On Rational Pairings of Functors

ON RA TIONAL P AIRINGS OF FUNCTORS BACHUKI MESABLISHVILI AND ROBER T WISBAUER Abstract. In the theory of coalgebras C o v er a ring R , the rational functor relates the category of mo dules o ver the algebra C ∗ (with conv olution product) with the category of como dules ov er C . It is based on the pairing of the algebra C ∗ with the coalgebra C prov ided b y the ev al uation map ev : C ∗ ⊗ R C → R . W e generalise this situation by deﬁning a p airing b et w een endofunctors T and G on any category A as a map, natural in a, b ∈ A , β a,b : A ( a, G ( b )) → A ( T ( a ) , b ) , and we call it r ational if these all are injective. In case T = ( T , m T , e T ) is a monad and G = ( G, δ G , ε G ) is a comonad on A , additional compatibility conditions are imp osed on a pairing b et ween T and G . If such a pairing i s give n and is rational, and T has a right adjoint monad T ⋄ , w e construct a r ational functor as the functor-part of an i dempoten t comonad on the T -mo dules A T which generalises the crucial prop erties of the rational functor f or coalgebras. As a special case w e consider pairi ngs on monoidal categories. Contents 1. Int ro duction 1 2. Preliminarie s 3 3. Pairings o f functors 5 4. Rational functors 12 5. Pairings in monoidal categories 15 6. Ent wining s in monoidal categ ories 19 References 31 1. Introduction The pairing of a k -v ector space V with its dual spa ce V ∗ = Hom( V , k ) provided b y the ev aluation ma p V ∗ ⊗ V → k can b e extended from base ﬁelds k to arbitr a ry base r ing s A . Then it can b e a pplied to the study o f A -corings C to o btain a faithful functor from the category o f C -como dules to the categor y of C ∗ -mo dules. The pur p ose o f this pap er it to extend these results to (endo)functors on arbitrary ca tegories. W e b egin by recalling s ome facts from mo dule theory . 1.1 . P airing of mo dules . Let C b e a bimo dule over a ring A and C ∗ = Hom A ( C, A ) the right dual. Then C ⊗ A − and C ∗ ⊗ A − a re endofunctors on the category A M of left A -mo dules and the ev aluation ev : C ∗ ⊗ A C → A, f ⊗ c 7→ f ( c ) , induces a pair ing b et w een these functors. F or left A -modules X, Y , the ma p α Y : C ⊗ A Y → A Hom( C ∗ , Y ) , c ⊗ y 7→ [ f 7→ f ( c ) y ] , induces the ma p β X,Y : A Hom( X , C ⊗ A Y ) − → A Hom( X , A Hom( C ∗ , Y )) , X f → C ⊗ A Y 7− → X f → C ⊗ A Y α Y − → A Hom( C ∗ , Y ) . 1 2 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER Clearly β X,Y is injective for all left A -modules X, Y if and only if α Y is a monomorphism (injectiv e) for any left A -mo dule Y , that is, C A is lo cally pro jective (see [1], [8, 42 .10]). Now consider the situatio n a bov e with so me additiona l structure. 1.2 . Pairings for cori ng s. Let C = ( C, ∆ , ε ) be a coring over the ring A , that is, C is a n A -bimo dule with bimo dule morphisms copro duct ∆ : C → C ⊗ A C and counit ε : C → A . Then the right dual C ∗ = Hom A ( C, A ) has a ring structure b y the con volution pro duct for f , g ∈ C ∗ , f ∗ g = f ◦ ( g ⊗ A I C ) ◦ ∆ (conv en tion o pp osite to [8, 17 .8]) with unit ε , and w e hav e a pairing b e t ween the co monad C ⊗ A − and the mo nad C ∗ ⊗ A − on A M . In this case, Hom A ( C ∗ , − ) is a comonad on A M and α Y considered in 1.1 induces a comonad morphism α : C ⊗ − → Hom A ( C ∗ , − ). W e hav e the commutative diagra ms (1.1) C ∗ ⊗ A C ∗ ⊗ A C I ⊗ I ⊗ ∆ / / ∗⊗ I   C ∗ ⊗ A C ∗ ⊗ A C ⊗ A C I ⊗ ev ⊗ I / / C ∗ ⊗ A C ev   C ε ⊗ I o o ε z z v v v v v v v v v v C ∗ ⊗ A C ev / / A . The Eilenberg- Moor e categor y M Hom( C ∗ , − ) of Hom( C ∗ , − )-como dules is equiv alent to the category C ∗ M of left C ∗ -mo dules (e.g. [5, Section 3 ]) and thus α induces a functor C M → M Hom( C ∗ , − ) ≃ C ∗ M which is fully faithful if a nd o nly if the pairing ( C ∗ , C, ev ) is ra tional, that is α Y is monomor ph for a ll Y ∈ A M (see [8, 19.2 and 19.3 ]). Moreov er, α is an isomor phis m if and only if the categorie s C M a nd C ∗ M a re equiv alent a nd this is tan tamount to C A being ﬁnitely generated and pro jective. In Section 2 w e recall the notions and some basic facts on natura l trans fo rmations b et w een endofuctors needed fo r our in vestigations. W eakening the conditions for a n adjoint pair of functors , a p airing of tw o functors T : A → B and G : B → A is deﬁned a s a ma p β a,b : A ( a, G ( b )) → A ( T ( a ) , b ), natur a l in a ∈ A , b ∈ B of tw o functors betw een arbitra ry categ ories is deﬁned in Section 3 (see 3.1) a nd it is calle d r ational if all the β a,b are injective maps. F or pair ing of mona ds T with como na ds G on a category A , a dditional conditions a re imp o sed on the deﬁning natural tr ansformations (see 3.2). These imply the existence of a functor Φ P : A G → A T from the G -co modules to the T -mo dules (see 3.5), which is full and faithful provided the pairing is r ational (see 3.7). Of sp ecial int erest is the situation that the monad T has a right adjoint T ⋄ and the la st pa rt of Section 3 is dea ling with this case. Referring to these results, a r ational fun ctor Rat P : A T → A T is asso ciated with any rational pair ing in Section 4. This leads to the deﬁnition of r ational T - modules and under some additional conditio ns they form a c o reﬂectiv e sub catgeory of A T (see 4.8). The a pplication of the genera l notions of pairings to monoidal ca tegories is o utlined in Section 5. The resulting formalism is very close to the mo dule cas e considered in 1.2. In Section 6, we apply our results to ent wining structures ( A , C , λ ) on monoidal ca tegories V = ( V , ⊗ , I ). The ob jects A and C incude a functor V ( − ⊗ C , A ) : V op → Set and if this is representable we call the ent w ining r epr esentable , that is if V ( − ⊗ C, A ) ≃ V ( − , E ) for so me ob ject E ∈ V . This E allows for an algebr a structure, a n alg ebra morphism A → E (see Prop osition 6.3) a nd a functor from the categ ory C A V ( λ ) of ent wined mo dules to the categor y E V of left E -mo dules. In case the tensor functor s hav e right adjoints a pairing on V is rela ted to the ent wining (see 6.9) and its prop erties a re studed. Several res ults known for the rationa l functors for o rdinary ent wined mo dules (see, for example, [1], [12] and [13] ) can b e obtained as coro llaries from the main result of this s ection (see Theo rem 6.9). ON RA TIONAL P AIRINGS OF FUNCTORS 3 2. Preliminaries In this section we re c all some notation a nd basic facts fr om catego ry theory . Througho ut A a nd B will de no te any categories . By I a , I A or just by I w e denote the identit y morphism of an ob ject a ∈ A , resp ectiv ely the identit y functor of a ca tegory A . 2.1 . Monads and comonads. F or a monad T = ( T , m T , e T ) on A , we write A T for the Eilenber g-Moo re category of T -mo dules; U T : A T → A , ( a, h a ) → a, for the under lying (forg etful) functor; φ T : A → A T , a → ( T ( a ) , ( m T ) a ) , for the free T -mo dule functor, and η T , ε T : φ T ⊣ U T : A T → A for the forge tfu l-free adjunction. Dually , if G = ( G, δ G , ε G ) is a co monad on A , we write A G for the categ ory of the E ilen berg- Moore category of G -como dules; U G : A G → A , ( a, θ a ) → a , for the for getful functor; φ G : A → A G , a → ( G ( a ) , ( δ G ) a ), for the co free G -co module functor, and η G , ε G : U G ⊣ φ G : A → A G for the forg etful-cofree adjunction. 2.2 . Idemp oten t comonads. A c o monad H = ( H , ε, δ ) is said to be idemp otent if one of the following equiv alent conditio ns is satisﬁed (see, for ex ample, [9]): (a) the forg etful functor U H : A H → A is full and faithful; (b) the unit η H : I → φ H U H of the adjunction U H ⊣ φ H is an isomo rphism; (c) δ : H → H H is an isomorphism; (d) for any ( a, ϑ a ) ∈ A H , the morphism ϑ a : a → H ( a ) is an iso morphism and ( ϑ a ) − 1 = ε a ; (e) H ε (or εH ) is an isomor phism. When H is a n idempotent comonad, then an ob ject a ∈ A is the carr ier of an H -como dule if and only if there exists an isomorphism H ( a ) ≃ a , or , equiv alently , if and only if the mor phism ε a : H ( a ) → a is an isomor phism. In this case, the pair ( a, ( ε a ) − 1 ) is a n H -como dule. In fact, every H -co module is of this form. Thus the categ ory A H is isomorphic to the full sub categor y of A genera ted by thos e o b jects for which there e x ists an isomorphism H ( a ) ≃ a . In par ticular, any comonad H = ( H, ε, δ ) with ε a comp onent wis e monomorphism is idem- po ten t (e.g . [9]). In this case, there is at most one mo rphism from a n y comonad H ′ to H . When H ′ is also an idempo ten t comonad, then a na tural tra nsformation τ : H ′ → H is a morphism of co monads if and only if ε · τ = ε ′ , where ε ′ is the co unit o f the comonad H ′ . W e will need the following r esult whose pr oof is an easy diagram chase: 2.3. Lemma. Supp ose that in the c ommutative diagr am a h 1   k / / b h 2   g / / f / / c h 3   a ′ k ′ / / b ′ g ′ / / f ′ / / c ′ , the b ottom r ow is an e qualiser and the morphism h 3 is a monomorphism. Then the left squ ar e in the diagr am is a pul lb ack if and only if the top r ow is an e qu ali ser diagr am. 2.4. Prop ositio n. L et t : G → R b e a natur al tra nsformation b etwe en endofunctors of A with c omp onentwise m onomo rphisms and assume that R pr eserves e qualisers. Then the fol lowing ar e e quivalent: (a) the functor G pr eserves e qualisers; 4 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER (b) for any r e gular monomorphism i : a 0 → a in A , the fol lowing s quar e is a pul lb ack: G ( a 0 ) G ( i ) / / t a 0   G ( a ) t a   R ( a 0 ) R ( i ) / / R ( a ) . When A admits and G pr eserves pushout s , (a) and (b) ar e e quivalent to: (c) G pr eserves r e gular monomorphisms, i.e. G t ake s a r e gular monomorphi sm into a r e gular monomorphism. Pr o of. (b) ⇒ (a) Let a k / / b g / / f / / c be an equaliser dia gram in A and consider the co mmutative diagra m G ( a ) t a   G ( k ) / / G ( b ) t b   G ( g ) / / G ( f ) / / G ( c ) t c   R ( a ) R ( k ) / / R ( b ) R ( g ) / / R ( f ) / / R ( c ) . Since R preserves equalisers , the b ottom row of this diagram is an equa lis er. By (b), the left square in the diagra m is a pullbac k. t c being a mo no morphism, it follo ws from Lemma 2.3 that the top row of the diagr am is an equa liser. Th us G preserves eq ua lisers. (a) ⇒ (b) Reconsider the diagram in the ab ov e pro of and apply Lemma 2 .3. Suppo se no w that A admits and G preserves pushouts. The implicatio n (a) ⇒ (c) alwa ys holds and so it remains to show (c) ⇒ (b) Consider an arbitra ry re g ular mo nomorphism i : a 0 → a in A . Since A admits pushouts and i is a regular mono mo rphism in A , the diagra m a 0 i / / a i 2 / / i 1 / / a ⊔ a 0 a , where i 1 and i 2 are the canonical injections into the pusho ut, is an equaliser diagr a m (e.g. [2, Prop osition 11.22]). Consider now the commutativ e dia gram G ( a 0 ) t a 0   G ( i ) / / G ( a ) t a   G ( i 2 ) / / G ( i 1 ) / / G ( a ⊔ a 0 a ) t a ⊔ a 0 a   R ( a 0 ) R ( i ) / / R ( a ) R ( i 2 ) / / R ( i 1 ) / / R ( a ⊔ a 0 a ) , in which the b ottom row is an equaliser diagr am since R preserves equalisers. Since G takes regular monomor phisms into reg ular monomor phisms and G preserves pushouts, the top row of the dia gram is also an e q ualiser diagra m. No w, using that t a ⊔ a 0 a is a mo no morphism, one ca n apply Lemma 2.3 to conclude that the squar e in the diagra m is a pullbac k showing (c) ⇒ (b).  