pair ings of functor s (see end of Section 2). No w we wan t to reconsider the pairing s in view of these constr uctions. F or any pairing ( L, R , α, β ) b etw een the categor ies A and B , ( R L, RεL, η ) is a q -unital monad and ( LR , L η R, ε ) is a q -counital comona d. It is ea sy to see that (i) if β is r e gular, then for any B ∈ B , R ε : R LR ( B ) → R ( B ) is a c omp atible RL -mo dule. (ii) if α is r e gular, then for any A ∈ A , L ( A ) , Lη : L ( A ) → L RL ( A ) is a c omp atible LR -c omo dules. 4.10. Comparison functors. F or a r e gular p airing ( L , R , α, β ) b etwe en A and B , ( RL, RεL, η ) is an r -u nital monad on A with a (c omp arison) functor b R : B → A RL , B 7→ ( R ( B ) , Rε : R LR ( B ) → R ( B )) , ( LR, Lη R, ε ) is an r -c ounital c omonad on B with a (c omp arison) functor e L : A → B − → LR , A 7→ ( L ( A ) , Lη : L ( A ) → L RL ( A )) , inducing c ommut ativity of the diagr ams A L / / φ RL ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ B b R   R / / A A RL U RL = = ④ ④ ④ ④ ④ ④ ④ ④ , B R / / φ LR ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ A e L   L / / B B LR U LR > > ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ . REGULAR P AIRINGS 11 It follows from 3.2 that for the r -unital monad ( RL , R εL, η ), ( φ RL , U RL , α RL , β RL ) is a r e g ular pair ing b etw een A and A RL . Similarly , by 4.2, for the R - c o untial comonad ( LR, Lη R, ε ), ( U LR , φ LR , α LR , β LR ) is a regular pairing b etw een B and B LR . 4.11. Rel ating ( L, R ) with ( φ RL , U RL ) and ( U LR , φ LR ) . With the ab ove notions we form the diag ram Mor B ( L ( A ) , B ) b R − , −   α / / Mor A ( A, R ( B )) β / / Mor B ( L ( A ) , B ) b R − , −   Mor RL ( φ RL ( A ) , R ( B )) α RL / / Mor A ( A, U RL R ( B )) β RL / / Mor RL ( φ RL ( A ) , R ( B )) . This diagr am is commutativ e if and only if α is symmetric (see Definitions 2.5). Similar co nstructions apply for ( L , R ), ( U LR , φ LR ) and e L − , − . and β is sy mmetric if and only if e K − , − · α · β = α LR · β LR · e L − , − . 4.12. Corollary . Consider a p airing ( L, R, α, β ) (se e 2.2). (1) The fol lowing ar e e qu ivalent: (a) ( L , R , α, β ) is a r e gular p airing; (b) ( RL , R εL, η ) is an r -u nital monad on A and ( LR, Lη R, ε ) is an r -c ounital c omonad on B . (2) The fol lowing ar e e qu ivalent: (a) ( L , R , α, β ) is a r e gular p airing with α and β symmetric; (b) ( RL , R εL, η ) is a we ak monad on A and ( LR, Lη R, ε ) is a we ak c omonad on B . 5. Entwining monads and com onads 5.1. Lifting of functors to mo dule categorie s . Let ( F, µ, η ) and ( L, µ ′ , η ′ ) b e r - unital mo nads on the categor ies A and B , resp ectively , and A F , B L the catego ries of the corres p o nding compatible modules (se e 3.2). Given functors T : A → B and T : A F → B L , we say that T is a lifting of T pr ovided the diagra m (5.1) A F T / / U F   B L U L   A T / / B is commutativ e, where the U ’s denote the forgetful functors . 