### 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 ρ^{ra} 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 |
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Pages (from-to) | 237-286 |

Number of pages | 50 |

Journal | Applied Categorical Structures |

Volume | 28 |

Issue number | 2 |

DOIs | |

Publication status | 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)