ON RA TIONAL P AIRINGS OF FUNCTORS 5 3. P airings of functors Generalising the results sket ched in the in tro ductio n we deﬁne the notion of pairings of functors on a rbitrary catego ries. 3.1 . Pairing of functors. F o r a n y functors T : A → B and G : B → A , there is a bijection (e.g. [20, 2.1 ]) b et w een (the cla ss of ) natural transformations b et w een functors A op × B → Set, β a,b : A ( a, G ( b )) → A ( T ( a ) , b ) , a ∈ A , b ∈ B , and natura l transformatio ns σ : T G → I B , with σ a := β G ( a ) ,a ( I G ( a ) ) : T G ( a ) → a and β a,b : a f → G ( b ) 7− → T ( a ) T ( f ) → T G ( b ) σ b → b. W e call ( T , G, σ ) a p airi ng b et w een the functor s T and G and name it a r ational p airing provided the β a,b are monomor phisms for all a ∈ A and b ∈ B . Clearly , if all the β a,b are isomor phisms, then we hav e a n adjoint p airi ng , that is, the functor G is rig ht adjoin t to T a nd σ is just the co unit of the adjunction. Thu s rational pairings generalise adjoin tness. W e ment ion that, given a pairing ( T , G, σ ), Medvedev [17] calls T a left semiadjoint to G pr o vided there is a natural transfor ma tion ϕ : I A → GT such that σ T ◦ T ϕ = I T . This mea ns tha t β − , − is a bifunctoria l c oretraction (dual of [1 7, Prop osition 1]) in which case σ is a rational pair ing. In case all β a,b are epimorphisms, the functor G is said to be a we ak right adjoi nt to T in Kainen [1 5]. F or more a bout w eakened forms of adjo intness we refer to B¨ orger and Tholen [6]. Similar to the condition on an adjoint pa ir of a monad and a co monad (see [11]) we deﬁne 3.2 . Pa iring of monads and comonads. A p airing P = ( T , G , σ ) b e t ween a monad T = ( T , m, e ) a nd a c omonad G = ( G, δ, ε ) on a catego ry A is a pairing σ : T G → I b et w een the functors T and G inducing - for a, b ∈ A - commutativit y of the diagr ams (3.1) A ( a, b ) A ( a, G ( b )) A ( a,δ b )   β P a,b / / A ( a,ε b ) 7 7 n n n n n n n n n n n n A ( T ( a ) , b ) A ( m a ,b )   A ( e a , b ) h h P P P P P P P P P P P P A ( a, G 2 ( b )) β P a,G ( b ) / / A ( T ( a ) , G ( b )) β P T ( a ) ,b / / A ( T 2 ( a ) , b ) , where (3.2) β P a,b : A ( a, G ( b )) → A ( T ( a ) , b ) , f : a → G ( b ) 7→ σ b · T ( f ) : T ( a ) → b. The pairing P = ( T , G , σ ) is said to b e r ational if β P a,b is injective for a n y a, b ∈ A . By the Y o neda Lemma, co mm utativ ity of the diag r ams in (3.1) corresp ond to commuta- tivit y of the dia g rams (3.3) G eG / / ε B B B B B B B B T G σ   I , T 2 G T 2 δ / / mG   T 2 G 2 T σG / / T G σ   T G σ / / I . 3.3 . P airings and morphis ms. L et P = ( T , G , σ ) b e a p airing, T ′ = ( T ′ , m ′ , e ′ ) any monad, and t : T ′ → T a monad morphism. (i) The triple P ′ = ( T ′ , G , σ ′ := σ · tG ) is also a p airing. (ii) If P is r ational, t he n P ′ is also r ational pr ovid e d the natur al tra nsformation t is a c omp onentwise epimorphi sm. 6 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER Pr o of. (i). The diagra m G eG C C C C ! ! C C C C ε   e ′ G / / T ′ G tG   I T G σ o o commutes s inc e t is a monad morphism (th us t · e ′ = e ) a nd b ecause o f the commutativit y o f the triangle in the diagra m (3.3.) Thus σ ′ · e ′ G = σ · tG · e ′ G = σ · e G = ε . Consider now the diag ram T ′ T ′ G T ′ T ′ δ / / ttG   m ′ G z z v v v v v v v v v T ′ T ′ GG ttGG % % L L L L L L L L L L T ′ tGG / / T ′ T GG (4) tT GG y y s s s s s s s s s s T ′ σG / / T ′ G tG | | y y y y y y y y T ′ G tG # # H H H H H H H H H (1) T T G mG   (2) T T δ / / T T GG (3) T σG / / T G σ z z v v v v v v v v v v T G (5) σ / / I , in which diag ram (1) comm utes since t is a morphism of monads , diagrams (2), (3) and (4) commute by natura lit y o f comp osition, and diagram (5) commut es by commutativit y of the rectangle in (3.3). It then follows that σ ′ · T σ ′ G · T ′ T ′ δ = σ · tG · T ′ σ G · T ′ tGG · T ′ T ′ δ = σ · tG · m ′ G = σ ′ · m ′ G, proving that the triple P ′ = ( T ′ , G , σ ′ = σ · tG ) is a pairing. (ii). It is ea sy to check that the comp osite A ( a, G ( b )) β P a,b − − → A ( T ( a ) , b ) A ( t a ,b ) − − − − → A ( T ′ ( a ) , b ) takes f : a → G ( b ) to σ ′ b · T ′ ( f ) : T ′ ( a ) → b and th us − since T ( f ) · t a = t G ( a ) · T ′ ( f ) by naturality of t − β P ′ a,b = A ( t a , b ) · β P a,b . If t is a comp onen t wise epimorphism, then t a is an epimor phism, and then the map A ( t a , b ) is injective. It follows that β P ′ a,b is als o injective provided that β P a,b is injective (i.e. the pairing P = ( T , G , σ ) is rationa l).  Dually , o ne has 3.4. Prop osition. L et P = ( T , G , σ ) b e a p airing, G ′ = ( G ′ , δ ′ , ε ′ ) any c omonad, and t : G ′ → G a c omonad morphism. (i) The triple P ′ = ( T , G ′ , σ ′ := σ · T t ) is also a p airi ng. (ii) If P is r ational, t he n P ′ is also r ational pr ovid e d the natur al tra nsformation t is a c omp onentwise monomorphism. 3.5 . F unctors induced b y pairings . L et P = ( T , G , σ ) b e a p airi ng on a c ate gory A with β P a, b : A ( a, G ( b )) → A ( T ( a ) , b ) (s e e (3.2). (1) If ( a, θ a ) ∈ A G , then ( a, β P a,a ( θ a )) = ( a, σ a · T ( θ a )) ∈ A T . (2) The assignments ( a, θ a ) 7− → ( a, σ a · T ( θ a )) , f : a → b 7− → f : a → b, yield a c onservative functor Φ P : A G → A T inducing a c ommutative diagr am (3.4) A G Φ P / / U G ! ! C C C C C C C C A T U T } } { { { { { { { { A . ON RA TIONAL P AIRINGS OF FUNCTORS 7 Pr o of. (1) W e have to show tha t the diagrams a e a / / C C C C C C C C C C C C C C C C C C T ( a ) β P a,a ( θ a )   a, T 2 ( a ) m a / / T ( β P a,a ( θ a ))   T ( a ) β P a,a ( θ a )   T ( a ) β P a,a ( θ a ) / / a are commutativ e. In the diagr am a θ a / / e a   G ( a ) e G ( a )   ε a ! ! D D D D D D D D D T ( a ) T ( θ a ) / / T G ( a ) σ a / / a the sq uare commutes by na turalit y o f e : I → T , while the tr iangle commutes by (3.3). Thus β P a,a ( θ a ) · e a = σ a · T ( θ a ) · e a = ε a · θ a . B ut ε a · θ a = I a since ( a, θ a ) ∈ A G , implying tha t β P a,a ( θ a ) · e a = I a . This shows that the left hand diag ram co mm utes. Since β P a,a ( θ a ) = σ a · T ( θ a ), and T ( θ a ) · T ( σ a ) = T ( σ G ( a ) ) · T 2 G ( θ a ) by natura lit y of σ , the right hand diagram can b e rewritten as T 2 ( a ) m a / / T 2 ( θ a )   T ( a ) T ( θ a ) / / T G ( a ) σ a " " D D D D D D D D D T 2 G ( a ) T 2 G ( θ a ) / / T 2 G 2 ( a ) T ( σ G ( a ) ) / / T G ( a ) σ a / / a . It is easy to see that σ a · T ( θ a ) · m a = ( A ( m a , a ) · β P a,a )( θ a ) and it follows from the c o mm uta tivit y of the b ottom diagr am in (3.1) that σ a · T ( θ a ) · m a = ( β P T ( a ) ,a · β P a,G ( a ) · A ( a, δ a ))( θ a ) = σ a · T ( σ G ( a ) ) · T 2 ( δ a ) · T 2 ( θ a ) . Recalling that δ a · θ a = G ( θ a ) · θ a since ( a, θ a ) ∈ A G , w e get σ a · T ( θ a ) · m a = σ a · T ( σ G ( a ) ) · T 2 G ( θ a ) · T 2 ( θ a ) , proving that the right hand diagr am is co mm utative. Th us ( a, β P a,a ( θ a )) ∈ A T . (2) By (1), it suﬃces to s how that if f : ( a, θ a ) → ( b, θ b ) is a mo rphism in A G , then f is a morphism in A T from the T -mo dule ( a, σ a · T ( θ a )) to the T -mo dule ( b, σ b · T ( θ b )). T o say that f : a → b is a morphism in A T is to say that the outer diagr am of T ( a ) T ( θ a ) / / T ( f )   T G ( a ) σ a / / T G ( f )   a f   T ( b ) T ( θ b ) / / T G ( b ) σ b / / b is c omm utativ e, which is indeed the ca se since the left squar e commutes b ecause f is a mor- phism in A G , while the r ig h t square commutes b y naturality of σ . Clearly Φ P is co nserv ative. The commutativit y o f the diagram of functors is o b v ious.  3.6 . Prop erties of the functor Φ P . L et P = ( T , G , σ ) b e a p airing on a c ate gory A with induc e d fun ctor Φ P : A G → A T (se e 3.5). (1) The functor Φ P is c omonadic if and only if it has a right adjoint. 8 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER (2) L et C b e a sm all c ate gory such that any functor C → A has a c olimit t hat is pr eserve d by T . Then the functor Φ P pr eserves the c olimi t of any functor C → A G . (3) When A admits and T pr eserves al l s mall c olimits, the c ate gory A G has and the functor Φ P pr eserves al l smal l c olimits. (4) L et A b e a lo c al ly pr esentable c ate gory, su pp ose that G pr eserves ﬁlter e d c olimits and T pr eserves al l smal l c olimits. Then Φ P is c omonadic. Pr o of. (1) Since U G creates equalisers of G -split pair s, U T creates all equa lisers that exist in A , and U T Φ P = U G , A G admits equalisers of Φ P -split pairs and Φ P preserves them. The result now follows from 3.5(2). (2) Since T preserves the colimit of any functor C → A , the catego r y A T admits and the functor U T preserves the colimit of any functor C → A T (see [7]). No w, if F : C → A G is an arbitra ry functor, then the compos ition Φ P · F : C → A T has a colimit in A T . Since U T · Φ P = U G and U G preserves all colimits that exist in A G , it follows that the functor U T preserves the c o limit o f the comp osite Φ P · F . No w the asser tion follows fro m the fact that an arbitrar y conserv ativ e functor r eﬂects such colimits as it pr eserves. (3) is a cor ollary of (2). (4) Note ﬁrs t that Ad´ amek and Rosick´ y prov ed in [2] that the Eilenber g-Moo re categ ory with res p ect to a ﬁltered- c olimit pr eserving mo nad on a lo cally presentable categ ory is loc a lly presentable. T his proo f can b e adopted to show that if G preserves ﬁltered colimits and A is lo cally presentable, then A G is a lso lo cally pr esen table. Therefore A G is ﬁnitely complete, co complete, co- wellp owered and has a small set of gener a tors (see [2]). Since the functor Φ P preserves a ll small co limits by (3), it follows from the (dual of the) Sp ecial Adjoint F unctor Theorem (see [16]) that Φ P admits a right adjoint functor. Com bining this with (1.ii) g iv es that the functor Φ P is comona dic.  3.7 . Prop erties of rational pairings. L et P = ( T , G , σ ) b e r ational p airing on A . (1) The functor Φ P : A G → A T is ful l and faithful. (2) The functor G pr eserves monomorphisms. (3) If A is ab elian and G pr eserves c okernels, then G is left exact. Pr o of. (1) Obviously , Φ P is faithful. Let ( a, θ a ) , ( b, θ b ) ∈ A G and let f : Φ P ( a, θ a ) = ( a, σ a · T ( θ a )) → ( b, σ b · T ( θ b )) = Φ P ( b, θ b ) be a morphism in A T . W e hav e to show that the diagram (3.5) a f / / θ a   b θ b   G ( a ) G ( f ) / / G ( b ) commutes. Since f is a morphism in A T , the diagra m (3.6) T ( a ) T ( θ a ) / / T ( f )   T G ( a ) σ a / / a f   T ( b ) T ( θ b ) / / T G ( b ) σ b / / b commutes and we hav e β P a,b ( θ b · f ) = σ b · T ( θ b · f ) = σ b · T ( θ b ) · T ( f ) by (3.6) = f · σ a · T ( θ a ) σ is a n atural = σ b · T G ( f ) · T ( θ a ) = σ b · T ( G ( f ) · θ a ) = β P a,b ( G ( f ) · θ a ) . ON RA TIONAL P AIRINGS OF FUNCTORS 9 Since β P a,b is injectiv e by ass umption, it follows that G ( f ) · θ a = θ b · f , tha t is, (3.5) commutes. (2) Let f : a → b b e a mo nomorphism in A . Then for any x ∈ A , the ma p A ( T ( x ) , f ) : A ( T ( x ) , a ) → A ( T ( x ) , b ) is injective. Considering the commutativ e diagram A ( x, G ( a )) β P x,a / / A ( x,G ( f ))   A ( T ( x ) , a ) A ( T ( x ) ,f )   A ( x, G ( b )) β P x,b / / A ( T ( x ) , b ) , one sees tha t the map A ( x, G ( f )) is injectiv e for all x ∈ A , proving that G ( f ) is a monomor- phism in A . (3) is a conse quence of (2).  