Relative topos theory via stacks

TL;DR

This paper develops a new foundation for relative topos theory using stacks, generalizing classical concepts and providing geometric interpretations of sheaves, stacks, and morphisms, thus addressing longstanding questions in topos theory.

Contribution

It introduces a novel adjunction between toposes over sheaves and indexed categories, and defines relative sites to represent geometric morphisms, expanding the theoretical framework of topos theory.

Findings

Established an adjunction generalizing presheaves and bundles

Provided fibrational descriptions of sheaf and stack images

Addressed Grothendieck's problem on relating small and large topoi

Abstract

We introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site $({\mathcal{C}}, J)$ and that of ${\mathcal{C}}$-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a 'petit' topos associated with a 'gros' topos, thereby providing an answer to a problem posed by Grothendieck in the seventies. Another key ingredient in our theory is a notion of relative site, which allows one to represent arbitrary geometric morphisms towards a fixed base topos of sheaves on a site as structure morphisms induced by relative sites over that site.