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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. …

Moreover, there is always a way to define an "Artin stack"- these are those stacks which arise as torsors for a groupoid object in your site. … The same argument holds for all fibred categories- it's only true if we restrict to STACKS (and $X$ then becomes the weak colimit of this cover, but, never mind). …

This can be done in the $2$-categorical setting with stacks with no adjustment: $x$ induces an adjoint pair of $2$-functors $$x_*:St(pt) \to St(X)$$ $$St(pt) \stackrel{}{\longleftarrow} St(X):x^*$$ …

$\Omega G$ 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 …

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 …

Spelling this out, you get Proposition 3.19 out of Noohi's Foundations of Topological Stacks I. … Hence, going to stacks "makes all actions faithful up to homotopy" (more accurately, stacks allow us to encode the isotropy data that would otherwise be lost in such a way that any action becomes as good …

There are two equivalent ways of describing topological stacks. … Topological stacks are then the full sub-2-category all stacks on $Top$ consisting of those stacks with an atlas. One is a "groupoidy" definition. …

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. …

Since geometric stacks are in some sense a natural generalization of representable presheaves, it would seem natural to expect a similar characterization of geometric stacks (at least in the case when …

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, …

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. …

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 …

.$ by restricting its weak left Kan extension to stacks we get a 2-functor (by abuse of notation) $\text{St}(\text{Mfd}) \stackrel{T}{\rightarrow} \text{St}(\text{Mfd}).$ Now, suppose $\mathbf{X}$ is …

Whilst asking this, I nearly forgot that one application does come to mind: http://www.math.fsu.edu/~aluffi/archive/paper325.pdf In this paper Behrang Noohi shows how to use topological stacks to calculate …

Suppose $X:D \to C$ is a fibered category (I do not assume the fibers to be groupoids). Suppose that $X$ is actually left adjoint to a fully faithful embedding $C \hookrightarrow D$. Is there a specia …