INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER

For ℳ and N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} finite module categories over a finite tensor category C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document}, the category ℛexC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$\end{document}(ℳ, N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document}) of right exact module functors is a finite module category over the Drinfeld center Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{Z} $$\end{document}(C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document}). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document}-C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document}-bimodule functors to objects of Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{Z} $$\end{document}(C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document}), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} are exact C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document}-modules and C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document} is pivotal, then the Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{Z} $$\end{document}(C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document})-module ℛexC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$\end{document}(ℳ, N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document}) is exact. We compute its relative Serre functor and show that if ℳ and N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} are even pivotal module categories, then ℛexC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$\end{document}(ℳ, N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document}) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{Z} $$\end{document}(C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document}).


Introduction
Module categories over monoidal categories have been a prominent topic in representation theory in the past two decades.The theory is particularly well-developed for finite tensor categories and their finite module and bimodule categories.Indeed, many notions and results in the theory of finite-dimensional representations over finite-dimensional Hopf algebras have found their natural conceptual home in this setting.Examples of such notions include the unimodularity of a finite tensor category and factorizability of a braided finite tensor category.Results include Radford's S 4 -formula [ENO], including its generalization to bimodule categories [FSS1], the equivalence of various characterizations of the non-degeneracy of a braiding on a finite 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 twodimensional 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 category and M and N be finite C-modules.Then the category Rex C (M, N ) of right exact module functors 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 ∈ Rex C (M, N ).We denote these 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, the Yoneda lemma implies a useful formula in terms of an end over morphism spaces: (1.1) The structure morphisms Nat(G, H) → Hom N (G(m), H(m)) of this end just give the components of the natural transformation.One of the main results of this paper, Theorem 18, is a similar expression Nat(F, G) = 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 the end 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 given by ) 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 these functors, which we call central integration functors, is shown in Theorem 15.
Since the internal natural transformations are internal Homs, they come with associative compositions.It follows in particular that for any module functor F the object Nat(F, F ) has a natural structure of a unital associative algebra in Z(C).We show that the structure morphisms Nat(G, H) → Hom N (G(m), H(m)) of the end behave in the same way as the component maps of an ordinary natural transformation.This allows us to define horizontal and vertical compositions which obey the Eckmann-Hilton relation.As a consequence, the object Nat(Id M , Id M ) of internal natural endotransformations of the identity functor is a commutative algebra in the braided 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 related by the Nakayama functor N r M ∈ Rex(M, M) according to (1.5) Based on this result we show in Theorem 36 that for any pair M 1 , M 2 of exact module categories over a pivotal finite tensor category C, the category Rex C (M 1 , M 2 ) is an exact Z(C)-module.Specifically, we compute its relative Serre functor to be with D the distinguished invertible object of C. In Corollary 38 we then conclude that in case C is unimodular, this exact module category is pivotal (in the sense of Definition 9).It follows that in this case Nat(F, F ) has the structure of a Frobenius algebra, and in particular Nat(Id M , Id M ) has the structure of a commutative Frobenius algebra.In this way, C-module categories 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 study relations between bimodule functors with codomain C and the Drinfeld center of C, which leads us to the notion of central integration functors.Section 4 deals with internal natural transformations.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, in Section 5 we combine these results with the theory of relative Serre functors and pivotal module categories to examine exactness and pivotality of the the functor category Rex C (M, N ) as a module category over Z(C).
A direct application of our results (and, in fact, also a major motivation for our investigations) is in the description of bulk fields in rigid logarithmic two-dimensional conformal field theories, i.e. conformal field theories whose chiral data are described by a modular finite tensor category C.This application will be discussed in detail elsewhere.Here we content ourselves 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) in full 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 has e.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 share the same chiral data based on a given modular tensor category C. It has been established almost two decades ago that in case C is semisimple, the datum that in addition to the chiral data 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 additional datum.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], the algebra Hom(m, m) is a symmetric Frobenius algebra for any m ∈ M. A right exact module functor G ∈ Rex C (M 1 , M 2 ) describes a topological defect line between local conformal field theories characterized by M 1 and by M 2 , respectively.It is then natural to propose that the defect fields that change a defect line labeled by G to a defect line labeled by H are given by Nat(G, H) ∈ Z(C).In particular, Nat(G, G) ∈ Z(C) is a symmetric Frobenius algebra; as a special case, Nat(Id M , Id M ) ∈ Z(C) a commutative symmetric Frobenius algebra, as befits the space of bulk fields.This proposal also leads to natural candidates for operator product expansions and passes non-trivial consistency checks.

Background
In this section we fix our notation and mention some pertinent structures and concepts.

Basic concepts
Monoidal categories.We denote the tensor product of a monoidal category by ⊗ and the monoidal unit by 1, and the associativity and unit constraints by α, l and r, i.e. a monoidal category is a quintuple C = (C, ⊗, 1, α, l, r).For better readability of various formulas, we sometimes 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 action functor C × M → M, exact in its first variable, together with a mixed associator and a mixed unitor 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 stated otherwise.We denote the action morphism by a dot and the mixed associator by a, i. Finite categories.We fix an algebraically closed field k.A finite k-linear category is an abelian category that is equivalent as abelian category to the category of finite-dimensional modules over a finite-dimensional k-algebra.A finite tensor category is a finite k-linear category which is rigid monoidal with appropriate compatibility conditions among the structures, see e.g.[EO] or [Sh3,Sect. 2.5].Since a finite tensor category is rigid, its tensor product functor is exact.Our conventions concerning dualities of a rigid category C are as follows.The right dual of an object c is denoted by c ∨ , and the right evaluation and coevaluation are morphisms while the left evaluation and coevaluation are with ∨ 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-linear abelian category and the action of C on M is linear and right exact in both variables.The category of right exact module endofunctors of a finite module category M over a finite tensor category C is again a finite tensor category [EGNO,Prop. 7.11.6.];we denote it by C ⋆ M .A finite 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].Indecomposable exact module categories over H-mod, for H a finite-dimensional Hopf algebra, are classified in [AM,Sect. 3.2].For recent results see also [Sh4].
Drinfeld centers.For A a monoidal category, a half-braiding for an object a 0 ∈ A is a natural family σ = (σ a ) a∈A of morphisms σ a : a ⊗ a 0 → a 0 ⊗ a such that (suppressing the associator of A) σ a⊗a ′ = (σ a ⊗ id a ′ ) • (id a ⊗ σ a ′ ) for all a, a ′ ∈ A and σ 1 = id a 0 .The Drinfeld center Z(A) of A has as objects pairs (a, σ) consisting of an object of A and a half-braiding on it.The morphisms Hom Z(A) ((a, σ), (a ′ , σ ′ )) are those morphisms 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 coherent isomorphisms D ⊗ x ∼ = x ∨∨∨∨ ⊗ D. A unimodular finite tensor category is a finite tensor category A for which the distinguished invertible object is the monoidal unit.There are several equivalent 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 functor from the enveloping category of C, i.e. the Deligne product of C rev with C (which exists, as C is finite 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 The braided monoidal structure on the functor Ξ C is given by the coherent family id A finite tensor category C is non-degenerate iff the functor Ξ C is an equivalence.If C is even a ribbon category, then C rev is a ribbon category with the inverse twist.A non-degenerate finite ribbon category is a modular tensor category, or modular category, for short.Traditionally, the term modular category has been used under the additional assumption that the finite tensor category 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.
These adjunctions are (co)monadic and thus give rise to a monad An analogous co-Yoneda lemma holds for ends: a∈A Hom A (−, a) * ⊗ F (a) ∼ = F .The isomorphisms in these formulas are uniquely determined by universal properties; accordingly, from now on we will write them as equalities.
Eilenberg-Watts calculus.For any pair of finite linear categories A and B there are two pairs of two-sided adjoint equivalences

Lex(A, B)
A opp ⊠ B Rex(A, B) (2.7) given by [Sh1, FSS1] This provides a Morita invariant version of the classical Eilenberg-Watts description of right or left exact functors between the categories R-mod and S-mod of modules over unital rings in terms 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 endofunctor which is called the Nakayama functor of the finite linear category A [FSS1,Def. 3.14].Analogously, by applying Φ l • Ψ r to Id A regarded as a right exact functor we obtain a left exact analogue N l A = a∈A Hom A (a, −) ⊗ a ∈ Lex(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 natural structure of a twisted bimodule functor, in the sense that there are coherent isomorphisms FSS1,Thm. 4.5].

