Ternary semi-direct products in semi-abelian categories

Abstract

We study the construction of ternary semi-direct products in semi-abelian categories, following a definition of higher semi-direct product previously introduced by Carrasco and Cegarra for groups and Lie algebras. We show that these objects are determined by actions and equivariant morphisms satisfying a certain compatibility condition in the ambient category, and that actions by semi-direct products are a special case of this construction. We also show how their structure can be further simplified in algebraically coherent or locally algebraically cartesian closed categories. We also give a concrete description of the structure of ternary semi-direct products in concrete categories, and how our categorical interpretation of the necessary structure relates to the group and Lie-theoretic versions given by Carrasco and Cegarra.