3.8 . Como nad m o rphisms. Recall (e.g . [3]) that a morphism o f comonads t : G → G ′ induces a functor t ∗ : A G → A G ′ , ( a, θ a ) 7→ ( a, t a · θ a ) . The pa ssage t → t ∗ yields a bijection betw een comonad morphisms G → G ′ and functors V : A G → A G ′ with U G ′ V = U G . If V : A G → A G ′ is such a functor, then the image o f any cofree G -como dule ( G ( a ) , δ a ) under V has the for m ( G ( a ) , s a ) for some s a : G ( a ) → G ′ G ( a ). Then the collec tio n { s a | a ∈ A } constitute a natural transfor mation s : G → G ′ G such that G ′ ε · s : G → G ′ is a comonad morphism. 3.9 . R igh t adjoin t for T . Let P = ( T , G , σ ) b e a pairing and supp ose that there exists a comonad T ⋄ = ( T ⋄ , δ ⋄ , ε ⋄ ) that is rig h t adjoint to the monad T with unit η : 1 → T ⋄ T . In this situa tion, the functor K T ,T ⋄ : A T → A T ⋄ that takes any ( a, h a ) ∈ A T to ( a, T ⋄ ( h a ) · η a ), is an isomorphism of ca tegories and U T ⋄ K T ,T ⋄ = U T (e.g. [19]). Since U T Φ P = U G , one gets the commutativ e dia gram A G Φ P / / U G ' ' O O O O O O O O O O O O O A T K T ,T ⋄ / / A T ⋄ U T ⋄ w w o o o o o o o o o o o o o o A . It follows tha t ther e is a morphism of como nads t : G → T ⋄ with K T ,T ⋄ Φ P = t ∗ . 3.10. Lemma. In the situation given in 3.9, t is the c omp osite G ηG / / T ⋄ T G T ⋄ σ / / T ⋄ . Pr o of. Since • for any ( a, θ a ) ∈ A G , Φ P ( a, θ a ) = ( a, σ a · T ( θ a )), and • for any ( a, h a ) ∈ A T , K T ,T ⋄ ( a, h a ) = ( a, T ⋄ ( h a ) · η a ), it follows tha t for any cofree G -como dule ( G ( a ) , δ a ), K T ,T ⋄ Φ P ( G ( a ) , δ a ) = ( G ( a ) , T ⋄ ( σ G ( a ) ) · T ⋄ T ( δ a ) · η G ( a ) ) , th us t a = T ⋄ ( ε a ) · T ⋄ ( σ G ( a ) ) · T ⋄ T ( δ a ) · η G ( a ) . But since T ⋄ ( ε a ) · T ⋄ ( σ G ( a ) ) = T ⋄ ( σ a ) · T ⋄ T G ( ε a ) 10 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER by naturality of σ and G ( ε a ) · δ a = I G ( a ) , one has t a = T ⋄ ( σ a ) · T ⋄ T G ( ε a ) · T ⋄ T ( δ a ) · η G ( a ) = T ⋄ ( σ a ) · η G ( a ) .  3.11 . Rig ht adjoi nt functor of t ∗ . In the setting of 3.9, supp ose that the ca teg ory A G admits equalisers. Then it is well-known (e.g. [10]) that for a n y comonad morphism t : G → T ⋄ , the functor t ∗ : A G → A T ⋄ admits a right adjoint t ∗ : A T ⋄ → A G , which can b e calculated as follows. Reca ll from 2 .1 that, for an y co mo nad G , we denote b y η G and ε G the unit and counit of the a djo int pair ( U G , φ G ). W riting α t for the comp osite t ∗ φ G η T ⋄ t ∗ φ G − − − − − − → φ T ⋄ U T ⋄ t ∗ φ G = φ T ⋄ U G φ G φ T ⋄ ε G − − − − → φ T ⋄ , and β t for the comp osite φ G U T ⋄ η G φ G U T ⋄ − − − − − − → φ G U G φ G U T ⋄ = φ G U T ⋄ t ∗ φ G U T ⋄ φ G U T ⋄ α t U T ⋄ − − − − − − − − − → φ G U T ⋄ φ T ⋄ U T ⋄ , then t ∗ is the eq ua liser (we as sumed that A G has equaliser s) t ∗ i t / / φ G U T ⋄ β t / / φ G U T ⋄ η T ⋄ / / φ G U T ⋄ φ T ⋄ U T ⋄ . Note that the co unit ε t : t ∗ t ∗ → I o f the adjunction t ∗ ⊣ t ∗ is the unique natura l tr ansfor- mation that makes the square in the following diag ram commute, (3.7) t ∗ t ∗ ε t t ∗ i t / / t ∗ φ G U T ⋄ α t U T ⋄   t ∗ β t / / t ∗ φ G U T ⋄ η T ⋄ / / t ∗ φ G U T ⋄ φ T ⋄ U T ⋄ α t U T ⋄ φ T ⋄ U T ⋄   I η T ⋄ / / φ T ⋄ U T ⋄ η T ⋄ φ T ⋄ U T ⋄ / / φ T ⋄ U T ⋄ η T ⋄ / / φ T ⋄ U T ⋄ φ T ⋄ U T ⋄ . It is not hard to see that for any a ∈ A , the a -co mponent of the natural transfor mation α t is just the morphism t a : G ( a ) → T ⋄ ( a ) seen a s a mor phism t ∗ φ G ( a ) = ( G ( a ) , t G ( a ) · δ a ) → φ T ⋄ ( a ) = ( T ⋄ ( a ) , δ ⋄ a ) in A T ⋄ . Indeed, since for any a ∈ A , ( ε G ) a = ε a , while fo r any ( a, ν a ) ∈ A T ⋄ , ( η T ⋄ ) ( a,ν a ) = ν a , ( α t ) a is the comp osite T ⋄ ( ε a ) · t G ( a ) · δ a . Considering now the diagram G ( a ) δ a / / H H H H H H H H H H H H H H H H H H GG ( a ) t G ( a ) / / G ( ε a )   T ⋄ G ( a ) T ⋄ ( ε a )   G ( a ) t a / / T ⋄ ( a ) , in which the square co mmutes by natur alit y of comp osition, while the triangle commutes by the deﬁnition of a comonad, one see s that ( α t ) a = t a . 3.12. Theorem. L et P = ( T , G , σ ) b e a p airing on A . Supp ose that A G admits e qualisers and that the monad T has a right adjoi nt c omonad T ⋄ . Then (1) the functor Φ P : A G → A T (and henc e also t ∗ : A G → A T ⋄ ) is c omonadic; (2) if the p airing P is r ational, t hen A G is e quivalent to a r eﬂe ctive sub c ate gory of A T . Pr o of. By 3.6(1), Φ P is comona dic if and o nly if it has a right adjo int. (1) Since the catego ry A G admits eq ua lisers, the functor t ∗ : A G → A T ⋄ has a right a djo int t ∗ : A T ∗ → A G (b y 3.1 1 ). Then evidently the functor t ∗ K T ,G is rig h t a djo int to the functor Φ P . Since K T ,T ⋄ Φ P = t ∗ , it is clear that t ∗ is als o co monadic. This c ompletes the pro of o f the ﬁrst part. ON RA TIONAL P AIRINGS OF FUNCTORS 11 (2) If P is rationa l, Φ P is full and faithful by 3.7(1), and when Φ P has a right adjoint, the unit of the a djunction is a comp onen t wise isomorphism (see [16]).  Given tw o functor s F , F ′ : A → B , we write Nat( F, F ′ ) for the collection of all natural transformatio ns from F to F ′ . As a co nsequence o f the Y o neda Lemma recall: 3.13. Lemm a. L et T = ( T , m, e ) b e a monad on the c ate gory A with right adjoint c omonad T ⋄ = ( T ⋄ , δ ⋄ , ε ⋄ ) , u nit η : I → T ⋄ T and c ounit ε : T T ⋄ → I . Then for any endofunctor G : A → A , ther e is a bije ct io n χ : Nat( T G, I ) → Nat( G, T ⋄ ) , T G σ → I 7− → G ηG → T ⋄ T G σ → T ⋄ , with the inverse given by the assignment G s → T ⋄ 7− → T G T s → T T ⋄ ε → I . 3.14. Prop osi ti on. L et T b e a monad on a c ate gory A with right adjoint c omonad T ⋄ (as in 3.13). Then, for a c omonad G = ( G, δ, ε ) on A and a natur al tr ansformation σ : T G → I , the fol lowing ar e e quivalent: (a) The triple P = ( T , G , σ ) is a p airing. (b) χ ( σ ) : G → T ⋄ is a morphism of c omonads. Pr o of. (a) ⇒ (b) follows from Lemma 3.10, while (b) ⇒ (a) follows from the dual of 3.3.  3.15. Prop osition. L et T b e a monad on A with right adjoint c omonad T ⋄ (as in 3.13) and let G = ( G, δ, ε ) b e a c omonad on A . (1) Ther e ex ists a bije ction b etwe en (i) natur al tr ansformations σ : T G → I for which the triple P = ( T , G , σ ) is a p airi ng; (ii) c omonad morphisms G → T ⋄ ; (iii) functors V : A G → A T such that U T V = U G . (2) A fu n ctor V : A G → A T with U T V = U G is an isomorphi sm if and only if ther e exists an isomorphism of c omonad s G ≃ T ⋄ . Pr o of. (1) By Pro position 3.14, it is enough to s ho w that there is a bijective co rresp ondence betw een comona d mor phisms G → T ⋄ and functors V : A G → A T such that U T V = U G . But to g iv e a como nad morphism G → T ⋄ is to g iv e a functor W : A G → A T ⋄ with U T ⋄ W = U G , which is in turn eq uiv alent - since K − 1 T ,T ⋄ is an iso morphism of categor ie s with U T K − 1 T ,T ⋄ = U T ⋄ (see 3.9) - to g iving a functor V : A G → A T with U T V = U G . (2) According to (1), to give a functor V : A G → A T with U T V = U G is to give a c o monad morphism t : G → T ⋄ such that t ∗ = K T ,T ⋄ V . It fo llows - since K T ,T ⋄ is a n isomor phism o f categorie s - that V is an isomorphism of categories if and only if t ∗ is, o r, eq uiv alently , if t is an isomorphism o f co mo nads.  3.16. Prop osition. L et P = ( T , G , σ ) b e a p airing on a c ate gory A and T ⋄ = ( T ⋄ , δ ⋄ , ε ⋄ ) a c omonad right adjoint t o T . Consider the statements: (i) the p airi ng P is r ational; (ii) χ ( σ ) : G → T ⋄ is c omp onentwise a monomorphism; (iii) the functor Φ P : A G → A T is ful l and faithful. Then one has the implic ations (i) ⇔ (ii) ⇒ (iii) . Pr o of. (i) ⇒ (iii) is just 3.7(1). (i) ⇔ (ii) F or any a ∈ A , cons ider the natura l transfo r mation A ( a, G ( − )) β P a, − / / A ( T ( a ) , − ) α a, − / / A ( a, T ⋄ ( − )) , 12 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER where α denotes the isomorphism of the adjunction T ⊣ T ⋄ . It is ea sy to see that the induced natural transfor ma tion G → T ⋄ is just χ ( σ ). Since ea ch co mponent o f α a, − is a bijection, it follows that χ ( σ ) is a co mponent wise monomorphism if and only if ea c h compo nen t of β P a, − is monomorphism for a ll a ∈ A .  4. Ra tional functors Let A b e an arbitrar y categor y admitting pullbacks. 