Questions tagged [differentiable-stacks]
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30 questions
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Morita equivalence of Lie groupoids and isomorphism of differentiable stacks
It's a well known fact two Lie groupoids are Morita-equivalent iff they induce isomorphic differentiable stacks (I'll call this statement "(1)").
It's also well known that there is a ...
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61
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Stack of smooth fiber bundles with fiber $F$
I'd like to premise that while I know the definition of (differentiable) stack, I'm not really into the language of schemes so my understanding of what is a moduli stack is pretty concrete and ...
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131
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Bibundle induced by a morphism of stacks
[This is a repost, because I've written the wrong page number in the previous version of this question. I'm sorry]
I'm currently reading "Orbifolds as stacks" by Eugene Lerman and I'm stuck ...
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Cohomology of differentiable stacks: should the sheaf be fine?
I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fifth page.
Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...
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Cohomology of a differentiable stack: evaluation at a point
I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fourth page.
Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...
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5
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1
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What does it mean for a space to be a differentiable stack?
(I'd like to premise that I'm not an expert about these topics (just a student), so many of my doubts and perplexities are probably symptoms of my mathematical immaturity)
I'm currently studying ...
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6
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1
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402
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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
$$G(M) := \text{...
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3
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Categorifying the definition of a principal $G$ bundle
For a Lie group $G$, we can define a principal $G$ bundle as a submersion of manifolds $\pi:P \to X$ equipped with a free right $G$-action on $P$ that is transitive on the fibres over $X$.
What goes ...
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On the exactness of the restriction to small etale site in the case of differentiable stacks
Let $X$ be a manifold considered as a stack on the big etale site of manifolds. Consider the functor $$ F \mapsto F_X$$ which sends a sheaf on the stack $X$ to the sheaf on the manifold $X$.
In Lemma ...
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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 ...
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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 ...
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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 $...
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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 ...
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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\...
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3
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184
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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 ...
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4
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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; ...
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190
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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 ...
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1
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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 ...
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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 ...
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6
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132
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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 ...
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2
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1
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558
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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 ...
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348
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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....
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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?
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530
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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 ...
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156
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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 ...
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1
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576
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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 ...
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3
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1
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271
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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 ...
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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 $\...
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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
...
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3
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2
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578
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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 $\...
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