# Questions tagged [differentiable-stacks]

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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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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 $X_1\xrightarrow{\phi} Y_1\... 0answers 59 views ### Lie group (topological group) action on differentiable stack (topological stack) Let$G$be a Lie group and$\mathcal{D}$be a differentiable stack (I am also ok to start with a topological group and topological stack). I have seen someone mentioning somewhere that the notion of ... 0answers 173 views ### connection on principal bundles over algebraic/geometric stacks Is there a notion of connection on a principal bundle over an algebraic or geometric stack? By a geometric stack, I mean a stack over category of manifolds that is representable by a Lie groupoid; ... 0answers 134 views ### Compactly supported cohomology of a topological DM stack Consider a separated topological Deligne-Mumford stack$\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently,$\mathfrak X$is locally ... 1answer 491 views ### What is the local structure of a general Artin stack? Let$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$by open ... 1answer 364 views ### Stacks as local quotients or via atlases If one looks up the definition of a Deligne--Mumford stack or an Artin stack, one usually finds something like: A DM (resp. Artin) stack is a stack$X$satisfying [insert condition in the diagonal ... 0answers 115 views ### On the weak homotopy type of a differentiable (Chen) space Suppose that$M$is a differentiable space in the sense of Chen (cf. https://ncatlab.org/nlab/show/Chen+space ). Assume that$M$also has the structure of a topological space and that the two ... 1answer 228 views ### Stack associated to Groupoid object in category$\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}$. There is a way to associate a stack (over the ... 0answers 246 views ### Notions of algebraic/differential geometry of scheme/manifolds extended to algebraic/differential stacks Given a manifold, one can associate a stack over the category of manifolds, which is a differential geometric stack. This gives a functor$\text{Man}\rightarrow \text{D.Stacks}$. This is an embedding.... 0answers 107 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? 2answers 398 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 ... 0answers 138 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 ... 1answer 313 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 ... 1answer 198 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 ... 0answers 232 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$\...
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 ...
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 \$\...