On the representability of actions in the topos context

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.