Initial Normal Covers in Bi-heyting Toposes

The dual of the category of pointed objects of a topos is semiabelian, thus is provided with a notion of semi-direct product and a corresponding notion of action. In this paper, we study various conditions for representability of these actions. First, we show this to be equivalent to the existence of initial normal covers in the category of pointed objects of the topos. For Grothendieck toposes, actions are representable provided the topos admits an essential Boolean covering. This contains the case of Boolean toposes and toposes of presheaves. In the localic case, the representability of actions forces the topos to be bi-Heyting: the lattices of subobjects are both Heyting algebras and the dual of Heyting algebras. 1. Introducing the problem Given a semi-abelian category V (see [3] or [12]), consider for every object G ∈ V the category Pt(G) of points over G, that is, of split epimorphisms with codomain G. The ‘kernel functor’ Ker : Pt(G) V , (q, s : A ⇆ G, qs = idG) Ker q is monadic (see [11]); let us write TG for the corresponding monad on V . For every object X ∈ V we have thus a notion of G-action on X , namely, a structure (X, ξ) of TG-algebra on X . This yields a functor ActX : V op Set mapping an object G to the set of G-actions on X . By monadicity, this functor is thus isomorphic to the functor SplExtX : V op Set mapping an object G ∈ V to the set of isomorphism classes of points over G with kernel X , that is, the set of isomorphism classes of split exact sequences