Composites of Central Extensions Form a Relative Semi-Abelian Category

We consider trivial and central extensions, in the sense of G. Janelidze and G. M. Kelly, which are defined with respect to an adjunction between a Barr-exact category C and a Birkhoff subcategory X of C. Assuming in addition that C is a pointed Mal'tsev category with cokernels, and that X is protomodular, we prove that: (a) the class of all trivial extensions and the class of all finite composites of central extensions form relative homological category structures on C; (b) if C has finite coproducts, then the class of all finite composites of central extensions forms a relative semi-abelian category structure on C.