Fibred sites and existential toposes

Abstract

In the context of relative topos theory via stacks, we introduce the notion of existential fibred site and of existential topos of such a site. These notions allow us to develop relative topos theory in a way which naturally generalizes the construction of toposes of sheaves on locales, and also provides a framework for investigating the connections between Grothendieck toposes as built from sites and elementary toposes as built from triposes. Then we focus on fibred preorder sites and establish a fibred generalisation of the ideal-completion of a preorder site. Lastly, we provide an explicit description of the hyperconnected-localic factorization of a geometric morphism in terms of internal locales.