Derived Stone Embedding

TL;DR

This paper extends the classical Stone embedding to a derived setting using $ty$-topoi, providing a partial characterization of the image of the pro-category of $inite spaces within pyknotic spaces.

Contribution

It introduces a derived Stone embedding for $inite spaces and characterizes its essential image using $ty$-topoi, extending classical results to the derived context.

Findings

Partial characterization of the derived Stone embedding's image

Extension of classical Stone embedding to derived setting

Use of $ty$-topoi machinery in the analysis

Abstract

A classical result, the Stone embedding, characterizes profinite sets as totally disconnected, compact Hausdorff spaces. Building on "Pyknotic objects, I. Basic notions", which introduced a derived Stone embedding of the pro-category of $\pi$-finite spaces into pyknotic spaces, this paper uses the $\infty$-topoi machinery to partially characterize the essential image of this embedding, extending the classical characterization to the derived setting.