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]](http://arxiv.org/abs/math/0306176) .)



Diffeology as a sheaf on the site of smooth manifolds
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](http://www.math.harvard.edu/~lurie/papers/croppedtopoi.pdf), 6.4.5.


