19 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 \...
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 $...
5 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 ...
  • 31.8k
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 ...
  • 26.2k
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 ...
  • 31.8k
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 $...
  • 31.8k
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 ...
  • 43.4k
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 $(...
  • 31.8k
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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