Internal Hom
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 ( ?; ?) : such that for every m ∈ M the functor Hom M (m, −) : M → C is right adjoint to the action functor H m , i.e. such that for any two objects m, m ′ ∈ M there is a natural family of isomorphisms (see e.g.[Os]).
Being a right adjoint, the internal Hom is left exact.When it is clear from the context which 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 are functorial isomorphisms (2.13) Proof.By the definition of the internal Hom and the fact that F is a module functor, we have for every c ∈ C, m ∈ M and n ∈ N .Thus (2.13) implies that F and G are adjoint.The converse holds by the Yoneda lemma.
Algebra structure on internal Ends.The counits of the adjunctions (2.15) in M given by the image of the identity morphism in End C (Hom(m, m ′ )) under the defining isomorphism (2.12).The composition ev m on internal Homs.Moreover, the component Taken together, these imply the natural isomorphisms 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 monoidal category and M a C-module, an internal coHom of M in C is a functor coHom( ?; ?) : of isomorphisms.By exactness of the action functors H m , also internal coHoms exist.Being a left 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 Hom and coHom are are indeed dual to each other: (2.23) By taking left duals in (2.16) we obtain a coassociative comultiplication (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 And analogously to 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 the family (2.12) reduces to the natural isomorphism that is furnished by the right duality.Similarly we then have coHom(c, c ′ ) = c ′ ⊗ ∨ c.

Pivotal module categories
An additional structure that a finite tensor category C may admit is a pivotal structure, i.e. a monoidal 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 of a Frobenius algebra makes sense in any monoidal category C, the one of a symmetric Frobenius algebra 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 of left and right exact module functors: we have [FSS3,Prop. 4.1] where M # and # M are the right modules with underlying linear category M opp described above.
Remark 6.Any pivotal tensor category is equivalent, as a pivotal category, to a strict pivotal category [NgS, Thm.2.2].Thus in case the finite tensor category C of our interest has a pivotal structure, for many purposes we may replace it by a strict pivotal one in which c ∨ = ∨ c holds for every 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 Furthermore, 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).Thus if C is pivotal, then Z(C) inherits a distinguished pivotal structure [EGNO,Exc. 7.13.6].For pivotal C we will consider Z(C) with this pivotal structure.
Exact module categories over a pivotal tensor category turn out to have a particularly interesting 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 family Remark 8.According to [FSS1,Thm. 4.26] the Nakayama and relative Serre functors of an exact module M over a finite tensor category C are related by N l 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 and only if it is an exact module category.In this case the relative Serre functor is an equivalence of categories.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 In short, a pivotal structure is an isomorphism, as module functors, from the identity functor to the Serre functor, where the pivotal structure on C has been used to turn them into module functors of the same type.If the module category M is indecomposable, then the identity functor Id M is a simple object in the category of right exact module endofunctors.Thus Schur's lemma implies Lemma 10. [Sh4, Lemma 3.12] Let M be an indecomposable exact module category over a pivotal finite tensor category.A pivotal structure on M, if it exists, is unique up to a scalar multiple.
Remark 11. (i) As a special case of Lemma 4.23 of [FSS1], consider a pivotal finite tensor category C as a module category over itself.We have (2.30) so in this case S l C coincides with the bidual functor.The pivotal structure of C then endows the 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 finite tensor 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: Denote by coev c,m : c → Hom(m, c.m) the unit of the adjunction H m ⊢ Hom(m, −).Let S r be a relative Serre functor on M. The internal trace [Sh4,Def. 3.7] is the composition tr m : Hom(m, S r (m)) where the first isomorphism is a component of the inverse of the defining structural morphism of S r .The internal trace is related to a non-degenerate pairing Hom(n, S r (m)) ⊗ Hom(m, n) → 1; 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. Proof.Abbreviate m∈M G(m, m) =: g.To obtain a candidate δ g for the coaction of Z on g we first concatenate the structure morphisms j g of the end and the bimodule structure of G (which will be suppressed in the considerations below), which gives us a family of morphisms.Since j g is dinatural, this constitutes a dinatural transformation from the object g to the functor Invoking the behavior of (co)ends over Deligne products [FSS1,Sect. 3.4] this, in turn, gives rise to a morphism in C that is induced by the functoriality of the end is even a morphism of Z-comodules.(ii) Let H 1 , H 2 : M # ⊠ M → C be bimodule functors and ν : H 1 → H 2 be a bimodule natural transformation.Then the morphism m∈M ν m,m : m∈M H 1 (m, m) → m∈M H 2 (m, m) in C induced by the functoriality of the coend is even a morphism of Z-modules.
Proof.We prove claim (i); the proof of (ii) is dual.Consider two copies of the diagram (3.5), one for G 1 and one for G 2 .We can connect the top lines of these two diagrams by the morphisms ν c.m,c.m : G 1 (c.m, c.m) → G 2 (c.m, c.m) and c .ν m,m .c ∨ : c .G 1 (m, m) .c ∨ → c .G 2 (m, m) .c ∨ .The resulting square commutes because ν is required to be a bimodule natural transformation.Similarly, we can connect the second line of the diagram for G 1 to the second line of the diagram for G 2 by the morphisms ν : g 1 → g 2 and c .ν .c ∨ : c .g 1 .c ∨ → c .g 2 .c ∨ .The two resulting squares that involve the dinatural structure morphisms for g 1 and g 2 , respectively, commute, owing to the definition of ν and the functoriality of the C-actions.Thus in short, in the diagram the left, right, front, back and top squares commute for every m ∈ M. As a consequence, the square 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 1 and G 2 can be combined to The two triangles in this diagram commute by construction, the top square is just the bottom square of (3.10), and the square on the right commutes by the functoriality of the end in the central 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 respectively.
We call the functors (3.13) the central integration functors for the C-module M.
4 Internal natural transformations

