(Co)ends for representations of tensor categories
aa r X i v : . [ m a t h . QA ] O c t (CO)ENDS FOR REPRESENTATIONS OF TENSORCATEGORIES NOELIA BORTOLUSSI AND MART´IN MOMBELLI
Abstract.
We generalize the notion of ends and coends in categorytheory to the realm of module categories over finite tensor categories. Wecall this new concept module (co)end . This tool allows us to give differentproofs to several known results in the theory of representations of finitetensor categories. As a new application, we present a description of therelative Serre functor for module categories in terms of a module coend,in a analogous way as a Morita invariant description of the Nakayamafunctor of abelian categories presented in [4].
Introduction
Throughout this paper, k will denote a field, all categories will be finite (inthe sense of [3]) abelian k -linear categories, and all functors will be additive k -linear. Given categories M , A , and a functor S : M op × M → A thenotion of the end R M ∈M S and coend R M ∈M S is a standard and very usefulconcept in category theory. The end of the functor S is an object in A together with dinatural transformations π M : Z M ∈M S .. −→ S ( M, M )with the following universal property; for any pair (
B, d ) consisting of anobject B ∈ A and a dinatural transformation d M : B .. −→ S ( M, M ), thereexists a unique morphism h : B → E in A such that d M = π M ◦ h for any M ∈ M . The notion of coend is defined dually.If M is a finite abelian, k -linear category, M can be thought of as amodule category over vect k , the tensor category of finite dimensional vector k -spaces. If M = m A is the category of finite dimensional right A -modules,where A is a finite dimensional k -algebra, then M has a left vect k -actionvect k × m A → m A ( V, M ) V ⊗ k M, Date : October 26, 2020.2010
Mathematics Subject Classification.
Key words and phrases. tensor category; module category. where the right action on V ⊗ k M is given on the second tensorand. If S :( m A ) op × m A → A is any functor, it posses a canonical natural isomorphism β VM,N : S ( M, V ⊗ k N ) → S ( V ∗ ⊗ k M, N ) , for any V ∈ vect k , M, N ∈ m A . The existence of β essentially follows fromthe additivity of the functor S .If in addition p M : E .. −→ S ( M, M ) is any dinatural transformation, itsatisfies equation(0.1) S (ev V ⊗ k id M , id M ) p M = S ( m V ∗ ,V,M , id M ) β VV ⊗ k M,M p V ⊗ k M , for any V ∈ vect k . This equation follows from the dinaturality of p . Hereev V : V ∗ ⊗ k V → k is the evaluation map, and m W,V,M : ( W ⊗ k V ) ⊗ k M → W ⊗ k ( V ⊗ k M ) is the canonical associativity of vector spaces. This impliesthat the end of S is the universal object among all dinatural transformationsthat satisfy (0.1). A similar observation can be made for the coend. This isthe starting point to generalize the notion of (co)end, where we will replacethe category vect k with an arbitrary tensor category.Let C be a finite tensor category, and M be a left C -module category withaction given by ⊲ : C × M → M . Assume S : M op × M → A is a functorthat comes equipped with natural isomorphisms β XM,N : S ( M, X ⊲ N ) → S ( X ∗ ⊲ M, N ) . We call this isomorphism a pre-balancing of S . In this general case, thepre-balancing is an extra structure of the functor S . We define the moduleend of S to be an object E ∈ A that comes with dinatural transformations π M : E .. −→ S ( M, M ) such that the equation(0.2) S (ev X ⊲ id M , id M ) π M = S ( m X ∗ ,X,M , id M ) β XX⊲M,M π X⊲M , is fulfilled, and it is universal among all objects in A with dinatural transfor-mations that satisfy (0.2). Unlike the case C = vect k , it may happen that adinatural transformation does not satisfy (0.2). We denote the module endas H M ∈M ( S, β ) , or sometimes simply as H M ∈M S whenever the pre-balancing β is undertstood from the context.An analogous definition can be made to define module coend , and also todefine module ends and coends starting from right C -module categories.In Section 3 we introduce the module (co)ends, and we prove severalresults that extend known properties of (co)ends. We prove that when thetensor category C = vect k our definition coincides with the usual (co)ends.We also study what happens when we restrict the module (co)ends to atensor subcategory.In Section 4 we give several applications. If M , N are left C -modulecategories, and F, G : M → N are C -module functors, the functorHom N ( F ( − ) , G ( − )) : M op × M → vect k CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 3 has a canonical pre-balancing γ , and we prove that there is an isomorphismNat m ( F, G ) ≃ I M ∈M (Hom N ( F ( − ) , G ( − )) , γ ) . Here Nat m ( F, G ) is the space of natural module transformations between F and G . Using this result we can set up a triangle of equivalences of categories M op ⊠ C N Fun C ( M bop , N ) Fun C ( M , N ) , L M , N e L M , N Θ M , N χ M , N Υ M , N generalizing the triangle presented in [4]. Here it is required that M , N are exact module categories. These equivalences are: L M , N : M op ⊠ C N →
Fun C ( M bop , N ) ,M ⊠ C N Hom M op ( − , M ) ⊲ N,χ M , N : Fun C ( M bop , N ) → M op ⊠ C N ,F I U ∈M op U ⊠ C F ( U ) , and on the other side of the triangle we have equivalences e L M , N : M op ⊠ C N →
Fun C ( M , N ) M ⊠ C N Hom M op ( M, − ) ∗ ⊲ N, Υ M , N : Fun C ( M , N ) → M op ⊠ C N ,F I M ∈M M ⊠ C F ( M ) . Here Θ M , N = e L M , N ◦ χ M , N . As a consequence of these equivalences, if A ∈ C is an algebra such that the module category C A is exact, we obtain akind of Peter-Weyl theorem for the regular A -bimodule A = A A A : A ≃ I M ∈M ∗ M ⊗ M. We also prove that the module functor Θ M , M bop (Id ) is equivalent as modulefunctors to the relative Serre functor of M . This description is an analogousform of the Morita invariant description of the Nakayama functor presentedin [4].If C and D are Morita-equivalent tensor categories, this means that thereexists an invertible ( C , D )-bimodule category B ; we prove that the corre-spondence M 7→
Fun C ( B , M ) , N 7→
Fun D ( B op , N ) BORTOLUSSI AND MOMBELLI is in fact part of a 2-equivalence between the 2-categories of C -module cat-egories and D -module categories. This result was proven in [3]. We showthat, for any D -module category N , the functorFun C ( B , Fun D ( B op , N )) → N H I B ∈B H ( B )( B )is an equivalence of D -module categories.In the last Section we show that the functor Υ : ( C ∗M ) ∗M → C defined asΥ( G ) = I M ∈M Hom(
M, G ( M ))is a quasi-inverse of the canonical functor can : C → ( C ∗M ) ∗M , can ( X )( M ) = X ⊲ M. Preliminaries and Notation.
We denote by vect k the category of finite-dimensional k -vector spaces. If M , N are categories, and F : M → N is afunctor, we shall denote by F r.a. , F l.a. : N → M its right and left adjoint,respectively.For any category M , the opposite category will be denoted by M op . Weshall denote by M , f objects and morphisms in M op that correspond to M and f . We shall also denote by F op : M op → N op the opposite functor to F ; that is, the functor defined as F op ( M ) = F ( M ), F op ( f ) = F ( f ) for anyobject M and any morphism f .1. Finite tensor categories
For basic notions on finite tensor categories we refer to [2], [3]. Let C be afinite tensor category over k ; that is a rigid monoidal category with simpleunit object , such that the underlying category is finite.If C has associativity constraint given by a X,Y,Z : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ) , we shall denote by C rev , the tensor category whose underlying abelian cate-gory is C , with reverse monoidal product ⊗ rev : C × C → C , X ⊗ rev Y = Y ⊗ X, and associativity constraints a rev X,Y,Z : ( X ⊗ rev Y ) ⊗ rev Z → X ⊗ rev ( Y ⊗ rev Z ) ,a rev X,Y,Z := a − Z,Y,X , for any X, Y, Z ∈ C . It is well known that for any pair of objects
X, Y ∈ C there are canonical isomorphisms φ rX,Y : ( X ⊗ Y ) ∗ → Y ∗ ⊗ X ∗ ,φ lX,Y : ∗ ( X ⊗ Y ) → ∗ Y ⊗ ∗ X. (1.1) CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 5
For any X ∈ C we shall denote byev X : X ∗ ⊗ X → , coev X : → X ⊗ X ∗ the evaluation and coevaluation. Abusing of the notation, we shall alsodenote by ev X : X ⊗ ∗ X → , coev X : → ∗ X ⊗ X the evaluation and coevaluation for the left duals. If f : X → Y is anisomorphism in C thenev Y ( f ⊗ id Y ) = ev X (id X ⊗ ∗ f ) . (1.2)For any X, Y ∈ C the following identities holdev X ⊗ Y = ev X (id X ⊗ ev Y ⊗ id ∗ X )(id X ⊗ Y ⊗ φ lX,Y ) , ( φ lX,Y ⊗ id X ⊗ Y )coev X ⊗ Y = (id ∗ Y ⊗ coev X ⊗ id Y )coev Y . (1.3)Off course that similar identities hold for the right duals, but they won’t beneeded.1.1. Algebras in tensor categories.
In this subsection we assume that C is a strict tensor category, this means in particular that the associativityconstraints are the identities. Let A, B ∈ C be algebras. We shall denote by C A , A C , A C B the categories of right A -modules, left A -modules and ( A, B )-bimodules in C , respectively. If V ∈ C A is a right A -module with action given by ρ V : V ⊗ A → V , and W ∈ A C is a left A -module with action given by λ W : A ⊗ W → W , we shall denote by π AV,W : V ⊗ W → V ⊗ A W the coequalizer ofthe maps ρ V ⊗ id W , id V ⊗ λ W : ( V ⊗ A ) ⊗ W −→ V ⊗ W. An object in the category A C B will be denoted as ( V, λ V , ρ V ) ∈ A C B , where λ V : A ⊗ V → V is the left action, and ρ V : V ⊗ B → V is the right action.Since the tensor product is exact in both variables, then π AV,W ⊗ U = π AV,W ⊗ id U , for any V ∈ C A , W ∈ A C , U ∈ C . We are going to freely use this fact withoutfurther mention. Lemma 1.1.
