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

Research output: Contribution to journalArticle

### Abstract

Let C be a finite tensor category, and let M be an exact left C-module category. The action of C on M induces a functor ρ: C→ Rex (M) , where Rex (M) is the category of k-linear right exact endofunctors on M. Our key observation is that ρ has a right adjoint ρra given by the end ρra(F)=∫M∈MHom̲(M,F(M))(F∈Rex(M)).As an application, we establish the following results: (1) We give a description of the composition of the induction functor CM∗→Z(CM∗) and Schauenburg’s equivalence Z(CM∗)≈Z(C). (2) We introduce the space CF (M) of ‘class functions’ of M and initiate the character theory for pivotal module categories. (3) We introduce a filtration for CF (M) and discuss its relation with some ring-theoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that ExtC∙(1,ρra(idM)) is isomorphic to the Hochschild cohomology of M. As an application, we show that the modular group acts projectively on the Hochschild cohomology of a modular tensor category.

Original language English 237-286 50 Applied Categorical Structures 28 2 https://doi.org/10.1007/s10485-019-09577-7 Published - 2020 Apr 1

### Keywords

• Finite tensor category
• Hochschild cohomology
• Modular tensor category
• Module category

### ASJC Scopus subject areas

• Theoretical Computer Science
• Computer Science(all)