Further Results on the Structure of (Co)Ends in Finite Tensor Categories

Abstract

Let $${\mathcal {C}}$$ be a finite tensor category, and let $${\mathcal {M}}$$ be an exact left $${\mathcal {C}}$$-module category. The action of $${\mathcal {C}}$$ on $${\mathcal {M}}$$ induces a functor $$\rho : {\mathcal {C}} \rightarrow \mathrm {Rex}({\mathcal {M}})$$, where $$\mathrm {Rex}({\mathcal {M}})$$ is the category of k-linear right exact endofunctors on $${\mathcal {M}}$$. Our key observation is that $$\rho $$ has a right adjoint $$\rho ^{\mathrm {ra}}$$ given by the end $$\begin{aligned} \rho ^{\mathrm {ra}}(F) = \int _{M \in {\mathcal {M}}} \underline{\mathrm {Hom}}(M, F(M)) \quad (F \in \mathrm {Rex}({\mathcal {M}})). \end{aligned}$$As an application, we establish the following results: (1) We give a description of the composition of the induction functor $${\mathcal {C}}_{{\mathcal {M}}}^* \rightarrow {\mathcal {Z}}({\mathcal {C}}_{{\mathcal {M}}}^*)$$ and Schauenburg’s equivalence $${\mathcal {Z}}({\mathcal {C}}_{{\mathcal {M}}}^*) \approx {\mathcal {Z}}({\mathcal {C}})$$. (2) We introduce the space $$\mathrm {CF}({\mathcal {M}})$$ of ‘class functions’ of $${\mathcal {M}}$$ and initiate the character theory for pivotal module categories. (3) We introduce a filtration for $$\mathrm {CF}({\mathcal {M}})$$ and discuss its relation with some ring-theoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that $$\mathrm {Ext}_{{\mathcal {C}}}^{\bullet }(1, \rho ^{\mathrm {ra}}(\mathrm {id}_{{\mathcal {M}}}))$$ is isomorphic to the Hochschild cohomology of $${\mathcal {M}}$$. As an application, we show that the modular group acts projectively on the Hochschild cohomology of a modular tensor category.