Topological semi-abelian algebras

Given an algebraic theory T whose category of models is semi-abelian, we study the category TopT of topological models of T and generalize to it most classical results on topological groups. In particular, TopT is homological, which includes Barr regularity and forces the Mal'cev property. Every open subalgebra is closed and every quotient map is open. We devote special attention to the Hausdorff, compact, locally compact, connected, totally disconnected and profinite T-algebras.