A foundation of relative topos theory via fibrations and comorphisms of sites

The present PhD thesis deals with the topic of relative topos theory over a base Grothendieck topos: the main goal is to show how the notions of comorphism of sites and of fibration provide an efficient formalism to study toposes over a base topos as toposes of relative sheaves. The main result in this work is the proof that fibrations over a base site (C,J) are 2-adjoint to toposes over the topos of sheaves Sh(C,J), and that this adjunction provides a 2-categorical generalization of the well known presheaf-bundle adjunction for topological spaces. The same adjunction is then specialized to presheaves, and provides a new description of the process of sheafification. Associating geometric morphisms to fibrations (seen as comorphisms of sites) also allows for an interpretation of toposes over a base as toposes of relative sheaves.