I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. **My question is:** what are the correct citations for these versions? **Pre-question:** Are the results correctly stated?


**Version 1**


Let $\mathrm{Man}\_d$ denote the category of $d$-dimensional smooth manifolds and local diffeomorphisms — what are often called the "étale" maps of manifolds. Let us say that a presheaf $F: \mathrm{Man}\_d^{\mathrm{op}} \to \mathrm{Set}$ is a *topological sheaf* if it is local for the usual (or "étale") topology on $\mathrm{Man}\_d$ and also takes homotopic maps in $\mathrm{Man}\_d$ to equal functions in $\mathrm{Set}$. Any topological sheaf $F$ then induces a $\mathbb Z/2 = \pi\_0(\hom(\mathbb R^d,\mathbb R^d)$-action on $F(\mathbb R^d)$. The claim is that $F$ is determined by the set $F(\mathbb R^d)$ with its $\mathbb Z/2$-action.


**Version 2**


Same thing, but with embeddings: replace $\mathrm{Man}\_d$ by the category $\mathrm{Emb}\_d$ of $d$-dimensional manifolds and open embeddings; use the Weiss topology.


**Version 1.$\infty$ and 2.$\infty$**


Same thing but replace $\mathrm{Set}$ with $\mathrm{Spaces}$, use homotopy sheaves, and classify in terms of homotopy $\hom(\mathbb R^d,\mathbb R^d) \simeq O(d)$-actions.



Reference request: sheaves on the site of d-manifolds
Version 1 is a simple corollary of the existence
of [differentiable good open covers](http://ncatlab.org/nlab/show/good+open+cover#DifferentiablyGoodOpenCover), i.e., covers whose finite intersections are diffeomorphic to R^n.


This shows that sheaves on the site of n-manifolds and etale maps are equivalent to sheaves on the site with a single object R^n.
The fact that the space of endomorphisms of R^n in this category is homotopy equivalent to O(n) completes the proof.


