The universal property of the category of condensed sets
The new theory of "condensed mathematics" being developed by Clausen and Scholze (which is closely related to the "pyknotic formalism" of Barwick and Haine) promises to make analytic geometry amenable to the powerful techniques of modern algebraic geometry. The most basic objects of this theory are the "condensed sets", which are akin to topological spaces, but which form a much nicer category.
The main goal of this talk will be to prove the following universal property of the category of condensed sets: it is the infinitary-pretopos completion of the pretopos of compact Hausdorff spaces. To prove this, I will give a few descriptions of condensed sets as small sheaves on certain large sites, and show that these are equivalent to the (rather more convoluted) definition to be found in Scholze's lecture notes.
This talk will be approximately 90 minutes with a short break in the middle.
This is an informal seminar on arithmetic algebraic geometry and related fields.
The purpose is to present work in progress or to learn about some existing piece of mathematics. Typically the talks are parts of a longer series.