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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). …

I am reading Orbifolds as stacks? … I am not aware of seeing bibundles as relation of stacks. Can some one help me to see bibundles as relation of stacks. …

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 …

Unfortunately, I myself have seen exactly four research articles (Noohi - Foundations of topological stacks I; Carchedi - Categorical properties of topological and differentiable stacks; Noohi - Homotopy … types of topological stacks; Metzler - Topological and smooth stacks) talking about stacks over the category of topological spaces. …

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. …

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 …

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\ …

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 …

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. …

So, geometric stacks are same as Lie groupoids; the ”geometric” stacks of interest over the category of manifolds are precisely the stacks associated to groupoid objects in the category of manifolds. … Are stacks over $\text{Sch}/S$ associated to groupoids over $\text{Sch}/S$ of any interest? Do they cover “most” of Algebraic stacks over the stack $S$ with what ever topology on $\text{Sch}/S$? …

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. …

I am reading Differentiable Stacks and Gerbes to understand about (hyper) cohomology groups of a stack $\mathcal{X}$ with values in a complex $\mathcal{M}$ of abelian sheaves over $\mathcal{X}$. …

Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of topological spaces on $X$. Then, the map $U\mapsto \pi_1(\mathcal{F}(U))$ for $U\subseteq X$ open is a gerbe over $X$. I learned this e …

What categories fibered in groupoids over $\mathcal{C}$ corresponds to stacks? … Orbifolds as stacks? How is a Stack the generalisation of a sheaf from a 2-category point of view? …

Above definition is from the notes Introduction to the language of gerbe and stacks by Ieke Moerdijk. I have had a look at the notes on 1-gerbes and 2-gerbes by Lawrence Breen. …