Descent for sheaves on compact Hausdorff spaces

TL;DR

This paper establishes descent properties for sheaves on compact Hausdorff spaces, showing that certain sheaf categories satisfy proper descent and linking sheaf theory with condensed mathematics.

Contribution

It proves proper descent for Postnikov complete sheaves on compact Hausdorff spaces and connects sheaf cohomology with condensed objects.

Findings

Proper descent holds for sheaves on compact Hausdorff spaces.

Sheaf and condensed cohomologies agree for locally compact Hausdorff spaces.

The sheaf functor is a sheaf on the category of compact Hausdorff spaces.

Abstract

These notes explain some descent results for $\infty$-categories of sheaves on compact Hausdorff spaces and derive some consequences. Specifically, given a compactly assembled $\infty$-category $\mathcal{E}$, we show that the functor sending a locally compact Hausdorff space $X$ to the $\infty$-category $\operatorname{Sh}^{\operatorname{post}}(X;\mathcal{E})$ of Postnikov complete $\mathcal{E}$-valued sheaves on $ X $ satisfies descent for proper surjections. This implies proper descent for left complete derived $\infty$-categories and that the functor $\operatorname{Sh}^{\operatorname{post}}(-;\mathcal{E})$ is a sheaf on the category of compact Hausdorff spaces equipped with the topology of finite jointly surjective families. Using this, we explain how to embed Postnikov complete sheaves on a locally compact Hausdorff space into condensed objects. This implies that the condensed and sheaf cohomologies of a locally compact Hausdorff space agree.