5.2. Prop osi tion. With the data given in 5.1 , c onsider the functors T F, L T : A → B and a natur al tr ansformation λ : L T → T F . The non-unital F -m o dule ( F , µ ) induc es an L -action on T F , χ : L T F λF − → T F F T µ − → T F . (1) If ( T F , χ ) is a (non-unital) L -mo dule, then we get the c ommut ative diagr am (5.2) LLT Lλ / / µ ′ T   LT F LT ϑ / / LT F λF / / T F F T µ   LT λ / / T F T ϑ / / T F. (2) If ( T F , χ ) is a c omp atible L -mo dule, then (with ϑ ′ = µ ′ · F η ′ ) (5.3) LT ϑ ′ T / / LT λ / / T F T ϑ / / T F = LT λ / / T F T ϑ / / T F. 12 R OBER T W I S BA UER (3) If η is symmetric in ( F , µ, η ) and ( A, ϕ ) is a c omp atible F -mo dule, then (5.4) T ϕ · λ A = T ϕ · λ A · LT ϕ · LT η A . Pro of. The pro of follows esse ntially as in the monad cas e replacing the identit y on F at some places by ϑ = µ · F η (see 3.1). T o show (3), Prop os ition 3.5 is needed. ⊔ ⊓ 5.3. Prop ositio n. L et ( F , µ, η ) and ( L, µ ′ , η ′ ) b e r -unital monads on A and B , r esp e ctively, and T : A → B any functor. Then a natu r al t r ansformatio n λ : LT → T F induc es a lifting to the c omp atible mo dules, T : A F → B L , ( A, ϕ ) 7→ ( T ( A ) , T ϕ · λ A : LT ( A ) → T ( A )) , if and only if the diagr am (5.2) is c ommu tative and e quation (5.3) holds. Pro of. O ne direction follows from Prop osition 5.2, the other o ne by a slight mo difica - tion of the pr o of in the monad case . ⊔ ⊓ T o show that the lifting pr op erty implies the exis tence of a natural transforma tion λ : L T → T F we need the symmetry of the units, that is , we r equire the r -unital monads to b e weak monads. Then we can extend Applegate’s lifting theor em for monads (and unital mo dules) (e.g. [13, Lemma 1 ], [22, 3 .3 ]) to w eak monads (and compatible mo dules ). 5.4. Theorem. L et ( F , µ, η ) and ( L, µ ′ , η ′ ) b e we ak monads on A and B , r esp e ctively. F or any functor T : A → B , t her e ar e bije ctive c orr esp ondenc es b etwe en (i) lif tings of T to T : A F → B L ; (ii) c omp atible L -mo dule structu r es  on T U F : A F → B ; (iii) natur al tr ansformations λ : LT → T F with c ommuting diagr ams (5.5) LLT Lλ / / µ ′ T   LT F λF / / T F F T µ   LT λ / / T F, LT ϑ ′ T / / λ   λ " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ LT λ   T F T ϑ / / T F. Pro of. (i) ⇔ (ii) follows by Prop osition 3.9. (ii) ⇒ (iii) Given the compatible L -mo dule structure map  , put λ := F · LT η : L T LT η − → LT F F − → T F . Notice that for λ we can take T ϑ · λ fr om Prop ositio n 5.2. (iii) ⇒ (i) Given λ with the commutativ e diag ram in (iii), it follows b y Prop ositions 5 .3 that  A := T ϕ · λ A induces a lifting. ⊔ ⊓ 5.5. Lifting of functors to como dules . Let ( G, δ, ε ) a nd ( H, δ ′ , ε ′ ) be r -unital co monads on the c a tegories A a nd B , res p ectively , and A G , B H the corr esp onding categories of the compatible co mo dules (see 4.2). Giv en a functor T : A → B , a functor b T : A G → B H , is said to b e is a lifting of T if the diagr am (5.6) A G b T / / U G   B H U H   A T / / B is commutativ e where the U ’s denote the forgetful functors . REGULAR P AIRINGS 13 5.6. Prop osition. Wit h the data given in 5 .5 , c onsider the functors T G, H T : A → B and a natur al t ra nsformation ψ : T G → H T . The (non- c oun ital) G -c omo dule ( G, δ ) induc es an H -c o action on T G , ζ : T G T δ − → T GG ψ G − → H T G. (1) If ( T G, ζ ) is a (n on- c oun ital) H -c omo dule, we get the c ommutative diagr am (5.7) T G T γ / / T δ   T G ψ / / H T δ ′ T   T GG ψ G / / H T G H T γ / / H T G H ψ / / H H T . (2) If H ( T G, ζ ) is a c omp atible H -mo dule, then (5.8) T G T γ / / T G ψ / / H T γ ′ T / / H T = T G T γ / / T G ψ / / H T . (3) If ε is symmetric and ( A, υ ) is a c omp atible G -c omo dule, then ψ · T υ = H T ε · H T υ · ψ · T υ . Pro of. The situation is dual to that o f Prop o s ition 5.2. ⊔ ⊓ 5.7. Prop osition. L et ( G, δ, ε ) and ( H, δ ′ , ε ′ ) b e r -c ounital c omonads on the c ate gories A and B , r esp e ctively, and T : A → B any functor. A natur al tr ansformation ψ : T G → H T induc es a lifting b T : A G → B H , ( A, υ ) 7→ ( T ( A ) , ψ · T υ : T ( A ) → H T ( A )) , if and only if the diagr am (5.7) is c ommu tative and e quation (5.8) holds. Pro of. The pro of is dual to that of Prop os ition 5.3. ⊔ ⊓ Dualising Theo rem 5.4, we obtain an extension o f Applegate’s lifting theorem for comon- ads (and como dules) (e.g. [2 2, 3.5]) to weak como nads (a nd compatible como dules). 5.8. Theorem. Le t ( G, δ, ε ) and ( H , δ ′ , ε ′ ) b e we ak c omonads on A and B , r esp e ctively. F or any funct or T : A → B , ther e ar e bije ct ive c orr esp ondenc es b etwe en (i) lif tings of T to b T : A G → B H ; (ii) c omp atible H -c omo dule stru ctur es υ : T U G → H T U G ; (iii) natur al tr ansformations ψ : T G → H T with c ommutative diagr ams T G T δ   ψ / / H T δ ′ T   T GG ψ G / / H T G H ψ / / H H T , T G T γ / / ψ   ψ # # ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ T G ψ   H T γ ′ T / / H T . Pro of. In view of 5.6 a nd 5 .7, the pro of is dual to that of Theorem 5.4. Here we take ψ as the co mpo s ition ψ · T γ (with ψ fr om 5.6). ⊔ ⊓ 6. Lifting of endofunctors to modul es and com odules Given a w eak monad ( F , µ, η ), or a weak como nad ( G δ, ε ), and a ny endofunctor T on the c ategory A , we have lear ned in the preceding sectio ns when T can b e lifted to an endofunctor of the compatible mo dules or como dules, resp ectively . Now, one may also ask if the lifting is a gain a weak monad or a weak comonad, resp ectively . 6.1. Ent wi ning r -unital m onads. F or we ak monads ( F , µ, η ) and ( T , ˇ µ, ˇ η ) on A and a natur al tra nsformation λ : F T → T F , the fol lowing ar e e quivalent: 14 R OBER T W I S BA UER (a) defining pr o duct and quasi-unit on T F by µ : T F T F T λF − → T T F F T T µ − → T T F ˇ µF − → T F , η : I A η − → F F ˇ η − → F T λ − → T F , yields a we ak m onad ( T F, µ, η ) on A ; (b) λ induc es c ommutativity of the diagr ams (6.1) F F T F λ / / µT   F T F λF / / T F F T µ   F T λ / / T F, F T ϑT / / λ   λ " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ F T λ   T F T ϑ / / T F, (6.2) F T T F ˇ µ   λT / / T F T T λ / / T T F ˇ µF   F T λ / / T F, F T F ˇ ϑ / / λ   λ " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ F T λ   T F ˇ ϑF / / T F ; (c) λ induc es c ommutativity of t he diagr ams in (6.1) and the squar e in ( 6 .2 ), and ther e ar e natu r al tr ansformations ˇ µF : T T F → T F and λ · F ˇ η : F → T F wher e ˇ µF is a left and right F -mo dule morphism and