# Questions tagged [differentiable-stacks]

The differentiable-stacks tag has no usage guidance.

24
questions

6
votes

1
answer

354
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### Anafunctors vs the plus construction

Given a Lie groupoid $G$, we can view it as representing a prestack on $\text{Mfld}$ by sending and manfold $M$ to the groupoid of smooth functors and smooth natural transformations
$$G(M) := \text{...

3
votes

3
answers

338
views

### Categorifying the definition of a principal $G$ bundle

For a Lie group $G$, we can define a principal $G$ bundle as a submersion of manifolds $\pi:P \to X$ equipped with a free right $G$-action on $P$ that is transitive on the fibres over $X$.
What goes ...

3
votes

0
answers

59
views

### On the exactness of the restriction to small etale site in the case of differentiable stacks

Let $X$ be a manifold considered as a stack on the big etale site of manifolds. Consider the functor $$ F \mapsto F_X$$ which sends a sheaf on the stack $X$ to the sheaf on the manifold $X$.
In Lemma ...

1
vote

0
answers

64
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### Is there an inverse image functor for sheaves on stacks?

I'm interested specifically in an inverse image functor between differentiable stacks, ie. stacks coming from Lie groupoids. Specifically, if I have a morphism of Lie groupoids $H\to G$ and I have a ...

5
votes

1
answer

393
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### Geometric realisation of smooth $\infty$-stacks

Let $Sh^\infty(\mathsf{Man})$ denote the $\infty$-category of sheaves of $\infty$-groupoids over the site $\mathsf{Man}$ of smooth manifolds (if you prefer, that's the model category of simplicial ...

2
votes

1
answer

149
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### Necessary and sufficient conditions for a Lie groupoid to present a stack

Let $\mathcal{G} = G_1 \rightrightarrows G_0$ be a Lie Groupoid (although I am also interested in groupoids internal to other sites), the stack associated to $\mathcal{G}$, which is sometimes denoted $...

16
votes

2
answers

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### Understanding the definition of stacks

First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...

2
votes

0
answers

56
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### Lie groupoid $G$ extensions and principal $\text{Out}(G)$ bundles over Lie groupoids

I am reading the paper Non abelian differentiable gerbes by C. Laurent-Gengoux et.al.
The definition $3.4$ of the paper goes as follows:
Definition : Let $X_1\xrightarrow{\phi} Y_1\...

3
votes

0
answers

172
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### Lie group (topological group) action on differentiable stack (topological stack)

Let $G$ be a Lie group and $\mathcal{D}$ be a differentiable stack (I am also ok to start with a topological group and topological stack).
I have seen someone mentioning somewhere that the notion of ...

4
votes

0
answers

265
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### connection on principal bundles over algebraic/geometric stacks

Is there a notion of connection on a principal bundle over an algebraic or geometric stack?
By a geometric stack, I mean a stack over category of manifolds that is representable by a Lie groupoid; ...

5
votes

0
answers

182
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### Compactly supported cohomology of a topological DM stack

Consider a separated topological Deligne-Mumford stack $\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $\mathfrak X$ is locally ...

4
votes

1
answer

622
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### What is the local structure of a general Artin stack?

Let $X$ be an Artin stack over the complex numbers. What can one say about the local structure of $X$, i.e. what is the simplest class of stacks by which were can always find a cover of $X$ by open ...

3
votes

1
answer

526
views

### Stacks as local quotients or via atlases

If one looks up the definition of a Deligne--Mumford stack or an Artin stack, one usually finds something like:
A DM (resp. Artin) stack is a stack $X$ satisfying [insert condition in the diagonal ...

6
votes

0
answers

130
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### On the weak homotopy type of a differentiable (Chen) space

Suppose that $M$ is a differentiable space in the sense of Chen (cf. https://ncatlab.org/nlab/show/Chen+space ).
Assume that $M$ also has the structure of a topological space and that the two ...

2
votes

1
answer

464
views

### Stack associated to Groupoid object in category $\text{Sch}/S$

Consider the category of manifolds $\text{Man}$.
A groupoid object in the category of manifolds is called a Lie groupoid, denoted by $\mathcal{G}$. There is a way to associate a stack (over the ...

2
votes

0
answers

322
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### Notions of algebraic/differential geometry of scheme/manifolds extended to algebraic/differential stacks

Given a manifold, one can associate a stack over the category of manifolds, which is a differential geometric stack. This gives a functor $\text{Man}\rightarrow \text{D.Stacks}$. This is an embedding....

5
votes

0
answers

118
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### Maps between simplicial manifolds

Is it true that every continuous simplicial map between smooth simplicial manifolds can be approximate (in some adequate sense) by a smooth simplicial map?

2
votes

2
answers

505
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### 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 ...

3
votes

0
answers

150
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### 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 ...

4
votes

1
answer

513
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### Stack being represented by a scheme/manifold

On page $10$ of the survey article Algebraic stacks, by T. Gomez (arXiv:math/9911199), we have following result
If a stack has an object with an automorphism other than the identity, then the ...

3
votes

1
answer

246
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### Examples of of gerbe over stacks in terms of manifolds

I am looking for some examples of gerbes over stacks (as defined in Understanding definition of gerbe over a stack) that comes from manifolds.
Let $M$ be a manifold then $\underline{M}$ is a stack ...

1
vote

0
answers

287
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### Atlas of gerbe over stack

Suppose $F:\mathcal{X}\rightarrow\mathcal{Y}$ is gerbe over stack and $p:X\rightarrow \mathcal{X}$ is an atlas $\mathcal{X}$.
Does this imply $F\circ p:X\rightarrow \mathcal{Y}$ is an atlas for $\...

2
votes

1
answer

2k
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### To check if a stack is coming from a manifold

Let $\mathcal{D}$ be a stack. An atlas for stack $\mathcal{D}$ is given by
a smooth manifold $X$ and
a map of stacks $p:\underline{X}\rightarrow \mathcal{D}$
such that, for any
manifold $M$ and
...

3
votes

2
answers

537
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### Understanding definition of gerbe over a stack

I am reading Differentiable Stacks and Gerbes by Kai Behrend and Ping Xu.
They define gerbe over a stack as follows.
Let $\mathfrak{X}$ be a differentiable stack. An $\mathfrak{S}$-stack $\...