Finite Tensor 2-Categories
Abstract
We propose a definition of a finite linear 2-category, as a 2-categorical analogue of the notion of a finite abelian category of Etingof-Ostrik. We go on to prove that finite linear 2-categories with adjunctible projectives are precisely the Morita 2-categories of (all) algebras in a finite multitensor category. Thus, finite linear 2-categories with adjunctible projectives can be viewed as `non-semisimple' generalizations of the finite semisimple 2-categories of Douglas-Reutter. Secondly, we study finite multitensor 2-categories, that is, finite linear 2-categories equipped with a monoidal structure for which every objects has duals. We extend several results about fusion 2-categories to our present setting. In particular, we investigate the properties of the 2-category of bimodules over a rigid algebra 𝒜 in a finite tensor 2-category C. We show that, if C has adjunctible projectives, then the 2-category of 𝒜-𝒜-bimodules in C is a finite multitensor 2-category. This construction provides many examples of finite tensor 2-categories. For instance, the Drinfeld center of a finite tensor 2-category with adjunctible projectives is a finite tensor 2-category.
Related
BibTeX
@misc{decoppet2026finite,
title = {{Finite Tensor 2-Categories}},
author = {Thibault D. Décoppet and Mateusz Stroiński and Tony Zorman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37016},
eprint = {2610.37016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37016},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Finite Tensor 2-Categories
AU - Thibault D. Décoppet
AU - Mateusz Stroiński
AU - Tony Zorman
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37016
UR - https://arxiv.org/abs/2610.37016
AB - We propose a definition of a finite linear 2-category, as a 2-categorical analogue of the notion of a finite abelian category of Etingof-Ostrik. We go on to prove that finite linear 2-categories with adjunctible projectives are precisely the Morita 2-categories of (all) algebras in a finite multitensor category. Thus, finite linear 2-categories with adjunctible projectives can be viewed as `non-semisimple' generalizations of the finite semisimple 2-categories of Douglas-Reutter. Secondly, we study finite multitensor 2-categories, that is, finite linear 2-categories equipped with a monoidal structure for which every objects has duals. We extend several results about fusion 2-categories to our present setting. In particular, we investigate the properties of the 2-category of bimodules over a rigid algebra 𝒜 in a finite tensor 2-category C. We show that, if C has adjunctible projectives, then the 2-category of 𝒜-𝒜-bimodules in C is a finite multitensor 2-category. This construction provides many examples of finite tensor 2-categories. For instance, the Drinfeld center of a finite tensor 2-category with adjunctible projectives is a finite tensor 2-category.
ER -