Compact sheaves on a locally compact space

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