Assume that C is a finite tensor category and A, B ∈ C arealgebras. The following statements hold: (i) If M ∈ C A then ∗ M ∈ A C . (ii) There are natural isomorphisms (1.4) Hom B ( M ⊗ A V, U ) ≃ Hom ( A,B ) ( V, ∗ M ⊗ U ) , (1.5) Hom A ( M, X ⊗ N ) ≃ Hom C ( M ⊗ A ∗ N, X ) , (1.6) Hom A ( M, X ⊗ N ) ≃ Hom A ( X ∗ ⊗ M, N ) , for any X ∈ C , M, N ∈ C A , V ∈ A C B , U ∈ C B . BORTOLUSSI AND MOMBELLI
Proof. (i). If M ∈ C A then ∗ M has structure of left A -module via λ M : A ⊗ ∗ M → ∗ M defined as(1.7) λ ∗ M = (id ∗ M ⊗ ev M )(id ∗ M ⊗ ρ M ⊗ id ∗ M )(coev M ⊗ id A ⊗ ∗ M ) . (ii). Let us prove only the first isomorphism. The others follow similarly.The object M ⊗ A V has a right B -module structure as follows. Consider φ : M ⊗ V ⊗ B → M ⊗ A V , φ = π M,V (id M ⊗ ρ V ). Then, ρ M ⊗ A V : M ⊗ A V ⊗ B → M ⊗ A V is defined as the unique morphism such that(1.8) ρ M ⊗ A V ( π M,V ⊗ id B ) = φ. Define Φ : Hom B ( M ⊗ A V, U ) → Hom ( A,B ) ( V, ∗ M ⊗ U ) as(1.9) Φ( f ) = (id ∗ M ⊗ f π M,V )(coev M ⊗ id V ) , for any f ∈ Hom B ( M ⊗ A V, U ) . Let us show Φ( f ) is a morphism of ( A, B )-bimodules. We need to prove that(1.10) ( λ ∗ M ⊗ id U )(id A ⊗ Φ( f )) = Φ( f ) λ V , and(1.11) (id ∗ M ⊗ ρ U )(Φ( f ) ⊗ id B ) = Φ( f ) ρ V , for any ( M, ρ M ) ∈ C A , ( V, λ V , ρ V ) ∈ A C B and ( U, ρ U ) ∈ C B . Here λ ∗ M isthe left action of A on ∗ M presented in (1.7).The left hand side of (1.10) is equal to( λ ∗ M ⊗ id U )(id A ⊗ Φ( f )) == (id ∗ M ⊗ ev M ⊗ id U )(id ∗ M ⊗ ρ M ⊗ id ∗ M ⊗ U )(coev M ⊗ id A ⊗ ∗ M ⊗ U )(id A ⊗ ∗ M ⊗ f π M,V )(id A ⊗ coev M ⊗ id V )= (id ∗ M ⊗ ev M ⊗ id U )(id ∗ M ⊗ ρ M ⊗ id ∗ M ⊗ U )(id ∗ M ⊗ M ⊗ A ⊗ ∗ M ⊗ f π M,V )(coev M ⊗ id A ⊗ ∗ M ⊗ M ⊗ V )(id A ⊗ coev M ⊗ id V )= (id ∗ M ⊗ ev M ⊗ id V )(id ∗ M ⊗ M ⊗ ∗ M ⊗ f π M,V )(id ∗ M ⊗ ρ M ⊗ id ∗ M ⊗ M ⊗ V )(id ∗ M ⊗ M ⊗ A ⊗ coev M ⊗ id V )(coev M ⊗ id A ⊗ V )= (id ∗ M ⊗ f π M,V )(id ∗ M ⊗ ev M ⊗ id M ⊗ V )(id ∗ M ⊗ M ⊗ coev M ⊗ id V )(id ∗ M ⊗ ρ M ⊗ id V )(coev M ⊗ id A ⊗ V )= (id ∗ M ⊗ f )(id ∗ M ⊗ π M,V ( ρ M ⊗ id V ))(coev M ⊗ id A ⊗ V )= (id ∗ M ⊗ f π M,V )(id ∗ M ⊗ M ⊗ λ V )(coev M ⊗ id A ⊗ V )= (id ∗ M ⊗ f π M,V )(coev M ⊗ id V ) λ V = Φ( f ) λ V . The first equality is by the definition of λ ∗ M and Φ( f ). The fifth equalityfollows from the rigidity axioms. The sixth equality is consequence of π M,V being the coequalizer of ρ M ⊗ id V , id M ⊗ λ V . The last equality follows by thedefinition of Φ( f ). CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 7
Since f is a B -module morphism,(1.12) ρ U ( f ⊗ id B ) = f ρ M ⊗ A V . Using (1.8), this equation implies(1.13) ρ U ( f π M,V ⊗ id B ) = f ρ M ⊗ A V ( π M,V ⊗ id B ) = f π M,V (id M ⊗ ρ V ) . Let us prove (1.11). The left hand side of (1.11) is equal to(id ∗ M ⊗ ρ U )(Φ( f ) ⊗ id B ) = (id ∗ M ⊗ ρ U ( f π M,V ⊗ id B ))(coev M ⊗ id V ⊗ B )= (id ∗ M ⊗ f π M,V (id M ⊗ ρ V ))(coev M ⊗ id V ⊗ B )= (id ∗ M ⊗ f π M,V )(coev M ⊗ id V ) ρ V = Φ( f ) ρ V . The first equality is by the definition of Φ( f ). The second equality followsfrom (1.13). And the last equality again follows from the definition of Φ( f ).Now, let us show that Φ has an inverse. Let us defineΨ : Hom ( A,B ) ( V, ∗ M ⊗ U ) → Hom B ( M ⊗ A V, U )as follows. Let g ∈ Hom ( A,B ) ( V, ∗ M ⊗ U ). Define Ψ( g ) = h where h : M ⊗ A V → U is the unique morphism such that(1.14) hπ M,V = (ev M ⊗ id U )(id M ⊗ g ) . Let us show Ψ( g ) is a B -module morphism. That is ρ U ( h ⊗ id B ) = hρ M ⊗ A V . For this, it is enough to prove ρ U ( hπ M,V ⊗ id B ) = hρ M ⊗ A V ( π M,V ⊗ id B ) . Starting from the left hand side ρ U ( hπ M,V ⊗ id B ) = ρ U (ev M ⊗ id U ⊗ B )(id M ⊗ g ⊗ id B )= (ev M ⊗ id U )(id M ⊗ ∗ M ⊗ ρ U )(id M ⊗ g ⊗ id B )= (ev M ⊗ id U )(id M ⊗ gρ V )= hπ M,V (id M ⊗ ρ V )= hρ M ⊗ A V ( π M,V ⊗ id B ) . The first equality is by (1.14). The third equality is consequence of g beinga B -module morphism. The fourth equality follows from (1.14) and thelast equality follows from (1.8). Let us show Φ and Ψ are inverses of eachanother. Let be f ∈ Hom B ( M ⊗ A V, U ). We haveΨΦ( f ) = Ψ((id ∗ M ⊗ f π M,V )(coev M ⊗ id V )) = h where hπ M,V = (ev M ⊗ id U )(id ∗ M ⊗ M ⊗ f π M,V )(id M ⊗ coev M id V )= f π M,V (ev M ⊗ id M ⊗ V )(id M ⊗ coev M ⊗ id V )= f π M,V
BORTOLUSSI AND MOMBELLI
The first equality is the definition of h , and the last equality follows fromthe rigidity axioms. Therefore, h = f and ΨΦ( f ) = f . The proof ofΦΨ = Id follows similarly.We shall only sketch the proof of isomorphism (1.5). DefineΦ AM,X,N : Hom A ( M, X ⊗ N ) → Hom C ( M ⊗ A ∗ N, X ) , Φ AM,X,N ( α ) π AM, ∗ N = (id X ⊗ ev N )( α ⊗ id ∗ N ) , (1.15)and Ψ AM,X,N : Hom C ( M ⊗ A ∗ N, X ) → Hom A ( M, X ⊗ N ) , Ψ AM,X,N ( α ) = ( απ AM, ∗ N ⊗ id N )(id M ⊗ coev N ) . (1.16)It follows by a direct calculation that Φ AM,X,N and Ψ
AM,X,N are well-definedand they are one the inverse of the other. (cid:3) Representations of tensor categories
A left module category over C is a category M together with a k -bilinearbifunctor ⊲ : C × M → M , exact in each variable, endowed with naturalassociativity and unit isomorphisms m X,Y,M : ( X ⊗ Y ) ⊲ M → X ⊲ ( Y ⊲ M ) , ℓ M : ⊲ M → M. These isomorphisms are subject to the following conditions:(2.1) m X,Y,Z⊲M m X ⊗ Y,Z,M = (id X ⊲ m Y,Z,M ) m X,Y ⊗ Z,M ( a X,Y,Z ⊲ id M ) , (2.2) (id X ⊲ ℓ M ) m X, ,M = r X ⊲ id M , for any X, Y, Z ∈ C , M ∈ M . Here a is the associativity constraint of C .Sometimes we shall also say that M is a C - module category or a represen-tation of C .Let M and M ′ be a pair of C -modules. A module functor is a pair ( F, c ),where F : M → M ′ is a functor equipped with natural isomorphisms c X,M : F ( X ⊲ M ) → X ⊲ F ( M ) ,X ∈ C , M ∈ M , such that for any X, Y ∈ C , M ∈ M :(id X ⊲ c Y,M ) c X,Y ⊲M F ( m X,Y,M ) = m X,Y,F ( M ) c X ⊗ Y,M (2.3) ℓ F ( M ) c ,M = F ( ℓ M ) . (2.4)There is a composition of module functors: if M ′′ is a C -module categoryand ( G, d ) : M ′ → M ′′ is another module functor then the composition(2.5) ( G ◦ F, e ) :
M → M ′′ , e X,M = d X,F ( M ) ◦ G ( c X,M ) , is also a module functor. CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 9 A natural module transformation between module functors ( F, c ) and(
G, d ) is a natural transformation θ : F → G such that d X,M θ X⊲M = (id X ⊲ θ M ) c X,M , (2.6)for any X ∈ C , M ∈ M . The vector space of natural module transformationswill be denoted by Nat m ( F, G ). Two module functors
F, G are equivalent if there exists a natural module isomorphism θ : F → G . We denote byFun C ( M , M ′ ) the category whose objects are module functors ( F, c ) from M to M ′ and arrows module natural transformations.Two C -modules M and M ′ are equivalent if there exist module functors F : M → M ′ , G : M ′ → M , and natural module isomorphisms Id M ′ → F ◦ G , Id M → G ◦ F .A module is indecomposable if it is not equivalent to a direct sum of twonon trivial modules. Recall from [3], that a module M is exact if for anyprojective object P ∈ C the object P ⊲ M is projective in M , for all M ∈ M .If M is an exact indecomposable module category over C , the dual category C ∗M = End C ( M ) is a finite tensor category [3]. The tensor product is thecomposition of module functors.A right module category over C is a finite category M equipped with anexact bifunctor ⊳ : M × C → M and natural isomorphisms e m M,X,Y : M ⊳ ( X ⊗ Y ) → ( M ⊳ X ) ⊳ Y, r M : M ⊳ → M such that(2.7) e m M⊳X,Y,Z e m M,X,Y ⊗ Z (id M ⊳ a X,Y,Z ) = ( e m M,X,Y ⊳ id Z ) e m M,X ⊗ Y,Z , (2.8) ( r M ⊳ id X ) e m M, ,X = id M ⊳ l X . If M , M ′ are right C -modules, a module functor from M to M ′ is a pair( T, d ) where T : M → M ′ is a functor and d M,X : T ( M ⊳ X ) → T ( M ) ⊳ X are natural isomorphisms such that for any X, Y ∈ C , M ∈ M :( d M,X ⊗ id Y ) d M⊳X,Y T ( m M,X,Y ) = m T ( M ) ,X,Y d M,X ⊗ Y , (2.9) r T ( M ) d M, = T ( r M ) . (2.10)The next result is well-known. See for example [1, Corollary 2.13.]. Lemma 2.1.
Let M , N be left C -module categories, and F, G : M → N are C -module functors. (i) The right and left adjoint of F , if they exists, have structure of C -module functor. (ii) If F ≃ G as C -module functors, then F l.a ≃ G l.a , F r.a ≃ G r.a as C -module functors. (iii) If F , F are composable C -module functors, there exists an isomor-phism of C -module functors ( F ◦ F ) l.a ≃ F l.a ◦ F l.a , ( F ◦ F ) r.a ≃ F r.a ◦ F r.a . (cid:3) Bimodule categories.
Assume that C , D , E are finite tensor cate-gories. A ( C , D ) − bimodule category is a category M with left C -modulecategory structure ⊲ : C × M → M , and right D -module category structure ⊳ : M × D → M , equipped with natural isomorphisms { γ X,M,Y : (
X ⊲ M ) ⊳ Y → X ⊲ ( M ⊳ Y ) , X ∈ C , Y ∈ D , M ∈ M} satisfying certain axioms. For details the reader is referred to [6], [7].If M is a right C -module category then the opposite category M op has aleft C -action given by C × M op → M op , ( X, M ) M ⊳ X ∗ , and associativity isomorphisms m op X,Y,M = m M,Y ∗ ,X ∗ (id M ⊳ φ rX,Y ) . Analo-gously, if M is a left C -module category then M op has structure of right C -module category, with action given by M op × C → M op , ( M, X ) X ∗ ⊲ M, with associativity constraints m op M,X,Y = m Y ∗ ,X ∗ ,M ( φ rX,Y ⊲ id M ) for all X, Y ∈C , M ∈ M . If M is a ( C , D )-bimodule category then M op is a ( D , C )-bimodule category.If M is a left C -module category, we shall denote by M bop = ( M op ) op .That is, M bop = M as categories, but the left action of C on M bop is ◮ : C × M bop → M bop ,X ◮ M = X ∗∗ ⊲ M, for any X ∈ C , M ∈ M .Assume that M is a ( C , D )-bimodule category, and N is a ( C , E )-bimodulecategory. The category Fun C ( M , N ) has a structure of ( D , E )-bimodulecategory. Let us briefly describe this structure. For more details, the readeris referred to [6]. The left and right actions are given by ⊲ : D ×
Fun C ( M , N ) → Fun C ( M , N ) ,⊳ : Fun C ( M , N ) × E → Fun C ( M , N ) , where(2.11) ( X ⊲ F )( M ) = F ( M ⊳ X ) , ( F ⊳ Y )( M ) = F ( M ) ⊳ Y, for any X ∈ D , Y ∈ E , F ∈ Fun C ( M , N ) and M ∈ M . CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 11
The internal Hom.
Let C be a tensor category and M be a left C -module category. For any pair of objects M, N ∈ M , the internal Hom isan object Hom(
M, N ) ∈ C representing the left exact functorHom M ( − ⊲ M, N ) : C op → vect k . This means that there are natural isomorphisms, one the inverse of eachother, φ XM,N : Hom C ( X, Hom(
M, N )) → Hom M ( X ⊲ M, N ) ,ψ XM,N : Hom M ( X ⊲ M, N ) → Hom C ( X, Hom(
M, N )) , (2.12)for all M, N ∈ M , X ∈ C . Sometimes we shall denote the internal Hom ofthe module category M by Hom M to emphasize that it is related to thismodule category. Similarly, if N is a right C -module category, for any pair M, N ∈ N the internal hom is the object Hom(
M, N ) ∈ C representing theleft exact functor Hom M ( M ⊳ − , N ) : C op → vect k . Lemma 2.2.
The following statements hold. Let M be a left C -module category. There are natural isomorphisms Hom M ( X ⊲ M, N ) ≃ Hom M ( M, N ) ⊗ X ∗ , Hom M ( M, X ⊲ N ) ≃ X ⊗ Hom M ( M, N ) . for any M, N ∈ M , X ∈ C . Analogously, if N is a right C -module category, there are naturalisomorphisms Hom N ( M ⊳ X, N ) ≃ ∗ X ⊗ Hom N ( M, N ) , Hom N ( M, N ⊳ X ) ≃ Hom N ( M, N ) ⊗ X. for any M, N ∈ N , X ∈ C .Proof. The functor Hom M ( M, − ) : M → C is the right adjoint of the functor R M : C → M , R M ( X ) = X ⊲ M . Since R M is a C -module functor then,it follows from Lemma 2.1 that, Hom M ( M, − ) is also a C -module functor.This implies in particular that there are natural isomorphismsHom M ( M, X ⊲ N ) ≃ X ⊗ Hom M ( M, N ) . The other three isomorphisms follow in a similar way. (cid:3)
Let M be a left C -module category. There is a relation between theinternal hom of M and M op , stated in the next Lemma. Lemma 2.3.
For any M ∈ M , the functors ∗∗ Hom M ( M, − ) , Hom M op ( − , M ) : M bop → C , are equivalent C -module functors. Also the functors Hom M op ( M, − ) ∗ , ∗ Hom M ( − , M ) : M → C are equivalent C -module functors. In particular, there are natural isomor-phisms ∗∗ Hom M ( M, N ) ≃ Hom M op ( N , M ) , for any M, N ∈ M .Proof.
The functors D : C → C bop , D ( X ) = X ∗∗ , and L M : C bop → M bop ,L M ( X ) = X ⊲ M , are C -module functors. A straightforward computationshows that ( L M ◦ D ) r.a. ≃ Hom M op ( − , M ) ,D r.a. ≃ ∗∗ ( − ) , ( L M ) r.a. ≃ Hom M ( M, − ) . Since D and L M are C -module functors, then, using Lemma 2.1 (i), it followsthat functors ∗∗ Hom M ( M, − ) , Hom M op ( − , M ) : M bop → C , are C -modulefunctors. Since ( L M ◦ D ) r.a. ≃ D r.a. ◦ ( L M ) r.a. , it follows from Lemma 2.1 (iii)that functors ∗∗ Hom M ( M, − ) , Hom M op ( − , M ) : M bop → C , are equivalentas C -module functors. The proof that functorsHom M op ( M, − ) ∗ , ∗ Hom M ( − , M ) : M → C are equivalent is done by showing that both functors are left adjoint of L M : C → M , L M ( X ) = X ⊲ M . (cid:3) Proposition 2.4.
Let A ∈ C be an algebra. The following statements hold. (i) For any
M, N ∈ C A , Hom C A ( M, N ) = ( M ⊗ A ∗ N ) ∗ . (ii) For any
M, N ∈ C A , Hom ( C A ) op ( M, N ) = ∗ ( N ⊗ A ∗ M ) .Proof. Both calculations of the internal hom follow from (1.5). (cid:3)
The following technical result will be needed later.
Lemma 2.5.
Let M be an exact module categories over C , and F : M → M be a C -module functor with left adjoint F l.a. : M → M . Then, there arenatural isomorphisms ξ M,N : Hom(
M, F ( N )) → Hom( F l.a. ( M ) , N ) . Proof.
Since F is a module functor, then F l.a. is also a module functor. Letus denote by b X,M : F l.a. ( X ⊲ M ) → X ⊲ F l.a. ( M )its module structure. Let Ω M,N : Hom M ( M, F ( N )) → Hom M ( F l.a. ( M ) , N )be natural isomorphisms. Take X ∈ C . The desired natural isomorphism isthe one induced by the composition of isomorphismsHom C ( X, Hom(
M, F ( N ))) ≃ Hom M ( X ⊲ M, F ( N )) ≃ Hom M ( F l.a. ( X ⊲ M ) , N ) ≃ Hom M ( X ⊲ F l.a. ( M ) , N ) ≃≃ Hom C ( X, Hom( F l.a. ( M ) , N ) . Using isomorphisms (2.12), one can describe explicitly this isomorphism as ξ M,N = ψ ZF l.a. ( M ) ,N (cid:0) Ω Z⊲M,N ( φ ZM,F ( N ) (id Z )) b − Z,M (cid:1) , (2.13)where Z = Hom( M, F ( N )). (cid:3) CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 13
The relative Serre functor.
Let M be a left C -module category.Following [10], [5] we recall the definition of the relative Serre functor of amodule category. The reader is also referred to [14]. Definition 2.6. A relative Serre functor for M is a pair ( S M , φ ), where S M : M → M is a functor equipped with natural isomorphisms(2.14) φ M,N : Hom(
M, N ) ∗ ≃ Hom( N, S M ( M )) , for any M, N ∈ M .In the next Proposition we summarize some known facts about relativeSerre functors that will be used later.
Proposition 2.7.
Let M be a left module category over C . The followingholds. (i) M posses a relative Serre functor if and only if M is exact. (ii) The functor S M : M → M bop is an equivalence of C -module cate-gories. (iii) The natural isomorphism φ M,N : Hom(
M, N ) ∗ → Hom( N, S M ( M )) , is an isomorphism of C -bimodule functors. (iv) The relative Serre functor is unique up to isomorphism of C -modulefunctors. (cid:3) Balanced tensor functors and Deligne tensor product.
We shallbriefly recall the definition of the relative Deligne tensor product over atensor category. The reader is referred to [6], [1] for more details. Assumethat M is a right C -module category and N a left C -module category. Let A be a category.A C - balanced functor is a pair (Φ , b ), where Φ : M × N → A is a functor,right exact in each variable, equipped with natural isomorphisms b M,X,N :Φ(
M ⊳ X, N ) → Φ( M, X ⊲ N ) such that it satisfies the pentagon(2.15) Φ(id M , m N X,Y,N ) b M,X ⊗ Y,N = b M,X,Y ⊲ N b M⊳X,Y,N Φ( m M M,X,Y , id N ) , for any X, Y ∈ C , M ∈ M , N ∈ N . The natural isomorphism b is called thebalancing of Φ.If (Φ , b ) , ( e Φ , e b ) : M×N → A are C -balanced functors, a C - balanced naturaltransformation α : Φ → e Φ is a natural transformation such that(2.16) α M,X⊲N b M,X,N = e b M,X,N α M⊳X,N , for any X ∈ C , M ∈ M , N ∈ N . The balanced tensor product (or sometimescalled relative Deligne tensor product ) is a category M ⊠ C N , equipped witha C -balanced functor ⊠ C : M × N → M ⊠ C N such that for any category A the functor Rex ( M ⊠ C N , A ) → Bal(
M × N , A ) F F ◦ ⊠ C is an equivalence of categories. Here Bal( M × N , A ) denotes the categoryof C -balanced functors and C -balanced natural transformations. Lemma 2.8.
Let M , f M be right C -module categories and N , e N be left C -module categories. If ( F, c ) : f M → M , ( G, d ) : e N → N are right exactmodule functors, and (Φ , b ) : M × N → A is a C -balanced functor, then Φ ◦ ( F × G ) : f M × e N → A is a C -balanced functor with balancing given by (2.17) e M,X,N = Φ(id F ( M ) , d − X,N ) b F ( M ) ,X,G ( N ) Φ( c M,X , id G ( N ) ) , for any M ∈ f M , N ∈ e N , X ∈ C .Proof. We must show that e satisfies (2.15). In this case we have to prove(2.18)Φ(id F ( M ) , G ( m e N X,Y,N )) e M,X ⊗ Y,N = e M,X,Y ⊲N e M⊳X,Y,N Φ( F ( m f M M,X,Y ) , id G ( N ) ) , for any X, Y ∈ C , M ∈ f M , N ∈ e N . The left hand side of (2.18) is equal to= Φ(id F ( M ) , G ( m e N X,Y,N ) d − X ⊗ Y,N ) b F ( M ) ,X ⊗ Y,G ( N ) Φ( c M,X ⊗ Y , id G ( N ) )= Φ(id F ( M ) , d − X,Y ⊲N (id X ⊲ d − Y,N ) m N X,Y,G ( N ) ) b F ( M ) ,X ⊗ Y,G ( N ) Φ( c M,X ⊗ Y , id G ( N ) )= Φ(id F ( M ) , d − X,Y ⊲N )Φ(id F ( M ) , id X ⊲ d − Y,N ) b F ( M ) ,X,Y ⊲G ( N ) b F ( M ) ⊳X,Y,G ( N ) Φ( m M F ( M ) ,X,Y , id G ( N ) )Φ( c M,X ⊗ Y , id G ( N ) ) . The first equality is by the definition of e . The second equality is a con-sequence of ( G, d ) being a module functor, and the last equality is because(Φ , b ) is a C -balanced functor. The right hand side of (2.18) is equal to= e M,X,Y ⊲N
Φ(id F ( M⊳X ) , d − Y,N ) b F ( M⊳X ) ,Y,G ( N ) Φ( c M⊳X,Y F ( m f M M,X,Y ) , id G ( N ) )= e M,X,Y ⊲N
Φ(id F ( M⊳X ) , d − Y,N ) b F ( M⊳X ) ,Y,G ( N ) Φ(( c − M,X ⊳ id Y ) m M F ( M ) ,X,Y c M,X ⊗ Y , id G ( N ) )= Φ(id F ( M ) , d − X,Y ⊲N ) b F ( M ) ,X,G ( Y ⊲N ) Φ( c M,X , id G ( Y ⊲N ) )Φ(id F ( M⊳X ) , d − Y,N ) b F ( M⊳X ) ,Y,G ( N ) Φ(( c − M,X ⊳ id Y ) m M F ( M ) ,X,Y c M,X ⊗ Y , id G ( N ) )= Φ(id F ( M ) , d − X,Y ⊲N ) b F ( M ) ,X,G ( Y ⊲N ) Φ( c M,X , id G ( Y ⊲N ) )Φ(id F ( M⊳X ) , d − Y,N )Φ( c − M,X , id Y ⊲G ( N ) ) b F ( M ) ⊳X,Y,G ( N ) Φ( m M F ( M ) ,X,Y c M,X ⊗ Y , id G ( N ) )= Φ(id F ( M ) , d − X,Y ⊲N ) b F ( M ) ,X,G ( Y ⊲N ) Φ(id F ( M⊳X ) , d − Y,N ) b F ( M ) ⊳X,Y,G ( N ) Φ( m M F ( M ) ,X,Y c M,X ⊗ Y , id G ( N ) )= Φ(id F ( M ) , d − X,Y ⊲N )Φ(id F ( M ) , id X ⊲ d − Y,N ) b F ( M ) ,X,Y ⊲G ( N ) b F ( M ) ⊳X,Y,G ( N ) Φ( m M F ( M ) ,X,Y c M,X ⊗ Y , id G ( N ) ) . The first and third equalities follow by the definition of e . The secondequality follows since ( F, c ) is a module functor. The fourth equality is
CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 15 consequence of the naturality of b for c M,X , and the sixth equality is thenaturality of b for d Y,N . Since both sides are equal, we get the result. (cid:3)
The next result is well-known.
Proposition 2.9.
Let
A, B ∈ C be algebras such that the module categories C A , C B are exact indecomposable. The following assertions hold. (i) The functor ∗ ( − ) : ( C A ) op → A C is an equivalence of right C -modulecategories. (ii) The restriction of the tensor product ⊗ : A C × C B → A C B is a C -balanced functor, and induces an equivalence of categories b ⊗ : A C ⊠ C C B → A C B , such that b ⊗ ◦ ⊠ C ≃ ⊗ , as C -balanced functors. (iii) The functor R : A C B → Fun C ( C A , C B ) , V
7→ −⊗ A V is an equivalenceof categories.Proof. (i) The duality functor ∗ ( − ) : ( C A ) op → A C has structure of modulefunctor with isomorphisms given by φ lX ∗ ,M : ∗ ( X ∗ ⊗ M ) → ∗ M ⊗ X, for any X ∈ C , M ∈ C A . Here φ l is the natural isomorphisms described in(1.1). Note that we are omitting the canonical natural isomorphism ∗ ( X ∗ ) ≃ X . For (ii) see [1]. The proof of (iii) can be found for example in [9, Prop.3.3]. (cid:3) The (co)end for module categories
Let C be a finite tensor category and M be a left C -module category.Assume that A is a category and S : M op × M → A a functor equippedwith natural isomorphisms(3.1) β XM,N : S ( M, X ⊲ N ) → S ( X ∗ ⊲ M, N ) , for any X ∈ C , M, N ∈ M . We shall say that β is a pre-balancing of thefunctor S . Definition 3.1.
The module end of the pair (
S, β ) is an object E ∈ A equipped with dinatural transformations π M : E .. −→ S ( M, M ) such that(3.2) S (ev X ⊲ id M , id M ) π M = S ( m X ∗ ,X,M , id M ) β XX⊲M,M π X⊲M , for any X ∈ C , M ∈ M , universal with this property. This means that if e E ∈ A is another object with dinatural transformations ξ M : e E .. −→ S ( M, M ),such that they verify (3.2), there exists a unique morphism h : e E → E suchthat ξ M = π M ◦ h . Sometimes we will denote the module end as H M ∈M ( S, β ), or simply as H M ∈M S , when the pre-balancing β is understood from the context.The module coend of the pair ( S, β ) is defined dually. This is an object C ∈ A equipped with dinatural transformations π M : S ( M, M ) .. −→ C suchthat(3.3) π M = π X ∗ ⊲M β XM,X ∗ ⊲M S (id M , m X,X ∗ ,M ) S (id M , coev X ⊲ id M ) , for any X ∈ C , M ∈ M , universal with this property. This means that if e C ∈ A is another object with dinatural transformations λ M : S ( M, M ) .. −→ e C such that they satisfy (3.3), there exists a unique morphism g : C → e C suchthat g ◦ π M = λ M . The module coend will be denoted H M ∈M ( S, β ), orsimply as H M ∈M S .A similar definition can be made for right C -module categories. Let B be a category, and N be a right C -module category endowed with a functor S : N op × N → B with a pre-balancing γ XM,N : S ( M ⊳ X, N ) → S ( M, N ⊳ ∗ X ) , for any M, N ∈ N , X ∈ C . Definition 3.2.
The module end for S is an object E ∈ B equipped withdinatural transformations λ N : E .. −→ S ( N, N ) such that(3.4) λ N = S (id N , id N ⊳ ev X ) S (id N , m − N,X, ∗ X ) γ XN,N⊳X λ N⊳X , for any N ∈ N , X ∈ C . We shall also denote this module end by H N ∈N ( S, γ ).Similarly, the module coend is an object C ∈ B with dinatural transfor-mations λ N : S ( N, N ) .. −→ C such that(3.5) λ N S (id N ⊳ coev X , id N ) = λ N⊳ ∗ X γ XN⊳ ∗ X,N S ( m − N, ∗ X,X , id N ) , for any N ∈ N , X ∈ C . We shall also denote this module coend by H N ∈N ( S, γ ).In the next Proposition we collect some results about module ends thatgeneralize well-known results in the theory of (co)ends. The proofs followthe same lines as the ones in usual ends. For the sake of completeness weinclude the proofs.
Proposition 3.3.
Assume that M , N are left C -module categories, and S, e S : M op × M → A are functors equipped with pre-balancings β XM,N : S ( M, X ⊲ N ) → S ( X ∗ ⊲ M, N ) , e β XM,N : e S ( M, X ⊲ N ) → e S ( X ∗ ⊲ M, N ) ,X ∈ C , M, N ∈ M . The following assertions holds CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 17 (i)
Assume that the module ends H M ∈M ( S, β ) , H M ∈M ( e S, e β ) exist andhave dinatural transformations π, e π , respectively. If γ : S → e S isa natural transformation such that (3.6) e β XM,N γ ( M,X⊲N ) = γ ( X ∗ ⊲M,N ) β XM,N , then there exists a unique map b γ : H M ∈M ( S, β ) → H M ∈M ( e S, e β ) suchthat e π M b γ = γ ( M,M ) π M for any M ∈ M . If γ is a natural isomor-phism, then b γ is an isomorphism. (ii) If the end H M ∈M ( S, β ) exists, then for any object U ∈ A , the end H M ∈M Hom A ( U, S ( − , − )) exists, and there is an isomorphism I M ∈M Hom A ( U, S ( − , − )) ≃ Hom A ( U, I M ∈M ( S, β )) . Moreover, if H M ∈M Hom A ( U, S ( − , − )) exists, then the end H M ∈M ( S, β ) exists. (iii) Assume F : A → A ′ is a right exact functor. Then, there is anisomorphism F ( I N ∈N ( S, β )) ≃ I N ∈N ( F ◦ S, F ( β )) . (iv) If F : M → N is an equivalence of C -module categories, then thereis an isomorphism I N ∈N S ≃ I M ∈M S ( F ( − ) , F ( − )) . Proof. (i). For any M ∈ N define λ M : H N ∈N ( S, β ) → e S ( M, M ) as λ M = γ ( M,M ) π M . It follows straightforward that λ is dinatural and since γ satisfies(3.6), then λ satisfies (3.2). By the universality of the module end, thereexists a morphism b γ : H M ∈M ( S, β ) → H M ∈M ( e S, e β ) such that e π M b γ = λ M = γ ( M,M ) π M .(ii). Let us assume that H M ∈M ( S, β ) exists, and has associated to itdinatural transformations π N : H M ∈M ( S, β ) → S ( N, N ). For any U ∈ A ,the pre-balancing for the functor Hom A ( U, S ( − , − )) is defined as β UX,M,N : Hom A ( U, S ( M, X ⊲ N )) → Hom A ( U, S ( X ∗ ⊲ M, N )) ,β UX,M,N ( f ) = β XM,N ◦ f. Also define π UN : Hom A ( U, I M ∈M ( S, β )) → Hom A ( U, S ( N, N )) ,π UN ( f ) = π N ◦ f. It follows by a straightforward computation that π U is a dinatural trans-formation, and they satisfy (3.2) using β U . It also follows easily that Hom A ( U, H M ∈M ( S, β )) together with π U satisfy the universal property ofthe module end, thus I M ∈M Hom A ( U, S ( − , − )) ≃ Hom A ( U, I M ∈M ( S, β )) . Now, let us assume that H M ∈M Hom A ( U, S ( − , − )) exists for any U ∈ A .Using item (i), we can define a functor F : A op → vect k ,F ( U ) = I M ∈M Hom A ( U, S ( − , − )) . We shall prove that F es left exact, and thus it is representable. Theobject representing the functor F will be a candidate for the module end H M ∈M ( S, β ).For any M ∈ M , and any f : U → V in A , denote( α f ) M : Hom A ( V, S ( M, M )) → Hom A ( U, S ( M, M ))( α f ) M ( g ) = g ◦ f. To prove that F is left exact, we need to show that, for any morphism f : U → V in A , F (coKer ( f )) = Ker ( F ( f )). Let be q = coKer ( f ) : V → C ,and l : K → F ( V ) be a k -linear map such that F ( f ) ◦ l = 0. Then( α f ) M ◦ π VM ◦ l = π UM ◦ F ( f ) ◦ l = 0 . The second equality follows from item (i). Since ker( α f ) = α q , there existsa map h M : K → Hom A ( C, S ( M, M )))such that ( α q ) M ◦ h M = π VM ◦ l . It is not difficult to prove that h is adinatural transformation, and they satisfy (3.2) (using the isomorphisms β C ). By the universal property of the module end, there exists a morphism φ : K → F ( C ) such that h M = π CM ◦ φ . It follows from item (i) that( α q ) M ◦ π CM ◦ φ = π VM ◦ F ( q ) ◦ φ. But also ( α q ) M ◦ π CM ◦ φ = ( α q ) M ◦ h M = π VM ◦ l, whence l = F ( q ) ◦ φ and therefore F ( q ) = ker( F ( f )). Hence F is representedby an object E ∈ A ; F ( U ) = Hom A ( U, E ). The maps δ M : E → S ( M, M ) ,δ M = π EM (id E ) are dinatural transformations, and they satisfy (3.2). Itfollows by a straightforward computation that E together with δ satisfy theuniversal property of the module end, thus E ≃ H M ∈M ( S, β ).The proof of (iii) and (iv) follows straightforward. (cid:3)
Remark . Off course that, similar results to those presented in Proposi-tion 3.3 can be stated for module coends, and also for module (co)ends forright module categories.
CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 19
Relation between module (co)ends for right and left modulecategories.
Let A be a category. Let M be a left C -module category,and a functor S : M op × M → A equipped with a pre-balancing β XM,N : S ( M, X ⊲N ) → S ( X ∗ ⊲M, N ). Then N = M op is a right C -module category.We can consider the functor S op : N op × N → A op . It posses a pre-balancing γ XM,N : S op ( M ⊳ X, N ) → S op ( M, N ⊳ ∗ X ) ,γ XM,N = β XM,N . Note that the pre-balancing γ is considered as a morphism in A op . The nextresult follows straightforward. Lemma 3.5.
There are isomorphisms I M ∈M ( S, β ) ≃ I M ∈N ( S op , γ ) , I M ∈M ( S, β ) ≃ I M ∈N ( S op , γ ) . (cid:3) A similar result also holds starting from a right C -module category N ,and a functor T : N op × N → A equipped with a pre-balancing γ XM,N : T ( M ⊳ X, N ) → T ( M, N ⊳ ∗ X ) . If M = N op , then M is a left C -module category, and we can consider thefunctor T op : M op × M → A op together with a pre-balancing β XM,N : T op ( M, X ⊲ N ) → T op ( X ∗ ⊲ M, N ) β XM,N = γ X ∗∗ M,N . The next result is a straightforward consequence of the definitions of module(co)end.
Lemma 3.6.
There are isomorphisms I N ∈N ( T, γ ) ≃ I M ∈M ( T op , β ) , I M ∈N ( T, γ ) ≃ I M ∈M ( T op , β ) . (cid:3) Parameter theorem for module ends.
Let C be a finite tensorcategory and M a left C -module category. Also, let A , B be categories. Westart with a functor S : M op ×M → Fun( A , B ) equipped with pre-balancing β XM,N : S ( M, X ⊲ N ) → S ( X ∗ ⊲ M, N ) , for any X ∈ C , M, N ∈ M . If theend H M ∈M ( S, β ) exists, it is an object in the category Fun( A , B ); we denotethis functor as (cid:0) I M ∈M ( S, β ) (cid:1) ( − ) : A → B . Alternatively, we can do the following construction. For any A ∈ A weget a functor S A : M op × M → B , S A ( M, N ) = S ( M, N )( A ). This functorcomes with pre-balancing β AX,M,N : S A ( M, X ⊲ N ) → S A ( X ∗ ⊲ M, N ) ,β AX,M,N = ( β XM,N ) A , for any X ∈ C , M, N ∈ M . If the module end H M ∈M ( S A , β A ) exists, it isan object in B , and it defines a functor S : A → B . The proof of the nextresult follows straightforward.
Theorem 3.7.
Provided all ends H M ∈M ( S A , β A ) exist, the functor S has acanonical structure of module end for the functor S . We write S = (cid:0) I M ∈M ( S, β ) (cid:1) ( − ) . (cid:3) Remark . Similar results can be obtained for module coends, and alsofor right C -module categories.3.3. Restriction of module (co)ends to tensor subcategories.
In thisSection, we shall show that the module (co)end coincides with the usual(co)end in the case the tensor category is vect k . We also study what happenswith the module (co)end when we restrict to a tensor subcategory.Let C be a tensor category and D ⊆ C be a tensor subcategory of C .Assume also that M is a left C -module category. We can consider therestricted D -module category Res DC M . The next result is a straightforwardconsequence of the definition of module (co)ends. Proposition 3.9.
Let S : M op × M → A be a functor equipped with pre-balancing β XM,N : S ( M, X ⊲ N ) → S ( X ∗ ⊲ M, N ) . (i) There exists a monomorphism in A I M ∈M ( S, β ) ֒ → I M ∈ Res CD M ( S, β ) . (ii) There exists an epimorphism in A I M ∈ Res DC M ( S, β ) ։ I M ∈M ( S, β ) . CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 21 (cid:3)
The next result says that the module (co)end coincides with the usualone in the case C = vect k . Proposition 3.10.
Let M , A be abelian k -linear categories, and S : M op ×M → A be a functor. In particular M is a left vect k -module category. Thefunctor S has a canonical pre-balancing β such that there are isomorphisms Z M ∈M S ≃ I M ∈M ( S, β ) , Z M ∈M S ≃ I M ∈M ( S, β ) . Proof.
We shall prove the first isomorphism concerning the usual end andthe module end. The other isomorphism for the coend follows similarly. Forthis, we will show that for such a functor S there exists a canonical pre-balancing β such that any dinatural transformation π M : E .. −→ S ( M, M )satisfies (3.2).Since M is a finite abelian k -linear category, there exists a finite di-mensional k -algebra A such that M is equivalent to the category of finitedimensional right A -modules m A . The action of vect k on m A is ⊲ : vect k × m A → m A ,X ⊲ M = X ⊗ k M, for any X ∈ vect k , M ∈ m A . The right action of A on X ⊗ k M is on thesecond tensorand. For any X, Y ∈ vect k , M ∈ m A the associativity of thismodule category is m X,Y,M : ( X ⊗ k Y ) ⊗ k M → X ⊗ k ( Y ⊗ k M ) ,m X,Y,M (( x ⊗ y ) ⊗ m ) = x ⊗ ( y ⊗ m ) . For any X ∈ vect k , x ∈ X , we denote by δ x : X → k the unique lineartransformation that sends x to 1, and any element of a direct complementof < x > to 0. If M ∈ m A , X ∈ vect k , x ∈ X we shall denote by ι Mx : M → X ⊲ M, p Mx : X ⊲ M → M,ι Mx ( m ) = x ⊗ m, p Mx ( y ⊗ m ) = δ x ( y ) m, for any y ∈ X, m ∈ M . Let ( x i ) , ( f i ) be a pair of dual basis of X and X ∗ respectively. For any x ∈ X, f ∈ X ∗ it is easy to verify that X i δ x i ( x ) δ f i ( f ) = f ( x ) . This equality implies that(3.7) ev X ⊗ id M = X i p Mx i p X ⊗ k Mf i m X ∗ ,X,M . Also, one can verify that(3.8) X i ι Mx i p Mx i = id X ⊗ k M , p Mx ι My = δ x ( y )id M . For any
M, N ∈ m A let us denote β XM,N : S ( M, X ⊲ N ) → S ( X ∗ ⊲ M, N ) ,β XM,N = ⊕ i S ( p Mf i , p Nx i ) , where ( x i ) , ( f i ) is a pair of dual basis of X and X ∗ respectively. One cancheck, using (3.8), that β XM,N is an isomorphism by showing that ⊕ i S ( ι Mf i , ι Nx i ) : S ( X ∗ ⊲ M, N ) → S ( M, X ⊲ N )is its inverse. Let E ∈ A be an object and π M : E .. −→ S ( M, M ) a dinaturaltransformation. Let us show that π satisfies equation (3.2). Let X ∈ vect k , M ∈ m A and let ( x i ) , ( f i ) be a pair of dual basis of X and X ∗ . The righthand side of equation (3.2) is S ( m X ∗ ,X,M , id M ) β XX⊲M,M π X⊲M = ⊕ i S ( m X ∗ ,X,M , id M ) S ( p X⊲Mf i , p Mx i ) π X⊲M = ⊕ i S ( m X ∗ ,X,M , id M ) S ( p Mx i p X⊲Mf i , id M ) π M = ⊕ i S ( p Mx i p X⊲Mf i m X ∗ ,X,M , id M ) π M = S (ev X ⊗ id M , id M ) π M . The second equality follows from the dinaturality of π , and the last equalityfollows from (3.7). (cid:3) Remark . Similar result to those obtained in Propositions 3.9, 3.10 arevalid for right module categories.4.
Applications to the theory of representations of tensorcategories
Throughout this section C will denote a finite tensor category.4.1. Natural module transformations as an end.
For a pair of functors
F, G : A → B between two abelian categories A , B , it is well known thatthere is an isomorphismNat ( F, G ) ≃ Z A ∈A Hom B ( F ( A ) , G ( A )) . In this Section, we generalize this result when F and G are C -module func-tors.Let M , N be C -module categories, and ( F, c ) , ( G, d ) :
M → N be mod-ule functors. Define S F,G : M op × M → vect k the functor S ( M, N ) =Hom N ( F ( M ) , G ( N )). For any pair of functions f : M → M ′ , g : N → N ′ in M , S F,G ( f, g )( α ) = G ( g ) ◦ α ◦ F ( f ), for any α ∈ Hom N ( F ( M ) , G ( N )). CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 23
The functor S F,G has a pre-balancing defined as follows. Set(4.1) β XM,N : Hom N ( F ( M ) , G ( X ⊲ N )) → Hom N ( F ( X ∗ ⊲ M ) , G ( N )) β XM,N ( α ) = ( ev X ⊲ id G ( N ) ) m − X ∗ ,X,G ( N ) (id X ∗ ⊲ d X,N α ) c X ∗ ,M , for any X ∈ C , M, N ∈ M . It follows straightforward that β XM,N are naturalisomorphisms with inverses given by(4.2) (cid:0) β XM,N (cid:1) –1 ( α ) = d − X,N (id X ⊲ αc − X ∗ ,M ) m X,X ∗ ,F ( M ) (coev X ⊲ id F ( M ) ) . Proposition 4.1.
For any pair of C -module functors F, G there is an iso-morphism
Nat m ( F, G ) ≃ I M ∈M ( S F,G , β ) . Proof.
For any M ∈ M , define π M : Nat m ( F, G ) → Hom N ( F ( M ) , G ( M ))by π M ( α ) = α M . It follows easily that π is a dinatural transformation. Letus show that π satisfies (3.2). Let α ∈ Nat m ( F, G ), M ∈ M , then the lefthand side of (3.2) evaluated in α is equal to α M F (ev X ⊲ id M ) . The right hand side of (3.2) evaluated in α is equal to= ( ev X ⊲ id G ( M ) ) m − X ∗ ,X,G ( M ) (id X ∗ ⊲ d X,M α X⊲M ) c X ∗ ,X⊲M F ( m X ∗ ,X,M )= ( ev X ⊲ id G ( M ) ) m − X ∗ ,X,G ( M ) (id X ∗ ⊲ (id X ⊲ α M ) c X,M ) c X ∗ ,X⊲M F ( m X ∗ ,X,M )= ( ev X ⊲ id G ( M ) )(id X ∗ ⊗ X ⊲ α M ) c X ∗ ⊗ X,M = α M ( ev X ⊲ id F ( M ) ) c X ∗ ⊗ X,M = α M F (ev X ⊲ id M ) . The second equality follows since α is a module natural transformation andsatisfies (2.6), the third equality follows by the naturality of m and since c satisfies (2.3) and the last one follows from the naturality of c .Let E be a vector space equipped with a dinatural transformation ξ M : E → Hom N ( F ( M ) , G ( M )) such that (3.2) is satisfied. Define h : E → Nat m ( F, G ) as follows. For any v ∈ E , M ∈ M , h ( v ) M = ξ M ( v ). It is clear,by definition, that π ◦ h = ξ . We must prove that for any v ∈ E , h ( v ) is anatural module transformation, that is, we must show that equation (2.6)is fulfilled, which in this case is(4.3) d X,M ξ X⊲M ( v ) = (id X ⊲ ξ M ( v )) c X,M , for any X ∈ C , M ∈ M . Since ξ satisfies (3.2), then (cid:0) β XX⊲M,M (cid:1) − (cid:0) ξ M ( v ) F ( ev X ⊲ id M ) F ( m − X ∗ ,X,M ) (cid:1) = ξ X⊲M ( v ) , for any v ∈ E . Using the definition of (cid:0) β XX⊲M,M (cid:1) − given in (4.2), thisequation is equivalent to d X,M ξ X⊲M ( v ) = (cid:0) id X ⊲ ξ M ( v ) F (( ev X ⊲ id M ) m − X ∗ ,X,M ) c − X ∗ ,X⊲M (cid:1) m X,X ∗ ,F ( X⊲M ) (coev X ⊲ id F ( X⊲M ) )= (cid:0) id X ⊲ ξ M ( v ) F ( ev X ⊲ id M ) (cid:1) (id X ⊲ c − X ∗ ⊗ X,M m − X ∗ ,X,F ( M ) (id X ∗ ⊲ c X,M )) m X,X ∗ ,F ( X⊲M ) (coev X ⊲ id )= (cid:0) id X ⊲ ξ M ( v ) (cid:1) (id X ⊗ ev X ⊲ id M )(id X ⊗ X ∗ ⊲ c X,M )(coev X ⊲ id )= (cid:0) id X ⊲ ξ M ( v ) (cid:1) c X,M . The second equality follows from (2.3), the third equality follows from thenaturality of c , and the last one follows from the rigidity of C . Hence,Nat m ( F, G ) satisfies the required universal property. (cid:3)
On the category of module functors.
Assume that C , E , D , are fi-nite tensor categories. Assume also that M is a ( C , E )-bimodule category,and that N is a ( C , D )-bimodule category. Then, we can consider the func-tors(4.4) L = L M , N : M op ⊠ C N →
Fun C ( M bop , N ) ,L ( M ⊠ C N ) = Hom M op ( − , M ) ⊲ N, (4.5) e L = e L M , N : M op ⊠ C N →
Fun C ( M , N ) e L ( M ⊠ C N ) = Hom M op ( M, − ) ∗ ⊲ N, Both functors are equivalences of ( E , D )-bimodule categories. This fact wasproven in [6, Thm. 3.20], see also [1]. The bimodule structure on the functorcategory Fun C ( M , N ) is described in (2.11). We will give another proof ofthe fact that L and e L are category equivalences, and we shall show an explicitdescription of a quasi-inverse using the module end of some functor in ananalogous way as [12, Lemma 3.5].For later use, let us explain explicitly what it means that e L is a bimod-ule functor. For any Z ∈ D , W ∈ E , M ∈ M , N ∈ N we have naturalisomorphisms(4.6) e L ( W ⊲ M ⊠ C N ) ≃ e L ( M ⊠ C N ) ◦ ( − ⊳ W ) , (4.7) e L ( M ⊠ C N ⊳ Z ) ≃ ( − ⊳ Z ) ◦ e L ( M ⊠ C N ) . Assume that M , N are exact indecomposable as left C -module categories,then there exist algebras A, B ∈ C such that
M ≃ C A , N ≃ C B as modulecategories. Recall that if M ∈ C A then, by Lemma 1.1 (i), ∗ M has structureof left A -module. Lemma 4.2.
Assume as above that M = C A , N = C B . Denote by ( S M , φ ) a relative Serre functor associated to M . Then, the following statementshold. CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 25 (i)
The functor e L M , N : M op ⊠ C N →
Fun C ( M , N ) is equivalent to thecomposition of functors ( C A ) op ⊠ C C B ∗ ( − ) ⊠ Id −−−−−−→ A C ⊠ C C B ⊗ −→ A C B R −→ Fun C ( C A , C B ) . Recall the definition of the functor R given in Proposition 2.9, R : A C B → Fun C ( C A , C B ) , R ( V )( X ) = X ⊗ A V . In particular, it followsthat e L is a category equivalence. (ii) For any M ∈ M , N ∈ N , there exists a natural isomorphism ofmodule functors (4.8) e L M , N ( M ⊠ C N ) ≃ L M , N ( M ⊠ C N ) ◦ S M . In particular L is also an equivalence of categories.Proof. Part (i) follows from the computation of the internal hom given inProposition 2.4 (ii). Let us prove (ii). It follows from Lemma 2.3 thatfunctors Hom M op ( M, − ) ∗ , ∗ Hom M ( − , M ) : M → C , are equivalent as C -module functors. Also, it follows from Lemma 2.3 thatthe functors ∗∗ Hom M ( M, S M ( − )) , Hom M op ( S M ( − ) , M ) : M → C are equivalent as C -module functors. This implies that e L M , N ( M ⊠ C N ) isequivalent to the C -module functor ∗ Hom M ( − , M ) ⊲ N : M → N , and L M , N ( M ⊠ C N ) ◦ S M is equivalent to the C -module functor ∗∗ Hom M ( M, S M ( − )) ⊲ N : M → N . The natural isomorphisms φ U,V : Hom(
U, V ) ∗ → Hom( V, S M ( U )) induce anisomorphism of C -module functors ∗∗ φ − ,M ⊲ id N : ∗ Hom M ( − , M ) ⊲ N → ∗∗ Hom M ( M, S M ( − )) ⊲ N. And this finishes the proof of the Lemma. (cid:3)
In what follows, we shall give an explicit description of a quasi-inverseof the functor e L using the module end. For any module functor ( F, c ) ∈ Fun C ( C A , C B ) we introduce some functors S F , D F , L F , R F that, later, wewill compute its module end.Define S F : ( C A ) op × C A → ( C A ) op ⊠ C C B ,S F ( M, N ) = M ⊠ C F ( N ) , (4.9)endowed with a pre-balancing β XM,N : S F ( M, X ⊲ N ) → S F ( X ∗ ⊲ M, N ) β XM,N = b − M,X,N (id M ⊠ C c X,N ) . Also D F : ( C A ) op × C A → ( C B ) op ⊠ C C A ,D F ( M, N ) = F ( M ) ⊠ C N, (4.10)endowed with a pre-balancing δ XM,N : D F ( M, X ⊲ N ) → D F ( X ∗ ⊲ M, N ) δ XM,N = ( c − X ∗ ,M ⊠ C id N ) b − M,X,N . Here b M,X,N : X ∗ ⊲ M ⊠ C N → M ⊠ C X ⊲ N is the balancing associated tothe Deligne tensor product ⊠ C , see Section 2.4.We also have functors L F , R F : ( C A ) op × C A → A C B , R F ( M , N ) = ∗ M ⊗ F ( N ) , L F ( M , N ) = ∗ F ( M ) ⊗ N, (4.11)equipped with pre-balancing γ XM,N : R F ( M, X ⊲ N ) → R F ( X ∗ ⊲ M, N ) ,γ XM,N = (( φ lX ∗ ,M ) − ⊗ id F ( N ) )(id ∗ M ⊗ c X,N ) , (4.12) η XM,N : L F ( M, X ⊲ N ) → L F ( X ∗ ⊲ M, N ) ,η XM,N = ∗ ( c X ∗ ,M )( φ lX ∗ ,F ( M ) ) − ⊗ id N . (4.13)Here we are omitting the isomorphisms X ≃ ∗ ( X ∗ ), for any X ∈ C , andisomorphisms φ l are those presented in (1.1). Lemma 4.3.
The following statements hold. (i)
There exists an equivalence of categories ∗ ( − ) ⊠ C Id : C op A ⊠ C C B → A C ⊠ C C B such that ( ∗ ( − ) ⊠ C Id ) ◦ ⊠ C ≃ ⊠ C ◦ ( ∗ ( − ) × Id ) as C -balanced functors. (ii) If the module end H M ∈C A ( S F , β ) exists, then b ⊗ ◦ ( ∗ ( − ) ⊠ C Id ) (cid:0) I M ∈C A ( S F , β ) (cid:1) ≃ I M ∈C A ( R F , γ ) . (iii) If the module end H M ∈C A ( D F , δ ) exists, then b ⊗ ◦ ( ∗ ( − ) ⊠ C Id ) (cid:0) I M ∈C A ( D F , δ ) (cid:1) ≃ I M ∈C A ( L F , η ) . Here b ⊗ : A C ⊠ C C B → A C B is the induced functor from the tensor product,that we have presented in Proposition 2.9 (ii). CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 27
Proof.
Part (i) follows since ⊠ C ◦ ( ∗ ( − ) × Id ) is a C -balanced functor. Thereare isomorphisms of C -balanced functors b ⊗ ◦ ( ∗ ( − ) ⊠ C Id ) ◦ S F ≃ b ⊗ ◦ ( ∗ ( − ) ⊠ C Id ) ◦ ⊠ C ◦ (Id × F ) ≃ b ⊗ ◦ ⊠ C ◦ ( ∗ ( − ) × F ) ≃ ⊗ ◦ ( ∗ ( − ) × F ) = R F . The first isomorphism follows by the definition of S F , the second isomor-phism follows from part (i), and the third isomorphism is the one presentedin Proposition 2.9 (ii). Now, part (ii) follows by applying Proposition 3.3(i). The proof of (iii) follows similarly. (cid:3) Theorem 4.4.
Assume the notation given above. The functor
Υ : Fun C ( C A , C B ) → C op A ⊠ C C B , given by Υ( F ) = I M ∈C A ( S F , β ) = I M ∈C A M ⊠ C F ( M ) is well-defined and is a quasi-inverse of the functor e L .Proof. Recall the definition of the functor R given in Proposition 2.9, R : A C B → Fun C ( C A , C B ) , R ( V )( X ) = X ⊗ A V . It follows from Lemma 4.2 that,the composition of functors( C A ) op ⊠ C C B ∗ ( − ) ⊠ Id −−−−−−→ A C ⊠ C C B ⊗ −→ A C B R −→ Fun C ( C A , C B )is isomorphic to e L . Thus, it is enough to show that the functorΨ : Fun C ( C A , C B ) → A C B , given by(4.14) Ψ( F ) = I M ∈C A ( R F , γ ) = I M ∈C A ∗ M ⊗ F ( M )is well-defined and it is a quasi-inverse of R . Since we know that R isan equivalence, we denote by Ψ an adjoint equivalence to R . Take F ∈ Fun C ( C A , C B ), and V ∈ A C B , thenHom ( A,B ) ( V, Ψ( F )) ≃ Nat m ( R ( V ) , F ) ≃ I M ∈C A (Hom B ( M ⊗ A V, F ( M )) , β ) ≃ I M ∈C A (Hom ( A,B ) ( V, ∗ M ⊗ F ( M )) , δ ) ≃ Hom ( A,B ) ( V, I M ∈C A ∗ M ⊗ F ( M )) The second isomorphism follows from Proposition 4.1. Here, the isomor-phism β is the one described in (4.1). The third isomorphism follows fromLemma 1.1 (ii); one can easily verify that if δ XM,N : Hom ( A,B ) ( V, ∗ M ⊗ F ( X ⊗ N )) → Hom ( A,B ) ( V, ∗ M ⊗ X ⊗ F ( N ))is defined as δ XM,N ( h ) = (id ∗ M ⊗ c X,N ) ◦ h , then the naturality of Φ impliesthat δ XM,N (Φ( α )) = Φ( β XM,N ( α )) , for any α ∈ Hom B ( M ⊗ A V, F ( X ⊗ N )). HereΦ : Hom B ( M ⊗ A V, F ( X ⊗ N )) → Hom ( A,B ) ( V, ∗ M ⊗ F ( X ⊗ N ))is the natural isomorphism described in (1.9). Thus, the third isomorphismfollows by applying Proposition 3.3 (i). The last isomorphism follows fromProposition 3.3 (ii). (cid:3) As an immediate consequence of the above Theorem, we have the follow-ing results.
Corollary 4.5.
Let A ∈ C be an algebra such that C A is an exact modulecategory. There is an isomorphism of A -bimodules A ≃ I M ∈C A ∗ M ⊗ M. (cid:3) Corollary 4.6.
Let M , N be exact indecomposable C -module categories. If U ∈ M , V ∈ N and F ∈ Fun C ( M , N ) , there are isomorphisms (4.15) I M ∈M e L M , N ( M ⊠ C F ( M )) ≃ F, (4.16) I M ∈M M ⊠ C e L M , N ( U ⊠ C V )( M ) ≃ U ⊠ C V. (cid:3) In [4, Lemma 3.8] it was proven that for a right exact functor F : M → N ,where M , N are abelian categories, there is an isomorphism Z N ∈N F r.a. ( N ) ⊠ N ≃ Z M ∈M M ⊠ F ( M ) . The next result is a generalization of that result; essentially it says that, fora C -module functor F : M → N , there is an isomorphism I N ∈N F r.a. ( N ) ⊠ C N ≃ I M ∈M M ⊠ C F ( M ) . The proof, however, is more complicated than the proof of [4, Lemma 3.8],since in module ends there is a new ingredient (the pre-balancing β ) thathas to be taken into account. CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 29
Proposition 4.7.
Let M , N be exact indecomposable left C -module cate-gories. Assume that ( F, c ) ∈ Fun C ( M , N ) is a module functor and ( F r.a. , d ) ∈ Fun C ( N , M ) . Recall the functors D F , S F defined in (4.10) , (4.9) . There isan isomorphism (4.17) I N ∈N ( D F r.a. , δ ) ≃ I M ∈M ( S F , β ) . Proof.
Since M , N are exact indecomposable, we can assume that there arealgebras A, B ∈ C such that M = C A , N = C B . Using Lemma 4.3, it will beenough to prove that there are isomorphisms I N ∈C B ( L F r.a. , η ) ≃ I M ∈C A ( R F , γ ) , as objects in A C B . Here η, γ are defined in (4.13), (4.12). Since the functor R : A C B → Fun C ( C A , C B ) , R ( V )( X ) = X ⊗ A V is an equivalence of cate-gories, using Proposition 3.3 (iii), it will be enough to prove that there is anisomorphism(4.18) I N ∈C B (cid:0) R ( ∗ F r.a. ( N ) ⊗ N ) , R ( η ) (cid:1) ≃ I M ∈C A (cid:0) R ( ∗ M ⊗ F ( M )) , R ( γ ) (cid:1) . Since the functor R is a quasi-inverse of the functor Ψ : Fun C ( C A , C B ) → A C B , presented in (4.14), it follows that I M ∈C A (cid:0) R ( ∗ M ⊗ F ( M )) , R ( γ ) (cid:1) ≃ F, and I N ∈C B (cid:0) R ( ∗ N ⊗ N ) , R ( γ ) (cid:1) ≃ Id C B . Hence, to prove isomorphism (4.18) of functors, it is sufficient to prove thatthere is an isomorphism I N ∈C B (cid:0) R ( ∗ F r.a. ( N ) ⊗ N ) , R ( η ) (cid:1) ( U ) ≃ I M ∈C B (cid:0) R ( ∗ M ⊗ M ) , R ( γ ) (cid:1) ( F ( U ))For any U ∈ C A . Applying Theorem 3.7, it will be enough to prove thatthere is an isomorphism I N ∈C B (cid:0) U ⊗ A ∗ F r.a. ( N ) ⊗ N, b η (cid:1) ≃ I M ∈C B (cid:0) F ( U ) ⊗ B ∗ M ⊗ M ) , b γ (cid:1) , where b η XM,N = R ( η ) U = id U ⊗ A ∗ ( d X ∗ ,M )( φ lX ∗ ,F r.a. ( M ) ) − ⊗ id N , b γ XM,N = R ( γ ) F ( U ) = id F ( U ) ⊗ B ( φ lX ∗ ,M ) − ⊗ id N , for any X ∈ C , M, N ∈ C B . For this purpose, we shall construct naturalisomorphisms a U,M : F ( U ) ⊗ B ∗ M → U ⊗ A ∗ F r.a. ( M ) such that(4.19) b η XM,N ( a U,M ⊗ id X ⊗ N ) = ( a U,X ∗ ⊗ M ⊗ id N ) b γ XM,N . It will follow then from Proposition 3.3 (i) the desired isomorphism betweenmodule ends, and this will finish the proof of the Proposition.Recall the isomorphisms Φ
AM,X,N , Ψ AM,X,N defined in (1.15), (1.16)Φ
AM,X,N : Hom A ( M, X ⊗ N ) → Hom C ( M ⊗ A ∗ N, X ) , Φ AM,X,N ( α ) π AM, ∗ N = (id X ⊗ ev N )( α ⊗ id ∗ N ) , Ψ AM,X,N : Hom C ( M ⊗ A ∗ N, X ) → Hom A ( M, X ⊗ N ) , Ψ AM,X,N ( α ) = ( απ AM, ∗ N ⊗ id N )(id M ⊗ coev N ) . We shall also denote natural isomorphisms ω M,N : Hom B ( F ( M ) , N ) → Hom A ( M, F r.a. ( N )) , comming from the adjunction ( F, F r.a. ). Naturality of ω implies that forany morphism f : N → e N in C B , and any α ∈ Hom B ( F ( M ) , N ) we havethat(4.20) ω M, e N ( f α ) = F r.a. ( f ) ω M,N ( α ) . This equation implies in particular that(4.21) ω U,Y ⊗ N (Ψ BF ( U ) ,Y,M (id )) = F r.a. (Ψ BF ( U ) ,Y,M (id )) ω U,F ( U ) (id ) . Define isomorphisms a U,M : F ( U ) ⊗ B ∗ M → U ⊗ A ∗ F r.a. ( M )induced by the natural isomorphismsHom C ( F ( U ) ⊗ B ∗ M, Z ) Ψ B −−→ Hom B ( F ( U ) , Z ⊗ M ) ω −→ Hom A ( U, F r.a. ( Z ⊗ M )) −→ Hom A ( U, Z ⊗ F r.a. ( M )) Φ A −−→ Hom C ( U ⊗ A ∗ F r.a. ( M ) , Z ) , for any Z ∈ C . This means that a − U,M = Φ
AU,Y,F r.a. ( M ) (cid:0) d Y,M ω U,Y ⊗ N (Ψ BF ( U ) ,Y,M (id )) (cid:1) , where Y = F ( U ) ⊗ B ∗ M. Using the definition of Φ A one gets that a − U,M π AU, ∗ F r.a. ( M ) = (id Y ⊗ ev F r.a. ( M ) ) (cid:0) d Y,M ω U,Y ⊗ M (Ψ BF ( U ) ,Y,M (id )) ⊗ id ∗ F r.a. ( M ) (cid:1) . (4.22)Here we are again denoting Y = F ( U ) ⊗ B ∗ M . Equation (4.19) is equivalentto ( a − U,X ∗ ⊗ M ⊗ id N ) b η XM,N ( π AU, ∗ F r.a. ( M ) ⊗ id X ⊗ N ) == b γ XM,N ( a − U,M ⊗ id X ⊗ N )( π AU, ∗ F r.a. ( M ) ⊗ id X ⊗ N ) , CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 31 which in turn (forgetting the last id N ) is equivalent to( a − U,X ∗ ⊗ M π AU, ∗ F r.a. ( X ∗ ⊗ M ) )(id U ⊗ ∗ ( d X ∗ ,M )( φ lX ∗ ,F r.a. ( M ) ) − ) == (id F ( U ) ⊗ B ( φ lX ∗ ,M ) − )( a − U,M π AU, ∗ F r.a. ( M ) ⊗ id X ) . (4.23)Using (4.22), the right hand side of (4.23) is equal to= (id F ( U ) ⊗ B ( φ lX ∗ ,M ) − )( a − U,M π AU, ∗ F r.a. ( M ) ⊗ id X )= (id F ( U ) ⊗ B ( φ lX ∗ ,M ) − ) (cid:0) id F ( U ) ⊗ B ∗ M ⊗ ev F r.a. ( M ) ⊗ id X (cid:1)(cid:0) d F ( U ) ⊗ B ∗ M,M ω U,F ( U ) ⊗ B ∗ M ⊗ M (Ψ BF ( U ) ,F ( U ) ⊗ B ∗ M,M (id )) ⊗ id ∗ F r.a. ( M ) ⊗ X (cid:1) = (id F ( U ) ⊗ B ( φ lX ∗ ,M ) − ) (cid:0) id F ( U ) ⊗ B ∗ M ⊗ ev F r.a. ( M ) ⊗ id X (cid:1)(cid:0) d F ( U ) ⊗ B ∗ M,M F r.a. (Ψ BF ( U ) ,F ( U ) ⊗ B ∗ ( M ) ,M (id )) ω U,F ( U ) (id ) ⊗ id ∗ F r.a. ( M ) ⊗ X (cid:1) The last equality follows from (4.21). It follows from (4.22), that the lefthand side of (4.23) is equal to= (cid:0) id F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ ev F r.a. ( X ∗ ⊗ M ) (cid:1)(cid:0) d F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ,X ∗ ⊗ M ω U,F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ X ∗ ⊗ M (Ψ BF ( U ) ,F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ,X ∗ ⊗ M (id )) ⊗ id ∗ F r.a. ( X ∗ ⊗ M ) (cid:1)(cid:0) id U ⊗ ∗ ( d X ∗ ,M )( φ lX ∗ ,F r.a. ( M ) ) − (cid:1) = (cid:0) id F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ ev F r.a. ( X ∗ ⊗ M ) (cid:1)(cid:0) id F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ F r.a ( X ∗ ⊗ M ) ⊗⊗ ∗ ( d X ∗ ,M )( φ lX ∗ ,F r.a. ( M ) ) − (cid:1)(cid:0) d F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ,X ∗ ⊗ M F r.a. (Ψ BF ( U ) ,F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ,X ∗ ⊗ M (id )) ⊗ id (cid:1)(cid:0) ω U,F ( U ) (id ) ⊗ id (cid:1) = (cid:0) id F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ ev X ∗ ⊗ F r.a. ( M ) ( d X ∗ ,M ⊗ ( φ lX ∗ ,F r.a. ( M ) ) − ) (cid:1)(cid:0) d F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ,X ∗ ⊗ M F r.a. (Ψ BF ( U ) ,F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ,X ∗ ⊗ M (id ) ⊗ id (cid:1)(cid:0) ω U,F ( U ) (id ) ⊗ id (cid:1) = (cid:0) id F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ ev X ∗ ⊗ F r.a. ( M ) (id ⊗ ( φ lX ∗ ,F r.a. ( M ) ) − ) (cid:1)(cid:0) d F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ X ∗ ,M ⊗ id ∗ F r.a. ( M ) ⊗ X (cid:1)(cid:0) F r.a. (Ψ BF ( U ) ,F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ,X ∗ ⊗ M (id ) ⊗ id (cid:1)(cid:0) ω U,F ( U ) (id ) ⊗ id (cid:1) . The second equation follows from (4.21), the third equality follows from(1.2), the fourth equality follows from (2.3) for the module functor ( F r.a. , d ),which in this case implies that(id ⊗ d X ∗ ,M ) d F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ,X ∗ ⊗ M = d F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ X ∗ ,M . At last, using the definition of Ψ B , (1.3) and the rigidity axioms one can seethat (cid:0) id F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ ev X ∗ ⊗ F r.a. ( M ) (id ⊗ ( φ lX ∗ ,F r.a. ( M ) ) − ) (cid:1)(cid:0) d F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ⊗ X ∗ ,M ⊗ id (cid:1)(cid:0) F r.a. (Ψ BF ( U ) ,F ( U ) ⊗ B ∗ ( X ∗ ⊗ M ) ,X ∗ ⊗ M (id )) ⊗ id (cid:1) ( ω U,F ( U ) (id ) ⊗ id ∗ F r.a ( M ) ⊗ X )= (id F ( U ) ⊗ B ( φ lX ∗ ,M ) − ) (cid:0) id F ( U ) ⊗ B ∗ M ⊗ ev F r.a. ( M ) ⊗ id X (cid:1)(cid:0) d F ( U ) ⊗ B ∗ M,M F r.a. (Ψ BF ( U ) ,F ( U ) ⊗ B ∗ ( M ) ,M (id ))( ω U,F ( U ) (id ) ⊗ id ∗ F r.a ( M ) ⊗ X ) . This implies (4.23), and finishes the proof of the Proposition. (cid:3)
A formula for the relative Serre functor.
Let M , N be exactindecomposable left C -module categories, and recall the functors L = L M , N : M op ⊠ C N →
Fun C ( M bop , N ) , e L = e L N , M bop : N op ⊠ C M bop → Fun C ( N , M bop )described in (4.4) and (4.5). Note that subindices of e L are different to thosepresented in (4.5). Lemma 4.8.
Use the above notation. For any M ∈ M , N ∈ N there existsan equivalence of module functors (4.24) L M , N ( M ⊠ C N ) l.a. ≃ e L N , M bop ( N ⊠ C M ) Proof. If B is an exact indecomposable right C -module category, define thefunctors H B B : B op → C , R N N : C → N ,H B B = Hom B ( − , B ) , R N N = − ⊲ N, for any B ∈ B , N ∈ N . A straightforward computation shows that( H B B ) l.a. ( X ) = B ⊳ X ∗ , ( R N N ) l.a. ( N ′ ) = ∗ Hom N ( N ′ , N )for any X ∈ C , N ′ ∈ N . Since L ( M ⊠ C N ) = R N N ◦ H M op M , then L ( M ⊠ C N ) l.a. ≃ ( H M op M ) l.a. ◦ ( R N N ) l.a. ≃ M ⊳
Hom N ( − , N ) = Hom N ( − , N ) ∗ ⊲ M (2.3) ≃ Hom N op ( − , N ) ∗∗∗ ⊲ M = e L ( N ⊠ C M ) . In the second equivalence, we are using the canonical isomorphisms ∗ X ∗ ≃ X . (cid:3) The next result is a formula for the relative Serre functor similar to theformula for the Nakayama functor given in [5]. Let M be an exact indecom-posable left C -module category. Let us denote by ◭ : M op × C → M op theaction of the opposite module category, that is, the one determined by(4.25) M ◭ X = X ∗ ⊲ M , CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 33 for any M ∈ M , X ∈ C . For any M ∈ M the functor T M : ( M op ) op × M op → M ,T M ( U, V ) = Hom M ( M, V ) ∗ ⊲ U, has a pre-balancing γ XU,V : T M ( U ◭ X, V ) → T M ( U, V ◭ ∗ X ) , given as the composition T M ( U ◭ X, V ) = Hom M ( M, V ) ∗ ⊲ ( X ∗ ⊲ U ) m − −−−→ (Hom M ( M, V ) ∗ ⊗ X ∗ ) ⊲ U −→ ( X ⊗ Hom M ( M, V )) ∗ ⊲ U → Hom M ( M, X ⊲ V ) ∗ ⊲ U = T M ( U, V ◭ ∗ X ) . Thus we can consider the coend I U ∈M op ( T M , γ ) . Since T can be thought of as a functor T : ( M op ) op × M op → Fun( M , M ), T ( U, V )( M ) = T M ( U, V ), then using the parameter theorem described inSection 3.2, we have a functor M I U ∈M op ( T M , γ ) . We shall denote this functor as I U ∈M op ( T − , γ ) = I U ∈M op Hom M ( − , U ) ∗ ⊲ U. It follows from Lemma 2.2 that H U ∈M op ( T − , γ ) : M → M bop is a C -modulefunctor. Theorem 4.9.
Let M be an exact indecomposable left C -module category.There exists an equivalence of C -module functors S M ≃ I U ∈M op Hom M ( − , U ) ∗ ⊲ U, Proof.
Let M , N be a pair of exact indecomposable left C -module categories.To prove the expression for the relative Serre functor, we will first computea quasi-inverse of the functor L = L M , N and then use equivalence (4.8).Recall that ◭ : M op × C → M op is the action of the opposite modulecategory. That is, M ◭ X = X ∗ ⊲ M , for any M ∈ M , X ∈ C . Let us denote by D : ( M op ⊠ C N ) op → N op ⊠ C M bop ,the functor determined by D ( M ⊠ C N ) = N ⊠ C M , M ∈ M , N ∈ N . Thefunctor D is an equivalence of categories.Take ( F, c ) ∈ Fun C ( M bop , N ). This means that c X,M : F ( X ∗∗ ⊲ M ) → X ⊲ F ( M ), for any M ∈ M , X ∈ C . DefineΘ F : ( M op ) op × M op → M op ⊠ C N , Θ F ( U , V ) = V ⊠ C F ( U ) . The functor Θ F has a pre-balancing ν XU,V : Θ F ( U ◭ X, V ) → Θ F ( U , V ◭ ∗ X ) ,ν XU,V = b − V, ∗ X,F ( U ) (id V ⊠ C c ∗ X,U ) . Here b V,X,U : V ◭ X ⊠ C U → V ⊠ C X ⊲ U is the balancing of the C -balancedfunctor ⊠ C . Define χ : Fun C ( M bop , N ) → M op ⊠ C N the functor given by χ ( F ) = I U ∈M op (Θ F , ν ) = I U ∈M op U ⊠ C F ( U ) . The existence of these coends follows from the existence of the ends presentedin Theorem 4.4 and the relation between ends and coends for left and rightmodule categories given in Lemma 3.5. Let us prove that χ is a quasi-inverseof L . Since we already know that L is a category equivalence, it is enoughto prove that χ ( L ( M ⊠ C N )) ≃ M ⊠ C N for any M ∈ M , N ∈ N . Since D is a category equivalence, this is equivalentto prove that(4.26) D ( χ ( L ( M ⊠ C N ))) ≃ D ( M ⊠ C N ) = N ⊠ C M . for any M ∈ M , N ∈ N . The left hand side of (4.26) is equal to D ( χ ( L ( M ⊠ C N ))) = D (cid:0)I U ∈M op U ⊠ C L ( M ⊠ C N )( U ) (cid:1) ≃ D (cid:0) I U ∈M bop U ⊠ C L ( M ⊠ C N )( U ) (cid:1) ≃ I U ∈M bop L ( M ⊠ C N )( U ) ⊠ C U ≃ I V ∈N V ⊠ C L ( M ⊠ C N ) l.a. ( V ) (4.24) ≃ I V ∈N V ⊠ C e L ( N ⊠ C M )( V ) (4.16) ≃ N ⊠ C M .
The first isomorphism follows from Lemma 3.6, the second one follows fromProposition 3.3 (iii), and the third isomorphism follows from Proposition4.17.Taking N = M bop and using (4.8), it follows that e L M , M bop ( χ (Id )) ≃ L M , M bop ( χ (Id )) ◦ S M ≃ S M , and we obtain the desired description of the relative Serre functor. (cid:3) CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 35
Correspondence of module categories for Morita equivalenttensor categories.
Assume that C , D are Morita equivalent tensor cate-gories. This means that there is an invertible exact ( C , D )-bimodule cate-gory B . We can assume that D = End C ( B ) rev , and the right action of D on B is given by evaluation ⊳ : B × D → B , B ⊳ F = F ( B ) . It was proven in [3, Theorem 3.31] that the maps
M 7→
Fun C ( B , M ) , N 7→
Fun D ( B op , N )are bijections, one the inverse of the other, between equivalence classes ofexact C -module categories and exact D -module categories. We shall giveanother proof of this fact by showing an explicit equivalence of D -modulecategories N ≃
Fun C ( B , Fun D ( B op , N )) , for any exact indecomposable D -module category N .For any ( H, d ) ∈ Fun C ( B , Fun D ( B op , N )) , define S H : B op × B → N , S H ( B, C ) = H ( C )( B ) . This functor comes with isomorphisms β XB,C : S H ( B, X ⊲ C ) → S H ( X ∗ ⊲ B, C ) ,β XB,C = (cid:0) d X,C (cid:1) B , for any X ∈ C , B, C ∈ B . Lemma 4.10.
The functor S H is a C -balanced functor with balancing givenby b B,X,C : S H ( X ∗ ⊲ B, C ) → S H ( B, X ⊲ C ) , b B,X,C = (cid:0) d X,C (cid:1) − B . In par-ticular, there exists a right exact functor b S H : B op ⊠ C B → N such that b S H ◦ ⊠ C ≃ S H as C -balanced functors.Proof. Since (
H, d ) is a module functor, the natural isomorphism d satisfy(2.3). This axiom implies that b satisfy (2.15). (cid:3) We can consider the functorΨ : Fun C ( B , Fun D ( B op , N )) → N , Ψ( H ) = I B ∈B ( S H , β ) = I B ∈B H ( B )( B ) . Proposition 4.11.
The functor Ψ is well-defined.Proof. The existence of the module end Ψ( H ) follows from applying thefunctor b S H to the module end H B ∈B B ⊠ C B , whose existence follow fromProposition 4.4, and using Proposition 3.3 (iii). (cid:3) Let us consider the functor e L = e L B , B : B op ⊠ C B →
End C ( B ) introduced inSection 4.2. Define also the functorΦ : N →
Fun C ( B , Fun D ( B op , N )) , Φ( N )( B )( C ) = e L ( C ⊠ C B ) ⊲ N, for any B, C ∈ B , N ∈ N . Theorem 4.12.
The functors Φ and Ψ are well-defined, and they establishan equivalence of left D -module categories N ≃
Fun C ( B , Fun D ( B op , N )) . Proof.
Take N ∈ N , B ∈ B . It follows immediately that Φ( N ) is a C -modulefunctor. That Φ and Φ( N )( B ) are D -module functors follow from the bi-module structure of the functor e L (4.6), (4.7). Let us show that the pair offunctors Φ, Ψ is an adjoint equivalence. Take H ∈ Fun C ( B , Fun D ( B op , N )) ,C , C ∈ B , thenΦ(Ψ( H ))( C )( C ) = e L ( C ⊠ C C ) ⊲ I B ∈B H ( B )( B ) ≃ I B ∈B H ( B )( e L ( C ⊠ C C ) ∗ ( B )) ≃ I B ∈B b S H (cid:0) e L ( C ⊠ C C ) ∗ ( B ) ⊠ C B (cid:1) ≃ b S H (cid:0) I B ∈B e L ( C ⊠ C C ) ∗ ( B ) ⊠ C B (cid:1) ≃ b S H (cid:0) I B ∈B B ⊠ C e L ( C ⊠ C C )( B ) (cid:1) ≃ b S H ( C ⊠ C C ) ≃ H ( C )( C ) . The first isomorphism follows since H ( B ) is a D -module functor, the secondisomorphism follows from the definition of b S H given in Lemma 4.10, thethird one follows from Proposition 3.3 (iii), the fourth isomorphism followsfrom (4.17), and the fifth isomorphism follows from (4.16).Now, let us take N ∈ N , thenΨ(Φ( N )) = I B ∈B Φ( N )( B )( B ) = I B ∈B e L ( B ⊠ C B ) ⊲ N ≃ Id ⊲ N ≃ N. The isomorphism follows from (4.15). One can verify, in the above proof ofΦ(Ψ( H )) ≃ H and in the proof of Ψ(Φ( N )) ≃ N , that each pre-balancingis used properly. (cid:3) CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 37
The double dual tensor category.
Let M be an exact indecompos-able left C -module category. Then the dual tensor category C ∗M = End C ( M )is again a finite tensor category [3]. The category C ∗M acts on M by evalu-ation: C ∗M × M → M , ( F, M ) F ( M ) . The category M is exact indecomposable over C ∗M , see [3, Lemma 3.25].Whence, we can consider the tensor category ( C ∗M ) ∗M = End C ∗M ( M ) . Thereis a canonical tensor functor can : C → ( C ∗M ) ∗M ,can ( X )( M ) = X ⊲ M, for any X ∈ C , M ∈ M . One can see that can ( X ) is a C ∗M -module functor.It was proven in [3, Theorem 3.27] that the functor can is an equivalence ofcategories. We shall give an expression of a quasi-inverse of this functor.Take ( G, d ) ∈ ( C ∗M ) ∗M . This means that we have natural isomorphisms d F,M : G ( F ( M )) → F ( G ( M )) , for any F ∈ C ∗M , M ∈ M . Let us denote H ( G,d ) : M op × M → C , H ( G,d ) ( M, N ) = Hom(
M, G ( N )) . The functor H ( G,d ) has a pre-balancing γ (seeing M as a left module categoryover C ∗M ). For any F ∈ C ∗M define γ FM,N : H ( G,d ) ( M, F ( N )) → H ( G,d ) ( F l.a. ( M ) , N ) , (Recall that F ∗ = F l.a. ) as the compositionHom( M, G ( F ( N ))) Hom( id ,d F,N ) −−−−−−−−−→ Hom(
M, F ( G ( N ))) → (2.13) −−−→ Hom( F l.a. ( M ) , G ( N )) . Explicitly, using (2.13), this isomorphism is γ FM,N = ψ ZF l.a. ( M ) ,G ( N ) (cid:0) Ω Z⊲M,G ( N ) ( φ ZM,F ( G ( N )) (id Z )) b − Z,M (cid:1) ◦ Hom(id , d
F,N )where Z = Hom( M, F ( G ( N ))), and isomorphism b is the module structureof the functor F l.a. . Recall the isomorphisms presented in (2.12) φ XM,N : Hom C ( X, Hom(
M, N )) → Hom M ( X ⊲ M, N ) ,ψ XM,N : Hom M ( X ⊲ M, N ) → Hom C ( X, Hom(
M, N )) , associated to the pair of adjoint functors ( − ⊲ M, Hom( M, − )). Theorem 4.13.
The functor
Υ : ( C ∗M ) ∗M → C given by Υ( G ) = I M ∈M (Hom( M, G ( M )) , γ ) is well-defined and it is a quasi-inverse of the functor can : C → ( C ∗M ) ∗M . Proof.
We shall prove that ( can,
Υ) is an adjoint equivalence, that is, thatthere are natural isomorphismsNat m ( can ( X ) , G ) ≃ Hom C ( X, Υ( G )) . Let us fix X ∈ C and ( G, d ) ∈ ( C ∗M ) ∗M . Then, using Proposition 4.1 wehave that Nat m ( can ( X ) , G ) ≃ I M ∈M (cid:0) Hom M ( X ⊲ M, G ( M )) , β (cid:1) . (4.27)According to (4.1), the pre-balancing β (recall that M is thought of as amodule category over C ∗M ) in this case is β FM,N : Hom M ( X ⊲ M, G ( F ( N ))) → Hom M ( X ⊲ F l.a. ( M ) , G ( N )) ,β FM,N ( α ) = (ev F ) G ( N ) F l.a. ( d F,N α ) b − X,M . Here ev F : F l.a. ◦ F → Id is the evaluation of the adjoint pair ( F l.a. , F ).If we denote by Ω M,N : Hom M ( M, F ( N )) → Hom M ( F l.a. ( M ) , N ) naturalisomorphisms, then (ev F ) M = Ω F ( M ) ,M (id F ( M ) ) , for any M ∈ M .Using Proposition 3.3 (ii) we can consider the module end I M ∈M (cid:0) Hom C ( X, Hom(
M, G ( M )) , b γ (cid:1) , where the pre-balancing in this case is b γ FM,N : Hom C ( X, Hom(
M, G ( F ( N ))) → Hom C ( X, Hom( F l.a. ( M ) , G ( M ))) , b γ FM,N ( α ) = γ FM,N ◦ α. Claim 4.1.
Isomorphisms ψ XM,G ( N ) : Hom M ( X ⊲ M, G ( N )) → Hom C ( X, Hom(
M, G ( N ))) commutes with the pre-balancings, that is (4.28) b γ FM,N ◦ ψ XM,G ( F ( N )) = ψ XF l.a. ( M ) ,G ( N ) ◦ β FM,N . As a consequence of this claim, using Proposition 3.3 (i), we get an iso-morphism of module ends I M ∈M (cid:0) Hom M ( X ⊲ M, G ( M )) , β (cid:1) ≃ I M ∈M (cid:0) Hom C ( X, Hom(
M, G ( M )) , b γ (cid:1) ≃ Hom C ( X, I M ∈M (cid:0) Hom(
M, G ( M )) , γ (cid:1) = Hom C ( X, Υ( G )) . The second isomorphism follows from Proposition 3.3 (ii). Combining thisisomorphism with (4.27) we get the result.
CO)ENDS FOR REPRESENTATIONS OF TENSOR CATEGORIES 39
It remains to prove the claim. Naturality of ψ, φ and b implies that(4.29) ψ XM,N ( α ( h ⊲ id M )) = ψ YM,N ( α ) ◦ h, (4.30) Hom(id , f ) ψ XM,N ( α ) = ψ XM,N ′ ( f ◦ α ) , (4.31) φ XM,N ( α ◦ h ) = φ YM,N ( α )( h ⊲ id M ) , (4.32) b − Y,M ( h ⊲ id F l.a. ( M ) ) = F l.a. ( h ⊲ id M ) b − X,M , for any morphism h : X → Y in C and any f : N → N ′ , M, N ′ , N ∈ M .Let α ∈ Hom M ( X ⊲ M, G ( F ( N ))), and Z = Hom( M, F ( G ( N ))), then theleft hand side of (4.28) evaluated in α is equal to γ FM,N ◦ ψ XM,G ( F ( N )) ( α ) == ψ ZF l.a. ( M ) ,G ( N ) (cid:0) Ω Z⊲M,G ( N ) ( φ ZM,F ( G ( N )) (id Z )) b − Z,M (cid:1)
Hom(id , d
F,N ) ψ XM,G ( F ( N )) ( α )= ψ ZF l.a. ( M ) ,G ( N ) (cid:0) Ω Z⊲M,G ( N ) ( φ ZM,F ( G ( N )) (id Z )) b − Z,M (cid:1) ψ XM,F ( G ( N )) ( d F,N α )= ψ XF l.a. ( M ) ,G ( N ) (cid:0) Ω Z⊲M,G ( N ) ( φ ZM,F ( G ( N )) (id Z )) b − Z,M ( ψ XM,F ( G ( N )) ( d F,N α ) ⊲ id F l.a. ( M ) ) (cid:1) = ψ XF l.a. ( M ) ,G ( N ) (cid:0) Ω Z⊲M,G ( N ) ( φ ZM,F ( G ( N )) (id Z )) F l.a. ( ψ XM,F ( G ( N )) ( d F,N α ) ⊲ id M ) b − X,M (cid:1) = ψ XF l.a. ( M ) ,G ( N ) (cid:0) Ω F ( G ( N )) ,G ( N ) (id ) F l.a. ( h ) b − X,M (cid:1)
The second equality follows from (4.30), the third equality follows from(4.29), the fourth follows from (4.32), the fifth equality follows from thenaturality of Ω. In the last equality the map h is h = φ ZM,F ( G ( N )) (id Z )( ψ XM,F ( G ( N )) ( d F,N α ) ⊲ id M ) . The right hand side of (4.28) evaluated in α is equal to ψ XF l.a. ( M ) ,G ( N ) (cid:0) Ω F ( G ( N )) ,G ( N ) (id ) F l.a. ( d F,N α ) b − X,M (cid:1) . It remains to observe that h = d F,N α , which follows from (4.31). (cid:3) Acknowledgments.
This work was partially supported by CONICET andSecyt (UNC), Argentina.
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