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 ...
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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