Descent for sheaves on 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.