Codensity and Stone spaces

TL;DR

This paper computes two codensity monads related to canonical functors involving finite sets and Sierpinski spaces, revealing their images as categories of Stone and sober spaces, thus providing new motivation for these classes.

Contribution

It explicitly computes codensity monads for key functors, linking them to Stone and sober spaces, and offers a new motivation for studying these spaces.

Findings

The codensity monad of the inclusion of FinSet into Top has Stone spaces as its essential image.

The codensity monad of the inclusion of powers of the Sierpinski space into Top has sober spaces as its essential image.

These results connect categorical constructions with classical topological spaces.

Abstract

We present a detailed computation of two codensity monads associated to two canonical functors -- the inclusion functor of FinSet into Top and the inclusion functor of the category of the powers of the Sierpinski space into Top. We show that the essential images of the two monads are the categories of Stone spaces and of sober spaces, respectively. A new motivation for defining these classes of spaces is therefore obtained.