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We will also drop the distinction between stacks isomorphic to manifolds and manifolds. … If it helps, I am trying to read about geometric stacks which are defined to be stacks over manifolds which possesses an atlas. …

is a way to associate a stack $\underline{M}$ with it and this gives an embedding of category of smooth manifolds into category of Categories fibered in groupoids which can be found in Orbifolds as Stacks …

Suppose $p:X\rightarrow \mathcal{X}$ is representable, then, for any scheme $M$ with a map of stacks $M\rightarrow \mathcal{X}$, the $2$-fiber product $X\times_{\mathcal{X}}M$ is a scheme. … Then, I want to see that $X\rightarrow \mathcal{X}$ is representable i.e., hen, for any scheme $M$ with a map of stacks $M\rightarrow \mathcal{X}$, the $2$-fiber product $X\times_{\mathcal{X}}M$ is a scheme …

Let $(\Gamma_1\rightrightarrows \Gamma_0)$ be a Lie groupoid. There are many places which define the notion of connection on a Lie groupoid. As far as I have seen, there is no mention of these not …

Following lemma is from Kai Behrend and Ping Xu's paper (page $8$, lemma $2.11$) Differentiable Stacks and Gerbes. Let $f:\mathcal{X}\rightarrow \mathcal{Y}$ be a morphisms of stacks. … So, Consider a morphism of stacks $f:X\rightarrow \mathcal{Y}$. Suppose that $X\times_\mathcal{Y}X$ is representable. …

gerbe territory (for smooth manifolds) if any one of the following is being considered a cohomology class in $H^3(X,\mathbb{Z})$ ** ** In similar manner, When reading about gerbes over stacks … Can some one give me some outline of how and what cohomology comes in when studying about gerbes over stacks? …

An atlas for a stack $\mathcal{D}\rightarrow Man$ is a smooth manifold $X$ and a map of stacks $p:\underline{X}\rightarrow \mathcal{D}$ such that, given a smooth manifold … $M$ and a map of stacks $f:\underline{M}\rightarrow \mathcal{D}$ the fibered product stack $\underline{M}\times_{\mathcal{D}}\underline{X}\rightarrow Man$ is isomorphic to a stack coming …

These days I am trying to understand about stacks and gerbes. Most of the articles that has something to do with gerbes cite this work Cohomologie non abélienne by Jean Giraud. …

"As locally any map $T\rightarrow G$ can be lifter to $\tilde{G}$", the map of stacks $[X/\hat{G}]\rightarrow [X/G]$ is a gerbe over stack. …

We have corresponding stacks associated : $(G\rightrightarrows *)$ gives stack $B(G\rightrightarrows *)$, usually denoted by $BG$. … Feel free to (I request you to) relate this to algebraic geometry version of stacks. …

Let $\mathcal{C}$ be a category. Fixing an object $U$ of $\mathcal{C}$, there are some obvious functors we can associate to it, for example: $h_U:\mathcal{C}^{op}\rightarrow \text{Set}$ given by $V\ …

Given Lie groupoids $\mathcal{G},\mathcal{H}$ a morphism of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$ comes from what is called a $\mathcal{G}-\mathcal{H}$ bibundle $P$. … This can be found in proposition $3.36$ of Orbifold as stacks. I think similar result in case of Algebraic geometry can be said. …

This definition can be found online and in particular in the paper Orbifolds as Stacks by Eugene Lerman. The category of these principal $\mathcal{G}$ bundles is denoted by $B\mathcal{G}$. …

I am familiar with the notion of Lie groupoids. But, only easy examples of Lie groupoids I am familiar with are the following: Lie groupoids coming from manifolds; that are of the form $(M\rightri …

I am reading "Some notes on Differentiable stacks" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows. Let $G$ be a Lie group action on a manifold $X$ (left action). …