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.