Online course on the geometry of toposes

Next week, I shall give an online course entitled The geometry of morphisms and equivalences of toposes, as part of the lecture series Topics in contemporary mathematics organized by Alexander Cruz.

I 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 I 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, I shall introduce the notion of weak morphism of toposes and characterize the functors which induce such morphisms.

In the second part, I 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. I 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, I 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, I 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.

This material is taken from the monograph draft

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