Jens Hemelaer on Presheaves & Generalized Topological Spaces 🌐

Talk at the school and conference “Toposes online” (24-30 June 2021):
https://aroundtoposes.com/toposesonline/

Slides:
https://aroundtoposes.com/wp-content/uploads/2021/07/HemelaerSlidesToposesOnline.pdf

Various ideas from topology have been generalized to toposes, for example surjections and inclusions, local homeomorphisms, or the fundamental group. Another interesting concept, that is less well-known, is the notion of a complete spread, that was brought from topology to topos theory by Bunge and Funk. We will discuss these concepts in the special case of toposes of presheaves on monoids. The aim is to gain geometric intuition about things that are usually thought of as algebraic.
Special attention will go to the underlying topos of the Arithmetic Site by Connes and Consani, corresponding to the monoid of nonzero natural numbers under multiplication. The topological concepts mentioned earlier will be illustrated using this topos and some of its generalizations corresponding to maximal orders.
The talk will be based on joint work with Morgan Rogers and joint work with Aurélien Sagnier.