λ · F ˇ η is an F -mo dule morphism. If these c onditions hold, we obtain morphisms of q -unital monads, λ · F ˇ η : F → T F and λ · η T : T → T F . Pro of. The ass ertions follow from the general r esults in Section 5 and s ome routine computations. ⊔ ⊓ 6.2. W eak cross ed pro ducts. Given ( F , µ, η ) and T : A → A , the comp ositio n T F may hav e a weak monad structure without requir ing s uch a structur e o n T . F or example, replacing the natural trans formations ˇ µF and λ · F ˇ η in 6.1(c) by some natural trans forma- tions ν : T T F → T F, ξ : F → T F , similar to 6.1(a), a multiplication a nd a quasi-unit can b e defined on T F . T o make this a weak mona d on A , sp ecia l conditions are to be imp osed o n ν and ξ w hich can b e obtained by ro utine c omputations. Having ν a nd ξ , one a ls o has natural transforma tio ns ¯ ν : T T T T η / / T T F ν / / T F, η : I A η / / F ξ / / T F, and it is easy to see that ¯ ν lea ds to the s ame pro duct on T F as ν do es. Thus ¯ ν and η may be used to define a weak monad structure on T F a nd the conditions required co me out as c o cycle and twiste d c onditions . F or more details we refer, e.g., to [1], [1 1, Section 3]. F or a weak co monad ( G, δ, ε ) a nd a n endofunctor T : A → A , we now co nsider liftings to the category o f compatible G -co mo dules, b T : A G → A G . The ca se when T has a weak comonad structure is dual to 6 .1: 6.3. Ent wi ning w eak comonads. F or we ak c omonads ( F, δ, ε ) , ( T , ˇ δ , ˇ ε ) , and a natu r al tr ansformatio n ψ : T G → GT , the fol lowing ar e e quivalent: (a) defining a c opr o duct and quasi-c ounit on T G by b δ : T G ˇ δG − → T T G T T δ − → T T GG T ψ G − → T GT G, b ε : T G ψ − → GT G ˇ ε − → G ε − → I A , yields a we ak c omonad ( T G, b δ , b ε ) on A ; REGULAR P AIRINGS 15 (b) ψ induc es c ommutativity of the diagr ams, wher e γ = T ε · δ , ˇ γ = T ˇ ε · ˇ δ , (6.3) T G T δ   ψ / / GT δT   T GG ψ G / / GT G Gψ / / GGT , T G T γ / / ψ   ψ " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ T G ψ   GT γ T / / GT , (6.4) T G ψ / / ˇ δ G   GT G ˇ δ   T T G T ψ / / T GT ψ T / / GT T , T G ˇ γ G / / ψ   ψ " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ T G ψ   GT G ˇ γ / / GT , (c) ψ induc es c ommu tativity of the diagr ams (6.3 ) and the squar e in (6.4) and we have natur al tr ansformations ˇ δ G : T G → T T G, G ˇ ε · ψ : T G → G, wher e ˇ δ G is a left and right G -c omo dule morphism and G ˇ ε · ψ is a left G -c omo dule morphism. If these c onditions hold, we obtain morphisms of q -unital c omonads, G ˇ ε · ψ : T G → G and εT · ψ : T G → T . 6.4. W eak c rossed copro ducts. In the situation of 6.3, the copro duct o n T G can also be e xpressed by r eplacing the natura l tra nsformations ˇ δ G and G ˇ ε · ψ by any natural transformatio ns ν : T G → T T G a nd ζ : T G → G , sub ject to cer tain conditions to obtain a weak co mo nad structure on T G . Given ν and ζ as ab ove, one may form b ν : T G ν / / T T G T T ε / / T T , b ζ : T G ζ / / G ε / / I A , and it is e a sy to see that these induce a w eak comonad s tr ucture on T G . This leads to the we ak cr osse d c opr o duct as considered (for co algebras ) in [11] and [1 2], for example. 