Compact sheaves on a locally compact space

TL;DR

This paper characterizes compact objects in the sheaf category over a locally compact space and explores conditions for compact generation, linking topology with categorical properties.

Contribution

It describes the compact objects in sheaf categories over locally compact spaces and establishes when these categories are compactly generated.

Findings

Compact objects in sheaf categories are characterized explicitly.

For connected manifolds, the sheaf category recovers known results of Neeman.

Sheaf categories are compactly generated if and only if the space is totally disconnected.

Abstract

We describe the compact objects in the $\infty$-category of $\mathcal C$-valued sheaves $\text{Shv} (X,\mathcal C)$ on a hypercomplete locally compact Hausdorff space $X$, for $\mathcal C$ a compactly generated stable $\infty$-category. When $X$ is a non-compact connected manifold and $\mathcal C$ is the unbounded derived category of a ring, our result recovers a result of Neeman. Furthermore, for $X$ as above and $\mathcal C$ a nontrivial compactly generated stable $\infty$-category, we show that $\text{Shv} (X,\mathcal C)$ is compactly generated if and only if $X$ is totally disconnected.