Inner autoequivalences in general and those of monoidal categories in particular

TL;DR

This paper develops a comprehensive 2-categorical framework for inner autoequivalences, unifying and extending existing theories, with specific focus on monoidal categories and their Picard 2-groups.

Contribution

It introduces a general theory of inner autoequivalences in 2-categories, connecting isotropy groups with Picard 2-groups in monoidal categories.

Findings

Isotropy 2-group of a monoidal category equals its Picard 2-group.

Dense subcategories facilitate isotropy computations with coproducts.

Unified various known results in a two-dimensional categorical setting.

Abstract

We develop a general theory of (extended) inner autoequivalences of objects of any 2-category, generalizing the theory of isotropy groups to the 2-categorical setting. We show how dense subcategories let one compute isotropy in the presence of binary coproducts, unifying various known one-dimensional results and providing tractable computational tools in the two-dimensional setting. In particular, we show that the isotropy 2-group of a monoidal category coincides with its Picard 2-group, i.e., the 2-group on its weakly invertible objects.