Inner automorphisms as 2-cells

TL;DR

This paper explores how inner automorphisms induce 2-categorical structures, analyzing the existence and construction of various limits and colimits within these enriched categories.

Contribution

It provides a detailed study of two-dimensional limits and colimits in 2-categories formed via inner automorphisms, highlighting conditions for their existence.

Findings

Connected limits and colimits become two-dimensional limits and colimits.

Many notable 2-categorical colimits can be constructed from underlying category colimits.

Disconnected colimits and certain 2-categorical limits require trivial automorphisms for their existence.

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.