7. Mixed entwinings and liftings Throughout this section let ( F , µ, η ) denote a weak monad and ( G, δ, ε ) a weak c o monad on any categor y A . In this section we inv estigate the lifting pro pe rties to compatible F -mo dules and compatible G -como dules, resp ectively . 7.1. Liftings of m onads and como nads. Consider the dia grams A F G / / U F   A F U F   A G / / A , A G b F / / U G   A G U G   A F / / A . In b oth cases the lifting prop erties are related to a natural tr a nsformation ω : F G → GF. 16 R OBER T W I S BA UER The lifting in the left hand ca se r equires commutativit y of the diag rams (Prop osition 5.3) (7.1) F F G F ω / / µG   F GF ω F / / GF F Gµ   F G ω / / GF, F G ω / / ϑG   ω " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ GF Gϑ   F G ω / / GF, whereas the lifting to A G needs commutativit y of the diagrams (Prop os ition 5.7) (7.2) F G F δ   ω / / GF δF   F GG ω G / / GF G Gω / / GGF, F G ω / / F γ   ω " " ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ GF γ F   F G ω / / GF. T o make G a non-counital co monad with copro duct δ , the latter has to b e an F - mo dule morphism, in par ticular, δ F : GF → GGF has to b e an F -morphism and this follows by commutativit y of the rectangle in (7.2) pr ovided the square in (7.1) is co mm utative. T o make the lifting b F a no n- unital monad with multiplication µ , the latter has to b e a G -como dule morphism, in par ticula r, µG : F F G → F G has to be a G - mo dule mo rphism and this follows b y co mmutativit y of the rectangle in (7.1) provided the sq ua re in (7.2) is commutativ e. 7.2. Natural transformations. The da ta given in 7.1 allow for natura l transforma tions ξ : G ηG / / F G ω / / GF εF / / F , b κ : GF ηG F / / F GF ω F / / GF F Gµ / / GF , b τ : F G F δ / / F GG ω G / / GF G εF G / / F G, with the prop erties Gµ · b κF = b κ · Gµ, b τ G · F δ = F δ · b τ , µ · ξ F = εF · b κ, ξ G · δ = b τ · η G. (i) If the r e ctangle in (7.1) is c ommutative, then b κ is idemp otent. (ii) If the r e ctangle in (7.2) is c ommutative, then b τ is idemp otent. T o ma ke the liftings weak comonads or weak monads, r esp ectively , we have to find pre-units or pre-c o units, resp ectively . In what fo llows we consider these ques tions. 7.3. Lemma. (Pr e-c ounits for G ) Assu m e the diagr ams in (7.1) to b e c ommu tative. Then the fol lowing ar e e quivale nt: (a) for any ( A, ϕ ) ∈ A F , ε A : G ( A ) → A is an F -mo dule morphism; (b) εF : GF → F is an F -morphism; (c) ϑ = µ · F η induc es c ommutativity of the diagr am (7.3) F G F ε / / ω   F ϑ   GF εF / / F. If these c onditions ar e satisfie d, then (with γ = Gε · ϑ ) µG · F b τ = b τ · µG and b τ = ϑγ . Pro of. This is shown by s traightforw ard verification. ⊔ ⊓ REGULAR P AIRINGS 17 7.4. Prop os i tion. Assum e the diagr ams in (7.1), (7.2) and (7.3) to b e c ommutative. Then ( G, δ, ε ) is a we ak c omonad on A F . Pro of. This follows from the preceding observ ations. ⊔ ⊓ Dual to Lemma 7.3 and 7 .4 we obtain for the quas i-units for b F : 7.5. Lemma. (Pr e-units for b F ) Assume the diagr ams in (7.2) t o b e c ommutative. Then the fol lowing ar e e quivale nt: (a) for any ( A, υ ) ∈ A G , η A : A → F ( A ) is a G -c omo dule morphism; (b) ηG : G → F G is G -c oline ar; (c) γ = Gε · δ induc es c ommutativity of the diagr am (7.4) G γ   ηG / / F G ω   G Gη / / GF. If these c onditions ar e satisfie d, then G b κ · δF = δ F · b κ and b κ = γ ϑ. Summing up the ab ove o bserv ations yields the 7.6. Prop os i tion. Assum e the diagr ams in (7.1), (7.2) and (7.4) to b e c ommutative. Then ( b F , µ, η ) is a we ak monad on A G . One may consider a lternative choices for a pr e-counit for G or a pre-unit for b F . 7.7. Lemma. Assu me t he diagr ams in (7.1) to b e c ommutative. With the notations fr om 7.2 , the fol lowing ar e e quivalent: (a) for any ( A, ϕ ) ∈ A F , ε A : G ( A ) ξ A / / F ( A ) ϕ / / A is an F -m o dule morphism; (b) εF : GF ξF / / F F µ / / F (= GF b κ / / GF εF / / F ) is an F -morphism; (c) c ommutativity of t he diagr am (7.5) F F G F ω / / F GF F εF / / F F µ   F G F η G O O ω / / GF εF / / F. If these c onditions ar e satisfie d, then b τ = µG · F b τ · F η G. Pro of. The pro of can b e obta ined b y some diag r am co nstructions. ⊔ ⊓ Notice that commutativit y of (7.3) implies commutativit y of (7.5). 7.8. Lemma. Assume t he diagr ams in (7.2 ) to b e c ommutative. Then the fol lowing ar e e qu ivalent: (a) for any ( A, υ ) ∈ A G , b η : A υ / / G ( A ) ξ A / / F ( A ) is a G -c omo dule m orphism; (b) b η G : G ηG / / F G b τ / / F G (= G δ / / GG ξG / / F G ) is G -c oline ar; 18 R OBER T W I S BA UER (c) c ommutativity of t he diagr am (7.6) G δ   ηG / / F G ω / / GF GG Gη G / / GF G Gω / / GGF. GεF O O If these c onditions ar e satisfie d, then b κ = GεF · G b κ · δ F . Pro of. The situation is dual to Lemma 7.7. ⊔ ⊓ Notice that commutativit y of (7.4) implies commutativit y of (7.6). 7.9. Prop ositio n . With the data given in 7.1 , assu me the diagr ams in (7.1), (7.2) and (7.5) to b e c ommutative. (1) If (7.6) is c ommutative, then ε fr om 7.7 is r e gular for δ , and for δ : G → GG with δ F : GF δF / / GGF G b κ / / GGF, ( G, δ , ε ) is an r -c ounital c omonad on A F . (2) If (7.4) is c ommutative, then δ F = δ F · b κ and ( G, δ , ε ) is a we ak c omonad on A F . Pro of. This can b e shown b y suitable diagr am co nstructions. ⊔ ⊓ 7.10. Prop osi tion. With the data given in 7.1 , assu me the diagr ams in (7.1), (7.2) , and (7.6) to b e c ommutative. (1) If (7.5) is c ommutative, then b η in 7.8 is r e gular for µ , and for b µ : F F → F with b µG : F F G F b τ / / F F G µG / / F G, ( b F , b µ, b η ) is an r -unital monad on A G . (2) If (7.3) is c ommutative, then b µG = b τ · µG and ( b F , b µ, b η ) is a we ak m onad on A G . Pro of. This is dual to Pr op osition 7.9. ⊔ ⊓ Ac kno wledgmen ts. The author wan ts to tha nk Gabriella B¨ ohm, T omasz Brzezi ´ ns ki and Bach uki Mesa blishvili for their interest in a pre v ious version of this pap er and for helpful comments o n the sub ject. References [1] Alonso ´ Alv arez, J.N., F ern´ andez Vilaboa, J.M. , Gonz´ alez Ro dr ´ ıguez, R., and Rodr ´ ıguez Rap oso, A.B. , Cr osse d pr o ducts in we ak c ontexts , Appl. Categ. Struct. 