Definition
Let C be a finite tensor category and M and N be left C-modules.It is well known that the category Rex C (M, N ) of right exact module functors is a finite category, and in fact has a natural structure of a finite Z(C)-module as follows: For (c 0 , β 0 ) ∈ Z(C) and is again right exact.Also, it acquires the structure of a C-module functor by the composition where the first isomorphism is furnished by the module functor structure on F , and the second and forth use the mixed associativity constraint for N .We write (c 0 , β 0 ) .F ∈ Rex C (M, N ) for the module functor obtained this way from the object (c 0 , β 0 ) ∈ Z(C) and the functor F ∈ Rex C (M, N ).It is straightforward to check that this prescription endows the finite functor category Rex 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 the functor category Rex C (M, N ) with the structure of a module over the finite tensor category Z(C) as described above.Given G, H ∈ Rex C (M, N ), we call the internal Hom the object of internal natural transformations from G to H.

Dually we set coNat(G, H)
Remark 17.The Yoneda lemma in the form of formula (2.6) allows one to express internal natural transformations as a coend: by the adjunction defining the internal Hom (4.3).We denote the dinatural structure morphisms of the coend (4.6) by

Description as an end: component morphisms for relative natural transformations
An ordinary natural transformation is a family of morphisms.As a consequence the set of ordinary natural transformations between any two functors G, H : C → D can (for C essentially small) be expressed as an end: The structure morphisms of this end are just the projections to the components of the natural transformation.As a special case, the natural transformations of the identity functor of a category C give the center End(Id C ) = c∈C Hom C (c, c) of C. The latter provides a Morita invariant formulation of the center 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 express internal natural transformations as an end as well.We start by noticing that in situations which 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 7 provide an illustrative example.It is thus natural to consider for any pair G, H : M → N of module functors the end (4.10) We denote the members of the dinatural family for this end by (4.12) We denote its dinatural family by A crucial observation is now that, since the internal Hom and internal coHom are bimodule functors, by Theorem 15 we can (and will) for G, H ∈ Rex 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 Proof.We prove (i); the proof of (ii) is dual.By the adjunction that defines the internal Hom Nat(F, G), proving (i) is equivalent to showing the adjunction in N commutes, where the morphism Φ c 0 .G is the module functor datum φ G for G, followed by a half-braiding.
An element of the morphism space on the left hand side of (4.16) is a morphism to an end in C.After post-composing with the structure morphisms j G,H m of that end, this amounts to a dinatural family of morphisms in C, labeled by objects m ∈ M. The data of this family are (D ′ ) : (4.21) and they are subject to two constraints: (C2 ′ ) Compatibility with the half-braiding: For every c ∈ C, the diagram commutes, where the right downwards arrow is the component at c of the distinguished half-braiding on Nat ′ (G, H) ∈ Z(C).
