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.
[Math] Applications of topological and diferentiable stacks
applicationsat.algebraic-topologydg.differential-geometrylie-groupoidsstacks
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I think the right thing to do with category-valued stacks is to keep the notion of "representability" the same (that is, use (pseudo) 2-pullbacks rather than comma objects), but to replace representability of the diagonal by representability of $X^2 \to X\times X$, where $X^2$ is the power (cotensor) of X by the free-living arrow (which is equivalent to X in case X is groupoid-valued). This does imply that $(j/j)$ is representable whenever the domain of $j$ is so, since $(j/j)$ is the pullback of $X^2 \to X\times X$ along $j\times j$.
One reason I think this is the right thing is that in the well-developed theory of "indexed categories" over a topos S regarded as "large categories relative to S as a universe of sets", representability of $X^2 \to X\times X$ is equivalent to the standard notion of "local smallness". (More generally, various kinds of "comprehensibility" for indexed categories can be rephrased as the representability of certain functors in the above sense.) Therefore, the resulting notion of "geometricity" would coincide with "essential smallness", as one would expect.
The notion of fine moduli space requires the existence of a universal family. In this case, you want a scheme $M$ equipped with a rank $n$ vector bundle $V$ on $M \times X$, such that pullback induces a natural bijection between the set of maps from any other scheme $Y$ to $M$ and the set of rank $n$ vector bundles on $X \times Y$. You can view $V$ as a family of vector bundles on $X$, parametrized by $M$. Vector bundles of positive rank do not admit universal families, in part due to the existence of automorphisms (and the existence of schemes with nontrivial fundamental group that can act as bases of nontrivial families). I don't have a precise grasp on what your question is asking, but depending on the application of choice, one can sometimes work with a coarse moduli space (which is roughly a way to ignore automorphisms), and one can sometimes rigidify the moduli problem to get a natural cover of the stack by a scheme. If $X$ is a general scheme (instead of, e.g., a point or a projective curve) the stack of vector bundles is unlikely to be algebraic, and nether simplifying option looks promising.
My wild guess is that you intend the Euclidean site to be an analogue of the category of affine schemes of finite type. You can form a notion of stack in topological spaces, by following the usual fibered category route, and you can certainly restrict to the subcategory of open subsets of Euclidean space. As Johannes Ebert mentioned in the comments, Noohi has some papers online that describe topological stacks. Some names that show up in the smooth setting include Alan Weinstein, Cristian Blohmann, and Chenchang Zhu (but I am relatively unfamiliar with this area).
Right now, there is an upper bound on the information content in mathematical abstraction given by the finite size of the human brain. Even if the robots take over, there is the finite size of the observable universe. More to the point at hand, objects more abstract than stacks were already considered in algebraic geometry during the 20th century. For example Grothendieck's Pursuing Stacks is one of the early attempts to apply homotopy theory techniques to work with more abstract objects like $n$-categories and $n$-stacks. I am not qualified to answer your revised question about fundamental ideas.
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I would like to point out that stacks are "just" higher analogues of sheaves - a very basic tool to arrange structure. The same is true for topological or differentiable stacks. So I think everybody who expects amazing applications of stacks should be able to name an equally amazing application of a sheaf. (I am not saying that those don't exist!)
That said, let me mention an application. In view of the fact that you didn't get any answers so far (apart from your own), I hope it's not too inappropriate to take one from my own research. It applies abelian gerbes with connection to lifting problems for principal bundles.
I hope the following specifications qualify the theorem below as application: its statement does not involve any stacks or gerbes, just "basic" differential geometry. Its proof, however, is a simple composition of two gerbe-theoretical theorems.
Of course some concepts that appear here would need some more explanation - but that's not the point. Let me better point out how gerbes with connection come into the picture. We employ two results from gerbe theory:
Associated to every lifting problem posed by a bundle $P$ is an $A$-gerbe over $M$, called the "lifting gerbe" and denoted $\mathcal{G}_P$. This gerbe represents geometrically the obstruction against lifts. Moreover, the actual lifts are in equivalence with trivializations of $\mathcal{G}_P$. The same works if one wants to include connections into the lifting problem. These are results of Murray and Gomi.
The category of $A$-gerbes with connection over $M$ is equivalent to a certain category of principal $A$-bundles with connection over $LM$ which are additionally equipped with "fusion products". The equivalence is established by a transgression functor, which has been introduced by Brylinski and McLaughlin. It takes trivializations of gerbes to sections of bundles.
Now, define $\mathcal{L}_P$ as the transgression of $\mathcal{G}_P$. Since transgression is an equivalence of categories, it is a bijections on Hom-sets, and this bijection is exactly the statement of the theorem.
Ok, in order to complete my claim that this is an application, I should probably mention an example where the theorem is useful. That's the case for $spin$ and $spin^c$ structures on manifolds, and I have learned about it from Stephan Stolz and Peter Teichner. In the case of $spin$ structures, $\mathcal{L}_P$ is a $\mathbb{Z}_2$-bundle over $LM$ and plays the role of the orientation bundle of $LM$. Since $\mathbb{Z}_2$ is discrete, all the connections disappear and forms are identically zero. So, the theorem says that isomorphism classes of $spin$ structures on $M$ are in bijection to "fusion-preserving orientations" of $LM$. In the $spin^c$ case, a similar statement follows that additionally includes the scalar curvature of the $spin^c$-structures.