18(3) (2010) , 231-258. [2] Bec k, J., Distributive laws , [ in:] Se minar on T riples and Cate goric al Homolo gy Theo ry , B. Ec kmann (ed.), Springer LNM 80 (1969), 119-140. [3] B¨ ohm, G., The we ak the ory of monads , Adv. Math. 225(1) (2010), 1-32. [4] B¨ ohm, G., Lack , S. and Street, R. , On the 2-c ate gory of we ak distributive la ws , Commun. Al gebra 39(12) (2011), 4567-4583. [5] B¨ ohm, G., Lac k, S. and Street, R., Idemp otent splitti ngs, c olimit co mpletion, and we ak asp e cts of the the ory of monads , J. Pure Appl. Al gebra 216 (2012) , 385-403. [6] B¨ ohm, G., N ill, F. and Szlac h´ anyi, K. , Wea k Hopf algebr as I: Inte gra l the ory a nd C ∗ -structur e , J. Algebra 221(2) (1999), 385-438. [7] Brzezi´ nski, T. , The structur e of c orings. Induction functors, Maschke-typ e the or em, and F r ob enius and Galois-typ e pr op erties, Alg. Rep. Theory 5 (2002) , 389-410. [8] Brzezi´ nski, T. and Wisbauer, R. , Corings and Co mo dules , London Math. So c. Lecture Note Series 309, Cambridge Universit y Pr ess (2003). REGULAR P AIRINGS 19 [9] Caenepeel, S. and De Groot, E., Mo dules over we ak e ntwining structur es , Andruskiewitsc h, N. (ed.) et al., New trends in Hopf algebra theory . Pro c. Coll. quant um groups and Hopf algebras, La F alda, Argen tina, 1999. Pr o vidence, Amer . Math. So c., Conte mp. Math. 267 (2000), 31-54. [10] Eilenberg, S. and Moor e, J.C., A djoint functors and triples , Ill. J. Math. 9 (1965), 381-398. [11] F ern´ andez Vilab oa, J.M., Gonz´ alez Ro dr ´ ıguez, R. and Rodr ´ ıguez Rap oso, A.B., Pr eunits and we ak cr osse d pr o ducts , J. Pure Appl. A l gebra 213(12) (2009), 2244-2261. [12] F ern´ andez Vilab oa, J.M., Gonz´ alez Rodr ´ ıguez, R. and Ro dr ´ ıguez Raposo, A.B., We ak Cr osse d Bipr o d- ucts and Wea k Pr oje ctions , arXiv:0906.1693 (2009). [13] Johnstone, P .T., A djoint lifting t he or ems for c ate g ories of mo dules , Bull. Lond. M ath. So c. 7 (1975), 294-297. [14] Kasch, F. and M ader, A., R e gularity and substruct ur e s of H om , F rontiers in Mathematics, Birkh¨ auser Basel (2009) [15] Lack , S. and Street, R., The formal the ory of monad s II , J. Pure Appl. Algebra 175(1-3) (2002), 243-265. [16] Medved ev, M.Y a., Semiadjoint functors and Kan extensions , Si b. Math. J. 15 (1974), 674-676; trans- lation fr om Sib. Mat. Zh. 15 (1974), 952-956. [17] Mesablishvili, B. and Wisbauer, R., Bimonads and Hopf monads on c ate g ories , J. K-Theory 7(2) (2011), 349-388. [18] Mesablishvili, B. and Wisbauer, R., On R ational Pairings o f F unctors , arXiv:1003.3221 (2010), to appear in Appl. Cat. Struct., DOI: 10.1007/s1048 5-011-9264-1 [19] Pareigis, B., Kate gorien und F unktor en , M athematisc he Leitf¨ aden, T eubner V erl ag, Stuttga rt (1969). [20] Street, R., The formal the ory of monads , J. Pure Appl. A lgebra 2 (1972), 149168. [21] Wisbauer, R., We ak co rings , J. Algebra 245(1) (2001), 123-160. [22] Wisbauer, R., Algebr as versus c o algebr as , Appl. Categ. Struct. 16(1-2) (2008), 255-295. [23] Wisbauer, R., Lift ing the or ems for tensor functors on mo dule ca te gories , J. Algebra Appl. 10(1) (2011), 129-155. Dep a r tm ent of Ma thema tics, Heinrich Heine University D ¨ usseldorf, Germ any e-mail: wisbauer@ma th.uni-duesseld orf.de