On the representability of actionsin a semi-abelian category
Francis Borceux, George Janelidze, G. M. Kelly · Theory and applications of categories · 2005
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.