To compare the two sides of (4.16), first notice that the adjunction defining the internal Hom for the C-module N gives natural isomorphisms for all m ∈ M.This adjunction maps data of type (D) to data of type (D ′ ).We are going to show that also the respective conditions on these data are mapped to each other.Thus consider condition (C1), i.e. the equality is the image of η m under the adjunction (4.24).A similar argument applies to pre-composition, showing that η m ′ • c 0 .G(f ) is mapped by (4.24) to Hom(G(f ), H(m ′ ) • η m .Together it follows that indeed condition (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 commuting diagram (C2 ′ ) with the canonical morphism c ⊗ j G,H m of the end, thereby obtaining morphisms in Hom C (c 0 ⊗ c, c ⊗ Hom(G(m), H(m)).Now take the upper-right composite morphism in (4.23).We can use the right dual of c to consider equivalently a morphism (4.25) Here the second morphism can be recognized as the one we used to get the structure of a comodule over the central comonad on Nat ′ (G, H).Hence the morphism (4.25) can be written as (here we suppress the bimodule functor structure of Hom).By the definition of η m , this morphism is nothing but Under the internal-Hom adjunction this morphism is mapped to Applying similar arguments to the lower-left composite morphism in (4.23) gives the morphism where the first morphism is obtained by combining the half-braiding β 0 with the coevaluation of c.This is nothing but and is thus under the internal-Hom adjunction mapped to the morphism Recalling the definition of Φ c 0 .G in terms of φ G and the half-braiding β 0 , we see that equality of the morphisms (4.28) and (4.31) is precisely the commuting diagram (C2).Thus we have established also the equivalence (C2) ⇔ (C2 ′ ).
It is instructive to express the situation considered in the proof of Theorem 18 schematically: We have a commuting diagram Here the left downwards arrow is post-composition by the structure morphisms of the end (4.10)i.e. maps f to j F,G m • f for some m ∈ M. The right downwards arrow comes from the fact that a natural transformation is a family of morphisms.The lower horizontal arrow is component-wise the internal Hom adjunction.
Example 19.For N = M and G = H = Id M , the object of internal natural transformations is (4.33) In particular, for C as a module category over itself, this is If C is semisimple, this is the object i x i ⊗ x ∨ i , with the summation being over the finitely many isomorphism 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 ∈ Rex C (M, N ), define a functor In the special case that M = N = C C is C seen as a module over itself, we have a canonical identification Rex C (C, C) ∼ = C, under which L Id C : Z(C) → C is the forgetful functor.It follows from the proof of Theorem 18 that the right adjoint of L F is the functor In the special case M = N = C C as well as F = Id C the functor (4.36) is given by 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 To this end we use the fact that a right exact functor admits a right adjoint, and that the right adjoint of the forgetful functor where F r.a. is the right adjoint of F and is the right adjoint of the forgetful functor U M .
Proof.Applying the composition of θ M with the forgetful functor U M : Z(C ⋆ M ) → C ⋆ M to the object (c 0 , β 0 ) ∈ Z(C) of the Drinfeld center gives the C-module endofunctor (c 0 , β 0 ) .Id M , which as a functor is given by acting with c 0 and has a module functor structure given by β 0 .Precomposing with F ∈ Rex C (M, M) yields with L F as introduced in Remark 20.By the adjunction in Remark 20 we thus have where the last isomorphism holds because θ M is an equivalence.It follows that for all ϕ ∈ Z(C ⋆ M ) we have Hom )) . (4.43) Here the first isomorphism uses the definition (4.36) of R F together with the adjunction we just 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 Comparing this formula with (4.34), we can rephrase this by saying that after application of Schauenburg's equivalence θ M , the center of any module category is diagonal.(In the application to two-dimensional conformal field theory alluded to at the end of the Introduction, this implies that the bulk state space of any full conformal field theory becomes diagonal when regarded not as an object of Z(C), but as an object in the equivalent category Z(C ⋆ M ).)

Compositions
Ordinary natural transformations can be composed horizontally as well as vertically.Both compositions are conveniently described component-wise.A vertical composition of internal natural transformations clearly exists, being just a particular instance the multiplication of internal Homs.In this subsection we introduce in addition a horizontal composition of internal natural transformations.We also describe their vertical composition from a different perspective.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 natural transformations, including an Eckmann-Hilton argument.We start by observing that for the Z(C)-module Rex C (M, N ) the natural evaluation ev (2.15) of internal Homs, which is used to obtain their multiplication, is a natural transformation ev F,G : Nat(F, G) .F → G (4.45) between module functors in Rex C (M, N ).Under the defining adjunction isomorphism of the internal 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 dinatural family Nat(F, G) → Hom(F (m), G(m)) m∈M , and this family is just the structure morphism of the end.Thus the components of the evaluation ev F,G are just the images in of the structure morphisms of the end under the internal Hom adjunction for the module category M over C. The associative multiplication of internal Homs Nat(−, −) is a family of morphisms in Z(C) obeying the standard associativity condition, and as described in (2.17) we have units ) that is introduced in (4.47) is called the vertical composition of internal natural transformations.
The following result justifies this terminology: (4.49) We have 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 nothing but the ordinary vertical composition of natural transformations.

Proof. Under the adjunction, the image in Hom
(we suppress the mixed associator of the module category N ).The components of this module natural 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 of that diagram.We must show that they are related by applying the internal-Hom adjunction to 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-Hom adjunction.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 Rex C (M,N ) (1 .F, F ).In terms of the diagram (4.32), we have the identity morphism in the component Hom N (1 .F (m), F (m)) of the natural isomorphism for every m ∈ M.
Next we note that a k-linear category D can be seen as a module category over the monoidal category 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 evaluation as the image of the identity map under the linear isomorphism Here in the equality we use that the action functor is exact, and by z∈Z(C) ev k z.F,G we denote the morphism out of the coend z∈Z Next we define modified vertical and horizontal compositions of relative natural To set the stage 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 ∈ Rex C (M, N ).Then for any pair z, z ′ of objects in Z(C) the module functor constraint of G induces an isomorphism Proof.For any m ∈ M there is an isomorphism of functors.Due to the fact that the braiding is natural and thus compatible with the module functor 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 ∈ Rex C (M, N ) the modified vertical composition of natural transformations is defined by (ii) Let M 1 , M 2 and M 3 be finite module categories over a finite tensor category C. For of natural transformations is defined to be the composition of the ordinary horizonal composition with the isomorphism given in Lemma 25.
Remark 27.Admittedly, the modified vertical composition µ ver looks somewhat unnatural.But it should be appreciated that by using the duality we can identify Proof.We write µ ver for the morphism out of the coend (4.62) that is defined by (4.63).We have to compare the morphisms Nat(G, H) ⊗ Nat(F, G) .F  where the first morphism is determined by the unit η ver F ∈ Hom Rex C (M,N ) (1 .F, F ), while the second is the structure morphism of the coend.
We are now in a position to introduce also a horizontal composition of internal natural transformations: Definition 31.Let M 1 , M 2 and M 3 be finite module categories over a finite tensor category C, and let F, F ′ ∈ Rex C (M 1 , M 2 ) and G, G ′ ∈ Rex C (M 2 , M 3 ).The horizontal composition is the morphism µ hor out of the coend (4.70) that is given by the family 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 the Eckmann-Hilton property: Proposition 32.Let M 1 , M 2 and M 3 be finite module categories over a finite tensor category C. Then for any two triples F, G, H ∈ Rex C (M 1 , M 2 ) and F ′ , G ′ , H ′ ∈ Rex C (M 2 , M 3 ), the diagram e. a has components a c,c ′ ,m : (c ⊗ c ′ ) .m → c .(c ′ .m) with c, c ′ ∈ C and m ∈ M. The natural isomorphism that defines the structure of a C-module functor G is denoted by φ G , i.e. φ G has components φ G c,m : G(c.m) → c .(G(m)) with c ∈ C and m ∈ M.
) 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 natural morphism Hom M (c.m, m ′ ) → Hom N (c.G(m), G(m ′ )) for any m, m ′ ∈ M and c ∈ C, and thus Hom C (c, Hom(m, m ′ )) → Hom C (c, Hom(G(m), G(m ′ )).By the Yoneda lemma this induces a morphism where on the left hand side one deals with composition of functors and on the right hand side with composition of morphisms 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 the monoidal 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 have a 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 thus has a left adjoint.Indeed we have Hom C (γ, Hom(m, c.m ′ )) ∼ = Hom M (γ .m, c .m ′ ) ∼ = Hom M (c ∨ .(γ.m), m ′ ) .29) Similarly there are coherent natural isomorphisms S l M (c.m) ∼ = ∨∨ c .S l M (m).These results allow one to give Definition 9. ([Sc2, Def.5.2] and [Sh4, Def.3.11]) A pivotal structure, or inner-product structure, on an exact module category M over a pivotal finite tensor category (C, π) is an isomorphism 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 morphisms from c .m to c ∨∨ .S r M (m) holds for every c ∈ C and every m ∈ M.

