Categories in Algebra, Geometry and Mathematical Physics
Conference in honour of Ross Street's sixtieth birthday
July 11-16 2005, Macquarie University, Sydney
Mon, 11 July: 10:30 - 11:20
On semi-direct products and the representability of actions.
Borceux, Francis (Université catholique de Louvain)
\thanks{This talk is a selection of results coming from recent joint works with Dominique Bourn, Maria-Manuel Clementino, George Janelidze and Max Kelly.}
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I consider a category $\cal V$ with semi-direct products and I write $\mathsf{Act}(G,X)$ for the set of actions of the object $G$ on the object $X$, in the sense of the theory of semi-direct products in $\cal V$. I investigate the representability of the functor $\mathsf{Act}(-,X)$. This is a very strong property.
Examples of categories with semi-direct products are the semi-abelian categories, but also the categories of topological models of a semi-abelian theory. And among the non-algebraic semi-abelian categories, we have the dual of the category of pointed objects in a topos.
When $\cal V$ is the category of groups, the functor $\mathsf{Act}(-,X)$ is represented by the group $\mathsf{Aut}(X)$ of automorphisms of $X$. When $\cal V$ is the category of Lie algebras, this functor is represented by the Lie algebra $\mathsf{Der}(X)$ of derivations of $X$. When $\cal V$ is the category of Boolean rings, or of commutative von Neumann regular rings, the functor is represented by the ring $\mathsf{End}(X)$ of $X$-linear endomorphisms of $X$, and so on.
A representability criterion for $\mathsf{Act}(-,X)$ is established in the case where $\cal V$ is semi-abelian, locally presentable, with finite limits commuting with filtered colimits. This contains all categories of models of a semi-abelian theory in a Grothendieck topos. For such categories, the representability of $\mathsf{Act}(-,X)$ reduces to the preservation of binary coproducts. A very simple necessary condition and a very simple sufficient condition can be given in terms of amalgamation properties. The precise form of the more involved ``if and only if'' amalgamation property is related to a new notion of ``normalization of a morphism''.
The study of normal coverings in the category of pointed objects of a topos $\cal E$ gives as corollary --- when these coverings exist --- the representability of the functor $\mathsf{Act}(-,X)$ for every object $X$ of the dual of the category of pointed objects of $\cal E$. This is the case for a wide class of toposes, which contains all boolean toposes and all toposes of presheaves.
Finally the case of topological models of a semi-abelian theory is investigated and transfer results from the $\mathsf{Set}$ case to the topological case are established.