On fibred products of toposes

Abstract

In the setting of relative topos theory, we show that the pullback of a relative presheaf topos on an arbitrary fibration is the relative presheaf topos on its inverse image. To this end, we develop and exploit a notion of extension with base change of a morphism of sites along the canonical functor. This provides a tool to compare the canonical relative site of the direct (resp. inverse) image with the direct (resp. inverse) image of the canonical relative site: although these operations do not commute in general, we show that in the case of the direct image they are related by an indexed weak geometric morphism, while in the case of the inverse image they can be compared via a cartesian functor that induces a suitable topology making them Morita-equivalent.