4.1 . Throug hout this section w e ﬁx a ra tional pairing P = ( T , G , σ ) on the categ ory A with a co monad T ⋄ = ( T ⋄ , δ ⋄ , ε ⋄ ) right adjoint to T , unit η : 1 → T ⋄ T and counit ε : T T ⋄ → I . Let t = χ ( σ ) (see Lemma 3 .13). F or any ( a, ϑ a ) ∈ A T ⋄ , write Υ( a, ϑ a ) for a chosen pullback (4.1) Υ( a, ϑ a ) p 2 / / p 1   G ( a ) t a   a ϑ a / / T ⋄ ( a ) . Since mono morphisms a r e s table under pullbacks in any categor y and s inc e t a and ϑ a are b oth monomorphisms, it follows that p 1 and p 2 are also monomor phisms. As a rig ht adjoint functor, T ⋄ preserves a ll limits existing in A and th us the categ ory A T ⋄ admits those limits existing in A . Moreover, the for g etful functor U T ⋄ : A T ⋄ → A creates them; hence these limits (in par ticular pullbacks) ca n b e computed in A . Now, ϑ a : a → T ⋄ ( a ) is the ( a, ϑ a )-comp onen t of the unit η T ⋄ : I → φ T ⋄ U T ⋄ and thus it can b e seen as a morphism in A T ⋄ from ( a, ϑ a ) to ( T ⋄ ( a ) , ( δ ⋄ ) a ), while t a : G ( a ) → T ⋄ ( a ) is the U T ⋄ ( a, ϑ a )-comp onen t of the natural transformation α t : t ∗ φ T ⋄ → φ T ⋄ (see 3 .1 1) a nd th us it can b e seen as a mo rphism in A T ⋄ from t ∗ ( G ( a ) , δ a ) = ( G ( a ) , t G ( a ) · δ a ) to ( T ⋄ ( a ) , δ ⋄ a ). It follows that the diagram (4.1) underlies a pullback in A T ⋄ . In other words, there exists exactly one T ⋄ -coalge br a structure ϑ Υ( a,ϑ a ) : Υ( a, ϑ a ) → T ⋄ (Υ( a, ϑ a )) o n Υ ( a, ϑ a ) making the diagram (4.2) (Υ( a, ϑ a ) , ϑ Υ( a,ϑ a ) ) p 2 / / p 1   ( G ( a ) , t G ( a ) · ( δ G ) a ) t a   ( a, ϑ a ) ϑ a / / ( T ⋄ ( a ) , ( δ ⋄ ) a ) a pullback in A T ⋄ . Mor eo v er, since in any functor category , pullbacks ar e computed comp o- nent wise, it follows that the diagr am (4.2) is the ( a, ϑ a )-comp onen t of a pullback diag r am in A T ⋄ , (4.3) Υ P 2 / / P 1   t ∗ φ G U T ⋄ α t U T ⋄   I η T ⋄ / / φ T ⋄ U T ⋄ . Since the forg etful functor U T : A T → A r e s pects monomorphisms a nd U T ⋄ K T ,T ⋄ = U T , it follows that the forgetful functor U T ⋄ : A T ⋄ → A also resp ects monomorphisms. Thus, the natural tr ansformations α t U T ⋄ and η T ⋄ are b oth compo nen t wise monomorphisms a nd hence so to o is the natura l transfo rmation P 1 : Υ → I . ON RA TIONAL P AIRINGS OF FUNCTORS 13 Summing up, we ha ve seen that for a n y ( a, ϑ a ) ∈ A T ⋄ , Υ( a, ϑ a ) is a n ob ject of A T ⋄ yielding an endofunctor Υ : A T ⋄ → A T ⋄ , ( a, ϑ a ) 7→ Υ( a, ϑ a ) . As we s hall see later on, the e ndo functor Υ is - under s ome assumptions - the functor -part of an idemp oten t comonad on A T ⋄ . 4.2. Prop osition. Under the assumptions fr om 4.1, supp ose that A admits and G pr eserves e qualisers. Then t he c ate gory A G also admits e qualisers and the functor t ∗ : A G → A T ⋄ pr eserves t hem. Pr o of. Since the functor G pres erv es equa lisers, the for getful functor U G : A G → A crea tes and prese r v es equalisers . Thus A G admits equalis ers. Since the forg etful functor U T ⋄ : A T ⋄ → A also cr eates and pres erv es equalisers, it follows from the commutativit y of the diag r am A G t ∗ / / U G " " D D D D D D D D A T ⋄ U T ⋄   A that the functor t ∗ : A G → A T ⋄ preserves equa lisers.  Note that when A is an ab elian categor y , and G preserves cokernels, then G preserves equalisers by Pr opositio n 2.4 and by 3.7(3). Note also that it follows fro m the previous propo sition that if A a dmits and G preserves equalisers, then the category A G admits eq ua lisers. In view of Prop osition 4.2, it then follows from 3.11 that the functor t ∗ : A G → A T ⋄ has a right a djoin t t ∗ : A T ⋄ → A G . 4.3. Prop osition. Under the c onditions of Pr op osition 4.2, the functor Υ : A T ⋄ → A T ⋄ is isomorphi c to the functor p art t ∗ t ∗ of the A T ⋄ -c omonad gener ate d by the adjunction t ∗ ⊣ t ∗ : A T ⋄ → A G . Pr o of. Since the functor G prese r v es equalisers, the functor t ∗ : A G → A T ⋄ also preserves equalisers by Pro position 4.2. Then the top row in the dia g ram (3.7) is an equaliser, a nd since the bottom ro w is also an equaliser and the natural transfo r mation α t : t ∗ φ G → φ T ⋄ is a comp onen t wise monomo rphism (since ( α ) a = t a for all a ∈ A , see 3.11), it follows fro m Lemma 2.3 that the square in the diagram (3.7) is a pullback. Compa r ing this pullback with (4.3), one se e s that Υ is isomor phic to t ∗ t ∗ (and P 1 to ε t ).  Note that in the s ituation of the pr evious prop o sition, ε t : Υ → I is co mponent wise a monomorphism. F or the next results we will assume that the A ha s and G pres erv es equaliser s. This implies in particular that the categor y A G admits equalis ers. 4.4. Prop osition. With the data given in 4.1 assu me that A has and G pr eserves e qualisers. L et ( a, ϑ a ) b e an any obje ct of A T ⋄ . Then (i) Υ(Υ( a, ϑ a )) ≃ Υ( a, ϑ a ) . (ii) F or every r e gu la r T ⋄ -sub c omo dule ( a 0 , ϑ a 0 ) of ( a, ϑ a ) , the fol lowing diag r am is a pu l lb ack, Υ( a 0 , ϑa 0 ) ( ε t ) ( a 0 ,ϑa 0 )   Υ( i ) / / Υ( a, ϑ a,ϑ a ) ( ε t ) ( a,ϑ a )   a 0 i / / a. 14 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER Pr o of. (i) As we have observed a fter Pr oposition 4.3, the na tural tr a nsformation ε t : Υ → I is comp onen t wise monomorphis m. Thus Υ is an idemp otent endofunctor, that is, Υ (Υ( a, ϑ a )) ≃ Υ( a, ϑ a ) . (ii) F or any regula r monomorphism i : ( a 0 , ϑ a 0 ) → ( a, ϑ a ) in A T ⋄ , co ns ider the commutativ e diagram (4.4) Υ( a 0 , ϑ a 0 ) ( ε t ) ( a 0 ,ϑ a 0 )   ( i t ) ( a 0 ,ϑ a 0 ) % % L L L L L L L L L L Υ( i ) / / Υ( a, ϑ a ) ( ε t ) ( a,ϑ a )   ( i t ) ( a,ϑ a ) % % K K K K K K K K K G ( a 0 ) t a 0   G ( i ) / / G ( a ) t a   a 0 i / / ϑ a 0 & & M M M M M M M M M M M a ϑ a % % T ⋄ ( a 0 ) T ⋄ ( i ) / / T ⋄ ( a ) . W e claim that the squar e ( ε t ) ( a,ϑ a ) · Υ( i ) = i · ( ε t ) ( a 0 ,ϑ a 0 ) is a pullback. Indeed, if f : x → a 0 and g : x → Υ( a, ϑ a ) are morphisms such that i · f = ( ε t ) ( a 0 ,ϑ a 0 ) · g , then we hav e T ⋄ ( i ) · ϑ a 0 · f = ϑ a · i · f = ϑ a · ( ε t ) ( a 0 ,ϑ a 0 ) · g = t a · ( i t ) ( a,ϑ a ) · g , and since the squa re t a · G ( i ) = T ⋄ ( i ) · t a 0 is a pullba c k, there exis ts a unique morphism k : x → G ( a 0 ) with t a 0 · k = ϑ a 0 · f and ( i t ) ( a,ϑ a ) · g = G ( i ) · k . Since the functor T ⋄ , as a right adjoint functor , preserves regular monomorphisms, the forgetful functor U T ⋄ : A T ⋄ → A also preserves r egular monomorphisms. Th us i : a 0 → a is a regular monomorphism in A . Then the square t a 0 · ( i t ) ( a 0 ,ϑ a 0 ) = ϑ a 0 · ( ε t ) ( a 0 ,ϑ a 0 ) is a pullback by Pr opositio n 2.4. Therefore, there exists a unique morphism k ′ : x → Υ( a 0 , ϑ a 0 ) with k = ( i t ) ( a 0 ,ϑ a 0 ) · k ′ and ( ε t ) ( a 0 ,ϑ a 0 ) · k ′ = f . T o show that Υ( i ) · k ′ = g , co ns ider the comp osite ( ε t ) ( a,ϑ a ) · Υ ( i ) · k ′ = i · ( ε t ) ( a 0 ,ϑ a 0 ) · k ′ = i · f = ( ε t ) ( a,ϑ a ) · g . Since ( ε t ) ( a,ϑ a ) is a mo nomorphism, we get k · k ′ = g . This completes the proo f of the fact that the squa re ( ε t ) ( a,ϑ a ) · Υ( i ) = i · ( ε t ) ( a 0 ,ϑ a 0 ) is a pullback.  4.5. Prop osition. Assume the same c onditions as in Pr op osition 4.4. The functor t ∗ : A G → A T ⋄ c or estricts to an e quivalenc e b etwe en A G and the ful l sub c ate gory of A T ⋄ gener ate d by those T ⋄ -c o algebr as ( a, ϑ a ) for which ther e ex ists an isomorphism Υ( a, ϑ a ) ≃ ( a, ϑ a ) . This holds if and only if ther e is a (ne c essarily un ique) morphism x : a → G ( a ) with t a · x = ϑ a . Pr o of. Since the categ o ry A G admits eq ualisers, it follows from Theorem 3 .12 that the functor t ∗ is c omonadic. Thus A G is equiv alen t to ( A T ⋄ ) Υ . But since ε t : Υ → I is a comp onen t wise monomorphism, the r esult follows from 2.2. Next, Υ( a, ϑ a ) ≃ ( a, ϑ a ) if a nd only if the morphism p 1 in the pullback diagr am (4.2) is an isomorphism. In this case the comp osite x = p 2 · ( p 1 ) − 1 : a → G ( a ) s atisﬁes the co ndition of the prop osition. Since t a is a monomorphis m, it is clear that such an x is unique. Conv ersely , suppose that there is a morphism x : a → G ( a ) with t a · x = ϑ a . Then it is easy to see – using that t a is a monomorphis m– that the square a I   x / / G ( a ) t a   a ϑ a / / T ⋄ ( a ) is a pullback. It follows that Υ( a, ϑ a ) ≃ ( a, ϑ a ).  ON RA TIONAL P AIRINGS OF FUNCTORS 15 4.6 . Rational functor. Assume the data fr om 4.1 to b e given and that A has and G pr eserves e qualisers. Deﬁ ne the functor Rat P : A T K T ,T ⋄ / / A T ⋄ Υ / / A T ⋄ K − 1 T ,T ⋄ / / A T . Then the t r iple (Rat P , ε P , δ P ) , wher e ε P = K − 1 T ,T ⋄ · ε t · K T ,T ⋄ and δ P = K − 1 T ,T ⋄ · δ t · K T ,T ⋄ , is a c omonad on A T . Mor e over, for any obje ct ( a, h a ) of A T , (i) Rat P (Rat P ( a, h a )) ≃ Rat P ( a, h a ); (ii) for any r e gular T -submo dule ( a 0 , h a 0 ) of ( a, h a ) , the fol lowi ng diag r am is a pul lb ack, Rat P ( a 0 , ϑa 0 ) ( ε P ) ( a 0 ,ϑa 0 )   Rat P ( i ) / / Rat P ( a, ϑ a,ϑ a ) ( ε P ) ( a,ϑ a )   a 0 i / / a. Pr o of. Observing that (Rat P , ε P , δ P ) is the c omonad o btained from the como nad (Υ , ε t , δ t ) along the isomorphism K − 1 T ,T ⋄ : A T ⋄ → A T (see [22]), the results follow fr om Prop osition 4.4.  W e c a ll a T -mo dule ( a, h a ) r ational if Rat P ( a, h a ) ≃ ( a, h a ) (which is the cas e if and only if ( ε P ) ( a,h a ) is an is omorphism, see 2.2), and write Rat P ( T ) for the co rresp o nding full sub c ategory o f A T . Applying P r opositio n 4.5 gives: 4.7. Prop osition. Under the assumptions of Pr op osition 4.6, let ( a, h a ) ∈ A T . Then ( a, h a ) ∈ Rat P ( T ) if and only if ther e exists a (n e c essarily unique) morphism x : a → G ( a ) inducing c ommut ativ ity of the diagr am G ( a ) t a   a x = = { { { { { { { { { ϑ a / / T ⋄ ( a ) . Putting together the infor mation o btained so far , we obtain as main res ult of this section: 4.8. Theorem . L et P = ( T , G , σ ) b e a r ational p airing on a c ate gory A with a c omonad T ⋄ = ( T ⋄ , δ ⋄ , ε ⋄ ) right adjoi nt t o T . Supp ose that A admits and G pr eserves e qualisers. (1) Rat P ( T ) is a c or eﬂe ct ive sub c ate gory of A T , i.e. the inclusion i P : Rat P ( T ) → A T has a right adjoint rat P : A T → Rat P ( T ) . (2) The idemp otent c omonad on A T gener ate d by the adjunction i P ⊣ r at P is just the idem- p otent m onad (Rat P , ε P ) . (3) The functor Φ P : A G → A T , ( a, ϑ a ) 7→ ( a, σ a · T ( ϑ a )) , c or estricts to an e quivalenc e of c ate gories R P : A G → Rat P ( T ) . As a spec ial cas e , consider for A the categ ory A M o f left A -modules, A an y r ing . F or an A -coring C there is a pairing ( C ∗ , C, ev ). If this is rationa l it follows by Theorem 4 .8 that the C - comodules for m a coreﬂective sub category of C ∗ M (see 1.2). 