Inner automorphisms as 2-cells
Abstract
Abstract inner automorphisms can be used to promote any category into a 2-category, and we study two-dimensional limits and colimits in the resulting 2-categories. Existing connected colimits and limits in the starting category become two-dimensional colimits and limits under fairly general conditions. Under the same conditions, colimits in the underlying category can be used to build many notable two-dimensional colimits such as coequifiers and coinserters. In contrast, disconnected colimits or genuinely 2-categorical limits such as inserters and equifiers and cotensors cannot exist unless no nontrivial abstract inner automorphisms exist and the resulting 2-category is locally discrete. We also study briefly when an ordinary functor can be extended to a 2-functor between the resulting 2-categories.
Cite
@article{arxiv.2406.13647,
title = {Inner automorphisms as 2-cells},
author = {Pieter Hofstra and Martti Karvonen},
journal= {arXiv preprint arXiv:2406.13647},
year = {2025}
}