Questions tagged [differentiable-stacks]

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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{...
Connor Grady's user avatar
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 ...
Spai's user avatar
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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 ...
Spai's user avatar
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1 vote
0 answers
64 views

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 ...
Josh Lackman's user avatar
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5 votes
1 answer
393 views

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 ...
André Henriques's user avatar
2 votes
1 answer
149 views

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 $...
Emilio Minichiello's user avatar
16 votes
2 answers
2k 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 ...
Bumblebee's user avatar
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2 votes
0 answers
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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\...
Praphulla Koushik's user avatar
3 votes
0 answers
172 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 ...
Praphulla Koushik's user avatar
4 votes
0 answers
265 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; ...
Praphulla Koushik's user avatar
5 votes
0 answers
182 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 ...
Mohan Swaminathan's user avatar
4 votes
1 answer
622 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 ...
John Pardon's user avatar
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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 ...
John Pardon's user avatar
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6 votes
0 answers
130 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 ...
John Klein's user avatar
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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 ...
Praphulla Koushik's user avatar
2 votes
0 answers
322 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....
Praphulla Koushik's user avatar
5 votes
0 answers
118 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?
User's user avatar
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2 votes
2 answers
505 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 ...
Praphulla Koushik's user avatar
3 votes
0 answers
150 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 ...
Praphulla Koushik's user avatar
4 votes
1 answer
513 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 ...
Praphulla Koushik's user avatar
3 votes
1 answer
246 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 ...
Praphulla Koushik's user avatar
1 vote
0 answers
287 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 $\...
Praphulla Koushik's user avatar
2 votes
1 answer
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 ...
Praphulla Koushik's user avatar
3 votes
2 answers
537 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 $\...
Praphulla Koushik's user avatar