TY - JOUR

T1 - Further Results on the Structure of (Co)Ends in Finite Tensor Categories

AU - Shimizu, Kenichi

N1 - Funding Information:
The author thanks Taro Sakurai for discussion. The author is supported by JSPS KAKENHI Grant Number JP16K17568.
Publisher Copyright:
© 2019, Springer Nature B.V.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - 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.

AB - 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.

KW - Finite tensor category

KW - Hochschild cohomology

KW - Modular tensor category

KW - Module category

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U2 - 10.1007/s10485-019-09577-7

DO - 10.1007/s10485-019-09577-7

M3 - Article

AN - SCOPUS:85082299905

SN - 0927-2852

VL - 28

SP - 237

EP - 286

JO - Applied Categorical Structures

JF - Applied Categorical Structures

IS - 2

ER -