5. P airings in monoidal ca tegories W e b egin b y revie wing some standar d deﬁnitions asso ciated with mo noidal ca tegories. Let V = ( V , ⊗ , I ) b e a monoidal categor y with tenso r pro duct ⊗ and unit ob ject I . W e will freely app eal to MacL a ne’s co herence theo rem (see [16]); in par ticular, we write as if the asso ciativity a nd unitality isomorphisms were identities. Thus X ⊗ ( Y ⊗ Z ) = ( X ⊗ Y ) ⊗ Z 16 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER and I ⊗ X = X ⊗ I for all X , Y , X ∈ V . W e sometimes collapse ⊗ to concatenation, to sav e space. Recall that an a lgebra A in V (or V -algebra ) is an ob ject A of V equipp ed with a multipli- cation m A : A ⊗ A → A and a unit e A : I → A inducing commut ativity of the diagrams A ⊗ A ⊗ A A ⊗ m A   m A ⊗ A / / A ⊗ A m A   A ⊗ A m A / / A, I ⊗ A I I I I I I I I I I I I I I I I I I I I e A ⊗ A / / A ⊗ A m A   A ⊗ I A ⊗ e A o o u u u u u u u u u u u u u u u u u u A. Dually , a co algebra C in V (or V -coalge br a) is an ob ject C of V equipp ed with a c o m ulti- plication δ C : C → C ⊗ C , a counit ε C : C → I and commutativ e dia grams C ⊗ C ⊗ C C ⊗ C δ C ⊗ C o o C ⊗ C C ⊗ δ C O O C, δ C o o δ C O O I ⊗ C J J J J J J J J J J J J J J J J J J J J C ⊗ C ε C ⊗ C o o C ⊗ ε C / / C ⊗ I t t t t t t t t t t t t t t t t t t C. δ C O O 5.1. Deﬁniti on. A triple P = ( A , C , t ) consisting of a V -a lgebra A , a V -coalg ebra C and a morphism t : A ⊗ C → I , for which the diagra ms (5.1) A ⊗ A ⊗ C A ⊗ A ⊗ δ C / / m A ⊗ C   A ⊗ A ⊗ C ⊗ C A ⊗ t ⊗ C / / A ⊗ C t   C e A ⊗ C o o ε C | | x x x x x x x x x x A ⊗ C t / / I commute, is ca lled a left p airing . A left action o f a monoidal ca tegory V = ( V , ⊗ , I ) on a c ategory X is a functor −♦− : V × X → X , called the action of V on X , alo ng with invertible natura l transfo r mations α A,B ,X : ( A ⊗ B ) ♦ X → A ♦ ( B ♦ X ) and λ X : I ♦ X → X, called the asso ciativity and unit isomorphisms, resp ectiv ely , satisfying tw o cohere nce a xioms (see B´ enab ou [4]). Ag a in we wr ite as if α and λ were identit ies. 5.2 . Example. Recall that if B is a bicategory (in the sense of B´ enabo u [4]), then for an y A ∈ Ob ( B ), the tr iple ( B ( A, A ) , ◦ , I A ), wher e ◦ deno tes the ho rizon tal comp osition op eration, is a mo noidal category , and that, for a n arbitrary B ∈ B , there is a c a nonical left action of B ( A, A ) on B ( B , A ), given b y f ♦ g = f ◦ g for all f ∈ B ( A, A ) and a ll g ∈ B ( B , A ). In particular, since monoida l ca tegories are nothing but bicatego ries with exactly one ob ject, any monoidal categor y V = ( V , ⊗ , I ) has a ca nonical (left) action on the category V , given by A ♦ B = A ⊗ B . 5.3 . Actions and pairings. Giv en a left a c tio n −♦− : V × X → X of a monoidal categor y V on a ca tegory X and an algebra A = ( A, e A , m A ) in V , o ne has a monad T X A on V deﬁned on any X ∈ V by • T X A ( X ) = A ♦ X , • ( e T X A ) X = e A ♦ X : X = I ♦ X → A ♦ X = T X A ( X ), • ( m T X A ) X = m A ♦ X : T X A ( T X A ( X )) = A ♦ ( A ♦ X ) = ( A ⊗ A ) ♦ X → A ♦ X = T X A ( X ) , ON RA TIONAL P AIRINGS OF FUNCTORS 17 and we write A X for the Eilenber g-Moo re categor y of T A -algebra s. Note that in the case of the canonica l left actio n of V on itself, A V is just the catego ry of (left) A -mo dules. Dually , for a V -coalg e bra C = ( C, ε C , δ C ), a comona d G C X = ( C ♦ − , ε C ♦− , δ C ♦− ) is deﬁned o n X and one has the corres ponding Eilenberg-Mo ore catego ry C X ; for X = V this is just the categ o ry of (left) C - comodules. It is easy to see that if P = ( A , C , t ) is a left pairing in V , then the triple P X = ( T X A , G C X , σ t ), where σ t is the natural tra nsformation t ♦− : T A · G C = A ♦ ( C ♦− ) = ( A ⊗ C ) ♦− → I ⊗ − = I , is a pair ing b et ween the mona d T X A and comona d G C X . W e say that a left pa iring ( A , B , σ ) is X - r ational , if the cor r esponding pa iring P X is rational, i.e. if the ma p β P X X,Y : X ( X , C ♦ Y ) → X ( A ♦ X , Y ) taking f : X → C ♦ Y to the compo site A ♦ X A ♦ f − − − → A ♦ ( C ♦ Y ) = A ⊗♦ Y σ ♦ Y − − − → Y , is injective. W e will generally drop the X from the no tations T X A , G C X and P X when ther e is no danger of confusion. 5.4 . Clos ed categories. A monoidal categor y V = ( V , ⊗ , τ , I ) is s aid to be right close d if each functor − ⊗ X : V → V has a right a djoin t [ X , − ] : V → V . So there is a bijection (5.2) π Y ,X,Z : V ( Y ⊗ X , Z ) ≃ V ( Y , [ X , Z ]) , with unit η Y X : X → [ Y , X ⊗ Y ] and counit e Y Z : [ Y , Z ] ⊗ Y → Z. W e write ( − ) ∗ for the functor [ − , I ] : V op → V that takes X ∈ V to [ X , I ] and f : Y → X to the mor phism [ f , I ] : [ X , I ] → [ Y , I ] that corr esponds under the bijection (5.2) to the co mposite [ X , I ] ⊗ Y [ X, I ] ⊗ f / / [ X , I ] ⊗ X e X I / / I . Symmetrically , a mo noidal catego ry V = ( V , ⊗ , τ , I ) is said to b e left close d if ea ch functor X ⊗ − : V → V has a r ig h t adjoint { X, − } : V → V . W e write η Y X : X → [ Y , Y ⊗ X ] and e Y Z : Y ⊗ { Y , Z } → Z for the unit and counit of the adjunction X ⊗ − ⊣ { X , − } . One calls a mono ida l c a tegory close d w he n it is b oth left and right closed. A t y pical example is the category of bimo dules ov er a no n-comm utativ e r ing R, with ⊗ R as ⊗ . 5.5 . Actions with right adjoin ts. Suppose now that −♦− : V × X → X is a left ac tio n of a monoidal categor y V on a catego ry X and that A = ( A, e A , m A ) is an algebra in V such that the functor A ♦− : X → X has a rig h t a djoin t { A, −} X : X → X (as it surely is when V = B ( A, A ) and X = B ( B , A ) for some ob jects A, B o f a close d bicategory B , see Example 5.2.) Recall (e.g. [11]) that g iv en a mona d T = ( T , m T , e T ) on a categ ory X and an endo functor G : X → X right adjoint to T , ther e is a unique way to make G into a c o monad G = ( G, δ G , ε G ) such that G is right a djoin t to the monad T . Since the functor { A, −} X : X → X is right adjoint to the functor A ♦− : X → X and since A ♦ − is the functor-part of the monad T A , there is a unique wa y to make { A, − } X int o a comona d G ( A ) = ( { A, −} X , δ G ( A ) , ε G ( A ) ) suc h that the como nad G ( A ) is r igh t adjoint to the monad T A . Dually , for any co a lgebra C = ( C, δ C , ε C ) in V such that the functor C ♦− : X → X has a right adjoint { C , − } X : X → X , there exists a monad T ( C ) whos e functor-par t is { C, −} X and which is right adjoint to the comonad G C . 5.6 . P airings with ri g h t adjoints. Consider a left a ction of a monoidal category V o n a category X and a left pairing P = ( A , C , σ ) in V , such tha t ther e exist adjunctions A ♦− ⊣ { A, −} X , −♦ A ⊣ [ A, − ] X , C ♦− ⊣ { C , −} X , and − ♦ C ⊣ [ C , − ] X . Since the comonad G ( A ) is right adjoint to the monad T A , the following are eq uiv alent b y Prop osition 3.16: 18 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER (a) the pairing P is X -rationa l; (b) for every X ∈ X , the comp osite α P X : C ♦ X η A C ♦ X / / { A, A ♦ ( C ♦ X ) } X = { A, ( A ⊗ C ) ♦ X } X { A, t ♦ X } X / / { A, X } X , where η A C ♦ X is the C ♦ X - component of the unit of the adjunction A ♦− ⊣ { A, −} X , is a monomorphism. If (one o f ) these conditions is satis ﬁe d, then the functor Φ P : C X → A X , ( X , X θ X − − → C ♦ X ) 7− → ( X, A ♦ X A ♦ θ X − − − − → A ♦ ( C ♦ X ) = ( A ⊗ C ) ♦ X t ♦ X − − − → X ) is full and faithful. W riting Rat P (resp. Rat P ( A )) for the functor Rat P X (resp. the ca teg ory Rat P X ( T A )), o ne gets: 5.7. Prop osition. Under t he c onditions given in 5.6, assume t he c ate gory X to admit e qualis- ers. If P is an X -r ational p airing in V such that either (i) the functor C ♦− : X → X pr eserves e qualisers, or (ii) X admits pu s ho uts and the functor C ♦− : X → X pr eserves r e gular monomorphisms, or (iii) X admits pushouts and every monomorphism in X is r e gular, then (1) the inclusion i P : Rat P ( A ) → A X has a right adjoint ra t P : A X → Rat P ( A ) ; (2) the idemp otent c omonad on A X gener ate d by t he adjunction is just Rat P : A X → A X ; (3) the functor Φ P : C X → A X c or estricts t o an e quivalenc e R P : Rat P ( A ) → C X . Pr o of. F or condition (i), the asser tion follows from Theor e m 4.8. W e show that (iii) is a particular case of (ii), while (ii) is itself a particular case of (i). Indeed, since P is an X -ratio na l pairing in V , it follows from 3.7(2) that the functor C ♦− : X → X preserves monomo rphisms, and if ev ery monomorphism in X is r egular, then C ♦− : X → X c le a rly pre s erv es reg ular monomorphisms. Next, since the functor C ♦ − : X → X admits a righ t adjoint, it pr e serv es pushouts, and then it follows from P r opositio n 2.4 that C ♦− : X → X preserves equaliser s . This completes the pro of.  5.8 . Nuclear ob jects. W e ca ll an o b ject V ∈ V is (left) X - pr enu cle ar (r esp. X - nucle ar ) if • the functor − ⊗ V : V → V ha s a rig h t adjoint [ V , − ] : V → V , • the functor V ∗ ♦− : X → X , with V ∗ = [ V , I ], has a rig h t adjoint { V ∗ , −} X : X → X , a nd • the comp osite α X : V ♦ X ( η X ) V ♦ X / / { V ∗ , V ∗ ♦ ( V ♦ X ) } = { V ∗ , ( V ∗ ⊗ V ) ♦ X } { V ∗ , e V I ♦ X } / / { V ∗ , X } , is a mo no morphism (resp. an isomo rphism), where (( η ) X ) V ♦ X is the V ♦ X -comp onen t of the unit η X : − → { V ∗ , V ∗ ♦−} of the a djunction V ∗ ♦− ⊣ { V ∗ , −} . Note that the morphism α X : V ♦ X → { V ∗ , X } is the tra nspose of the morphism e V I ♦ X : ( V ∗ ⊗ V ) ♦ X → X under the adjunction V ♦− ⊣ { V , −} X . Applying Prop osition 5.7 and 6.1 0 , we get: 5.9. Prop osi ti on. L et V b e a monoidal close d c ate gory and C = ( C, ε C , δ C ) a V -c o algebr a with C X -pr enucle ar, and assum e X to admit e qualisers. If either (i) the functor C ♦− : X → X pr eserves e qualisers, or (ii) X admits pu s ho uts and the functor C ♦− : X → X pr eserves r e gular monomorphisms, or (iii) X admits pushouts and every monomorphism in X is r e gular, ON RA TIONAL P AIRINGS OF FUNCTORS 19 then Rat P ( C ) ( C ∗ ) is a fu l l c or eﬂe ctive su b c ate gory of C ∗ X and t he functor Φ P ( C ) : C X → C ∗ X c or estricts to an e quivalenc e R P ( C ) : C X → Rat P ( C ) ( C ∗ ) . Spec ia lising the previo us result to the ca s e of the left action of the monoidal ca tegory C ∗ M C ∗ of C ∗ -bimo dules on the categor y of left C ∗ -mo dules, o ne see s tha t the equiv a lence of the category of como dules C M and a full sub categor y of C ∗ M for A C lo cally pr o jective addressed in 1.2 is a sp ecial case of the preceding theorem. Recall (e.g. from [1 6]) that a monoida l category V = ( V , ⊗ , I ) is said to be symmetric if for all X, Y ∈ V , there exists functorial isomorphisms τ X,Y : X ⊗ Y → Y ⊗ X obey ing certa in ident ities. It is clear that if V is closed, then { X , −} ≃ [ X, − ] for all X ∈ V . 5.10 . Pa irings i n symm etric monoidal close d categories. Let V = ( V , ⊗ , I , τ , [ − , − ]) b e a symmetric monoidal closed categor y and P = ( A , C , t ) a left pairing in V . F or any X ∈ V , we write γ X : A ∗ ⊗ X → [ A, X ] for the morphis m that corresp onds under π (see (5 .2) ) to the comp osition A ∗ ⊗ X ⊗ A τ A ∗ , X ⊗ A / / X ⊗ A ∗ ⊗ A e A I / / X . Since the functor [ A, − ], as a left adjoint, preserves colimits, it follows fr om [14, Theorem 2.3] that there exists a unique mor phism γ ( t ) : C → A ∗ such that the diagram (5.3) C ⊗ X γ ( t ) ⊗ X / / α X ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q A ∗ ⊗ X γ X   [ A, X ] commutes. Hereb y the morphism γ ( t ) c o rresp onds under π to the comp osite σ · τ C, A . 