On the representability of actions in the topos context

TL;DR

This paper investigates when the functor representing actions in a topos context is representable, finding it holds in Boolean and presheaf toposes but not universally across all topos types.

Contribution

It extends the understanding of action representability from groups to semi-abelian categories within topos theory, identifying specific conditions for representability.

Findings

Representability holds in Boolean toposes.

Representability holds in toposes of presheaves.

Representability does not hold in general Grothendieck toposes.

Abstract

The notion of a group G acting on a group X is well-known. Fixing X, the corresponding functor Act(-,X) is representable by the group [X] of automorphisms of X. The notion of G-action on X has been generalized to the context of a semi-abelian category, but in this general context, the functor Act(-,X) is generally not representable. We investigate the representability of the functor Act(-,X) for the semi-abelian category, dual of the category of pointed objects of a topos E. The representability holds in particular when E is a Boolean topos or a topos of presheaves of sets, but does not hold in general, not even for a Grothendieck topos E.