3
[Sh4, Thm.3.15] Let (M, π M ) be a pivotal exact module category over a pivotal finite tensor category (C, π).Then the algebra Hom(m, m) in C has the structure of a symmetric Frobenius algebra with Frobenius form λ m : Hom(m, m) Integrating bimodule functors to objects in the Drinfeld center Recall from the paragraph before Lemma 4 the right C-module structures # M and M # that are 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 m∈M G(m, m) has a natural structure of a comodule over the central comonad Z of C, and can thus be seen as an object in the Drinfeld center Z(C).(ii)Let H : M # ⊠ M → C be a bimodule functor.Then the coend m∈M H(m, m) has a natural structure of a module over the central monad Z of C, and can thus be seen as an object in the Drinfeld center Z(C).
structure morphism of the central comonad.Writing δc := 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 ∨ not directly use this diagram here, but it will be instrumental in the proof of Lemma 14 below.)Now denote by ∆ : Z(c) → Z 2 (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 2 (g) x .Z(g) .x ∨ (x⊗y) .g .(x⊗y) ∨ (x⊗y) .G(m, m) .(x⊗y) , while the lower square commutes owing to the relation (3.4).The second diagram is g Z(g) Z 2 (g) x .g .x ∨ x .Z(g) .x ∨ (x⊗y) .G(m, m) .(x⊗y) ∨ (x⊗y) .g .(x⊗y) and the lower right square commute again due to (3.4), while the upper right square 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 commutative diagram analogous to (3c : c ⊗ x ⊗ ∨ c → Z(x) the structure morphism of the central monad, h := m∈M H(m, m) and ρc := ρ h • ζ c .An analogous result holds for natural transformations: Lemma 14.Let M be a C-module.

(
situation for ordinary natural transformations, the morphism j G,H m in C plays the role of projecting to the mth 'component' Hom N (G(m), H(m)) of the object Nat ′ (G, H).Dually, we have for G, H : M → N the coend coNat ′ (G, H) = m∈M coHom N (G(m), H(m)) .
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 transformations: Analogously, the internal coHom (4.4) is canonically isomorphic to a coend: coNat(F, G) = m∈M coHom(F (m), G(m)) .(4.15) N ) and z ∈ Z(C).Writing z = (c 0 , β 0 ) with c 0 ∈ C and β 0 a half-braiding on c 0 , the right hand side of (4.16) can be described as follows.The data characterizing a module natural transformation from z.G to H are a family (D) : η m : c 0 .G(m) → H(m) 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 ) Module natural transformation: With φ H the datum turning H into a module functor, for every c ∈ C and m ∈ M the diagram c 0 .G(c .m) H(c .m) which is exact, is the coinduction functor associated with the central comonad on C ⋆ M .Proposition 21.Let C be a finite tensor category and M a C-module.For F, G ∈ Rex C (M, M) we have θ M (Nat(F, G)) ∼ = I(G • F r.a. ) ∈ Z(C ⋆ M ) , (4.39) .54) It should be appreciated that the composition of Hom D as an internal Hom for D as a vect kmodule and the ordinary composition of morphisms in D coincide.This observation allows us to give the following convenient description of the evaluation ev F,G : Nat(F, G) .F → G in (4.45): Invoking from Remark 17 the expression (4.6) for Nat(F,
of dualities, µ ver boils down to the ordinary vertical composition (see Remark 27), the two composites (4.66) and (4.67) coincide.It thus follows that µ ver indeed describes the vertical composition µ ver of internal natural transformations.Remark 30.We describe again the unit of the vertical composition.It is the morphism in Hom Z(C) (1 Z(C) , Nat(F, F )) that is given by the composite1 Z(C) − − → Hom Rex C (M,N ) (1 Z(C) .F, F ) ⊗ k 1 Z(C) ı F,F 1 Z(C) −−−−→ z∈Z(C) Hom Rex C (M,N ) (z .F, F ) ⊗ k z ,(4.68) commutes.Here the horizontal arrow is the braiding in Z(C).