Compact sheaves on a locally compact space

Abstract:

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