On the representability of actionsin a semi-abelian category

By Francis Borceux, George Janelidze. Theory and Applications of Categories, 2005

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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.

Subfields: Artificial Intelligence, Computer Networks and Communications, Management of Technology and Innovation

View Paper: https://www.semanticscholar.org/paper/4bc95ee481df181be28843f67026d0469fff9372