On bornological semi-abelian algebras

If \Bbb T is a semi-abelian algebraic theory, we prove that the category {\rm Born}^{\Bbb T} of bornological \Bbb T-algebras is homological with semi-direct products. We give a formal criterion for the representability of actions in {\rm Born}^{\Bbb T} and, for a bornological \Bbb T-algebra X, we investigate the relation between the representability of actions on X as a \Bbb T-algebra and as a bornological \Bbb T-algebra. We investigate further the algebraic coherence and the algebraic local cartesian closedness of {\rm Born}^{\Bbb T} and prove in particular that both properties hold in the case of bornological groups.

Keywords:

Semi-abelian algebraic theory , bornology , bornological algebra , bornological group , action representability , algebraic coherence , local algebraic cartesian closedness