5.11. Prop osition. Consider the situation given in 5.10. (1) If P is a r ational p airing, then the morphism γ ( t ) : C → A ∗ is pur e, that is, for any X ∈ V , the morphism γ ( t ) ⊗ X is a monomorphism. (2) If A is V -nucle ar, then P is r at ional if and only if γ ( t ) : C → A ∗ is pur e. (3) Φ P : C V → A V is an isomorp hism if and only if A is V -nu cle ar and γ ( t ) : C → A ∗ is an isomorphism. In this c ase C is also V - n ucle ar. Pr o of. (1) T o say that P is a rational pairing is to say that α X is a monomorphism for all X ∈ V (see Prop osition 5.6). Then it follows from the commutativit y of the diagr am (5.3) that γ ( t ) ⊗ X is a lso a mo nomorphism, thus γ ( t ) : C → A ∗ is pure. (2) follows from the commut ativity of dia gram (5.3). (3) One direction is clea r, s o supp ose that the functor Φ is an isomorphism of categ ories. Then it follows from Pr oposition 3.15(2) that α X is an isomorphism for all X ∈ V . It implies that the functor [ A, − ] pre serv es colimits and thus the morphism γ X is an isomo rphism (see [14, Theorem 2 .3]). Therefor e A is V - n uclear. Since α X and γ X are b oth isomor phisms, it follows from the commutativit y of dia gram (5.3) that the morphism γ ( t ) ⊗ X is also a n isomorphism. This clear ly implies that γ ( t ) : C → A ∗ is an isomo rphism. It is prov ed in [21, Coro lla ry 2.2] that if A is V -nuclear, then so is A ∗ .  6. Entwinings in monoidal ca tegories Recall (for exa mple, from [18]) that an ent wining in a monoidal ca tegory V = ( V , ⊗ , I ) is a triple ( A , C , λ ), where A = ( A, e A , m A ) is a V -alg e br a, C = ( C, ε C , δ C ) is a V - c oalgebra, and 20 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER λ : A ⊗ C → C ⊗ A is a morphism inducing c o mm uta tivit y o f the diagrams C e A ⊗ C   C ⊗ e A % % K K K K K K K K K K A ⊗ C λ   A ⊗ ε C # # F F F F F F F F F A ⊗ C λ / / C ⊗ A , C ⊗ A ε C ⊗ A / / A, A ⊗ C λ   A ⊗ δ C / / A ⊗ C ⊗ C λ ⊗ C / / C ⊗ A ⊗ C C ⊗ λ   A ⊗ A ⊗ C m A ⊗ C   A ⊗ λ / / A ⊗ C ⊗ A λ ⊗ A / / C ⊗ A ⊗ A C ⊗ m A   C ⊗ A δ C ⊗ A / / C ⊗ C ⊗ A, A ⊗ C λ / / C ⊗ A . F or any en twinin g ( A , C , λ ), the natura l tr a nsformation λ ′ = λ ⊗ − : T A ◦ G C = A ⊗ C ⊗ − → C ⊗ A ⊗ − = G C ◦ T A is a mixed dis tributiv e law fro m the monad T A to the co monad G C . W e wr ite e C for the A V -comonad g G C , that is, for any ( V , h V ) ∈ A V , e C ( V , h V ) = ( C ⊗ V , A ⊗ C ⊗ V λ ⊗ V − − − → C ⊗ A ⊗ V C ⊗ h V − − − − → C ⊗ V ) , and wr ite C A V ( λ ) for the categor y V G C T A ( λ ′ ). An o b ject of this category is a three-tuple ( V , θ V , h V ), where ( V , θ V ) ∈ C V and ( V , h V ) ∈ A V , with commuting diagram (6.1) A ⊗ V h V / / A ⊗ θ V   V θ V / / C ⊗ V A ⊗ C ⊗ V λ ⊗ V / / C ⊗ A ⊗ V . C ⊗ h V O O The assig nmen t (( V , h V ) , θ ( V ,h V ) ) 7− → ( V , θ ( V ,h V ) , h V ) yields an isomorphism of catego ries Λ : ( A V ) e C → C A V ( λ ) . 6.1 . Repres en table en t wi nings. F or ob jects A , C in a monoidal category V = ( V , ⊗ , I ), consider the functor V ( − ⊗ C , A ) : V op → Set taking an arbitr a ry ob ject V ∈ V to the set V ( V ⊗ C, A ) . Supp ose there is a n ob ject E ∈ V that repres en ts the functor, i.e. there is a natural bijection ω : V ( − ⊗ C , A ) ≃ V ( − , E ) . W riting β : E ⊗ C → A fo r the mor phism ω − 1 ( I E ), it follows that for any o b ject V ∈ V and any morphism f : V ⊗ C → A , there exists a unique β f : V → E making the diagr am V ⊗ C f / / β f ⊗ C % % J J J J J J J J J A E ⊗ C β < < x x x x x x x x x commute. It is c lear tha t ω − 1 ( β f ) = f . W e call an ent wining ( A , C , λ ) r epr esentable if the functor V ( − ⊗ C, A ) : V op → Set is representable. 6.2 . Examples. (i) If the functor − ⊗ C : V → V has a r igh t a djoin t [ C, − ] : V → V , then it follows from the bijection V ( V ⊗ C , A ) ≃ V ( V , [ C , A ]) that the ob ject [ C, A ] represents the functor V ( − ⊗ C, A ). In pa rticular, when V is right closed, each en t wining in V is r epresen table. ON RA TIONAL P AIRINGS OF FUNCTORS 21 (ii) If C is a right V -nuclear ob ject, the functor V ( − ⊗ C , A ) : V op → Set is representable. Indeed, to say that C is r igh t V -nuclear is to say that the functor C ⊗ − : V → V ha s a right adjoin t { C, −} : V → V , the functor − ⊗ ∗ C = − ⊗ { C, I } : V → V also has a right adjoint [ ∗ C , − ] : V → V , and the mor phis m α ′ V : V ⊗ C → [ ∗ C , V ] is a natural isomo rphism. Considering then the comp osition of bijections V ( − ⊗ ∗ C , ?) ≃ V ( − , [ ∗ C , ?]) ≃ V ( − , ? ⊗ C ) , one sees that the functor − ⊗ C admits the functor − ⊗ ∗ C as a left adjoint. The same arguments as in the pro of of Theorem X.7.2 in [16] then show that there is a natural bijection V ( − ⊗ C , A ) ≃ V ( − , A ⊗ ∗ C ) . Thu s the ob ject A ⊗ ∗ C repres en ts the functor V ( − ⊗ C , A ) : V op → Set. 6.3. Prop osition. L et ( A , C , λ ) b e a r epr esentable entwining in V = ( V , ⊗ , I ) with r epr esent- ing obje ct E (se e 6.1). L et e E = β τ : I → E and m E = β  : E ⊗ E → E , wher e τ is the c omp osite I ⊗ C ≃ C ε C − − → I e A − − → A, while  is the c omp osite E ⊗ E ⊗ C E ⊗ E ⊗ δ C − − − − − − → E ⊗ E ⊗ C ⊗ C E ⊗ β ⊗ C − − − − − → E ⊗ A ⊗ C E ⊗ λ − − − → E ⊗ C ⊗ A β ⊗ A − − − → A ⊗ A m A − − → A. (i) The triple ( E , e E , m E ) is a V -algebr a. (ii) The morphism i := β A ⊗ ε C : A → E is a morphism of V -algebr as. Pr o of. (i) First obser v e commutativit y o f the diagrams (6.2) E ⊗ E ⊗ C m E ⊗ C    % % J J J J J J J J J J I ⊗ C = C e E ⊗ C   e A · ε C $ $ J J J J J J J J J J E ⊗ C β / / A, E ⊗ C β / / A. T o prove that m E is ass o ciative, i.e. m E · ( m E ⊗ E ) = m E · ( E ⊗ m E ), it is to show β · ( m E ⊗ C ) · ( m E ⊗ E ⊗ C ) = β · ( m E ⊗ C ) · ( E ⊗ m E ⊗ C ) . F or this, consider the diag ram (deleting the ⊗ -symbols) E E E C E E E δ C / / m E E C   E E E C C E E β C / / m E E C C   E E AC E E λ / / m E C A   E E C A E E δ C A / / m E C A   E E C C A E β C A / / E AC A E λA   E E C (1) E E δ C / / E E C C (2) E β C / / E AC (3) E λ / / E C A β A   E C AA β AA   AA m A   (4) AAA Am A   m A A o o A (5) AA , m A o o in which the dia grams (1),(2) a nd (3) commute b y naturality of comp osition, dia gram (4 ) commutes by deﬁnition of m E , and diagr am (5) commutes b y asso ciativity of m A . It follows (6.3) β · ( m E C ) · ( m E E C ) = m A · Am A · β AA · E λA · E β C A · E E δ C A · E E λ · E E β C · E E E δ C . 22 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER Now we have β · m E C · E m E C = m A · β A · E λ · E β C · E E δ C · E m E C nat. of comp osition = m A · β A · E λ · E β C · E m E C C · E E E δ C by (6.2) = m A · β A · E λ · E m A C · E β AC · E E λC · E E β C C · E E E δ C C · E E E δ C coassoc iativit y of δ C = m A · β A · E λ · E m A C · E β AC · E E λC · E E β C C · E E E C δ C · E E E δ C nat. of comp osition = m A · β A · E λ · E m A C · E β AC · E E λC · E E Aδ C · E E β C · E E E δ C λ is an e ntwining = m A · β A · E C m A · E λA · E Aλ · E β AC · E E λC · E E Aδ C · E E β C · E E E δ C nat. of comp osition = m A · β A · E C m A · E λA · E β C A · E E C λ · E E λC · E E Aδ C · E E β C · E E E δ C λ is an e ntwining = m A · β A · E C m A · E λA · E β C A · E E δ C A · E E λ · E E β C · E E E δ C nat. of comp osition = m A · Am A · β AA · E λA · E β C A · E E δ C A · E E λ · E E β C · E E E δ C by (6.3) = β · m E C · m E E C. It follows that m E · ( m E ⊗ E ) = m E · ( E ⊗ m E ), and thus m E is a ssocia tiv e. T o prov e that e E is the unit for the m ultiplication m E , it is to show β · ( m E ⊗ C ) · ( e E ⊗ E ⊗ C ) = β and β · ( m E ⊗ C ) · ( E ⊗ e E ⊗ C ) = β . In the diag ram E C (1) e E E C   E δ C / / E C C (2) e E E C C   β C / / AC (3) e E AC   λ / / C A (4) e E C A   ε C A / / A e E A   3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 E E C (5) E E δ C / / E E C C (6) E β C / / E AC E λ / / E C A β A / / AA m A / / A E C E e E C O O E δ C / / E C C E e E C C O O E ε C C / / E C E e A C O O E C e A        B B        β / / A (7) Ae A        C C        A , (8)                           the diagrams (1), (2), (3), (5) and (7) are co mm utative by naturality of comp osition, the dia - grams (4) and (6) a re commutativ e by (6.2), diagr am (8) and the top triang le are commutativ e since e A is the unit for the multiplication m A , and the botto m triangle commutes since λ is an ent wining. Commutativit y of the diagra m implies β · ( m E ⊗ C ) · ( e E ⊗ E ⊗ C ) = ( ε C ⊗ A ) · λ · ( β ⊗ C )( E ⊗ δ C ) λ is an ent win ing = ( A ⊗ ε C ) · ( β ⊗ C ) · ( E ⊗ δ C ) nat. of comp osition = β · ( E ⊗ C ⊗ ε C ) · ( E ⊗ δ C ) since( C ⊗ ε C ) · δ C = I = β = β · ( m E ⊗ C ) · ( E ⊗ e E ⊗ C ) . Thu s e E is the unit for m E . This completes the pr oof of (i). (ii) W e have to show i · e A = i E and m E · ( i ⊗ i ) = i · m A . F or this consider the diagr am C ε C   e A ⊗ C / / A ⊗ C i ⊗ C / / A ⊗ ε C   E ⊗ C β y y t t t t t t t t t t I e A / / A ON RA TIONAL P AIRINGS OF FUNCTORS 23 Since the sq uare commutes by na turalit y of co mposition, while the triangle commutes by deﬁnition of i , the o uter diagr am is also commutativ e, meaning β e A · ε C = i · e A . But β e A · ε C = e E . Thu s i · e A = e E . Next, consider the diag ram A ⊗ A ⊗ C ⊗ C A ⊗ A ⊗ ε C ⊗ C / / i ⊗ A ⊗ C ⊗ C   A ⊗ A ⊗ C i ⊗ A ⊗ C   A ⊗ λ / / A ⊗ C ⊗ A i ⊗ C ⊗ A   A ⊗ ε C ⊗ A   < < < < < < < < < < < < < < < < < A ⊗ A ⊗ C A ⊗ A ⊗ δ C 6 6 m m m m m m m m m m m m m i ⊗ i ⊗ C   E ⊗ A ⊗ C ⊗ C E ⊗ A ⊗ ε C ⊗ C * * U U U U U U U U U U U U U U U U E ⊗ i ⊗ C ⊗ C   E ⊗ E ⊗ C E ⊗ E ⊗ δ C / / E ⊗ E ⊗ C ⊗ C E ⊗ β ⊗ C / / E ⊗ A ⊗ C E ⊗ λ / / E ⊗ C ⊗ A β ⊗ A / / A ⊗ A in whic h the tw o triangles commute by deﬁnition of i , and the quadrangles commute by naturality o f co mp osition. W e hav e ω − 1 ( m E · ( i ⊗ i )) = m A · ( β ⊗ A ) · ( E ⊗ λ ) · ( E ⊗ β ⊗ C ) · ( E ⊗ E ⊗ δ C ) · ( i ⊗ i ⊗ C ) = m A · ( A ⊗ ε C ⊗ A ) · ( A ⊗ λ ) · ( A ⊗ A ⊗ ε C ⊗ C ) · ( A ⊗ A ⊗ δ C ) since ( ε C ⊗ C ) · δ C = I = m A · ( A ⊗ ε C ⊗ A ) · ( A ⊗ λ ) by deﬁniti on of λ = m A · ( A ⊗ A ⊗ ε C ) nat. of comp osition = ( A ⊗ ε C ) · ( m A ⊗ C ) since A ⊗ ε C = β · ( i ⊗ C ) = β · ( i ⊗ C ) · ( m A ⊗ C ) = β · (( i · m A ) ⊗ C ) = ω − 1 ( i · m A ) . Thu s m E · ( i ⊗ i ) = i · m A . This completes the pro of.  6.4. Prop osition. With the data given in Pr op osition 6.3, ther e is a functor Ξ : C A V ( λ ) → E V with c ommutative diagr am (6.4) C A V ( λ ) Ξ / / C A U " " E E E E E E E E E V E U ~ ~ } } } } } } } } V , wher e C A U : C A V ( λ ) → V is the evident for getful fu n ctor. Pr o of. F or any ( V , θ V , h V ) ∈ C A V ( λ ), write ι V : E ⊗ V → V for the comp osite E ⊗ V E ⊗ θ V − − − − → E ⊗ C ⊗ V β ⊗ V − − − → A ⊗ V h V − − → V . W e claim that ( V , ι V ) ∈ E V . Indeed, to show that ι V · ( e E ⊗ V ) = 1 , consider the dia gram V e E ⊗ V   θ V / / C ⊗ V e E ⊗ C ⊗ V   ε C ⊗ V / / V e A ⊗ V   G G G G G G G G G G G G G G G G G G E ⊗ V E ⊗ θ V / / E ⊗ C ⊗ V β ⊗ V / / A ⊗ V h V / / V , in which the left squar es co mm utes by na tur alit y of comp osition, the r igh t one commutes by (6.2), while the triangle commutes since ( V , h V ) ∈ A V . It follows – since ( V , θ V ) ∈ C V , and hence ( ε C ⊗ V ) · θ V = I – tha t ι V · ( e E ⊗ V ) = ( ε C ⊗ V ) · θ V = I . 