Questions tagged [differentiable-stacks]

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14
votes
2answers
1k views

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 ...
5
votes
0answers
115 views

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 ...
4
votes
1answer
291 views

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 ...
4
votes
1answer
426 views

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 ...
4
votes
0answers
164 views

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; ...
4
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0answers
121 views

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
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0answers
105 views

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?
3
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1answer
332 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 ...
3
votes
1answer
185 views

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 ...
3
votes
2answers
388 views

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 $\...
3
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0answers
134 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 ...
2
votes
1answer
180 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
1answer
2k views

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 ...
2
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0answers
55 views

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 ...
2
votes
0answers
213 views

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....
1
vote
2answers
353 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 ...
1
vote
0answers
34 views

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\...
1
vote
0answers
215 views

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 $\...