The geometry of morphisms and equivalences of toposes
Olivia Caramello
University of Insubria and IHES (Gelfand Chair)
This online mini-course consisting of three lectures of two hours each will take place on October 26th, October 27th and October 30th at 11:00 am (Bogotá time) as part of the series "Topics in contemporary mathematics" organized by Alexander Cruz at Universidad Nacional de Colombia. Please write to jacruzmo@unal.edu.co in order to get the link to the zoom meeting.
Abstract: We will present a number of fundamental results and constructions on the theme of sites and morphisms of toposes, some of which generalize theorems from SGA4.
In the first part of the course we shall present constructions allowing to turn any morphism of sites into a a comorphism of sites inducing the same geometric morphism (up to equivalence) and conversely; moreover, we shall introduce the notion of weak morphism of toposes and characterize the functors which induce such morphisms.
In the second part, we shall discuss continuous comorphisms of sites, present an explicit characterization for them (also in terms of relative cofinality conditions), and show that this class of comorphisms includes all fibrations as well as morphisms of fibrations. We shall also present a characterization theorem for essential geometric morphisms and locally connected morphisms in terms of continuous functors, and a topos-theoretic interpretation of (a relative version) of the comprehensive factorization of a functor.
In the third part, we shall present a theorem providing necessary and sufficient explicit conditions for a morphism of sites to induce an equivalence of toposes; this generalizes Grothendieck’s Comparison Lemma. Lastly, we shall give an overview of results characterizing important properties of geometric morphisms of toposes (such as being an inclusion, a surjection, hyperconnected, localic, local etc.) in terms of properties of morphisms or comorphisms of sites.