Action representability of the category of internal groupoids

Abstract

When $\mathbb C$ is a semi-abelian category, it is well known that the category $\mathsf{Grpd}(\mathbb C)$ of internal groupoids in $\mathbb C$ is again semi-abelian. The problem of determining whether the same kind of phenomenon occurs when the property of being semi-abelian is replaced by the one of being action representable (in the sense of Borceux, Janelidze and Kelly) turns out to be rather subtle. In the present article we give a sufficient condition for this to be true: in fact we prove that the category $\mathsf{Grpd}(\mathbb C)$ is a semi-abelian action representable algebraically coherent category with normalizers if and only if $\mathbb C$ is a semi-abelian action representable algebraically coherent category with normalizers. This result applies in particular to the categories of internal groupoids in the categories of groups, Lie algebras and cocommutative Hopf algebras, for instance.