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Souriau's definition of diffeology may be phrased as defining a concrete sheaf on the category $\mathsf{Open}$ of open subsets of Euclidean/coordinate spaces. It seems to me, unless I am missing something, that any such sheaf extends to a sheaf on the site $\mathsf{Man}$ of all smooth manifolds. Is there a proof written down somewhere?

More generally it seems to me that the inclusion $\mathsf{Open} \hookrightarrow \mathsf{Man}$ should induce an equivalence of 2-categories between the 2-category of stacks on $\mathsf{Man}$ and the stacks on $\mathsf{Open}$. Is this true? If so, is there a reference? (Metzler seems to mentions something like this in passing in here: arXiv:math/0306176 [math.DG] .)

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up vote 10 down vote accepted

This would be the "comparison lemma" from SGA 4-1, Éxposé III, Thm. 4.1: if $C$ is a full subcategory of a site $D$, equipped with the induced topology, and if every object of $D$ is covered by objects of $C$, then the restriction functor $Shv(D)\to Shv(C)$ is an equivalence of categories.

More generally, for any finite $n$, the $n$-topoi of $n$-stacks on $C$ and $D$ are equivalent iff the $1$-topoi of $1$-stacks (=sheaves) are, so the answer to your second question is also yes. This is because the $n$-topos of $n$-stacks on a $1$-site is $1$-localic, which means that it belongs to the image of the fully faithful embedding of $1$-topoi into $n$-topoi. This is disussed in Higher Topos Theory, 6.4.5.

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Thanks. Would you know of a more "elementary" reference for part 2 of my question? –  Eugene Lerman Jul 26 '13 at 19:41
I don't know any other reference, sorry. –  Marc Hoyois Jul 27 '13 at 8:30
Actually Metzler references J. Giraud, Cohomologie non abélienne, p. 91, where the special case $n=2$ is proved (even for stacks of categories). –  Marc Hoyois Jul 27 '13 at 10:22
You can also piggyback from the $n=1$ case, by taking the Jardine model structure on simplicial sheaves (for hypersheaves), and left Bousfield localizing to recover the 1-truncated objects (the stacks of groupoids). Since the two categories of simplicial sheaves would become equivalent (as 1-categories), in particular you get a Quillen equivalence, and hence an equivalence between their 2-categories of stacks. –  David Carchedi Oct 4 '13 at 21:57
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