Olivia Caramello: “Introduction to relative topos theory”

Olivia Caramello: “Introduction to relative topos theory” Relativity techniques for schemes play a crucial role in Grothendieck’s refoundation of algebraic geometry. We shall present an analogous set of tools and techniques for investigating relative toposes, i.e. toposes defined over a given base topos. Our new foundations for relative topos theory are based on fibrations and stacks; in fact, our theory develops and extends the original approach introduced by Giraud in his paper “Classifying topos”. A central result 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. For the benefit of those who are not familiar with the language of topos theory, we shall review, in the first part of the course, the basic theory of sheaves on a site, as well as a number of fundamental results on functors between sites inducing morphisms between the corresponding toposes. The lecture by Riccardo Zanfa will discuss applications of our theory in the classical (discrete) setting of sheaves on a site. This is joint work with Riccardo Zanfa. https://utge.lakecomoschool.org/