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20 votes
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Understanding the definition of stacks

A canonical example of a sheaf of sets on a topological space $X$ is the sheaf that sends an open subset $U$ of $X$ to the set of continuous real-valued functions on $U$. The gluing property then says ...
Dmitri Pavlov's user avatar
9 votes

Understanding the definition of stacks

What categories fibered in groupoids over $\mathcal{C}$ corresponds to stacks? A category fibered in groupoids over $\mathcal{C}$ is given by a functor $p_{\mathcal{F}}:\mathcal{F}\rightarrow \...
Praphulla Koushik's user avatar
7 votes
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Categorifying the definition of a principal $G$ bundle

The definition you are looking for is precisely Def. 6.1.5 in: Nikolaus, Thomas; Waldorf, Konrad, Four equivalent versions of nonabelian gerbes, Pac. J. Math. 264, No. 2, 355-420 (2013). ZBL1286.55006....
Konrad Waldorf's user avatar
6 votes

Categorifying the definition of a principal $G$ bundle

A good place to start is : Larry Breen. Notes on 1-and 2-gerbes. In J. Baez and J. May, editors, Towards Higher Categories, volume 152 of The IMA Volumes in Mathematics and its Applications, pages ...
Tim Porter's user avatar
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6 votes
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Geometric realisation of smooth $\infty$-stacks

The case when $M$ is a smooth manifold follows from the smooth Oka principle. See there for an expository account of the argument and references to additional sources. Indeed, the left side of (*) is $...
Dmitri Pavlov's user avatar
6 votes

Stacks as local quotients or via atlases

I'm not entirely sure if this qualifies as an answer, but it is certainly too long for a comment. I hope that someone else will give a better answer. If you want to define an algebraic stack as ...
R. van Dobben de Bruyn's user avatar
5 votes
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Stack being represented by a scheme/manifold

If all objects of a stack have trivial automorphism groups then it is equivalent to a sheaf, as pointed out by Daniel Litt in the comments. Pick your favourite non-representable sheaf as a ...
David Roberts's user avatar
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5 votes
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Morita equivalence of Lie groupoids and isomorphism of differentiable stacks

The "well-known fact" is simply not true if you assume "isomorphic stacks" means literally isomorphic (say as fibred categories). My impression is that people who work in certain ...
David Roberts's user avatar
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4 votes
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What does it mean for a space to be a differentiable stack?

Stacks form an (∞,1)-category. The latter informal notion has many equivalent implementations: simplicial category, topological category, quasicategory (also known as ∞-category), Segal category, ...
Dmitri Pavlov's user avatar
4 votes
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Anafunctors vs the plus construction

The long-expected answer. $\DeclareMathOperator{\op}{op} \DeclareMathOperator{\Cat}{\mathbf{Cat}}\DeclareMathOperator{\Gpd}{\mathbf{Gpd}} \DeclareMathOperator{\disc}{disc}\DeclareMathOperator{\pr}{pr}$...
David Roberts's user avatar
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4 votes

What is the local structure of a general Artin stack?

The stack of curves of genus 0 with at most one node is a quotient stack (see The integral Chow ring of the stack of at most 1-nodal rational curves, but Edidin and Fulghesu), so you are fine in this ...
Angelo's user avatar
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4 votes

Categorifying the definition of a principal $G$ bundle

A modern presentation that fully covers the indicated cases can be found in the work of Nikolaus–Schreiber–Stevenson: Principal ∞-bundles – General theory. Principal ∞-bundles – Presentations. In ...
Dmitri Pavlov's user avatar
3 votes
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Examples of of gerbe over stacks in terms of manifolds

There are no other such gerbes. If $M$ and $N$ are manifolds, and $p\colon \underline{M}\to \underline{N}$ is a gerbe, then the corresponding map of manifolds is a diffeomorphism. The same holds if ...
David Roberts's user avatar
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3 votes
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Fibered product of stacks comes from a Lie groupoid

Think of $BG$ and $BH$ as topological stacks, whereby one can calculate a topological groupoid presenting the stack $BG\times_{BH} BG$, namely the following: the object space is the space underlying $...
David Roberts's user avatar
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3 votes

Fibered product of stacks comes from a Lie groupoid

Pullbacks of stacks coming from Lie groupoids are not always equivalent to Lie groupoids. Take $G=H=\mathbb{R}$. Define $F(x)=0$ if $x\leq 0$ and $F(x)=exp(−1/x^2)$ if $x>0$. The pullback is not ...
Dmitri Pavlov's user avatar
2 votes
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Necessary and sufficient conditions for a Lie groupoid to present a stack

Note that $\hat r\mathcal{G}$ is always a prestack. This is basically equivalent to the fact that $C^\infty(-,G_1)$ is a sheaf, and it means that your functor $$ \hat r\mathcal{G}(U) \to \mathrm{...
Konrad Waldorf's user avatar
2 votes

Understanding definition of gerbe over a stack

If R→X is an epimorphism, then each fiber F_x is nonempty. Then if R→R ×_X R is an epimorphism, this means that π_0(F_x) is a point (otherwise π_0(F_x)→π_0(F_x) × π_0(F_x) cannot be an epimorphism), i....
Dmitri Pavlov's user avatar
2 votes
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Stack associated to Groupoid object in category $\text{Sch}/S$

For any algebraic stack $X$, there is a groupoid in schemes whose fppf stackification is equivalent to $X$. You can construct such a groupoid following the stacks project, by choosing a smooth ...
S. Carnahan's user avatar
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2 votes
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To check if a stack is coming from a manifold

Take $M = X$ and $f=p$, so that $\underline{P} = \underline{X} \times_\mathcal{D} \underline{X}$. The Lie groupoid $P\rightrightarrows X$ you get should be proper, in the sense the source-target map $(...
David Roberts's user avatar
  • 34.7k
1 vote
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Understanding definition of gerbe over a stack

I am trying to write down what does it mean to say those two maps $\mathcal{D}\rightarrow \mathcal{C}$ and $\mathcal{D}\rightarrow \mathcal{D}\times_{\mathcal{C}}\mathcal{D}$ to be epimorphisms. I am ...
Praphulla Koushik's user avatar

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