How nice is the category of condensed sets?

Alexander Campbell – 16 December 2020

The new theory of "condensed mathematics" being developed by Clausen and Scholze promises to make analytic geometry amenable to the powerful techniques of modern algebraic geometry. The basic objects of this theory are the "condensed sets", which may be defined as the small sheaves on the large site of compact Hausdorff spaces with the coherent topology. (These are nearly the same as the pyknotic sets of Barwick and Haine, up to issues of size.)

In this talk we shall study several categorical properties of the category of condensed sets (which properties are surely known to the experts). We shall prove that this category is a locally small, well-powered, locally cartesian closed infinitary-pretopos, that it is neither a Grothendieck topos nor an elementary topos -- since it lacks both a small generator (indeed, it is not even total) and a subobject classifier -- but that it does have a large generator of finitely presentable projectives, and hence is algebraically exact. We shall also discuss the relationship of Spanier's quasi-topological spaces to condensed sets.