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If $f:M \to N$ is a submersion with connected fibers, then the fibers of $f$ foliate $M$. This is called a simple foliation of $M$ and the leaf space can be identified with $N$. Suppose that a foliati …

I have been playing with/thinking about diffeological spaces a bit recently, and I would like understand something rather crucial before going further. First a little background: Diffeological spaces …

Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to [G]$, where $[G]$ is the differentiable stack ass …

Recall that a morphism $f:C \to D$ in a category $\mathscr{C}$ is representable if for all maps $g:E \to D$ in $\mathscr{C},$ the pullback $C \times_{D} E$ exists. Let now $\mathscr{C}$ be the catego …

There is a theorem due to Losik which shows that the category of Frechet manifolds embeds fully-faithfully into diffeological spaces. (Diffeological spaces are concrete sheaves on the site of (Euclide …

I should update with a mention of some of my own results in http://arxiv.org/abs/1504.02394: There is a proof of Segal's theorem that the classifying space $B\Gamma^q$ of Haefliger's foliation grou …

What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well awar …

$\Omega G$ won't be a differentiable stack unless you are willing to go to infinite dimensions. Provided you are considering $S^1$ as a smooth manifold, Nerses answer above is correct- it gives you a …

In theory, most anything can be expressed with SDG, and there has been some work in expressing some of GR in this context, but I am not sure if much has been done beyond proof-of-concept. You can goog …

Actually, one doesn't need the comparison lemma in this case. As it turns out, $\mathbf{Man}$ is the Karoubi envelope of $\mathbf{Open},$ (see the Examples section of http://ncatlab.org/nlab/show/Karo …

I will show that stacks over diffeological spaces are "the same" (in the sense of equivalence of 2-categories) as ordinary stacks on manifolds. The Grothendieck pre-topology in question is the Grothe …

The answer is no. Claim: If $\mathfrak{DiffSt}$ were cocomplete, it would be reflective in $St\left(Mfd\right).$ Proof: Let $i:Mfd \hookrightarrow \mathfrak{DiffSt}$ be the full and faithful inclus …

How about this? Apply the tangent functor $T$ to $M_\bullet$ to get a new simplicial manifold $TM_\bullet,$ that is take the composite $$\Delta^{op} \stackrel{M_\bullet}{\longrightarrow} Mfd \stackre …

Ok, so, I will try to answer this best I can. first, I'll tell you a skewed-perspective of what an infinity topos is (or ought-to-be). As for how you can "do differential geometry"- this is a bold sta …

This is partly an answer (or maybe more of an extended comment) and partly a response to Christoph's answer, but too long for a comment. One needs to be careful what you mean by "associated 2-functors …