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.
Keywords: semi-abelian category, variety, semi-direct product, action
2000 MSC: 18C10, 18D35, 18G15
Theory and Applications of Categories,
Vol. 14, 2005,
No. 11, pp 244-286.