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 $\rho :\mathcal{C}\to \mathrm{Rex}(\mathcal{M})$, 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 ${\rho }^{\mathrm{ra}}(F)={\int }_{M\in \mathcal{M}}\underline{\mathrm{Hom}}(M,M)$. As an application, we establish the following results: (1) We give a description of the composition of the induction functor ${\mathcal{C}}_{\mathcal{M}}^{\ast }\to \mathcal{Z}({\mathcal{C}}_{\mathcal{M}}^{\ast })$ and Schauenburg's equivalence $\mathcal{Z}({\mathcal{C}}_{\mathcal{M}}^{\ast })\approx \mathcal{Z}(\mathcal{C})$. (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.

48 pages; v2: reorganized and rewritten. Question 6.10 in v1 has been answered

References in corpus (4)

• Eilenberg-Watts calculus for finite categories and a bimodule Radford $S^4$ theorem
• SL(2,Z)-action for ribbon quasi-Hopf algebras
• Central elements of the Jennings basis and certain Morita invariants
• Integrals for finite tensor categories

Cited by in corpus (1)

• The character algebra for module categories over Hopf algebras