20
votes

Accepted

### 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 ...

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 \...

7
votes

Accepted

### 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....

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 ...

6
votes

Accepted

### 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 $...

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 ...

5
votes

Accepted

### 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 ...

5
votes

Accepted

### 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 ...

4
votes

Accepted

### 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, ...

4
votes

Accepted

### Anafunctors vs the plus construction

The long-expected answer. $\DeclareMathOperator{\op}{op} \DeclareMathOperator{\Cat}{\mathbf{Cat}}\DeclareMathOperator{\Gpd}{\mathbf{Gpd}} \DeclareMathOperator{\disc}{disc}\DeclareMathOperator{\pr}{pr}$...

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 ...

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 ...

3
votes

Accepted

### 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 ...

3
votes

Accepted

### 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 $...

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 ...

2
votes

Accepted

### 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{...

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....

2
votes

Accepted

### 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 ...

2
votes

Accepted

### 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 $(...

1
vote

Accepted

### 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 ...

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