If T is a semi-abelian algebraic theory, we prove that the category Born T of bornological T-algebras is homological with semi-direct products. We give a formal criterion for the representability of actions in Born T and, for a bornological T-algebra X, we investigate the relation between the representability of actions on X as a T-algebra and as a bornological Talgebra. We investigate further the algebraic coherence and the algebraic local cartesian closedness of Born T and prove in particular

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... that both properties hold in the case of bornological groups. The category of bornological sets This section lists some useful basic properties of the category of bornological sets; various results are borrowed from the first chapter of [3]. Definition 2.1. A bornological set (X, B) is a set X provided with a family B (called a bornology) of so-called bounded subsets, in such a way that: