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In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+2\right)$ times, and in general, …

The "classical" definition of a representation of a Lie groupoid is rather similar to that of a Lie group. For a Lie group representation, you start with a vector space $V$ and define a representation …

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 an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a to …

Haefliger's foliation groupoid is homotopy equivalent to classifying space of the discrete monoid of embeddings of $\mathbb{R}^n$ into itself $B\mathbf{Emb}\left(\mathbb{R}^n\right)$ using differentiable stacks …

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 aware of several statements made more beautiful in the language of stacks, but, I'm looking for a concrete application. …

In DGA-V, Lurie remarks that what he defines as higher Deligne-Mumford stacks lacks separation axioms, but that they may be added in by hand later. …

$\Omega G$ 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 …

There is a canonical equivalence of $2$-categories $$St\left(Man/M\right) \simeq St\left(Man\right)/M$$ between stacks on the large site of $M$ and stacks on the site of manifolds equipped with a map … It doesn't embed into the $2$-category of stacks on the small site of $M$ however. …

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 …

1). In what way is the Grothendieck construction a genralization of the étalé space construction? Well, if $F$ is a sheaf on a topological space $X,$ this really means that $F$ is a sheaf on the pose …

I will show that stacks over diffeological spaces are "the same" (in the sense of equivalence of 2-categories) as ordinary stacks on manifolds. … In summary: Stacks over diffeological spaces are "the same" as stacks over manifolds. …

.$ Proof: Let $i:Mfd \hookrightarrow \mathfrak{DiffSt}$ be the full and faithful inclusion of manifolds into differentiable stacks. … It is easy to cook up examples like this that are not represented by differentiable stacks, since the limits must agree with point-wise limits, since $R$ would be a right-adjoint. …

First a couple corrections. If $M$ is represented by quotient stack, it better be a contravariant functor, and moreover, it should probably take values in groupoids, not set. Anyway, here's what going …

I could be wrong, but I am going to be brave and assume that coarse moduli spaces are defined in the analogous way as for topological stacks. … representable sheaves: Let $\pi_F:\int F \to Sch$ be the fibered category representing $F.$ Then $F$ is the colimit of $y \circ \pi_F,$ where $y:Sch \to St\left(Sch\right)$ is the Yoneda embedding into stacks …