Initial normal covers in bi-Heyting toposes

F. Borceux, D. Bourn and P. Johnstone

Address. Universit\'e de Louvain, Belgium
Universit\'e du Littoral, Calais, France
University of Cambridge, United Kingdom


Abstract. The dual of the category of pointed objects of a topos is semi-abelian, 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 {\em bi-Heyting}: the lattices of subobjects are both Heyting algebras and the dual of Heyting algebras.