# 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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### 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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### 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 $\...