Compact sheaves on a locally compact space

We describe the compact objects in the $\mathrm{\infty }$-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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-category, we show that $\text{Shv}(X,\mathcal{C})$ is compactly generated if and only if $X$ is totally disconnected.