Endomorphisms of cell 2-representations

TL;DR

This paper characterizes the endomorphism categories of cell 2-representations in certain fiat 2-categories, providing a complete description of J-simple fiat 2-categories with a single non-identity two-sided cell, and simplifies conditions for positively graded cases.

Contribution

It offers a detailed classification of endomorphism categories of cell 2-representations and describes J-simple fiat 2-categories with one two-sided cell, extending understanding in 2-category theory.

Findings

Endomorphism categories of cell 2-representations are explicitly determined.

J-simple fiat 2-categories with one two-sided cell are fully described.

Restrictions are shown to be redundant for positively graded 2-categories.

Abstract

We determine the endomorphism categories of cell 2-representations of fiat 2-categories associated with strongly regular two-sided cells under some natural assumptions. Along the way, we completely describe J-simple fiat 2-categories which have only one two-sided cell J apart from the identities, under the same conditions as above. For positively graded 2-categories, we show that the additional restrictions are redundant.