Inner automorphisms of groupoids

TL;DR

This paper characterizes the inner automorphisms of groupoids as those induced by conjugation with a bisection, extending the concept of inner automorphisms beyond groups to more complex structures.

Contribution

It computes the inner automorphisms of groupoids using the framework of comorphisms, and explores generalizations to topological, Lie groupoids, and related structures.

Findings

Inner automorphisms of groupoids are induced by conjugation with a bisection.

The result holds in the category of groupoids and comorphisms, not in the standard homomorphism category.

Connections with inverse semigroup theory are discussed.

Abstract

Bergman has given the following abstract characterisation of the inner automorphisms of a group $G$: they are exactly those automorphisms of $G$ which can be extended functorially along any homomorphism $G \rightarrow H$ to an automorphism of $H$. This leads naturally to a definition of "inner automorphism" applicable to the objects of any category. Bergman and Hofstra--Parker--Scott have computed these inner automorphisms for various structures including $k$-algebras, monoids, lattices, unital rings, and quandles---showing that, in each case, they are given by an obvious notion of conjugation. In this note, we compute the inner automorphisms of groupoids, showing that they are exactly the automorphisms induced by conjugation by a bisection. The twist is that this result is false in the category of groupoids and homomorphisms; to make it true, we must instead work with the less familiar category of groupoids and comorphisms in the sense of Higgins and Mackenzie. Besides our main result, we also discuss generalisations to topological and Lie groupoids, to categories and to partial automorphisms, and examine the link with the theory of inverse semigroups.