24 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER Next, consider the diag ram E E V (1) E E θ V / / E E θ V   E E C V (2) E β V / / E E C θ V   E AV (3) E h V / / E Aθ V   E V E θ V % % K K K K K K K K K E E C V (4) m E C V   E E δ C V / / E E C C V E β C V / / E AC V E λV / / E C AV β AV   E C h V / / E C V β V   AAV (5) m A V   Ah V / / AV h V   E C V β V / / AV (6) h V / / V , in which diagr am (1) comm utes since ( V , θ V ) ∈ C V , the diagr ams (2) and (5) are commutative by naturality of compos itions, diag ram (3) commutes by (6.1), diagram (4) co mm utes by deﬁnition of m E , and diagr am (6) commutes since ( V , h V ) ∈ A V . It follows that ι V · ( E ⊗ ι V ) = h V · ( β ⊗ V ) · ( E ⊗ θ V ) · ( E ⊗ h V ) · ( E ⊗ β ⊗ V ) · ( E ⊗ E ⊗ θ V ) = h V · ( β ⊗ V ) · ( m E ⊗ C ⊗ V ) · ( E ⊗ E ⊗ θ V ) nat. of comp osition = h V · ( β ⊗ V ) · ( E ⊗ θ V ) · ( m E ⊗ V ) = ι V · ( m E ⊗ V ) . Thu s, ( V , ι V ) ∈ E V . Next, if f : ( V , θ V , h V ) → ( V ′ , θ V ′ , h V ′ ) is in C A V ( λ ), then the diagram (6.5) A ⊗ V A ⊗ f   h V / / V f   θ V / / C ⊗ V C ⊗ f   A ⊗ V ′ h V ′ / / V ′ θ V ′ / / C ⊗ V ′ is commutativ e. In the diagram E ⊗ V E ⊗ f   E ⊗ θ V / / E ⊗ C ⊗ V E ⊗ C ⊗ f   β ⊗ V / / A ⊗ V A ⊗ f   h V / / V f   E ⊗ V ′ E ⊗ θ V ′ / / E ⊗ C ⊗ V ′ β ⊗ V ′ / / A ⊗ V ′ h V ′ / / V ′ , the middle square commutes by naturality of comp osition, while the other squar e s comm ute by (6.5). Thus, f ca n b e seen a s morphism in E V from ( V , ι V ) to ( V ′ , ι V ′ ). It follows tha t the assignment ( V , θ V , h V ) 7− → ( V , ι V = h V · ( β ⊗ V ) · ( E ⊗ θ V )) yields a functor Ξ : C A V ( λ ) → E V . It is clea r that Ξ ma kes the diagr am (6.4) commute.  In order to pro ceed, we need the following r esult. 6.5. Lemma. L et T = ( T , e T , m T ) and H = ( H, e H , m H ) b e monads on a c ate gory A , i : T → H a monad morphism and i ∗ : A H → A T the functor that takes an H -algebr a ( a, h A ) to the T -algebr a ( a, h a · i a ) . Supp ose that T ⋄ = ( T ⋄ , e ⋄ , m ⋄ ) (r esp. H ⋄ = ( H ⋄ , e H ⋄ , m H ⋄ ) ) is a c omonad that is right adjoint to T (r esp. H ) . Write i : H ⋄ → T ⋄ for the mate of i . Then (1) i is a morphism of c omonads. ON RA TIONAL P AIRINGS OF FUNCTORS 25 (2) We have c ommu tativity of the diagr am (6.6) A H K H,H ⋄   i ∗ / / A T K T ,T ⋄   A H ⋄ ( i ) ∗ / / A T ⋄ . (3) If A admits b oth e qualisers and c o e qualisers, then (i) the functors i ∗ and ( i ) ∗ admit b oth right and left adjoints; (ii) i ∗ is monadic and ( i ) ∗ is c omonadic. Pr o of. (1) This follows fro m the pro perties of mates (see, for exa mple, [19]). (2) An eas y calculation shows that for any ( a, h a ) ∈ A H , one has ( i ) ∗ ◦ K H,H ⋄ ( a, h a ) = ( a, ( i ) a · H ⋄ ( h a ) · σ a ) and K T ,T ⋄ ◦ i ∗ ( a, h a ) = ( a, T ⋄ ( h a ) · T ⋄ ( i a ) · τ a ) , where σ : I → H ⋄ H is the unit of the adjunction H ⊣ H ⋄ , while τ : I → T ⋄ T is the unit of the adjunction T ⊣ T ⋄ . Consider ing the dia gram a σ a / / τ a   H ⋄ H ( a ) i H ( a )   H ⋄ ( h a ) / / H ⋄ ( a ) ( i ) a   T ⋄ T ( a ) T ⋄ ( i a ) / / T ⋄ H ( a ) T ⋄ ( h a ) / / T ⋄ ( a ) , in which the right squa re c omm utes b y natura lit y of i , while the left one co mm utes by Theorem IV.7.2 of [16], since i is the mate of i . Thus ( i ) a · H ⋄ ( h a ) · σ a = T ⋄ ( h a ) · T ⋄ ( i a ) · τ a , and hence ( i ) ∗ ◦ K H,H ⋄ = K T ,T ⋄ ◦ i ∗ . (3)(i) If the categor y A admits b oth equalisers a nd co equalisers , then A H admits eq ualisers, while A H ⋄ admits co equalisers. Since K H,H ⋄ is a n isomor phism of categories , it follows that the categor y A H as well as the categ ory A H ⋄ admit b oth equa lisers and co equalisers . Then, according to 3.1 1 and its dual, the functor ( i ) ∗ admits a r igh t adjoint ( i ) ∗ , while the functor i ∗ admits a left a djoin t i ∗ . Then clearly the comp osite i ! = ( K H,H ⋄ ) − 1 · ( i ) ∗ · K T ,T ⋄ is rig h t adjoint to i ∗ , while the comp osite ( i ) ! = K H,H ⋄ · i ∗ · ( K T ,T ⋄ ) − 1 is left adjoint to ( i ) ∗ . (3)(ii) Since the functors i ∗ and ( i ) ∗ are clearly conser v a tive, the assertio n follows by a simple applicatio n of Beck’s monadicity theorem (see, [16]) and its dual.  6.6 . Left and rig h t adjoints to the functor i ∗ . In the setting considered in Prop osition 6.3, supp ose now that V admits b oth eq ua lisers a nd co equalisers and that there are adjunctions A ⊗ − ⊣ { A, −} : V → V and E ⊗ − ⊣ { E , −} : V → V . Then, accor ding to Lemma 6.5, the functor i ∗ : E V → A V has both left and rig h t adjoints i ∗ and i ! . It follows from 3.11 a nd its dual that for any ( V , h V ) ∈ A V , i ∗ ( V , h V ) is the co equaliser E ⊗ A ⊗ V E ⊗ h V / / h r E ⊗ V / / E ⊗ V q ( V ,h V ) / / i ∗ ( V , h V ) , while i ! ( V , h V ) is the equaliser (6.7) i ! ( V , h V ) e ( V ,h V ) / / { E , V } k / / { h l E ,V } / / { A ⊗ E , V } , where h r E = m E · ( E ⊗ i ), h l E = m E · ( i ⊗ E ) and k is the transp ose of the comp osition A ⊗ E ⊗ { E , V } A ⊗ e E V − − − − → A ⊗ V h V − − → V . 26 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER W e wr ite E ⊗ A − for the functor i ∗ (as well as for the A V -monad genera ted by the a djunction i ∗ ⊣ i ∗ ) and write { E , −} A for the functor i ! (as well a s for the A V -comonad generated b y the adjunction i ∗ ⊣ i ! ). According to Le mma 6.5, the comparis o n functor K i ∗ : E V → ( A V ) E ⊗ A − is a n equiv a lence of categ ories. It is easy to c heck that for an y ( V , h V ) ∈ E V , K i ∗ ( V , h V ) = (( V , ν V ) , κ ( V ,h V ) ), where ν V = h V · ( i ⊗ V ), while κ ( V ,h V ) : E ⊗ A V → V is the unique morphism with co mmutative diagram (6.8) E ⊗ V q ( V ,ν V ) / / h V # # F F F F F F F F F E ⊗ A V κ ( V ,h V ) { { v v v v v v v v v V . Such a unique morphism exists b ecause the morphism h V : E ⊗ V → V co equalises the pair of morphisms ( h E ⊗ V , E ⊗ ν V ) . 6.7 . Pa iring i nduced b y the adjunction E ⊗ A − ⊣ { E , − } A . W e refer to the s e tting considered in 6 .6. W riting F for the c o mposition ( A V ) e C Λ − → C A V ( λ ) Ξ − → E V K i ∗ − − → ( A V ) E ⊗ A − K E ⊗ A − , { E , −} A − − − − − − − − − − → ( A V ) { E , −} A , one easily sees tha t F makes the diagram ( A V ) e C F / / U e C " " F F F F F F F F ( A V ) { E , −} A { E , −} A U y y t t t t t t t t t t A V commute, were U e C is the ev ident forgetful functor. Then, a ccording to 3.8, there is a unique morphism of co monads α : e C → { E , −} A such that α ∗ = F . Since the triple ( E ⊗ A − , { E , − } A , b e E − ), where b e E − is the counit of the adjunction E ⊗ A − ⊣ { E , −} A , is a pair ing (see 3.1), it follows from Prop osition 3.4 that the triple (6.9) P ( λ ) = ( E ⊗ A − , { E , − } A , σ := b e E − · ( E ⊗ A α )) is a pair ing on the category A V . A direct insp ection shows that for a n y (( V , ν V ) , θ ( V ,ν V ) ) ∈ ( A V ) e C , ΞΛ(( V , ν V ) , θ ( V ,ν V ) ) = ( V , ξ ) , where ξ is the comp osite E ⊗ V E ⊗ θ ( V ,ν V ) − − − − − − − → E ⊗ C ⊗ V β ⊗ V − − − → A ⊗ V ν V − − → V . Then K i ∗ ΞΛ(( V , ν V ) , θ ( V ,ν V ) ) = K i ∗ ( V , ξ ) = (( V , ν V ) , κ ( V ,ξ ) : E ⊗ A V → V ) , and thus F (( V , ν V ) , θ ( V ,ν V ) ) = K E ⊗ A − , { E , −} A (( V , ν V ) , κ ( V ,ξ ) ) = (( V , ν V ) , θ ( V ,ν V ) ) , where θ ( V ,ν V ) is the comp osite V b η E V − − → { E , E ⊗ A V } A { E ,κ ( V ,ξ ) } A − − − − − − − − → { E , V } A . Here b η E − : I → { E , E ⊗ A −} A is the unit of the adjunction E ⊗ A − ⊣ { E , −} A . Since for any ob ject ( V , ν V ) ∈ V A , the pair ( e C ( V , ν V ) , ( δ e C ) ( V ,ν V ) ) = (( C ⊗ V , h ) , δ C ⊗ V ) , ON RA TIONAL P AIRINGS OF FUNCTORS 27 where h is the co mposite ( C ⊗ ν V ) · ( λ ⊗ V ) : A ⊗ C ⊗ V → C ⊗ V , is an ob ject of the ca tegory ( A V ) e C , one has F (( C ⊗ V , h ) , δ C ⊗ V ) = (( C ⊗ V , ( C ⊗ ν V ) · ( λ ⊗ V ) , θ ( C ⊗ V ,h ) ) , where θ ( C ⊗ V ,h ) : C ⊗ V → { E , E ⊗ A V } A is the comp osite C ⊗ V b η E C ⊗ V − − − − → { E , E ⊗ A ( C ⊗ V ) } A { E ,κ ( C ⊗ V,ξ ) } − − − − − − − − − → { E , C ⊗ V } A . Here ξ is the comp osite E ⊗ C ⊗ V E ⊗ δ C ⊗ V − − − − − − → E ⊗ C ⊗ C ⊗ V β ⊗ C ⊗ V − − − − − → A ⊗ C ⊗ V λ ⊗ V − − − → C ⊗ A ⊗ V C ⊗ ν V − − − − → C ⊗ V . Now, according to 3.8, the ( V , ν V )-comp onen t α ( V ,ν V ) of the c omonad mor phism α : e C → { E , −} A is the co mposite C ⊗ V b η E C ⊗ V − − − − → { E , E ⊗ A ( C ⊗ V ) } A { E ,κ ( C ⊗ V,ξ ) } − − − − − − − − − → { E , C ⊗ V } A { E ,ε C ⊗ V } A − − − − − − − − → { E , V } A , i.e. α ( V ,ν V ) is the transp ose of the comp osite ( ε C ⊗ V ) · κ ( C ⊗ V ,ξ ) . By (6.8), the diagra m E ⊗ C ⊗ V q ( C ⊗ CV ,ξ )   ξ ' ' O O O O O O O O O O O E ⊗ A ( C ⊗ V ) κ ( C ⊗ V,ξ ) / / C ⊗ V ε C ⊗ V / / V commutes. In the dia gram E ⊗ C ⊗ V E ⊗ δ c ⊗ V / / Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q E ⊗ C ⊗ C ⊗ V β ⊗ C ⊗ V / / E ⊗ ε C ⊗ C ⊗ V   A ⊗ C ⊗ V A ⊗ ε C ⊗ C ' ' O O O O O O O O O O O O λ ⊗ V / / C ⊗ A ⊗ V C ⊗ ν V / / ε C ⊗ A ⊗ V   C ⊗ V ε C ⊗ V   E ⊗ C ⊗ V β ⊗ V / / A ⊗ V ν V / / V , the left triangle commutes s ince C is a coa lgebra in V , the middle tr iangle commutes since the triple ( A , C , λ ) is an en t wining, and the trap eze and the re c tangle co mm ute by naturality of comp osition, hence ( ε C ⊗ V ) · ξ = ν V · ( β ⊗ V ) . Thus, κ ( C ⊗ V ,ξ ) is the unique morphism that makes the dia gram (6.10) E ⊗ C ⊗ V β ⊗ V / / q ( C ⊗ V,h )   A ⊗ V ν V   E ⊗ A ( C ⊗ V ) ( ε C ⊗ V ) · κ ( C ⊗ V,h ) / / V commute. (Recall that here h = ( C ⊗ ν V ) · ( λ ⊗ V ) : A ⊗ C ⊗ V → C ⊗ V .) 