I'm looking for a reference for the following result:

Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over $X$. Then the category of sheaf of $\mathcal{R}$-modules over $X$ is equivalent to the category of non-degenerate $\Gamma_c(\mathcal{R})$-modules, where $\Gamma_c(\mathcal{R})$ is the ring of continuous compactly supported functions on $X$ with value in $\mathbb{R}$.

I would also be interested in references in the case where $X$ is compact or under more general assumption (like when $\mathcal{R}$ is replaced by any c-soft sheaf of rings...)

  • $\begingroup$ I am not sure it's in it (but it's a book on my long reading list); perhaps it appears in Bredon's Sheaf Theory yellow book, or perhaps in the references. $\endgroup$ – Alan Nov 16 '16 at 14:31

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