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 (C,J) and that of 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.
We shall progressively release expanded versions of this text contaning new developments in the directions sketched in the introduction.
This work has been recently presented at the conference Toposes online:
The following video focuses on the specialization of the fundamental adjunction in the setting of presheaves (or discrete fibrations):
Thanks again to all our speakers for accepting our invitation or sending us a talk proposal. This has made it possible for us to set up a very rich and varied programme.
We are very happy to have attracted so many people (533 registered participants) to this event, coming from different areas of mathematics, physics, computer science, philosophy and engineering, and also from industry. This illustrates the wide-ranging impact of toposes, and the increasing interest in the subject, across different fields of knowledge.
We hope that this event will further stimulate interdisciplinary research inspired by topos-theoretic ideas, and its applications in different fields. The participants in the conference, as well as any other person interested in toposes, are warmly encouraged to join our community and exchange with each other through the Forum.
Many thanks to IHES and the University of Insubria for their support in organizing this event; the videos of all the talks and course lectures of "Toposes online" will be made available on the YouTube channel of IHES.
We look forward to seeing many of you online and on the forum!