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I am not an expert in the algebraic category, however, I know that 2) is an open problem in the differentiable category; it is not known if every smooth orbifold is a global quotient stack. It is true …

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). …

However, you should be a little careful with your proof: Behrang Noohi's develops a homotopy theory for topological stacks. … For example, paratopological and pseudotopological stacks (see: Homotopy Theory of Topological stacks by Behrang Noohi). …

$ sends holomorphic stacks (stacks coming from groupoid objects in complex manifolds) to differentiable stacks. … $, when restricted to holomorphic stacks agrees with the answer of David Roberts, only, it makes no explicit reference to atlases. …

I would take the view point that a monoidal category is a bicategory with one object. (Then a category fibered in monoidal categories should be the same thing as a weak functor into bicategories that …

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 …

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

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 …

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 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 …

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 …

P.S., everything I said will hold for stacks as well. …

If $X_1 \rightrightarrows X_0$ is a groupoid and $\mathcal{X}$ is the associated stack, consider the atlas $p:X_0 \to \mathcal{X}$. If you form the weak $2$-pullback of $p \times p:X_0 \times X_0 \to …

.$ 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. …

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. …