On toposes, algebraic theories, semi-abelian categories and compact Hausdorff spaces

In this paper we study the categories Cop ∗ and ( Cop ∗ )T of T-models in Cop ∗ for an arbitrary algebraic theory T, when C is a topos or the category CHaus of compact Hausdorff spaces. It is well-known that, when C is a topos, Cop ∗ is semi-abelian. We show that CHausop ∗ is semi-abelian, as well as ( CHausop ∗ )T, and that, when C is a topos having locales of subobjects, ( Cop ∗ )T is also semi-abelian. In addition, we prove the representability of actions in CHausop ∗ .