6.8. Prop osition. F or any ( V , ν V ) ∈ A V , c onsider the morphism α ′ ( V ,ν V ) : C ⊗ V → { E , V } that is the tr ansp ose of t he c omp osition E ⊗ C ⊗ V β ⊗ V − − − → A ⊗ V ν V − − → V . Then the diagr am { E , V } A e ( V ,ν V ) / / { E , V } C ⊗ V α ( V ,ν V ) e e K K K K K K K K K K α ′ ( V ,ν V ) 9 9 t t t t t t t t t is c ommut at ive. 28 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER Pr o of. W e show ﬁr st that (6.11) { h E , V } · α ′ ( V ,ν V ) = k · α ′ ( V ,ν V ) (see the eq ua liser diagr am (6.7)). Since the transp ose of the morphis m k is the comp osite A ⊗ E ⊗ { E , V } A ⊗ e E V − − − − → A ⊗ V h V − − → V , while the tra nspose of the mo rphism { h l E , V } is the co mp osite A ⊗ E ⊗ { E , V } A ⊗ E ⊗{ h l E ,V } − − − − − − − − − → A ⊗ E ⊗ { A ⊗ E , V } e A ⊗ E V − − − → V and this is e a sily se en to b e the comp osite A ⊗ E ⊗ { E , V } h l E ⊗{ E ,V } − − − − − − − → E ⊗ { E , V } e E V − − → V . Thu s, since ( h l E ⊗ { E , V } ) · ( A ⊗ E ⊗ α ′ ( V ,ν V ) ) = ( E ⊗ α ′ ( V ,ν V ) ) · ( h l E ⊗ C ⊗ V ) by naturality of comp osition, it is enough to show that the diag ram A ⊗ E ⊗ C ⊗ V A ⊗ E ⊗ α ′ ( V ,ν V ) / / h l E ⊗ C ⊗ V   A ⊗ E ⊗ { E , V } A ⊗ e E V / / h l E ⊗{ E ,V }      A ⊗ V ν V   E ⊗ C ⊗ V E ⊗ α ′ ( V ,ν V ) / / E ⊗ { E , V } e E V / / V is commutativ e. Since e E V · ( E ⊗ α ′ ( V ,ν V ) ) = ν V · ( β ⊗ V ), the diagra m can be rewritten as (6.12) A ⊗ E ⊗ C ⊗ V h l E ⊗ C ⊗ V   A ⊗ ( ν V · ( β ⊗ V )) / / A ⊗ V ν V   E ⊗ C ⊗ V ν V · ( β ⊗ V ) / / V . Consider now the diag ram A ⊗ E ⊗ C A ⊗ E ⊗ δ C / / i ⊗ E ⊗ C   A ⊗ E ⊗ C ⊗ C A ⊗ β ⊗ C / / i ⊗ E ⊗ C ⊗ C   A ⊗ A ⊗ C A ⊗ λ / / i ⊗ A ⊗ C   A ⊗ C ⊗ A i ⊗ C ⊗ A   A ⊗ ε C ⊗ A & & M M M M M M M M M M M E ⊗ E ⊗ C E ⊗ E ⊗ δ C / / E ⊗ E ⊗ C ⊗ C E ⊗ β ⊗ C / / E ⊗ A ⊗ C E ⊗ λ / / E ⊗ C ⊗ A β ⊗ A / / A ⊗ A , in which the three rectangles commute by naturality of comp osition, while the triangle c o m- m utes since ω − 1 ( A ⊗ ε C ) = β · ( i ⊗ C ). Recalling now that h l E = m E · ( i ⊗ E ) , we have β · ( h l E ⊗ C ) = β · ( m E ⊗ C ) · ( i ⊗ E ⊗ C ) = m A · ( β ⊗ A ) · ( E ⊗ λ ) · ( E ⊗ β ⊗ C ) · ( E ⊗ E ⊗ δ C ) · ( i ⊗ E ⊗ C ) = m A · ( A ⊗ ε C ⊗ A ) · ( A ⊗ λ ) · ( A ⊗ β ⊗ C ) · ( A ⊗ E ⊗ δ C ) since λ i s an e n t wining = m A · ( A ⊗ A ⊗ ε C ) · ( A ⊗ β ⊗ C ) · ( A ⊗ E ⊗ δ C ) nat. of comp osition = m A · ( A ⊗ β ) · ( A ⊗ E ⊗ C ⊗ ε C ) · ( A ⊗ E ⊗ δ C ) since ( C ⊗ ε C ) · δ C = I C = m A · ( A ⊗ β ) . Therefore, m A · ( A ⊗ β ) = β · ( h l E ⊗ C ), a nd henc e ( m A ⊗ V ) · ( A ⊗ β ⊗ V ) = ( β ⊗ V ) · ( h l E ⊗ C ⊗ V ) . Using now that ν V · ( A ⊗ ν V ) = ν V · ( m A ⊗ V ), since ( V , ν V ) ∈ A V , one has ν V · ( A ⊗ ν V ) · ( A ⊗ β ⊗ V ) = ν V · ( β ⊗ V ) · ( h l E ⊗ C ⊗ V ) . ON RA TIONAL P AIRINGS OF FUNCTORS 29 Thu s the diagram (6.12) co mm utes. It follows tha t { h l E , V } · α ′ ( V ,ν V ) = k · α ′ ( V ,ν V ) , and since the diagra m (6.7) is an equaliser , there exists a unique mor phism γ ( V ,ν V ) : C ⊗ V → { E , V } A that makes the diagra m { E , V } A e ( V ,ν V ) / / { E , V } C ⊗ V γ ( V ,ν V ) e e K K K K K K K K K K α ′ ( V ,ν V ) 9 9 t t t t t t t t t commute. W e claim that γ ( V ,ν V ) = α ( V ,ν V ) . T o s e e this, co nsider the diag ram E ⊗ C ⊗ V E ⊗ γ ( V ,ν V ) / / q ( C ⊗ V,h )   E ⊗ { E , V } A E ⊗ e ( V ,ν V ) / / q { E ,V } A   E ⊗ { E , V } e E V   E ⊗ A ( C ⊗ V ) E ⊗ A γ ( V ,ν V ) / / E ⊗ A { E , V } A b e E V / / V , where b e E − is the counit of the adjunction − ⊗ A E ⊣ [ E , − ] A . In this diagra m, the left rectangle commutes b y natura lit y o f q (recall that γ ( V ,ν V ) : C ⊗ V → { E , V } A is a morphism in A V ), while the rig h t one co mm utes by deﬁnition of b e E − . Since e E V · ( E ⊗ e ( V ,ν V ) ) · ( E ⊗ γ ( V ,ν V ) ) = e E V · ( E ⊗ ( e ( V ,ν V ) · γ ( V ,ν V ) )) = e E V · ( E ⊗ α ′ ( V ,ν V ) ) , and since e E V · ( E ⊗ α ′ ( V ,ν V ) ) = ν V · ( β ⊗ V ), it follows that the diagr a m E ⊗ C ⊗ V β ⊗ V / / q ( C ⊗ V,h )   A ⊗ V ν V   E ⊗ A ( V ⊗ C ) b e E V · ( E ⊗ A γ ( V ,ν V ) ) / / V commutes. Comparing this diagra m with (6.10), one see s that b e E V · ( E ⊗ A γ ( V ,ν V ) ) = ( ε C ⊗ V ) · κ ( C ⊗ V ,h ) . Thu s γ ( V ,ν V ) : C ⊗ V → { E , V } A is the transp ose of the mo rphism ( ε C ⊗ V ) · κ ( C ⊗ V ,h ) . Th us γ ( V ,ν V ) is just α ( V ,ν V ) . This co mpletes the pr oof.  When the pairing P ( λ ) (6.9) is rational, we wr ite Ra t P ( λ ) ( E ) for the full sub category of the catego ry E V gener ated by those o b jects whose images under the functor K i ∗ lie in the category Rat P ( λ ) ( E ⊗ A − ). The following r e sult extends [1, Theorem 3.10], [12, Pro p osition 2 .1 ], and [13, Theorem 2.6] from mo dule categories to monoida l catego ries. 6.9. Theorem. Le t V = ( V , ⊗ , I ) b e a monoidal c ate gory with V admitting b oth e qualisers and c o e qualisers, and ( A , C , λ ) a r epr esentable entwining with r epr esentable obje ct E . Supp ose that (1) the functors A ⊗ − , E ⊗ − : V → V have right adjoints { A, −} and { E , −} , (2) for any ( V , ν V ) ∈ A V , the tr ansp ose α ′ ( V ,ν V ) : C ⊗ V → { E , V } of the c omp osite E ⊗ C ⊗ V β ⊗ V − − − → A ⊗ V ν V − − → V is a monomorphi sm, and (3) (i) the functor C ⊗ − : V → V pr eserves e qualisers, or (ii) the c ate gory V admits pushouts and the fun ctor C ⊗ − : V → V pr eserves r e gular monomorphisms and has a right adjoint, or (iii) the c ate gory V admits pushouts, every monomorphism in V is r e gular and t he functor C ⊗ − : V → V has a right adjoi nt. 30 BA CHUKI MES ABLISHVILI AND R OBER T WISBAUER Then the p airing P ( λ ) is r ational and ther e is an e quivalenc e of c ate gories C A V ( λ ) ≃ Rat P ( λ ) ( E ) . Pr o of. Since e ( V ,ν V ) : { E , V } A → { E , V } is a n equa liser for all ( V , ν V ) ∈ A V , α ( V ,ν V ) is a monomorphism if and o nly if α ′ ( V ,ν V ) is so . Th us, the pairing P ( λ ) is rational if and only if for any ( V , ν V ) ∈ A V , the morphism α ′ ( V ,ν V ) : C ⊗ V → { E , V } is a monomor phism. Thus, condition (2) implies that the pair ing P ( λ ) is rationa l. Next, since the forgetful functor A U : A V → V preserves and crea tes equaliser s, the functor e C : A V → A V pr eserves eq ua lisers if and only if the comp osite A U e C : A V → V do es so . B ut for any ( V , ν V ) ∈ A V , A U e C ( V , ν V ) = C ⊗ V . It follo ws that if the functor C ⊗ − : V → V preserves equalis e rs, then the functor e C : A V → A V do es so. Since e a c h of the conditio ns in (3) implies that the functor C ⊗ − : V → V preser v es equalise r s (see the pro of of Pr opositio n 5.7) and since the functor E ⊗ A − : A V → E V ha s a right adjoint { E , −} A : A V → E V , one can apply Theorem 4.8 to get the desir ed result.  Since for any V -c o algebra C = ( C, ε C , δ C ), the identit y mor phism I C : C ⊗ I = C → C = I ⊗ C is an en twinin g fr om the trivial V -alge br a I = ( I , I I , I I ) to the V - c o algebra C , it follows from Example 6.2(1) that this ent wining is r epresen table with represe ntable ob ject C ∗ = [ C , I ]. Applying Pr o position 6.3 gives: 6.10 . Coalgebras in monoi dal clos ed categories. A ssume the monoidal c ate gory V t o b e close d and c onsider any V -c o algebr a C = ( C, ε C , δ C ) . Then the t rip le C ∗ = ( C ∗ = [ C , I ] , e C ∗ , m C ∗ ) is a V -algebr a, wher e e C ∗ = π ( ε C ) , while m C ∗ is the morphism C ∗ ⊗ C ∗ → C ∗ that c orr esp onds to the c omp osite C ∗ ⊗ C ∗ ⊗ C C ∗ ⊗ C ∗ ⊗ δ C − − − − − − − − → C ∗ ⊗ C ∗ ⊗ C ⊗ C C ∗ ⊗ e C I ⊗ C − − − − − − − → C ∗ ⊗ C e C I − − → I under the bije ction (s e e (5.2)) π = π C ∗ ⊗ C ∗ ,C, I : V ( C ∗ ⊗ C ∗ ⊗ C, I ) ≃ V ( C ∗ ⊗ C ∗ , C ∗ ) . 6.11. Prop osition. In the situation of 6.10, the triple P ( C ) = ( C ∗ , C , t = e C I : C ∗ ⊗ C → I ) is a left p airing in V . Pr o of. W e just note that the equalities π − 1 ( e C ∗ ) = ε C and π − 1 ( m C ∗ ) = e C I · ( C ∗ ⊗ e C I ⊗ C ) · ( C ∗ ⊗ C ∗ ⊗ δ C ) imply commutativit y o f the diagrams C ∗ ⊗ C ∗ ⊗ C C ∗ ⊗ C ∗ ⊗ δ C / / m C ∗ ⊗ C   C ∗ ⊗ C ∗ ⊗ C ⊗ C C ∗ ⊗ e C I ⊗ C / / C ∗ ⊗ C e C I   C e C ∗ ⊗ C o o ε C { { w w w w w w w w w w C ∗ ⊗ C e C I / / I .  Applying now either Prop osition 5 .9 or Pr opositio n 6.11 y ields 6.12. Theorem. L et L et V = ( V , ⊗ , I , [ − , − ]) b e a m onoidal close d c ate gory, C = ( C, ε C , δ C ) a V -c o algebr a with C V -pr enucle ar, and assume V to admit e qualisers. If either (i) the functor C ⊗ − : V → V pr eserves e qualisers, or (ii) C ⊗ − : V → V admits pushout s and the fun ctor C ⊗ − : V → V pr eserves r e gular monomorphisms, or ON RA TIONAL P AIRINGS OF FUNCTORS 31 (iii) V admits pushouts and every monomorphism in V is r e gular, then Rat P ( C ) ( C ∗ ) is a ful l c or eﬂe ctive sub c ate gory of C ∗ V and the fun ctor Φ P ( C ) : C V → C ∗ V c or estricts to an e quivalenc e R P ( C ) : C V → Rat( C ∗ ) . A sp ecial case of the situation describ ed in Theorem 6 .12 is giv en by a ﬁnite dimensional k -coalgebra C ov er a ﬁeld k and V the ca tegory o f k -vector spaces . Ac knowledgements. The work on this paper was sta rted dur ing a visit of the ﬁrst author at the Depar tmen t of Mathematics at the Heinrich Heine Universit y o f D¨ usseldorf supp orted by the Germa n Research F ounda tion (DFG) and contin ued with s upp ort by V o lksw agen F o un- dation (Ref.: I/8 4 328 ). The authors express their thanks to all these ins titutions. References [1] Abuhlail, J.Y. , Ra tional mo dules for c orings , Commun. Algebra 31(12) (2003) , 5793-5840. [2] Ad´ amek, J. and Rosick´ y, J., L o c al ly pr esentable and ac c essible ca te gories , Cambridge Universit y Press, 1994. [3] Barr, M. and W ells, C., T op oses, T riples, and The ories , Gr undleh ren der Math. Wissensc haften 278, Springer-V erlag, 1985. [4] B ´ ena bou, J., Intr o duction to bic ate gories, Lecture Notes i n Mathematics V ol. 40 (1967), 1–77. [5] B¨ ohm, G., Brzezi´ nski, T. and Wisbauer, R., Monads and c omona ds on mo dule c ate gories , J. Al geb ra 322 (2009), 1719-1747. [6] B¨ orger, R., Tholen, W., Abschw¨ achungen des A djunkt i ons b egriﬀs , M anuscr. Math. 19 (1976), 19-45. [7] Borceux, F., Handb o ok of Cate g orica l A lgebr a , vol. 2, Cam bridge Univ ersity Pr ess, 1994. [8] Brzezi´ nski, T. and Wisbauer, R., Corings and Como dules , London M at h. So c. LN 309, Cambridge Univ er- sity Press, 2003. [9] Clark, J. and Wisbauer, R., Idemp otent monad s and ⋆ -functors , J. Pure Appl. Algebra, [10] Dubuc, E., Kan ext ensions in enriche d c ate gory t he ory , LN Math. 145, Springer-V erlag, 1970. [11] Eilen berg, S. and Mo ore, J., A djoint functors and triples , Illinois J. Math. 9 (1965), 381-398. [12] El Kaoutit, L. and G´ omez -T orrecillas, J., Corings with exact r ational functors and inje ct ive obje ct s , in Mo dules and Comodules, Brzezinski, T. ; G´ omez Pardo, J. L. ; Shestak ov, I.; Smith, P . F. (Eds.), T r ends in Mathematics, Bi rkh¨ auser (2008), 185-201. [13] El Kaoutit, L., G´ omez-T orrecillas, J. and Lobillo, F.J. , Semisimple c orings , Algebra Coll oq. 11 (2004), 427–442. [14] Fisher-Pa lmquist, J. and New ell, D., T ri ples on functor ca te g ories , J. Algebra 25 (1973), 226-258. [15] Kainen, P .C., We ak adjoint funct ors , Math. Z. 122 (1971) , 1-9. [16] MacLane , S., Catego ries for the Working Mathematician , Gr adu ate T exts in Mathematics V ol. 5, Springer, Berlin-New Y ork, 1971. [17] Medv edev , M.Y a., Semiadjoint functors and Kan ex tensions , Sib. M ath . J. 15 (1974), 674-676; translation from Sib. Mat. Zh. 15 (1974), 952-956. [18] Mesablish vili, B. , Entwining Structur es in Monoidal Cate gories , J. Al gebra 319(6) (2008 ), 2496-2517. [19] Mesablish vili, B. and Wisbauer, R., Bimonads and Hopf monads on ca te g ories , J. K-Theory (2010), arXiv:0710.1163 (2008) [20] P areigis, B. , Kate g orien und F unktor e n , Mathematische Leitf¨ ade n, T eubner V erl ag, Stuttgart, 1969. [21] Ro w e, K. , N ucle arity , Can. M at h. Bull. 31 (1988), 227-235. [22] Sc hubert, H. , Categories , Berli n-Heidelberg-New Y ork, Springer-V erlag, 1972. Addresses: Razmadze Mathematical Institute, 1, M. Aleksidze s t., Tbilisi 0193 , and Tbilisi Centre fo r Mathematica l Sciences, Chavc hav adze Ave. 7 5, 3/ 35, Tbilisi 01 6 8, Republic of Geo rgia, bachi@rmi.acnet.ge Department of Mathematics of HHU, 4 0 225 D¨ usseldorf, Germany , wisbauer@math.un i-duesseldorf. de

On Rational Pairings of Functors

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment