Internal natural transformations and Frobenius algebras in the Drinfeld center
aa r X i v : . [ m a t h . C T ] A ug ZMP-HH/20-16Hamburger Beitr¨age zur Mathematik Nr. 865
August 2020
Internal natural transformationsand Frobenius algebras in the Drinfeld center
J¨urgen Fuchs a and Christoph Schweigert b a Teoretisk fysik, Karlstads UniversitetUniversitetsgatan 21, S – 651 88 Karlstad b Fachbereich Mathematik, Universit¨at HamburgBereich Algebra und ZahlentheorieBundesstraße 55, D – 20 146 Hamburg
Abstract
For M and N finite module categories over a finite tensor category C , the category R ex C ( M , N )of right exact module functors is a finite module category over the Drinfeld center Z ( C ). Westudy the internal Homs of this module category, which we call internal natural transformations.With the help of certain integration functors that map C - C -bimodule functors to objects of Z ( C ),we express them as ends over internal Homs and define horizontal and vertical compositions.We show that if M and N are exact C -modules and C is pivotal, then the Z ( C )-module R ex C ( M , N ) is exact. We compute its relative Serre functor and show that if M and N areeven pivotal module categories, then R ex C ( M , N ) is pivotal as well. Its internal Ends are thena rich source for Frobenius algebras in Z ( C ). Introduction
Module categories over monoidal categories have been a prominent topic in representationtheory in the past two decades. The theory is particularly well-developed for finite tensorcategories and their finite module and bimodule categories. Indeed, many notions and resultsin the theory of finite-dimensional representations over finite-dimensional Hopf algebras havefound their natural conceptual home in this setting. Examples of such notions include theunimodularity of a finite tensor category and factorizability of a braided finite tensor category.Results include Radford’s S -formula [ENO], including its generalization to bimodule categories[FSS1], the equivalence of various characterizations of the non-degeneracy of a braiding on afinite tensor category [Sh2], and the theory of ‘reflections’ of Hopf algebras [BLS]. Moreover,module and bimodule categories have been used intensively in the study of subfactors, of two-dimensional conformal field theory, and of three-dimensional topological field theory.The following fact about module categories is well known. Let C be a finite tensor categoryand M and N be finite C -modules. Then the category R ex C ( M , N ) of right exact module func-tors is a finite module category over the Drinfeld center Z ( C ) (which is a finite tensor category).In this paper we study the internal Homs Hom Z ( C ) ( G, H ) for
G, H ∈ R ex C ( M , N ). We denotethese internal Homs by Nat( G, H ) ∈ Z ( C ) and call them internal natural transformations .For the vector space of ordinary natural transformation between two linear functors, theYoneda lemma implies a useful formula in terms of an end over morphism spaces:Nat( G, H ) = Z m ∈M Hom N ( G ( m ) , H ( m )) . (1.1)The structure morphisms Nat( G, H ) → Hom N ( G ( m ) , H ( m )) of this end just give the compo-nents of the natural transformation. One of the main results of this paper, Theorem 18, is asimilar expression Nat( F, G ) = Z m ∈M Hom N ( F ( m ) , G ( m )) (1.2)for the internal natural transformations as objects in Z ( C ). In particular, we show that theend on the right hand side has a natural structure of an object in the Drinfeld center.A crucial ingredient that allows us to obtain this result are two functors R • : F un C|C ( M ⊠ M , C ) → Z ( C ) and R • : F un C|C ( M ⊠ M , C ) → Z ( C ) (1.3)given by R • : G Z m ∈M G ( m, m ) and R • : H Z m ∈M H ( m, m ) (1.4)for G ∈ F un C|C ( M ⊠ M , C ) and H ∈ F un C|C ( M ⊠ M , C ), respectively, where M and M are two right C -module structures on the opposite category M opp . The existence of thesefunctors, which we call central integration functors , is shown in Theorem 15.Since the internal natural transformations are internal Homs, they come with associativecompositions. It follows in particular that for any module functor F the object Nat( F, F ) has anatural structure of a unital associative algebra in Z ( C ). We show that the structure morphismsNat( G, H ) → Hom N ( G ( m ) , H ( m )) of the end behave in the same way as the component mapsof an ordinary natural transformation. This allows us to define horizontal and vertical compo-sitions which obey the Eckmann-Hilton relation. As a consequence, the object Nat(Id M , Id M )2f internal natural endotransformations of the identity functor is a commutative algebra in thebraided category Z ( C ).We also study the situation that the monoidal category C has the additional structure of a pivotal tensor category. (This endows its Drinfeld center Z ( C ) with a pivotal structure as well.)Moreover, we assume that the module categories under investigation are now exact C -modules.As we show in Proposition 34, in this case the two central integration functors (1.4) are relatedby the Nakayama functor N r M ∈ R ex ( M , M ) according to R • = R • ◦ (cid:0) Id M opp ⊠ N r M (cid:1) . (1.5)Based on this result we show in Theorem 36 that for any pair M , M of exact module categoriesover a pivotal finite tensor category C , the category R ex C ( M , M ) is an exact Z ( C )-module.Specifically, we compute its relative Serre functor to beS r R ex = N r N ◦ ( D . − ) ◦ N r M , (1.6)with D the distinguished invertible object of C . In Corollary 38 we then conclude that incase C is unimodular, this exact module category is pivotal (in the sense of Definition 9). Itfollows that in this case Nat( F, F ) has the structure of a Frobenius algebra, and in particularNat(Id M , Id M ) has the structure of a commutative Frobenius algebra. In this way, C -modulecategories become a rich source of Frobenius algebras in Z ( C ).This paper is organized as follows. After setting the stage in Section 2, in Section 3 we studyrelations between bimodule functors with codomain C and the Drinfeld center of C , whichleads us to the notion of central integration functors. Section 4 deals with internal naturaltransformations. In particular, in Section 4.2 we explain how they can be expressed as an end,and in Section 4.3 we introduce and study their horizontal and vertical compositions. Finally, inSection 5 we combine these results with the theory of relative Serre functors and pivotal modulecategories to examine exactness and pivotality of the the functor category R ex C ( M , N ) as amodule category over Z ( C ).A direct application of our results (and, in fact, also a major motivation for our inves-tigations) is in the description of bulk fields in rigid logarithmic two-dimensional conformalfield theories, i.e. conformal field theories whose chiral data are described by a modular finitetensor category C . This application will be discussed in detail elsewhere. Here we content our-selves with mentioning the basic idea. When C is modular, then we have a braided equivalence Z ( C ) ≃ C rev ⊠ C . The algebra of bulk fields (or, more generally, disorder and defect fields) infull local conformal field theory can therefore be regarded as an object in Z ( C ).The field algebras in local conformal field theories should be Frobenius algebras; this hase.g. been demonstrated for bulk algebras of rigid logarithmic conformal field theories in [FuS].It is also well known that there are different full local conformal field theories that sharethe same chiral data based on a given modular tensor category C . It has been establishedalmost two decades ago that in case C is semisimple, the datum that in addition to the chiraldata is needed to characterize a local conformal field theory is a (semisimple, indecomposable) C -module category [FFFS, FFRS]. The results of Section 5 show that for C not semisimple,a pivotal indecomposable module category M is a natural candidate for such an additionaldatum. Boundary conditions of the full conformal field theory are then described by objects m ∈ M and boundary fields by internal Homs Hom( m, m ′ ) ∈ C . By Theorem 3.15 of [Sh4], the3lgebra Hom( m, m ) is a symmetric Frobenius algebra for any m ∈ M . A right exact modulefunctor G ∈ R ex C ( M , M ) describes a topological defect line between local conformal fieldtheories characterized by M and by M , respectively. It is then natural to propose that thedefect fields that change a defect line labeled by G to a defect line labeled by H are givenby Nat( G, H ) ∈ Z ( C ). In particular, Nat( G, G ) ∈ Z ( C ) is a symmetric Frobenius algebra; asa special case, Nat(Id M , Id M ) ∈ Z ( C ) a commutative symmetric Frobenius algebra, as befitsthe space of bulk fields. This proposal also leads to natural candidates for operator productexpansions and passes non-trivial consistency checks. In this section we fix our notation and mention some pertinent structures and concepts.
Monoidal categories.
We denote the tensor product of a monoidal category by ⊗ and themonoidal unit by , and the associativity and unit constraints by α , l and r , i.e. a monoidalcategory is a quintuple C = ( C , ⊗ , , α, l, r ). For better readability of various formulas, wesometimes take, without loss of generality, the tensor product to be strict, i.e. take the associator α and unit constraints l and r to be identities. Module categories.
The notion of a (left) module category M over a monoidal category C , or C -module, for short, categorifies the notion of module over a ring: There is an actionfunctor C × M → M , exact in its first variable, together with a mixed associator and a mixedunitor that obey mixed pentagon and triangle relations. For background on module categories,as well as module functors and module natural transformations, see e.g. [EGNO, Ch. 7] or[Sh3, Sect. 2.3]. In the present paper, module categories will be left modules unless statedotherwise. We denote the action morphism by a dot and the mixed associator by a , i.e. a hascomponents a c,c ′ ,m : ( c ⊗ c ′ ) . m → c . ( c ′ . m ) with c, c ′ ∈ C and m ∈ M . The natural isomorphismthat defines the structure of a C -module functor G is denoted by φ G , i.e. φ G has components φ Gc,m : G ( c.m ) → c . ( G ( m )) with c ∈ C and m ∈ M . Finite categories.
We fix an algebraically closed field k . A finite k -linear category is anabelian category that is equivalent as abelian category to the category of finite-dimensionalmodules over a finite-dimensional k -algebra. A finite tensor category is a finite k -linear categorywhich is rigid monoidal with appropriate compatibility conditions among the structures, seee.g. [EO] or [Sh3, Sect. 2.5]. Since a finite tensor category is rigid, its tensor product functoris exact. Our conventions concerning dualities of a rigid category C are as follows. The rightdual of an object c is denoted by c ∨ , and the right evaluation and coevaluation are morphismsev r c ∈ Hom C ( c ∨ ⊗ c, ) and coev r c ∈ Hom C ( , c ⊗ c ∨ ) , (2.1)while the left evaluation and coevaluation areev l c ∈ Hom C ( c ⊗ ∨ c, ) and coev l c ∈ Hom C ( , ∨ c ⊗ c ) (2.2)4ith ∨ c the left dual of c .A module category M over a finite tensor category C is called finite iff M is a finite k -linearabelian category and the action of C on M is linear and right exact in both variables. Thecategory of right exact module endofunctors of a finite module category M over a finite tensorcategory C is again a finite tensor category [EGNO, Prop. 7.11.6.]; we denote it by C ⋆ M . Afinite C -module is called exact iff p . m is projective in M for each projective p ∈ C and each m ∈ M . In particular, C is an exact module category over itself [EO, Def. 3.1]. Indecomposableexact module categories over H -mod, for H a finite-dimensional Hopf algebra, are classifiedin [AM, Sect. 3.2]. For recent results see also [Sh4]. Drinfeld centers.
For A a monoidal category, a half-braiding for an object a ∈ A is anatural family σ = ( σ a ) a ∈A of morphisms σ a : a ⊗ a → a ⊗ a such that (suppressing the asso-ciator of A ) σ a ⊗ a ′ = ( σ a ⊗ id a ′ ) ◦ (id a ⊗ σ a ′ ) for all a, a ′ ∈ A and σ = id a . The Drinfeld center Z ( A ) of A has as objects pairs ( a, σ ) consisting of an object of A and a half-braiding onit. The morphisms Hom Z ( A ) (( a, σ ) , ( a ′ , σ ′ )) are those morphisms a f −→ a ′ in A that satisfy( f ⊗ id b ) ◦ σ b = σ ′ b ◦ (id b ⊗ f ) for all b ∈ A . For any monoidal category A , the Drinfeld center Z ( A ) has a natural braided monoidal structure. Unimodular categories.
In any finite tensor category there is (uniquely up to isomorphism)a distinguished invertible object D , an invertible object that comes [ENO, Thm 3.3] with co-herent isomorphisms D ⊗ x ∼ = x ∨∨∨∨ ⊗ D . A unimodular finite tensor category is a finite tensorcategory A for which the distinguished invertible object is the monoidal unit. There are severalequivalent characterizations of unimodularity [Sh1], e.g. the forgetful functor U : Z ( A ) → A from the Drinfeld center is a Frobenius functor. Modular categories.
For C a braided finite tensor category, we denote by C rev its reverse ,i.e. the same monoidal category, but with inverse braiding. There is a canonical braided functorΞ C : C rev ⊠ C → Z ( C ) (2.3)from the enveloping category of C , i.e. the Deligne product of C rev with C (which exists, as C isfinite abelian), to the Drinfeld center of C . As a functor, Ξ C maps the object u ⊠ v ∈ C rev ⊠ C to the tensor product u ⊗ v ∈ C endowed with the half-braiding γ u ⊗ v that has components γ c ; u ⊗ v := (id u ⊗ β − v,c ) ◦ ( β c,u ⊗ id v ) for c ∈ C , with β the braiding in C . The braided monoidalstructure on the functor Ξ C is given by the coherent family id u ⊗ β v,x ⊗ id y of isomorphismsfrom u ⊗ v ⊗ x ⊗ y to u ⊗ x ⊗ v ⊗ y .A finite tensor category C is non-degenerate iff the functor Ξ C is an equivalence. If C is evena ribbon category, then C rev is a ribbon category with the inverse twist. A non-degenerate finiteribbon category is a modular tensor category , or modular category, for short. Traditionally, theterm modular category has been used under the additional assumption that the finite tensorcategory C is semisimple, i.e. a fusion category; in our context, such a restriction is not natural.A modular category is in particular unimodular. The central monad and comonad.
The Drinfeld center comes with a forgetful functor U : Z ( C ) → C that omits the half-braiding. U is exact and hence has a left and a right adjoint.5hese adjunctions are (co)monadic and thus give rise to a monadZ : C −−→ C c R x ∈C x ⊗ c ⊗ ∨ x (2.4)on C and to a comonad Z : C −−→ C c R x ∈C ∨ x ⊗ c ⊗ x (2.5)(or, equivalently, c R x ∈C x ⊗ c ⊗ x ∨ ). These are called the central monad and central comonad ,respectively. Since the adjunctions are monadic and comonadic respectively, we have canonicalequivalences of the categories of Z-modules, Z-comodules and the Drinfeld center Z ( C ). Formore details about the central monad and comonad see e.g. [BV] and [TV, Ch. 9].When dealing with coends, such as the one defining the central monad, a convenient tool isthe following form of the Yoneda lemma (see e.g. [FSS1, Prop. 2.7]): Lemma 1.
Let A and B be finite linear categories and F : A → B a linear functor. Then thereis a natural isomorphism Z a ∈A Hom A ( a, − ) ⊗ F ( a ) ∼ = F (2.6)of linear functors.An analogous co-Yoneda lemma holds for ends: R a ∈A Hom A ( − , a ) ∗ ⊗ F ( a ) ∼ = F . The isomor-phisms in these formulas are uniquely determined by universal properties; accordingly, fromnow on we will write them as equalities. Eilenberg-Watts calculus.
For any pair of finite linear categories A and B there are twopairs of two-sided adjoint equivalences L ex ( A , B ) A opp ⊠ B R ex ( A , B ) Ψ l Φ l Φ r Ψ r (2.7)given by [Sh1, FSS1] Φ l ( a ⊠ b ) := Hom A ( a, − ) ⊗ b , Ψ l ( F ) := R a ∈A a ⊠ F ( a )and Φ r ( a ⊠ b ) := Hom A ( − , a ) ∗ ⊗ b , Ψ r ( G ) := R a ∈A a ⊠ G ( a ) . (2.8)This provides a Morita invariant version of the classical Eilenberg-Watts description of right orleft exact functors between the categories R -mod and S -mod of modules over unital rings interms of S - R -bimodules. The functors (2.8) are therefore called Eilenberg-Watts equivalences.Applying these equivalences to the identity functor on A , regarded as a left exact functor,yields a right exact endofunctorN r A := Φ r ◦ Ψ l (Id A ) = Z a ∈A Hom A ( − , a ) ∗ ⊗ a ∈ R ex ( A , A ) , (2.9)which is called the Nakayama functor of the finite linear category A [FSS1, Def. 3.14]. Anal-ogously, by applying Φ l ◦ Ψ r to Id A regarded as a right exact functor we obtain a left exact6nalogue N l A = R a ∈A Hom A ( a, − ) ⊗ a ∈ L ex ( A , A ). The functor N l A is left adjoint to N r A . For A and B finite tensor categories, the Nakayama functor of an A - B -bimodule M has a naturalstructure of a twisted bimodule functor, in the sense that there are coherent isomorphismsN r M ( a.m.b ) ∼ = ∨∨ a. N r M ( m ) .b ∨∨ (2.10)for all m ∈ M , a ∈ A and b ∈ B [FSS1, Thm. 4.5]. For any m ∈ M we denote the action functor by H m = − .m : C → M . As H m is (right) exact,the following functors exist: Definition 2.
Let C be a monoidal category and M be a C -module. An internal Hom of M in C is a functor Hom M (?; ?) : M opp ⊠ M → C (2.11)such that for every m ∈ M the functor Hom M ( m, − ) : M → C is right adjoint to the actionfunctor H m , i.e. such that for any two objects m, m ′ ∈ M there is a natural familyHom C ( c, Hom M ( m, m ′ )) ∼ = Hom M ( c . m, m ′ ) (2.12)of isomorphisms (see e.g. [Os]).Being a right adjoint, the internal Hom is left exact. When it is clear from the contextwhich module category M is concerned, we simply write Hom in place of Hom M .We note that there is no separate notion of a ‘relative adjoint’ functor: Lemma 3.
Let C be a monoidal category and M , N be C -modules. Let F : M → N and G : N → M be module functors. Then F and G are adjoint functors if and only if there arefunctorial isomorphisms Hom N ( F ( m ) , n ) ∼ = Hom M ( m, G ( n )) . (2.13) Proof.
By the definition of the internal Hom and the fact that F is a module functor, we haveHom M ( c . m, G ( n )) ∼ = Hom C (cid:0) c, Hom M ( m, G ( n )) (cid:1) andHom N ( F ( c.m ) , n ) ∼ = Hom N ( c . F ( m ) , n ) ∼ = Hom C (cid:0) c, Hom N ( F ( m ) , n ) (cid:1) (2.14)for every c ∈ C , m ∈ M and n ∈ N . Thus (2.13) implies that F and G are adjoint. The converseholds by the Yoneda lemma. Algebra structure on internal Ends.
The counits of the adjunctions H m ⊣ Hom( m, − )provide for any two objects m, m ′ ∈ M a canonical morphismev m,m ′ : Hom( m, m ′ ) . m → m ′ , (2.15)in M given by the image of the identity morphism in End C (Hom( m, m ′ )) under the definingisomorphism (2.12). The composition ev m ′ .m ′′ ◦ (id Hom( m ′ ,m ′′ ) ⊗ ev m.m ′ ) ◦ a Hom( m ′ ,m ′′ ) , Hom( m,m ′ ) ,m furnishes an associative multiplication µ m,m ′ ,m ′′ : Hom( m ′ , m ′′ ) ⊗ Hom( m, m ′ ) −→ Hom( m, m ′′ ) (2.16)7n internal Homs. Moreover, the component −→ Hom( m, .m ) = Hom( m, m ) (2.17)of the unit of the adjunction H m ⊣ Hom( m, − ) is a unit for the multiplication (2.16). Compatibility with module functors.
A module functor G : M → N induces a naturalmorphism Hom M ( c.m, m ′ ) → Hom N ( c.G ( m ) , G ( m ′ )) for any m, m ′ ∈ M and c ∈ C , and thusHom C ( c, Hom( m, m ′ )) → Hom C ( c, Hom( G ( m ) , G ( m ′ )). By the Yoneda lemma this induces amorphism G : Hom( m, m ′ ) −→ Hom( G ( m ) , G ( m ′ )) (2.18)of internal Hom objects in C . It is easy to check that G ◦ G ′ = G ◦ G ′ , where on the left handside one deals with composition of functors and on the right hand side with composition ofmorphisms in C . Internal Hom as a bimodule functor.
For a left C -module M the opposite category M opp can be endowed in many ways with the structure of a right C -module, which are related by themonoidal functor of taking biduals. Of relevance to us are the following two choices of the right C -action: either m . c := ∨ c . m for m ∈ M , or else m . c := c ∨ . m . We denote the former right C -module by M and the latter by M . Then in particular both M ⊠ M and M ⊠ M havea natural structure of a C -bimodule. It follows that Hom is naturally a bimodule functor, with C regarded as a bimodule over itself; Lemma 4. [Sh4, Lemma 2.7]
The functor
Hom(?; ?) : M ⊠ M → C is a bimodule functor.Proof.
For any c ∈ C the endofunctor F c of M defined by F c ( m ) = c . m is left exact and thushas a left adjoint. Indeed we haveHom C ( γ, Hom( m, c.m ′ )) ∼ = Hom M ( γ . m, c . m ′ ) ∼ = Hom M ( c ∨ . ( γ.m ) , m ′ ) ∼ = Hom C ( c ∨ ⊗ γ, Hom( m, m ′ )) ∼ = Hom C ( γ, c . Hom( m, m ′ )) (2.19)for any γ ∈ C . Similarly, Hom C ( γ, Hom( c.m, m ′ )) ∼ = Hom M ( γ, Hom( m, m ′ ) ⊗ c ∨ ) for γ ∈ C . Thusthere are isomorphismsHom( m, c ′ . m ′ ) ∼ = c ′ ⊗ Hom( m, m ′ ) and Hom( c . m, m ′ ) ∼ = Hom( m, m ′ ) ⊗ c ∨ . (2.20)Taken together, these imply the natural isomorphismsHom( c . m, c ′ . m ′ ) ∼ = c ′ ⊗ Hom( m, m ′ ) ⊗ c ∨ (2.21)that are required for Hom(?; ?) to be a bimodule functor. Internal coHom.
There is an obvious dual notion to the internal Hom: For C a monoidalcategory and M a C -module, an internal coHom of M in C is a functor coHom(?; ?) : M opp ⊠ M→ C such that for any m, m ′ ∈ M there is a natural familyHom C (coHom( m ′ , m ) , c ) ∼ = Hom M ( m, c . m ′ ) (2.22)8f isomorphisms. By exactness of the action functors H m , also internal coHoms exist. Being aleft adjoint, the internal coHom is right exact.On the left hand side of (2.22) we have Hom C (coHom( m ′ , m ) , c ) ∼ = Hom C ( c ∨ , coHom( m ′ , m ) ∨ ),while the right hand side is Hom M ( m, c . m ′ ) ∼ = Hom C ( c ∨ , Hom( m, m ′ )). Thus the internal Homand coHom are are indeed dual to each other:coHom( m ′ , m ) ∨ ∼ = Hom( m, m ′ ) and coHom( m ′ , m ) ∼ = ∨ Hom( m, m ′ ) . (2.23)By taking left duals in (2.16) we obtain a coassociative comultiplicationcoHom( m ′′ , m ) = ∨ Hom( m, m ′′ ) −→ ∨ (cid:0) Hom( m ′ , m ′′ ) ⊗ Hom( m, m ′ ) (cid:1) ∼ = coHom( m ′ , m ) ⊗ coHom( m ′′ , m ′ ) . (2.24)A counit for this comultiplication is given by the left dual of the unit for the multiplication(2.16)Analogously to the morphism (2.18), for any module functor G : M → N we get a morphism G : coHom( m, m ′ ) −→ coHom( G ( m ) , G ( m ′ )) of internal coHom objects in C . And analogouslyto Lemma 4 one shows that coHom(?; ?) : M ⊠ M → C is naturally a bimodule functor.
Remark 5.
For C as a module over itself, the internal Hom is Hom( c, c ′ ) = c ′ ⊗ c ∨ , and thefamily (2.12) reduces to the natural isomorphism that is furnished by the right duality. Similarlywe then have coHom( c, c ′ ) = c ′ ⊗ ∨ c . An additional structure that a finite tensor category C may admit is a pivotal structure, i.e. amonoidal isomorphism π : Id C → − ∨∨ from the identity functor to the right double-dual functor.The presence of a pivotal structure has important consequences; for instance, while the notion ofa Frobenius algebra makes sense in any monoidal category C , the one of a symmetric Frobeniusalgebra does so only if C is pivotal (and even depends on the choice of pivotal structure).Pivotality will be used extensively in Section 5; we therefore discuss it here in some detail.If M and N are left modules over a pivotal finite tensor category C , then the Eilenberg-Watts equivalences (2.7) of linear categories induce adjoint equivalences involving categories ofleft and right exact module functors: we have [FSS3, Prop. 4.1] L ex C ( M , N ) Z ( M ⊠ N ) Ψ l Φ l and Z ( M ⊠ N ) R ex C ( M , N ) Φ r Ψ r (2.25)where M and M are the right modules with underlying linear category M opp describedabove. Remark 6.
Any pivotal tensor category is equivalent, as a pivotal category, to a strict pivotalcategory [NgS, Thm. 2.2]. Thus in case the finite tensor category C of our interest has a pivotalstructure, for many purposes we may replace it by a strict pivotal one in which c ∨ = ∨ c holds forevery c ∈ C . When doing so, M and M are the same C -module; we denote it by the symbol M . The equivalences (2.25) then combine to L ex C ( M , N ) Z ( M ⊠ N ) R ex C ( M , N ) . Ψ l Φ l Φ r Ψ r (2.26)9urthermore, in this case the Nakayama functor N r M of M becomes an ordinary module functor,rather than a module functor twisted by a double dual.Next we note that the Drinfeld center Z ( C ) of a rigid category C is rigid as well. Moreover,the components π c : c → c ∨∨ of a pivotal structure on C are even morphisms in Z ( C ). Thusif C is pivotal, then Z ( C ) inherits a distinguished pivotal structure [EGNO, Exc. 7.13.6]. Forpivotal C we will consider Z ( C ) with this pivotal structure. Exact module categories over a pivotal tensor category turn out to have a particularlyinteresting theory. Let us first recall
Definition 7. [FSS1, Def. 4.22] Let M be a left C -module. A right relative Serre functor on M is an endofunctor S r M of M equipped with a familyHom( m, n ) ∨ ∼ = −−→ Hom( n, S r M ( m )) (2.27)of isomorphisms natural in m, n ∈ M . A left relative Serre functor S l M on M comes analogouslywith a family ∨ Hom( m, n ) ∼ = −−→ Hom(S l M ( n ) , m ) (2.28)of natural isomorphisms. Remark 8.
According to [FSS1, Thm. 4.26] the Nakayama and relative Serre functors of an ex-act module M over a finite tensor category C are related by N l M ∼ = D C . S l M and N r M ∼ = D − C . S r M .In particular the Nakayama and relative Serre functors coincide iff C is unimodular.It is known [FSS1, Prop. 4.24] that a finite left C -module admits a relative Serre functor if andonly if it is an exact module category. In this case the relative Serre functor is an equivalence ofcategories. A right relative Serre functor on M is a twisted module functor [FSS1, Lemma 4.23]in the sense that there are coherent natural isomorphisms φ S r c,m : S r M ( c.m ) ∼ = −−→ c ∨∨ . S r M ( m ) . (2.29)Similarly there are coherent natural isomorphisms S l M ( c.m ) ∼ = ∨∨ c . S l M ( m ). These results allowone to give Definition 9. ([Sc2, Def. 5.2] and [Sh4, Def. 3.11]) A pivotal structure , or inner-product struc-ture , on an exact module category M over a pivotal finite tensor category ( C , π ) is an isomor-phism π M : Id M ∼ = −→ S r M of functors such that the equality φ S r c,m ◦ π M c.m = π c . π M m of morphismsfrom c . m to c ∨∨ . S r M ( m ) holds for every c ∈ C and every m ∈ M .In short, a pivotal structure is an isomorphism, as module functors, from the identity functorto the Serre functor, where the pivotal structure on C has been used to turn them into modulefunctors of the same type. If the module category M is indecomposable, then the identityfunctor Id M is a simple object in the category of right exact module endofunctors. ThusSchur’s lemma implies Lemma 10. [Sh4, Lemma 3.12] Let M be an indecomposable exact module category over apivotal finite tensor category. A pivotal structure on M , if it exists, is unique up to a scalarmultiple. 10 emark 11. (i) As a special case of Lemma 4.23 of [FSS1], consider a pivotal finite tensorcategory C as a module category over itself. We haveS l C ( c. ) = ∨∨ c . S l C ( ) = ∨∨ c , (2.30)so in this case S l C coincides with the bidual functor. The pivotal structure of C then endowsthe module category C C with the structure of a pivotal module category.(ii) It follows [Sh4, Thm. 3.13] that for M an indecomposable pivotal exact C -module, the finitetensor category C ∗M of right exact C -module endofunctors is a pivotal tensor category.We are now in a position to introduce further structure on an exact C -module M : Denoteby coev c,m : c → Hom( m, c.m ) the unit of the adjunction H m ⊢ Hom( m, − ). Let S r be a relativeSerre functor on M . The internal trace [Sh4, Def. 3.7] is the compositiontr m : Hom( m, S r ( m )) ∼ = −−−→ Hom( m, m ) ∨ coev ∨ −−−−−→ ∨ ∼ = , (2.31)where the first isomorphism is a component of the inverse of the defining structural morphism ofS r . The internal trace is related to a non-degenerate pairing Hom( n, S r ( m )) ⊗ Hom( m, n ) → ;indeed this pairing factors into the composition of the internal Hom and the internal trace.Based on these structures the following has been shown recently: Theorem 12. [Sh4, Thm. 3.15]
Let ( M , π M ) be a pivotal exact module category over a pivotalfinite tensor category ( C , π ) . Then the algebra Hom( m, m ) in C has the structure of a symmetricFrobenius algebra with Frobenius form λ m : Hom( m, m ) ( π M m ) ∗ −−−−−→ Hom( m, S r ( m )) tr m −−−→ , (2.32) for any m ∈ M . Recall from the paragraph before Lemma 4 the right C -module structures M and M thatare defined on the opposite category M opp of the abelian category underlying a C -module M .We have Lemma 13.
Let M be a C -module. (i) Let G : M ⊠ M → C be a bimodule functor. Then the end R m ∈M G ( m, m ) has a naturalstructure of a comodule over the central comonad Z of C , and can thus be seen as an objectin the Drinfeld center Z ( C ) . (ii) Let H : M ⊠ M → C be a bimodule functor. Then the coend R m ∈M H ( m, m ) has a naturalstructure of a module over the central monad Z of C , and can thus be seen as an object inthe Drinfeld center Z ( C ) . roof. Abbreviate R m ∈M G ( m, m ) =: g . To obtain a candidate δ g for the coaction of Z on g wefirst concatenate the structure morphisms j g of the end and the bimodule structure of G (whichwill be suppressed in the considerations below), which gives us a family g j gc.m −−−→ G ( c.m, c.m ) φ G −−−→ ∼ = c . G ( m, m ) . c ∨ (3.1)of morphisms. Since j g is dinatural, this constitutes a dinatural transformation from the object g to the functor ( C ⊠ M ) opp ⊠ ( C ⊠ M ) −−→ C ( c ⊠ m ) ⊠ ( c ′ ⊠ m ′ ) c ′ . G ( m, m ′ ) . c ∨ . (3.2)Invoking the behavior of (co)ends over Deligne products [FSS1, Sect. 3.4] this, in turn, givesrise to a morphism δ g : g −→ Z c ⊠ m ∈C ⊠ M c . G ( m, m ) . c ∨ ∼ = Z c ∈C Z m ∈M c . G ( m, m ) . c ∨ ∼ = Z c ∈C c . Z m ∈M G ( m, m ) . c ∨ = Z( g ) , (3.3)such that ( c . j gm . c ∨ ) ◦ ζ gc ◦ δ g = φ G ◦ j gc.m : g −→ c . G ( m, m ) . c ∨ (3.4)with ζ xc : Z( x ) → c ⊗ x ⊗ c ∨ the structure morphism of the central comonad. Writing ˆ δ c := ζ c ◦ δ g for c ∈ C , together with (3.1) this gives the commutative diagram G ( c.m, c.m ) c . G ( m, m ) . c ∨ g c . g . c ∨ Z( g ) ∼ = j gc.m δ g ˆ δ c c.j gm .c ∨ ζ c (3.5)(We do not directly use this diagram here, but it will be instrumental in the proof of Lemma14 below.)Now denote by ∆ : Z( c ) → Z ( c ) the comultiplication of Z. To see the coaction property of δ g we compare two commutative diagrams. The first of these is g Z( g ) Z ( g ) x . Z( g ) . x ∨ ( x ⊗ y ) . g . ( x ⊗ y ) ∨ ( x ⊗ y ) . G ( m, m ) . ( x ⊗ y ) ∨ δ g j g ( x ⊗ y ) .m ∆ ζ gx ⊗ y ζ Z ( g ) x x.ζ gy .x ∨ ( x ⊗ y ) .j gm . ( x ⊗ y ) ∨ (3.6)12ere the upper square commutes by the defining propertyZ( c ) ( x ⊗ y ) . c . ( x ⊗ y ) ∨ Z ( c ) x . Z( c ) . x ∨ ζ cx ⊗ y ∆ ζ Z ( c ) x x.ζ cy .x ∨ (3.7)of the comultiplication, while the lower square commutes owing to the relation (3.4). Thesecond diagram is g Z( g ) Z ( g ) x . g . x ∨ x . Z( g ) . x ∨ ( x ⊗ y ) . G ( m, m ) . ( x ⊗ y ) ∨ ( x ⊗ y ) . g . ( x ⊗ y ) ∨ δ g j g ( x ⊗ y ) .m Z( δ g ) ζ gx ζ Z ( g ) x x.δ g .x ∨ x.j gy.m .x ∨ x.ζ gy .x ∨ ( x ⊗ y ) .j gm . ( x ⊗ y ) ∨ (3.8)Here the left and the lower right square commute again due to (3.4), while the upper rightsquare commutes by the definition of Z. Comparing the outer hexagons of the diagrams (3.6)and (3.8) establishes the comodule property of the morphism δ g and thus proves claim (i).Claim (ii) follows by applying claim (i) to the opposite category. For instance, the commutativediagram analogous to (3.5) reads c . H ( m, m ) . ∨ c H ( c.m, c.m ) c . h . ∨ c h Z( h ) ∼ = c.i hm . ∨ c i hc.m ζ c ˆ ρ c ρ h (3.9)with ζ xc : c ⊗ x ⊗ ∨ c → Z( x ) the structure morphism of the central monad, h := R m ∈M H ( m, m )and ˆ ρ c := ρ h ◦ ζ c .An analogous result holds for natural transformations: Lemma 14.
Let M be a C -module. (i) Let G , G : M ⊠ M → C be bimodule functors and ν : G → G be a bimodule naturaltransformation. Then the morphism ν := R m ∈M ν m,m : R m ∈M G ( m, m ) → R m ∈M G ( m, m ) in C that is induced by the functoriality of the end is even a morphism of Z -comodules. (ii) Let H , H : M ⊠ M → C be bimodule functors and ν : H → H be a bimodule naturaltransformation. Then the morphism R m ∈M ν m,m : R m ∈M H ( m, m ) → R m ∈M H ( m, m ) in C induced by the functoriality of the coend is even a morphism of Z -modules. roof. We prove claim (i); the proof of (ii) is dual. Consider two copies of the diagram (3.5), onefor G and one for G . We can connect the top lines of these two diagrams by the morphisms ν c.m,c.m : G ( c.m, c.m ) → G ( c.m, c.m ) and c . ν m,m . c ∨ : c . G ( m, m ) . c ∨ → c . G ( m, m ) . c ∨ . Theresulting square commutes because ν is required to be a bimodule natural transformation.Similarly, we can connect the second line of the diagram for G to the second line of the diagramfor G by the morphisms ν : g → g and c . ν . c ∨ : c . g . c ∨ → c . g . c ∨ . The two resulting squaresthat involve the dinatural structure morphisms for g and g , respectively, commute, owing tothe definition of ν and the functoriality of the C -actions. Thus in short, in the diagram G ( c.m, c.m ) c . G ( m, m ) . c ∨ G ( c.m, c.m ) c . G ( m, m ) . c ∨ g c . g . c ∨ g c . g . c ∨∼ = j g c . m ∼ = ν c.m c.ν m .c ∨ c.j g m .c ∨ νj g c . m c.ν.c ∨ c.j g m .c ∨ (3.10)the left, right, front, back and top squares commute for every m ∈ M . As a consequence, thesquare at the bottom of (3.10) commutes as well.Proceeding in the same way as above, the lower triangles in the two diagrams of type (3.5) for G and G can be combined to g c . g . c ∨ g c . g . c ∨ Z( g )Z( g ) ˆ δ g c ν ˆ δ g c c . ν . c ∨ ζ g c Z( ν ) ζ g c (3.11)The two triangles in this diagram commute by construction, the top square is just the bottomsquare of (3.10), and the square on the right commutes by the functoriality of the end in thecentral comonad Z. Thus the square on the left commutes as well. This is the desired result:it states that ν is a morphism of Z-comodules.We combine Lemma 13 and Lemma 14 to Theorem 15.
Let C be a finite tensor category and M be a C -module. Then the assignments R • : G Z m ∈M G ( m, m ) and R • : H Z m ∈M H ( m, m ) (3.12)14 or G ∈ F un C|C ( M ⊠ M , C ) and H ∈ F un C|C ( M ⊠ M , C ) provide functors R • : F un C|C ( M ⊠ M , C ) → Z ( C ) and R • : F un C|C ( M ⊠ M , C ) → Z ( C ) , (3.13) respectively. We call the functors (3.13) the central integration functors for the C -module M . Let C be a finite tensor category and M and N be left C -modules. It is well known that thecategory R ex C ( M , N ) of right exact module functors is a finite category, and in fact has anatural structure of a finite Z ( C )-module as follows: For ( c , β ) ∈ Z ( C ) and F ∈ R ex C ( M , N )the functor M −→ N ,m c . F ( m ) . (4.1)is again right exact. Also, it acquires the structure of a C -module functor by the composition c . F ( c . m ) ∼ = −−→ c . ( c . F ( m )) ∼ = −−→ ( c ⊗ c ) . F ( m ) ( β ) c . F ( m ) −−−−−−−−→ ∼ = ( c ⊗ c ) . F ( m ) ∼ = −−→ c . ( c . F ( m )) , (4.2)where the first isomorphism is furnished by the module functor structure on F , and the secondand forth use the mixed associativity constraint for N . We write ( c , β ) . F ∈ R ex C ( M , N )for the module functor obtained this way from the object ( c , β ) ∈ Z ( C ) and the functor F ∈ R ex C ( M , N ). It is straightforward to check that this prescription endows the finite functorcategory R ex C ( M , N ) with the structure of a module over the finite tensor category Z ( C ). Definition 16.
Let C be a finite tensor category and let M and N be C -modules. Endow thefunctor category R ex C ( M , N ) with the structure of a module over the finite tensor category Z ( C ) as described above. Given G, H ∈ R ex C ( M , N ), we call the internal HomNat( G, H ) := Hom R ex C ( M , N ) ( G, H ) ∈ Z ( C ) (4.3)the object of internal natural transformations from G to H .Dually we set coNat( G, H ) := coHom R ex C ( M , N ) ( G, H ) ∈ Z ( C ) . (4.4) Remark 17.
The Yoneda lemma in the form of formula (2.6) allows one to express internalnatural transformations as a coend:Nat(
F, G ) = Z z ∈Z ( C ) Hom Z ( C ) ( z, Nat(
F, G )) ⊗ k z ∈ Z ( C ) (4.5)15r, equivalently, Nat( F, G ) = Z z ∈Z ( C ) Hom R ex C ( M , N ) ( z.F, G ) ⊗ k z ∈ Z ( C ) (4.6)by the adjunction defining the internal Hom (4.3). We denote the dinatural structure morphismsof the coend (4.6) by ı F,Gz : Hom R ex C ( M , N ) ( z.F, G ) ⊗ k z −→ Z z ′ ∈Z ( C ) Hom R ex C ( M , N ) ( z ′ .F, G ) ⊗ k z ′ . (4.7) An ordinary natural transformation is a family of morphisms. As a consequence the set ofordinary natural transformations between any two functors
G, H : C → D can (for C essentiallysmall) be expressed as an end:Nat( G, H ) = Z c ∈C Hom D ( G ( c ) , H ( c )) . (4.8)The structure morphisms j G,Hc : Z c ′ ∈C Hom D ( G ( c ′ ) , H ( c ′ )) → Hom D ( G ( c ) , H ( c )) (4.9)of this end are just the projections to the components of the natural transformation. As aspecial case, the natural transformations of the identity functor of a category C give the center End(Id C ) = R c ∈C Hom C ( c, c ) of C . The latter provides a Morita invariant formulation of thecenter Z ( A ) of an algebra A , according to End(Id A -mod ) ∼ = Z ( A ).We are now going to show that the results of the previous subsection allow us to expressinternal natural transformations as an end as well. We start by noticing that in situationswhich involve module categories, it can be rewarding to replace morphism sets (or rather,morphism spaces) by internal Homs – the relative Serre functors introduced in Definition 7provide an illustrative example. It is thus natural to consider for any pair G, H : M → N ofmodule functors the end Nat ′ ( G, H ) := Z m ∈M Hom N ( G ( m ) , H ( m )) . (4.10)We denote the members of the dinatural family for this end by j G,Hm : Z m ′ ∈M Hom N ( G ( m ′ ) , H ( m ′ )) −→ Hom N ( G ( m ) , H ( m )) . (4.11)Similarly to the situation for ordinary natural transformations, the morphism j G,Hm in C playsthe role of projecting to the m th ‘component’ Hom N ( G ( m ) , H ( m )) of the object Nat ′ ( G, H ).Dually, we have for
G, H : M → N the coendcoNat ′ ( G, H ) = Z m ∈M coHom N ( G ( m ) , H ( m )) . (4.12)16e denote its dinatural family by b G,Hm : coHom N ( G ( m ) , H ( m )) −→ Z m ′ ∈M coHom N ( G ( m ′ ) , H ( m ′ )) . (4.13)A crucial observation is now that, since the internal Hom and internal coHom are bimodulefunctors, by Theorem 15 we can (and will) for G, H ∈ R ex C ( M , N ) regard the objects (4.10)and (4.12) as objects in Z ( C ). (It should be appreciated, though, that the structure morphisms j G,Hm and b G,Hm are morphisms in C and not in Z ( C ).) We are then ready to state Theorem 18.
Let C be a finite tensor category and M and N be finite C -modules. (i) The end
Nat ′ ( G, H ) ∈ Z ( C ) is canonically isomorphic to the internal natural transforma-tions: Nat(
F, G ) = Z m ∈M Hom( F ( m ) , G ( m )) . (4.14)(ii) Analogously, the internal coHom (4.4) is canonically isomorphic to a coend: coNat(
F, G ) = Z m ∈M coHom( F ( m ) , G ( m )) . (4.15) Proof.
We prove (i); the proof of (ii) is dual. By the adjunction that defines the internal HomNat(
F, G ), proving (i) is equivalent to showing the adjunctionHom Z ( C ) ( z, Nat ′Z ( C ) ( G, H )) ∼ = Hom R ex C ( M , N ) ( z.G, H ) (4.16)for G, H ∈ R ex C ( M , N ) and z ∈ Z ( C ). Writing z = ( c , β ) with c ∈ C and β a half-braidingon c , the right hand side of (4.16) can be described as follows. The data characterizing amodule natural transformation from z.G to H are a family(D) : (cid:0) η m : c .G ( m ) → H ( m ) (cid:1) m ∈M (4.17)of morphisms in N , indexed by elements in m ∈ M , and subject to two types of conditions:(C1) Naturality: For every morphism m f −→ m ′ in M the diagram c . G ( m ) H ( m ) c . G ( m ′ ) H ( m ′ ) η m c . G ( f ) H ( f ) η m ′ (4.18)in N commutes.(C2) Module natural transformation: With φ H the datum turning H into a module functor,for every c ∈ C and m ∈ M the diagram c . G ( c . m ) H ( c . m )( c ⊗ c ) . G ( m ) c . H ( m ) Φ c .G η c.m φ H c.η m (4.19)in N commutes, where the morphism Φ c .G is the module functor datum φ G for G , followedby a half-braiding. 17n element of the morphism space on the left hand side of (4.16) is a morphism e η : ( c , β ) → Nat ′Z ( C ) ( G, H ) (4.20)to an end in C . After post-composing with the structure morphisms j G,Hm of that end, thisamounts to a dinatural family of morphisms in C , labeled by objects m ∈ M . The data of thisfamily are (D ′ ) : (cid:0)e η m := j G,Hm ◦ e η : c → Hom( G ( m ) , H ( m ) (cid:1) m ∈M , (4.21)and they are subject to two constraints:(C1 ′ ) Dinaturality: For every morphism m f −→ m ′ in M the diagram c Hom( G ( m ) , H ( m ))Hom( G ( m ′ ) , H ( m ′ )) Hom( G ( m ) , H ( m ′ )) e η m e η m ′ Hom( G ( m ) ,H ( f ))Hom( G ( f ) ,H ( m ′ )) (4.22)in C commutes.(C2 ′ ) Compatibility with the half-braiding: For every c ∈ C , the diagram c ⊗ c Nat ′ ( G, H ) ⊗ cc ⊗ c c ⊗ Nat ′ ( G, H ) β − ˜ η c ⊗ cc ⊗ ˜ η c (4.23)commutes, where the right downwards arrow is the component at c of the distinguishedhalf-braiding on Nat ′ ( G, H ) ∈ Z ( C ).To compare the two sides of (4.16), first notice that the adjunction defining the internal Homfor the C -module N gives natural isomorphismsHom N ( c . G ( m ) , H ( m )) ∼ = −−→ Hom C ( c , Hom( G ( m ) , H ( m )) (4.24)for all m ∈ M . This adjunction maps data of type (D) to data of type (D ′ ). We are going toshow that also the respective conditions on these data are mapped to each other.Thus consider condition (C1), i.e. the equality H ( f ) ◦ η m = η m ′ ◦ c . G ( f ) of morphisms inHom N ( c . G ( m ) , H ( m ′ )) for all morphisms m f −→ m ′ . By definition of Hom( G ( m ) , H ( f )), theinternal Hom adjunction maps G ( f ) ◦ η m to Hom( G ( m ) , H ( f )) ◦ e η m with e η m is the image of η m under the adjunction (4.24). A similar argument applies to pre-composition, showing that η m ′ ◦ c . G ( f ) is mapped by (4.24) to Hom( G ( f ) , H ( m ′ ) ◦ e η m . Together it follows that indeedcondition (C1) is mapped to condition (C1 ′ ), so that (C1) ⇔ (C1 ′ ).Next we pick an object m ∈ M and post-compose the two composite morphisms in the com-muting diagram (C2 ′ ) with the canonical morphism c ⊗ j G,Hm of the end, thereby obtaining18orphisms in Hom C ( c ⊗ c, c ⊗ Hom( G ( m ) , H ( m )). Now take the upper-right composite mor-phism in (4.23). We can use the right dual of c to consider equivalently a morphism c e η −−→ Nat ′ ( G, H ) −→ c ⊗ Hom( G ( m ) , H ( m )) ⊗ c ∨ . (4.25)Here the second morphism can be recognized as the one we used to get the structure of acomodule over the central comonad on Nat ′ ( G, H ). Hence the morphism (4.25) can be writtenas c e η −−→ Nat ′ ( G, H ) j G,Hc.m −−−−→
Hom( G ( c . m ) , H ( c . m )) Hom(( φ G ) − ,φ H ) −−−−−−−−−−−→ c ⊗ Hom( G ( m ) , H ( m )) ⊗ c ∨ (4.26)(here we suppress the bimodule functor structure of Hom). By the definition of e η m , thismorphism is nothing but c e η c.m −−−→ Hom( G ( c . m ) , H ( c . m )) Hom(( φ G ) − ,φ H ) −−−−−−−−−−−→ c ⊗ Hom( G ( m ) , H ( m )) ⊗ c ∨ . (4.27)Under the internal-Hom adjunction this morphism is mapped to c . ( c . G ( m )) c . ( φ G ) − −−−−−−−→ c . G ( c . m ) η c.m −−−→ H ( c . m ) Φ H −−−→ c . H ( m ) . (4.28)Applying similar arguments to the lower-left composite morphism in (4.23) gives the morphism c −→ c ⊗ c ⊗ c ∨ c ⊗ e η ⊗ c ∨ −−−−−→ c ⊗ Nat ′ ( G, H ) ⊗ c ∨ −→ c ⊗ Hom( G ( m ) , H ( m )) ⊗ c ∨ , (4.29)where the first morphism is obtained by combining the half-braiding β with the coevaluationof c . This is nothing but c −→ c ⊗ c ⊗ c ∨ c ⊗ e η m ⊗ c ∨ −−−−−−→ c ⊗ Hom( G ( m ) , H ( m )) ⊗ c ∨ , (4.30)and is thus under the internal-Hom adjunction mapped to the morphism c . ( c . G ( m )) ∼ = ( c ⊗ c ) . G ( m ) β .G ( m ) −−−−−−→ ( c ⊗ c ) . G ( m ) c.η m −−−→ c . H ( m ) . (4.31)Recalling the definition of Φ c .G in terms of φ G and the half-braiding β , we see that equalityof the morphisms (4.28) and (4.31) is precisely the commuting diagram (C2). Thus we haveestablished also the equivalence (C2) ⇔ (C2 ′ ).It is instructive to express the situation considered in the proof of Theorem 18 schematically:We have a commuting diagramHom Z ( C ) (( c , β ) , Nat ′Z ( C ) ( F, G )) Hom R ex C ( M , N ) (( c , β ) .F, G ) Q m Hom C ( c , Hom( F ( m ) , G ( m ))) Q m Hom N ( c .F ( m ) , G ( m )) (4.16) ∼ = ∼ = (4.32)Here the left downwards arrow is post-composition by the structure morphisms of the end (4.10)i.e. maps f to j F,Gm ◦ f for some m ∈ M . The right downwards arrow comes from the fact that anatural transformation is a family of morphisms. The lower horizontal arrow is component-wisethe internal Hom adjunction. 19 xample 19. For N = M and G = H = Id M , the object of internal natural transformations isF M := Nat(Id M , Id M ) = Z m ∈M Hom M ( m, m ) ∈ Z ( C ) . (4.33)In particular, for C as a module category over itself, this isF C = Z c ∈C Hom C ( c, c ) = Z c ∈C c ⊗ c ∨ ∈ Z ( C ) . (4.34)If C is semisimple, this is the object L i x i ⊗ x ∨ i , with the summation being over the finitely manyisomorphism classes of simple objects, and with half-braiding as given e.g. in [BK, Thm. 2.3].It is natural to refer to the object F M in Z ( C ) as the center of the C -module M . Remark 20.
Given a functor F ∈ R ex C ( M , N ), define a functor L F : Z ( C ) −→ R ex C ( M , N ) ,z z . F (4.35)In the special case that M = N = C C is C seen as a module over itself, we have a canonicalidentification R ex C ( C , C ) ∼ = C , under which L Id C : Z ( C ) → C is the forgetful functor. It followsfrom the proof of Theorem 18 that the right adjoint of L F is the functor R F : R ex C ( M , N ) −→ Z ( C ) ,G R m ∈M Hom N ( F ( m ) , G ( m )) = Nat( F, G ) . (4.36)In the special case M = N = C C as well as F = Id C the functor (4.36) is given by c Z c ∈C Hom( c, c ⊗ c ) ∼ = Z c ∈C c ⊗ c ⊗ c ∨ , (4.37)where we use Remark 5 and the central comonad (2.5). In other words, the adjunction (4.16)can be regarded as a generalization of the adjunction that defines the central comonad.An interesting consequence of Theorem 18 is obtained when combining it with the fact [Sc1]that, for C a finite tensor category and M a C -module, there is an explicit braided equivalence θ M : Z ( C ) → Z ( C ⋆ M ) . (4.38)Let us compute the object θ M (Nat( F, G )) ∈ Z ( C ⋆ M )) for F, G ∈ R ex C ( M , M ). To this end weuse the fact that a right exact functor admits a right adjoint, and that the right adjoint of theforgetful functor U M : Z ( C ⋆ M ) → C ⋆ M , which is exact, is the coinduction functor associated withthe central comonad on C ⋆ M . Proposition 21.
Let C be a finite tensor category and M a C -module. For F, G ∈ R ex C ( M , M ) we have θ M (Nat( F, G )) ∼ = e I ( G ◦ F r . a . ) ∈ Z ( C ⋆ M ) , (4.39) where F r . a . is the right adjoint of F and e I : C ⋆ M −−→ Z ( C ⋆ M ) ϕ R ψ ∈C ⋆ M ψ r . a . ◦ ϕ ◦ ψ (4.40) is the right adjoint of the forgetful functor U M . roof. Applying the composition of θ M with the forgetful functor U M : Z ( C ⋆ M ) → C ⋆ M to theobject ( c , β ) ∈ Z ( C ) of the Drinfeld center gives the C -module endofunctor ( c , β ) . Id M , whichas a functor is given by acting with c and has a module functor structure given by β . Pre-composing with F ∈ R ex C ( M , M ) yields( U M ◦ θ M )( c , β ) ◦ F = L F ( c , β ) , (4.41)with L F as introduced in Remark 20. By the adjunction in Remark 20 we thus haveHom C ⋆ M (( U M ◦ θ M )( c , β ) ◦ F, G ) ∼ = Hom Z ( C ) (( c , β ) , R F ( G ))= Hom Z ( C ) (( c , β ) , Nat(
F, G )) ∼ = Hom Z ( C ⋆ M ) ( θ M ( c , β ) , θ M (Nat( F, G ))) , (4.42)where the last isomorphism holds because θ M is an equivalence. It follows that for all ϕ ∈ Z ( C ⋆ M )we have Hom Z ( C ⋆ M ) ( ϕ, θ M (Nat( F, G ))) ∼ = Hom C ⋆ M ( U M ( ϕ ) ◦ F, G ) ∼ = Hom C ⋆ M ( U M ( ϕ ) , G ◦ F r . a . ) ∼ = Hom Z ( C ⋆ M ) ( ϕ, e I ( G ◦ F r . a . )) . (4.43)Here the first isomorphism uses the definition (4.36) of R F together with the adjunction wejust derived (and again the fact that θ M is an equivalence). Remark 22.
Specifically for the case that F = G = Id M , we find that θ M (Nat(Id M , Id M )) ∼ = Z ψ ∈C ⋆ M ψ r . a . ◦ ψ . (4.44)Comparing this formula with (4.34), we can rephrase this by saying that after application ofSchauenburg’s equivalence θ M , the center of any module category is diagonal. (In the applicationto two-dimensional conformal field theory alluded to at the end of the Introduction, this impliesthat the bulk state space of any full conformal field theory becomes diagonal when regardednot as an object of Z ( C ), but as an object in the equivalent category Z ( C ⋆ M ).) Ordinary natural transformations can be composed horizontally as well as vertically. Bothcompositions are conveniently described component-wise. A vertical composition of internal natural transformations clearly exists, being just a particular instance the multiplication ofinternal Homs. In this subsection we introduce in addition a horizontal composition of internalnatural transformations. We also describe their vertical composition from a different perspec-tive. As we will see, these compositions are again naturally formulated in terms of components.Indeed, the constructions can largely be performed in analogy with those for ordinary naturaltransformations, including an Eckmann-Hilton argument.We start by observing that for the Z ( C )-module R ex C ( M , N ) the natural evaluation ev(2.15) of internal Homs, which is used to obtain their multiplication, is a natural transformationev F,G : Nat(
F, G ) . F → G (4.45)21etween module functors in R ex C ( M , N ). Under the defining adjunction isomorphism of theinternal Hom, ev F,G is induced from the identity morphism on Nat(
F, G ). Owing to Nat ∼ = Nat ′ the latter is a morphism whose codomain is an end, so that it can be described as a dinaturalfamily (cid:0) Nat(
F, G ) → Hom( F ( m ) , G ( m )) (cid:1) m ∈M , and this family is just the structure morphismof the end. Thus the components of the evaluation ev F,G are just the images in (cid:0)
Nat(
F, G ) .F ( m ) → G ( m ) (cid:1) m ∈M (4.46)of the structure morphisms of the end under the internal Hom adjunction for the modulecategory M over C . The associative multiplication of internal Homs Nat( − , − ) is a family ofmorphisms µ ver ≡ µ ver ( G , G , G ) : Nat( G , G ) ⊗ Nat( G , G ) −→ Nat( G , G ) (4.47)in Z ( C ) obeying the standard associativity condition, and as described in (2.17) we have unitsid F ∈ Hom Z ( C ) ( Z ( C ) , Nat(
F, F )) . (4.48) Definition 23.
Let M and N be C -modules and G , G , G ∈ R ex C ( M , N ). The morphism µ ver from Nat( G , G ) ⊗ Nat( G , G ) to Nat( G , G ) that is introduced in (4.47) is called the vertical composition of internal natural transformations.The following result justifies this terminology: Proposition 24.
Let M and N be C -modules and G , G , G ∈ R ex C ( M , N ) . Consider for m ∈ M the composition α m : Nat( G , G ) ⊗ Nat( G , G ) j G ,G m ⊗ j G ,G m −−−−−−−−−→ Hom N ( G ( m ) , G ( m )) ⊗ Hom N ( G ( m ) , G ( m )) µ G m ) ,G m ) ,G m ) −−−−−−−−−−−−→ Hom N ( G ( m ) , G ( m )) . (4.49) We have α m = j G ,G m ◦ µ ver (4.50) for any m ∈ M .Moreover, for any F ∈ R ex C ( M , N ) the unit id F ∈ Hom Z ( C ) ( Z ( C ) , Nat(
F, F )) satisfies j F,Fm ◦ id F = id F ( m ) ∈ Hom C (cid:0) , Hom N ( F ( m ) , F ( m )) (cid:1) . (4.51) for all m ∈ M Put differently, when thinking about the structure morphisms j G,G ′ m (4.11) of the end (4.10)as projections to components, the composition µ ver of internal natural transformations is noth-ing but the ordinary vertical composition of natural transformations. Proof.
Under the adjunction, the image in Hom R ex C ( M , N ) (Nat( G , G ) ⊗ Nat( G , G ) .G , G ) of µ ver ( G , G , G ) ∈ Hom Z ( C ) (Nat( G , G ) ⊗ Nat( G , G ) , Nat( G , G ) is given, by the definitionof composition of internal Homs, by the compositeNat( G , G ) ⊗ Nat( G , G ) . G ⊗ ev −−−−→ Nat( G , G ) . G −−→ G (4.52)22we suppress the mixed associator of the module category N ). The components of this modulenatural transformation are of the form appearing in the lower right corner of the diagram (4.32).On the other hand, the morphism α m is of the type that appears in the lower left corner ofthat diagram. We must show that they are related by applying the internal-Hom adjunctionto each component in the direct product. But this is indeed the case because, as noted above(see (4.46)), the evaluation is related to the structure maps of the end by the internal-Homadjunction. Hence the claim follows.The assertion about the unit id F follows directly from the definitions: By the adjunction (4.16)it is related to id F ∈ Hom R ex C ( M , N ) ( . F, F ). In terms of the diagram (4.32), we have theidentity morphism in the component Hom N ( . F ( m ) , F ( m )) of the natural isomorphism forevery m ∈ M .Next we note that a k -linear category D can be seen as a module category over the monoidalcategory vect k . For a vect k -module the Hom and internal Hom coincide, so that the internal-Hom adjunction gives, for each pair d, d ′ ∈ D of objects, a natural evaluationev k d,d ′ : Hom D ( d, d ′ ) ⊗ k d −→ d ′ (4.53)as the image of the identity map under the linear isomorphismHom D (Hom D ( d, d ′ ) ⊗ k d, d ′ ) ∼ = −−→ Hom D ( d, d ′ ) ∗ ⊗ k Hom D ( d, d ′ ) . (4.54)It should be appreciated that the composition of Hom D as an internal Hom for D as a vect k -module and the ordinary composition of morphisms in D coincide.This observation allows us to give the following convenient description of the evaluationev F,G : Nat(
F, G ) . F → G in (4.45): Invoking from Remark 17 the expression (4.6) for Nat( F, G ),we getev
F,G = Z z ∈Z ( C ) ev k z.F,G : (cid:16) Z z ∈Z ( C ) Hom R ex C ( M , N ) ( z.F, G ) ⊗ k z (cid:17) . F = Z z ∈Z ( C ) Hom R ex C ( M , N ) ( z.F, G ) ⊗ k z.F −→ G . (4.55)Here in the equality we use that the action functor is exact, and by R z ∈Z ( C ) ev k z.F,G we denote themorphism out of the coend R z ∈Z ( C ) Hom R ex C ( M , N ) ( z.F, G ) ⊗ k z.F that is defined by the family (cid:0) ev k z.F,G : Hom R ex C ( M , N ) ( z.F, G ) ⊗ k z.F → G (cid:1) z ∈Z ( C ) which is dinatural in z ∈ Z ( C ).Next we define modified vertical and horizontal compositions of relative natural To set thestage for the horizontal composition, we formulate Lemma 25.
Let M and N be module categories over a finite tensor category C , and let F, G ∈ R ex C ( M , N ). Then for any pair z, z ′ of objects in Z ( C ) the module functor constraintof G induces an isomorphism( z . G ) ◦ ( z ′ . F ) ∼ = −−→ ( z ⊗ Z ( C ) z ′ ) . ( G ◦ F ) (4.56)of C -module functors. 23 roof. For any m ∈ M there is an isomorphism( z . G ) ◦ ( z ′ . F )( m ) = z . G ( z ′ .F ( m )) ∼ = −−→ ( z ⊗ Z ( C ) z ′ ) . ( G ◦ F )( m ) (4.57)of functors. Due to the fact that the braiding is natural and thus compatible with the modulefunctor datum φ G , this is even an isomorphism of C -module functors.This result allows us to give Definition 26. (i) Let M and N be finite module categories over a finite tensor category C . For any triple F, G, H ∈ R ex C ( M , N ) the modified vertical composition g µ ver : Hom R ex C ( M , N ) ( z.G, H ) ⊗ k Hom R ex C ( M , N ) ( z ′ .F, G ) −→ Hom R ex C ( M , N ) (( z ⊗ z ′ ) . F, H ) (4.58)of natural transformations is defined by g µ ver ( β, α ) : ( z ⊗ z ′ ) . F ∼ = −−→ z . ( z ′ . F ) z.α −−−→ z . G β −−→ H (4.59)for z, z ′ ∈ Z ( C ).(ii) Let M , M and M be finite module categories over a finite tensor category C . For z, z ′ ∈ Z ( C ), F, F ′ ∈ R ex C ( M , M ) and G, G ′ ∈ R ex C ( M , M ) the modified horizontalcomposition g µ hor : Hom R ex C ( M , M ) ( z.G, G ′ ) ⊗ k Hom R ex C ( M , M ) ( z ′ .F, F ′ ) −→ Hom R ex C ( M , M ) (( z ⊗ z ′ ) . G ◦ F, G ′ ◦ F ′ ) (4.60)of natural transformations is defined to be the composition of the ordinary horizonalcomposition with the isomorphism given in Lemma 25. Remark 27.
Admittedly, the modified vertical composition g µ ver looks somewhat unnatural.But it should be appreciated that by using the duality we can identifyHom R ex C ( M , N ) ( z . G, H ) ∼ = Hom R ex C ( M , N ) ( G, z ∨ . H ) . (4.61)Upon this identification, g µ ver just becomes the ordinary vertical composition. Remark 28.
In the special case G = F and z ′ = Z ( C ) , the identity natural transformation inHom R ex C ( M , M ) ( Z ( C ) . F, F ) is a unit e η ver F for the modified vertical composition g µ ver .Similarly, for M = M , G = G ′ = Id M and z = Z ( C ) , the identity natural transformation inHom R ex C ( M , M ) ( Z ( C ) . Id M , Id M ) is a unit e η hor M for the modified horizontal composition g µ hor .It should be appreciated that whenever both units are defined, they are the same, e η hor M = e η verid M .24 roposition 29. Let M and N be finite module categories over a finite tensor category C , andlet F, G, H ∈ R ex C ( M , N ) . The vertical composition µ ver : Nat( G, H ) ⊗ Nat(
F, G ) → Nat(
F, H ) defined in (4.47) is the morphism out of the coend Nat(
G, H ) ⊗ Nat(
F, G )= Z z ∈Z ( C ) Hom R ex C ( M , N ) ( z.G, H ) ⊗ k z ⊗ Z ( C ) Z z ′ ∈Z ( C ) Hom R ex C ( M , N ) ( z ′ .F, G ) ⊗ k z ′ ∼ = Z z , z ′ ∈Z ( C ) Hom R ex C ( M , N ) ( z.G, H ) ⊗ k Hom R ex C ( M , N ) ( z ′ .F, G ) ⊗ k ( z ⊗ Z ( C ) z ′ ) (4.62) that is given by the family Hom R ex C ( M , N ) ( z.G, H ) ⊗ k Hom R ex C ( M , N ) ( z ′ .F, G ) ⊗ k ( z ⊗ Z ( C ) z ′ ) g µ ver −−−−→ Hom R ex C ( M , N ) ( z ⊗ Z ( C ) z ′ ) . F, H ) ⊗ k ( z ⊗ Z ( C ) z ′ ) ı F,Hz ⊗ z ′ −−−−→ Z ζ ∈Z ( C ) Hom R ex C ( M , N ) ( ζ . F, H ) ⊗ k ζ = Nat( F, H ) (4.63) of morphisms in Z ( C ) ( with ı F,H the dinatural family defined in (4.7)) which is dinatural in z, z ′ ∈ Z ( C ) .Proof. We write b µ ver for the morphism out of the coend (4.62) that is defined by (4.63). Wehave to compare the morphisms (cid:0) Nat(
G, H ) ⊗ Nat(
F, G ) (cid:1) . F b µ ver . id F −−−−−−→ Nat(
F, H ) . F ev −−→ H (4.64)and Nat( G, H ) . (cid:0) Nat(
F, G ) . F (cid:1) id . ev −−−−→ Nat(
G, H ) . G ev −−→ H . (4.65)Now the morphism (4.64) can be expressed in terms of the familyHom R ex C ( M , N ) ( z.G, H ) ⊗ k Hom R ex C ( M , N ) ( z ′ .F, G ) ⊗ k ( z ⊗ Z ( C ) z ′ ) . F g µ ver . id F −−−−−−→ Hom R ex C ( M , N ) (( z ⊗ Z ( C ) z ′ ) . F, H ) ⊗ k ( z ⊗ Z ( C ) z ′ . F ) ev ( z ⊗ z ′ ) .F,H −−−−−−−−→ H (4.66)that is dinatural in z, z ′ ∈ Z ( C ), while the morphism (4.65) is described by the familyHom R ex C ( M , N ) ( z.G, H ) ⊗ k z ⊗ Z ( C ) Hom R ex C ( M , N ) ( z ′ .F, G ) ⊗ k z ′ . F id ⊗ ev z ′ .F,G −−−−−−−−→ Hom R ex C ( M , N ) ( z.G, H ) ⊗ k z . G ev z.G,H −−−−−→ H . (4.67)Since upon use of dualities, g µ ver boils down to the ordinary vertical composition (see Remark27), the two composites (4.66) and (4.67) coincide. It thus follows that b µ ver indeed describesthe vertical composition µ ver of internal natural transformations. Remark 30.
We describe again the unit of the vertical composition. It is the morphism inHom Z ( C ) ( Z ( C ) , Nat(
F, F )) that is given by the composite Z ( C ) −−→ Hom R ex C ( M , N ) ( Z ( C ) . F, F ) ⊗ k Z ( C ) ı F,F Z ( C ) −−−−→ Z z ∈Z ( C ) Hom R ex C ( M , N ) ( z . F, F ) ⊗ k z , (4.68)25here the first morphism is determined by the unit e η ver F ∈ Hom R ex C ( M , N ) ( . F, F ), while thesecond is the structure morphism of the coend.We are now in a position to introduce also a horizontal composition of internal naturaltransformations: Definition 31.
Let M , M and M be finite module categories over a finite tensor category C , and let F, F ′ ∈ R ex C ( M , M ) and G, G ′ ∈ R ex C ( M , M ). The horizontal composition Nat R ex C ( M , M ) ( G, G ′ ) ⊗ Nat R ex C ( M , M ) ( F, F ′ ) → Nat R ex C ( M , M ) ( G ◦ F, G ′ ◦ F ′ ) (4.69)is the morphism µ hor out of the coendNat( G, G ′ ) ⊗ Nat(
F, F ′ )= Z z ∈Z ( C ) Hom R ex C ( M , M ) ( z.G, G ′ ) ⊗ k z ⊗ Z ( C ) Z z ′ ∈Z ( C ) Hom R ex C ( M , M ) ( z ′ .F, F ′ ) ⊗ k z ′ ∼ = Z z , z ′ ∈Z ( C ) Hom R ex C ( M , M ) ( z.G, G ′ ) ⊗ k Hom R ex C ( M , M ) ( z ′ .F, F ′ ) ⊗ k ( z ⊗ Z ( C ) z ′ ) (4.70)that is given by the familyHom R ex C ( M , M ) ( z.G, G ′ ) ⊗ k Hom R ex C ( M , M ) ( z ′ .F, F ′ ) ⊗ k ( z ⊗ Z ( C ) z ′ ) g µ hor −−−−→ Hom R ex C ( M , M ) (cid:0) ( z ⊗ Z ( C ) z ′ ) . ( G ◦ F ) , G ′ ◦ F ′ (cid:1) ⊗ k ( z ⊗ Z ( C ) z ′ ) ı G ◦ F,G ′◦ F ′ z ⊗ z ′ −−−−−−→ Z ζ ∈Z ( C ) Hom R ex C ( M , M ) ( ζ . ( G ◦ F ) , G ′ ◦ F ′ ) ⊗ k ζ = Nat( G ◦ F, G ′ ◦ F ′ ) (4.71)of morphisms in Z ( C ), which is dinatural in z, z ′ ∈ Z ( C ).The so defined horizontal composition µ hor and the vertical composition µ ver satisfy theEckmann-Hilton property: Proposition 32.
Let M , M and M be finite module categories over a finite tensor category C . Then for any two triples F, G, H ∈ R ex C ( M , M ) and F ′ , G ′ , H ′ ∈ R ex C ( M , M ) , thediagram Nat( G ′ , H ′ ) ⊗ Nat(
G, H ) ⊗ Nat( F ′ , G ′ ) ⊗ Nat(
F, G ) Nat( G ′ , H ′ ) ⊗ Nat( F ′ , G ′ ) ⊗ Nat(
G, H ) ⊗ Nat(
F, G )Nat( G ′ ◦ G, H ′ ◦ H ) ⊗ Nat( F ′ ◦ F, G ′ ◦ G ) Nat( F ′ , H ′ ) ⊗ Nat(
F, H )Nat( F ′ ◦ F, H ′ ◦ H ) µ hor ⊗ µ hor µ ver ⊗ µ ver µ ver µ hor (4.72) commutes. Here the horizontal arrow is the braiding in Z ( C ) . roof. When describing all horizontal and vertical compositions in the diagram in terms of di-natural families analogous to (4.63) and (4.71), the statement follows directly from the standardproperties of vertical and horizontal compositions of module natural transformations.As usual, the Eckmann-Hilton property allows one to derive commutativity.
Corollary 33.
Let M be a finite module category over a finite tensor category C . Then thealgebra Nat(Id M , Id M ) in Z ( C ) is braided commutative.Proof. Put all six functors in Proposition 32 to be the identity functor of M . Then comparisonwith Remark 28 tells us that the horizontal and vertical composition have the same unit. So far we have been dealing with finite tensor categories and their (bi)module categories. Interms of modular functors, these structures are naturally related to a framed modular functor[DSS,FSS2]. To have a relation with an oriented modular functor, additional algebraic structureis required, in particular the finite tensor categories should come with a pivotal structure. Asa consequence, the algebras arising as internal Ends will have the additional structure of aFrobenius algebra. In this section we study relative natural transformations in such a setting.Thus we now suppose that C is a pivotal finite tensor category and that M is an exact C -module. Without loss of generality we then further assume that C is strict pivotal, so thatthe Nakayama functor of M is an ordinary module functor and M = M =: M , see Remark6. In this situation, the central integration functors R • and R • appearing in Theorem 15 areboth functors from F un C|C ( M ⊠ M , C ) to Z ( C ). We now show that they are actually relatedby the module Eilenberg-Watts equivalences (2.26): Proposition 34.
Let C be a pivotal finite tensor category and M be an exact C -module. Thenthe two functors R • , R • : F un C|C ( M ⊠ M , C ) → Z ( C ) satisfy R • = R • ◦ (cid:0) Id M ⊠ N r M (cid:1) (5.1) with N r M the right exact Nakayama functor of M .Proof. Since M is exact, all module and bimodule functors with domain M ⊠ M are exact.For G ∈ F un C|C ( M ⊠ M , C ) we therefore have Z m ∈M G ( m ⊠ m ) ∼ = G (cid:0)R m ∈M m ⊠ m (cid:1) and Z m ∈M G ( m ⊠ m ) ∼ = G (cid:0)R m ∈M m ⊠ m (cid:1) (5.2)as isomorphisms of objects in C . Now for any object µ ∈ Z ( M ⊠ M ) with underlying object˙ µ ∈ M ⊠ M , the object G ( ˙ µ ) comes with a canonical half-braiding, given by G ( ˙ µ ) ⊗ c ∼ = G ( ˙ µ . c ) ∼ = G ( c . ˙ µ ) ∼ = c ⊗ G ( ˙ µ ) for c ∈ C . Moreover, the objects R m ∈M m ⊠ m and R m ∈M m ⊠ m of C areendowed with natural balancings [FSS1, Cor. 4.3], i.e. they are in fact objects of Z ( M ⊠ M ). Itfollows that the isomorphisms in (5.2) are even isomorphisms in Z ( C ). Analogous isomorphismsin Z ( C ) are valid when m ⊠ m is replaced by m ⊠ H ( m ) for any module endofuctor H .27ence it remains to show that there is an isomorphism Z m ∈M m ⊠ m ∼ = Z m ∈M m ⊠ N r M ( m ) (5.3)of objects in Z ( M ⊠ M ). That this is indeed the case follows from the two-sided adjointequivalences (2.26). Indeed, when considering the identity functor on M as a left exact functorwith trivial module structure, one has Φ l (Id M ) = R m ∈M m ⊠ m ∈ Z ( M ⊠ M ); the isomorphism(5.3) is then obtained by recalling the definition (2.9) of the Nakayama functor together withthe fact that the Eilenberg-Watts functors Φ r and Ψ r are quasi-inverses.Of particular interest to us is a statement that follows from Proposition 34 together withthe following result (for which C is not required to be pivotal): Lemma 35.
Let C be a finite tensor category and M and N be exact C -module categories. Let F, G : M → N be module functors. Then there is an isomorphism Z m ∈M coHom( F ( m ) , G ( m )) ∼ = Z m ∈M Hom(S l N ◦ F ( m ) , G ( m )) (5.4) as objects in Z ( C ) . In particular, the coends on both sides of this equality exist.Proof. For any pair of module functors
F, G : M → N , the two functors coHom(
F, G ) andHom(S l N ◦ F, G ) are bimodule functors from M ⊠ M to C . That they are actually bimodulefunctors follows from the properties of the internal Hom and coHom together with the twistedbimodule property of the left relative Serre functor [FSS1, Lemma 4.23] which is analogous to(2.29). By combining the defining property (2.28) of a left relative Serre functor and the relation(2.23) between internal Hom and coHom, we obtain, for any pair F, G of module functors, anisomorphism coHom(
F, G ) ∼ = Hom(S l N ◦ F, G ) (5.5)of bimodule functors. By Theorem 15, this isomorphism of functors implies an isomorphism oftheir coends (5.4) as objects in Z ( C ). Theorem 36.
Let C be a pivotal finite tensor category and M and N be exact C -modules.Then the functor category R ex C ( M , N ) is an exact module category over Z ( C ) .Proof. Applying Proposition 34 to the bimodule functor Hom(S l N ◦ F, G ) we obtain an isomor-phism Z m ∈M Hom(S l N ◦ F ( m ) , G ( m )) ∼ = Z m ∈M Hom(S l N ◦ F ( m ) , G ◦ N r M ( m )) (5.6)of objects in Z ( C ). The left and right Nakayama functors of a finite linear category form anadjoint pair [FSS1, Lemma 3.16]. Using that S l N = D − . N l N = D ∨ . N l N with D the distinguishedinvertible object of C (see Remark 8), it follows that there are isomorphismsHom(S l N ◦ F, G ◦ N r M ) ∼ = Hom(N l N ◦ F, D . G ◦ N r M ) ∼ = Hom( F, N r N ◦ D . G ◦ N r M ) (5.7)of functors. Combining this statement with Lemma 35 we find an isomorphism Z m ∈M coHom( F ( m ) , G ( m )) ∼ = Z m ∈M Hom( F ( m ) , N r N ◦ D . G ◦ N r M ( m )) (5.8)28s objects in Z ( C ). This means that the functor N r N ◦ ( D . − ) ◦ N r M is a relative Serre functorfor R ex C ( M , N ). This shows in particular that R ex C ( M , N ) is an exact module category over Z ( C ).We will use the notation S r R ex := N r N ◦ ( D . − ) ◦ N r M (5.9)for the relative Serre functor for R ex C ( M , N ) obtained in the proof. Remark 37. If C is unimodular, then S r R ex = N r N ◦ ( − ) ◦ N r M is the right Nakayama functorN r R ex ( M , N ) of the category R ex ( M , N ) of all functors from M to N [FSS1, Lemma 3.21].It is instructive to check explicitly that the functor (5.9) satisfies the following two propertieswhich befit the relative Serre functor of R ex C ( M , N ) (taking, for perspicuity, C just to bepivotal, rather than strict pivotal):(1) S r R ex is a twisted Z ( C )-module functor, i.e. there are coherent natural isomorphisms φ R exz,F : S r R ex ( z.F ) ∼ = −−→ z ∨∨ . S r R ex ( F ) (5.10)for z ∈ Z ( C ) and F ∈ R ex C ( M , N ).(2) S r R ex is an endofunctor of R ex C ( M , N ). That is, for F ∈ R ex C ( M , N ), the functor S r R ex ( F )is again in R ex C ( M , N ), i.e. is a (non-twisted) module functor: there are coherent naturalisomorphisms S r R ex ( F )( c . m ) ∼ = −−→ c . (cid:0) S r R ex ( F )( m ) (cid:1) (5.11)for c ∈ C and m ∈ M .Let us first check that the functor S r R ex sends C -module functors to C -module functors. For F ∈ R ex C ( M , N ), c ∈ C and m ∈ M we haveS r R ex ( F )( c . m ) ≡ N r N ◦ ( D . F ) ◦ N r M ( c . m ) ∼ = −−→ N r N ◦ ( D . F ) (cid:0) ∨∨ c . N r M ( m ) (cid:1) ∼ = −−→ N r N ◦ (cid:0) D . ( ∨∨ c . F ◦ N r M ( m ) (cid:1) ∼ = −−→ N r N (cid:0) ( D ⊗ ∨∨ c ) . F ◦ N r M ( m ) (cid:1) ∼ = −−→ N r N (cid:0) ( c ∨∨ ⊗ D ) . F ◦ N r M ( m ) (cid:1) ∼ = −−→ ∨∨ c ∨∨ . (cid:0) N r N ◦ ( D . F ) ◦ N r M ( m ) (cid:1) = c . (cid:0) N r N ◦ ( D . F ) ◦ N r M ( m ) (cid:1) ≡ c . (cid:0) S r R ex ( F )( m ) (cid:1) . (5.12)Here we first use that N r M is a twisted module functor, then that F is a module functor, thenthe module constraint, then the fact that by the Radford S -theorem the quadruple right dualof C is naturally isomorphic to conjugation by D , and finally that N r N is a twisted modulefunctor. The same type of calculation, only one step shorter, shows that S r R ex is a properlytwisted Z ( C )-module functor: For F ∈ R ex C ( M , N ) and z ∈ Z ( C ) we have (writing z = ( ˙ z, β ))S r R ex ( z . F ) ≡ N r N ◦ ( D . ( z . F )) ◦ N r M ∼ = −−→ N r N ◦ (cid:0) ( D ⊗ ˙ z ) . F (cid:1) ◦ N r M∼ = −−→ N r N ◦ ( ˙ z ∨∨∨∨ ⊗ D ) . F ) ◦ N r M∼ = −−→ ˙ z ∨∨ . (cid:0) N r N ◦ ( D . F ) ◦ N r M (cid:1) ≡ z ∨∨ . (cid:0) S r R ex ( F ) (cid:1) . (5.13)29o far, we have imposed the requirement of being pivotal on the finite tensor category C . Now assume further that the C -modules M and N are pivotal as well, with respectivepivotal structures π M and π N . The pivotal structure π M gives a family of isomorphisms π M m : m −→ S r M ( m ) = D . N r M ( m ) which are twisted module natural transformations, and analo-gously for π N . In particular, for any functor F ∈ R ex C ( M , N ) and any object m ∈ M we canform the composite F ( m ) F ( π M m ) −−−−−→ F ( D . N r M ( m )) π N F ( D. Nr M ( m )) −−−−−−−−−→ D . N r N ◦ F ◦ ( D . N r M )( m ) ∼ = D ⊗ ∨∨ D. N r N ◦ F ◦ N r M ( m ) . (5.14)In case C is unimodular , i.e. D = , this gives us a family of isomorphisms π N F ◦ N r M ( m ) ◦ F ( π M m ) : F ( m ) ∼ = −−→ N r N ◦ F ◦ N r M ( m ) = S r R ex ( F )( m ) (5.15)which provides an isomorphism Id R ex C ( M , N ) ∼ = −−→ S r R ex (5.16)of endofunctors of R ex C ( M , N ). We then arrive at Corollary 38.
Let C be a unimodular pivotal finite tensor category and M and N be pivotalmodule categories over C . Then the functor category R ex C ( M , N ) has a structure of a pivotalmodule category over the pivotal finite tensor category Z ( C ) .Proof. By unimodularity of C we have N r M = S r M and N r N = S r N . As in the proof of Proposition34 let us take, without loss of generality, the pivotal structure of C to be strict. Then upon usingthe isomorphisms S r M → Id M and S r N → Id N that come from the pivotal structure on M and N , respectively, to trivialize the relative Serre and thus Nakayama functors of M and N , theconsistency condition that according to Definition 9 has to be met in order for the isomorphism(5.16) to be a pivotal structure on R ex C ( M , N ) reduces to a combination of identities andis thus trivially satisfied. We refrain from spelling out the explicit form that the consistencycondition takes after inserting the unique isomorphisms involved in those trivializations, as itdoes not add any further insight.By combining Corollary 38 with Theorem 12 we further get Corollary 39.
Let C be a unimodular pivotal finite tensor category, let M and N be exact C -modules with pivotal structures, and let F : M → N a module functor. Then the algebra
Nat(
F, F ) has a natural structure of a symmetric Frobenius algebra in the Drinfeld center Z ( C ) .In particular, Nat(Id M , Id M ) has a natural structure of a commutative symmetric Frobeniusalgebra. Acknowledgements:
JF is supported by VR under project no. 2017-03836. CS is partially supported by the RTG1670 “Mathematics inspired by String theory and Quantum Field Theory” and by the DeutscheForschungsgemeinschaft (DFG, German Research Foundation) under Germany’s ExcellenceStrategy - EXC 2121 “Quantum Universe” - 390833306.30 eferences [AM] N. Andruskiewitsch and J.M. Mombelli,
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