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Questions tagged [differentiable-stacks]

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Fibered product of stacks comes from a Lie groupoid

Suppose $\mathcal{G},\mathcal{H}$ are Lie groupoids and $F:B\mathcal{G}\rightarrow B\mathcal{H}$ be a morphism of stacks. We can talk about the fibered product $B\mathcal{G}\times_{B\mathcal{H}}B\...
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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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1answer
230 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 ...
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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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Confusion in the definition of epimorphism of morphism of categories fibered in groupoids

I am reading Differentiable Stacks and Gerbes by Kai Behrend and Ping Xu. According to this paper, a morphism of groupoid fibrations (Categories fibered in groupoids over the category of manifolds) $...
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Lie groupoid with non trivial automorphisms does not come from genuine space

This question came from comments of this question. Suppose $\mathcal{G}$ is a Lie groupoid, we consider the category $B\mathcal{G}$ whose objects are principal $\mathcal{G}$ bundles and morphisms are ...
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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 a smooth manifold $X$ and a map of stacks $p:\underline{X}\rightarrow \mathcal{D}$ such that, for any manifold $M$ and a map of ...
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2answers
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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 $\...