All Questions
Tagged with dg.differential-geometry stacks
33 questions
22
votes
3
answers
1k
views
Applications of topological and diferentiable stacks
What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well ...
- 15.5k
20
votes
7
answers
3k
views
What are the occurrences of stacks outside algebraic geometry, differential geometry, and general topology?
What are the occurrences of the notion of a stack outside algebraic geometry, differential geometry, and general topology?
In most of the references, the introduction of the notion of a stack takes ...
- 6,023
12
votes
4
answers
2k
views
Motivation for definition of Quotient stack
I am reading "Some notes on Differentiable stacks" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows.
Let $G$ be a Lie group action on a manifold $X$ (left ...
- 6,023
11
votes
2
answers
1k
views
Diffeology as a sheaf on the site of smooth manifolds
Souriau's definition of diffeology may be phrased as defining a concrete sheaf on the category $\mathsf{Open}$ of open subsets of Euclidean/coordinate spaces. It seems to me, unless I am missing ...
- 1,710
11
votes
1
answer
1k
views
Teaching stacks to differential geometry students
Does anyone have any experience teaching stacks over the category of manifolds to students whose background is, say, a semester-long course on manifolds? Does anyone know of any publicly available ...
- 1,710
10
votes
3
answers
1k
views
Connections on principal bundles via stacks?
Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to [G]$, where $[G]$ is the differentiable stack ...
- 15.5k
8
votes
1
answer
524
views
Stacks over diffeologies
Konrad Waldorf shows in his paper one may realize a Grothendieck topology on the category of diffeological spaces. Is there any work exploring stacks over the category of diffeologies?
- 83
8
votes
1
answer
281
views
Stack associated to Lie group and manifold
Given a Lie group $G$, we have a Lie groupoid $(G\rightrightarrows *)$ and stack $BG=B\mathcal{G}$ of principal $G$ bundles.
Given a smooth manifold $M$, we have Lie groupoid $(M\rightrightarrows M)...
- 6,023
7
votes
3
answers
1k
views
How should one think about the band of a gerbe?
Let $X$ be a topological space. Let $\mathcal{F}$ be a fibered category over $X$; seen as an assignment of a category $\mathcal{F}(U)$ for each open $U\subseteq X$.
A fibered catgeory $\mathcal{F}$...
- 6,023
7
votes
4
answers
5k
views
Cotangent bundle of a differentiable stack
If you ever wanted to construct the tangent bundle of a differentiable stack, it's relatively simple:
First, if $\mathbf{X}$ is a stack coming from a Lie groupoid $\mathcal{G}$, you could just say $\...
- 15.5k
7
votes
1
answer
403
views
What is the local structure of a Lie groupoid?
A manifold is locally $\mathbb R^n$. An orbifold is locally $\mathbb R^n/\{\text{finite group}\}$. Is there a similar way to think about the local structure of a Lie groupoid $G_1 \rightrightarrows ...
- 54.6k
7
votes
3
answers
813
views
Is there a "geometric" language that describes the equivalence groupoid of a foliated manifold?
Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions.
Here, ...
- 54.6k
7
votes
1
answer
1k
views
Understanding the definition of $G$-gerbe
In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following.
Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\...
- 6,023
6
votes
1
answer
373
views
How to differentiate natural transformations?
Let $G_{\bullet} = (G_1 \rightrightarrows G_0)$ and $H_{\bullet} = (H_1 \rightrightarrows H_0)$ be Lie groupoids and $\varphi_\bullet, \psi_\bullet : G_\bullet \to H_\bullet$ be Lie groupoid morphisms,...
- 325
4
votes
2
answers
633
views
Based loop groups as stacks?
I have been stuck for some time, thinking about the following question.
Let $G$ be a Lie group. Its classifying space $BG$ can be seen as the differentiable stack $[pt/G]$, which is of dimension $-...
- 123
4
votes
1
answer
247
views
unit element under map of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$
Let $\mathcal{G}$ be a Lie groupoid. The target map $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ is a principal $\mathcal{G}$ bundle.
This article Orbifolds as Stacks? by Eugene Lerman calls (in page $...
- 6,023
4
votes
1
answer
334
views
Gluing together together differentiable stacks
I am trying to figure out the conditions under which you can glue together a collection of (differentiable) stacks by equivalences, and get a differentiable stack.
More precisely, I have a collection ...
- 43
4
votes
1
answer
154
views
Co-completeness of differential stacks?
I once heard a rumour that various nice categories of stacks were co-complete. Gepner and Henriques, working from the groupoids point of view, give a construction [link] of 2-colimits of topological ...
4
votes
1
answer
541
views
Intrinsic Characterization of when an orbifold (or more general stack) is effective?
Recall that an orbifold is an etale and proper differentiable stack $X$. Etale means that it admits an etale atlas $M \to X$ from a manifold $M$ (which is to say it is represented by an etale Lie ...
- 15.5k
3
votes
3
answers
522
views
Lie groupoids in practice
I am familiar with the notion of Lie groupoids.
But, only easy examples of Lie groupoids I am familiar with are the following:
Lie groupoids coming from manifolds; that are of the form $(M\...
- 6,023
3
votes
1
answer
1k
views
Diagonal is representable then any morphism is representable
Ariyan Javanpeykar said here in comments that,
If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable?
I could not find the statement (...
- 6,023
3
votes
1
answer
196
views
Can stabilizer groups in an orbifold have global twisting?
Can stabilizer groups in an orbifold have global twisting?
For example, consider the two groups $\mathbb Z/3\times\mathbb Z$ and $\mathbb Z/3\rtimes\mathbb Z$ (where $\mathbb Z\to\operatorname{Aut}(\...
- 18.7k
3
votes
0
answers
84
views
stack (in groupoids) over a site $\mathcal{C}$
Question : What is a stack (in groupoids) over a site $\mathcal{C}$ for you?
There are two a ways to think about it.
A stack over a site $\mathcal{C}$ is a category $\mathcal{D}$ with a functor $\...
- 6,023
3
votes
0
answers
156
views
Criterion for a sheaf $\mathfrak{S}^{op}\rightarrow (Set)$ to be representable
I am reading Differentiable stacks and gerbes by Kai Behrend and Ping Xu.
Let $\mathfrak{S}$ denote the category of smooth manifolds and smooth maps. Consider Grothendieck topology given by open ...
- 6,023
2
votes
2
answers
530
views
Fibered product of stacks comes from a Lie groupoid
I am adding some context here. I am reading Introduction to Differentiable Stacks by Gregory Ginot.
In page no $7$, just before the remark $2.2$ he says the following.
One shall be careful that ...
- 6,023
2
votes
1
answer
246
views
$C^*$-algebras appearance in study of Lie groupoids and differentiable stacks
I am reading Differentiable stacks, gerbes, and twisted K-Theory by Ping Xu.
To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All ...
- 6,023
2
votes
1
answer
839
views
Sheaf / de Rham cohomology of a stack with values in a complex of abelian sheaves
I am reading Differentiable Stacks and Gerbes to understand about (hyper) cohomology groups of a stack $\mathcal{X}$ with values in a complex $\mathcal{M}$ of abelian sheaves over $\mathcal{X}$.
...
- 6,023
2
votes
1
answer
401
views
Composition of bibundles
I am reading Orbifolds as stacks?
Given Lie groupoids $\mathcal{G}$ and $\mathcal{H}$ there is a notion of what is called a bibundle from $\mathcal{G}$ to $\mathcal{H}$ which is supposed to be a ...
- 6,023
2
votes
1
answer
229
views
Central extension gives a gerbe over stack
Consider a central extension of Lie groups $1\rightarrow S^1\rightarrow \hat{G}\xrightarrow{\pi} G\rightarrow 1$.
I understand that this mean $\pi:\hat{G}\rightarrow G$ is a surjective homomorphism ...
- 6,023
2
votes
0
answers
192
views
Diagonal is representable then composition is representable
Let $\mathcal{X}$ be a stack over $S$ i.e., a stack over category of schemes over $S$ (which we denote by $Sch/S$) which comes with a functor $\mathcal{X}\rightarrow Sch/S$. Consider the diagonal map ...
- 6,023
2
votes
0
answers
271
views
Einstein's field equation on orbifolds
I was wondering if there is some kind of Einstein's field equation for orbifolds (say semi-Riemannian of Lorentz signature if this make sense).
Here, by an orbifold I mean the "stacky" quotient of, ...
user95283
1
vote
0
answers
115
views
connections on Lie groupoids/differentiable stacks
Let $(\Gamma_1\rightrightarrows \Gamma_0)$ be a Lie groupoid.
There are many places which define the notion of connection on a Lie groupoid.
As far as I have seen, there is no mention of these ...
- 6,023
1
vote
0
answers
184
views
Pushforward for differentiable stacks/ Lie groupoids
Let $X$ be a differentiable stacks, and let $(G_{0}, G_{1}, s,t)$ be a Lie groupoid representing $X$. Let $NG_{\bullet}$ be the nerve of the above groupoid. The De rham complex of $X$ can be defined ...
- 1,424