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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.

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  • $\begingroup$ I guess I am also interested in versions 1.1 and 2.1, where you study stacks of (1-)groupoids rather than fully-infinitized homotopy sheaves/stacks. $\endgroup$
    – Theo Johnson-Freyd
    Commented Sep 30, 2015 at 16:13
  • $\begingroup$ Version 1 follows instantly from the existence of differentiable good open covers, which is definitely a folklore result. $\endgroup$
    – Dmitri Pavlov
    Commented Oct 1, 2015 at 17:18
  • $\begingroup$ @DmitriPavlov Ah, good, and googling "differentiable good open covers" results in an nlab page with textbook references. Do you want to make your comment an answer? $\endgroup$
    – Theo Johnson-Freyd
    Commented Oct 2, 2015 at 4:26

1 Answer 1

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Version 1 is a simple corollary of the existence of differentiable good open covers, 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.

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  • $\begingroup$ Dmitri: this is probably not helpful. Theo started his question by "I believe I know how to prove the following results". He was specifically asking for references. $\endgroup$
    – André Henriques
    Commented Oct 2, 2015 at 21:59
  • $\begingroup$ @AndréHenriques: There are references in the link I supplied. $\endgroup$
    – Dmitri Pavlov
    Commented Oct 2, 2015 at 22:03
  • $\begingroup$ @AndréHenriques: Although I can see why one might be confused: a visited link is almost invisible in the new style. $\endgroup$
    – Dmitri Pavlov
    Commented Oct 3, 2015 at 10:06
  • $\begingroup$ @AndréHenriques Given that I asked Dmitri to make his comment and answer, I think it reasonable to accept the answer, also since a few days is plenty for a question like this. In the best of worlds I would like original references, but Dmitri makes a convincing argument that Version 1, at least, is a folk theorem; the linked citations also answer my ulterior goal, which was to have something to cite that isn't obviously overkill. $\endgroup$
    – Theo Johnson-Freyd
    Commented Oct 4, 2015 at 5:30

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