On the Representability of Actions in a Semi-abelian Category

We consider a semi-abelian category V and we write Act(G,X) for the set of actions of the object G on the object X, in the sense of the theory of semi-direct products in V. We investigate the representability of the functor Act(−,X) in the case where V is locally presentable, with finite limits commuting with filtered colimits. This contains all categories of models of a semi-abelian theory in a Grothendieck topos, thus in particular all semi-abelian varieties of universal algebra. For such categories, we prove first that the representability of Act(−,X) reduces to the preservation of binary coproducts. Next we give both a very simple necessary condition and a very simple sufficient condition, in terms of amalgamation properties, for the preservation of binary coproducts by the functor Act(−,X) in a general semi-abelian category. Finally, we exhibit the precise form of the more involved “if and only if” amalgamation property corresponding to the representability of actions: this condition is in particular related to a new notion of “normalization of a morphism”. We provide also a wide supply of algebraic examples and counter-examples, giving in particular evidence of the relevance of the object representing Act(−,X), when it turns out to exist. 1. Actions and split exact sequences A semi-abelian category is a Barr-exact, Bourn-protomodular, finitely complete and finitely cocomplete category with a zero object 0. The existence of finite limits and a zero object implies that Bourn-protomodularity is equivalent to, and so can be replaced with, the following split version of the short five lemma: