Fibred sites and existential toposes

TL;DR

This paper introduces existential fibred sites and toposes within relative topos theory, generalizing sheaf constructions and connecting Grothendieck and elementary toposes through fibred preorder sites and internal locales.

Contribution

It develops the theory of existential fibred sites and toposes, extending classical constructions and providing new insights into the structure of geometric morphisms.

Findings

Generalization of sheaf toposes on locales

Fibred ideal-completion of preorder sites

Explicit description of hyperconnected-localic factorization

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.