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Ariyan Javanpeykar said here in comments that, $X\times_{\mathcal{X}}X$ being a scheme is equivalent to representability of $X\rightarrow \mathcal{X}$. Context is as in this question. Suppose $ …

Question is as in the title: Can some one point me (if there is any) to an English translation of Pierre Cartier's PhD thesis?

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. Suppose giv …

This is not a research level problem for sure. But, similar question was asked by some one else $2$ years back on Stack exchange has not received any attention. So, I thought it does not suit there …

Given a Lie group $G$, we have a Lie groupoid $(G\rightrightarrows *)$ and stack $BG=B\mathcal{G}$ of principal $G$ bundles. Given a smooth manifold $M$, we have Lie groupoid $(M\rightrightarrows M …

Let $\text{Sch}$ be the category of schemes. Let $S$ be an object of $\text{Sch}$. Consider the category $\text{Sch}/S$. Some interesting topologies on $\text{Sch}/S$ are Zariski, fpqc, étale, fppf. …

Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note: We will not distinguish between s …

Question: What are (some of) the stacks (occurring in algebraic/differential geometry) that are fibered in arbitrary categories and not necessarily in groupoids? In the notes Notes on Grothendieck t …

A "similar" result along with proof can be found as Lemma 2.14 of Differentiable Stacks and Gerbes. I would like to give more details if you want.

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 as …

This is not an answer, just too long for a comment. So, writing as an answer. It turns out that, one may not be able to see the correspondence between these three definitions as one of them is stated …

There is some result in the case of Lie groupoids and I believe this is related. Given Lie groupoids $\mathcal{G},\mathcal{H}$ a morphism of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$ comes from w …

Given a Lie group action $G$ on $X$ we have what is called a action groupoid (Translation groupoid) associated with the action usually denoted by $G\ltimes X$. Objects of this category are elements …

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). We …

I was reading David Carchedi's answer for a question on Grothendieck topology for a non-small category. It "reads" like people "choose" if they allow manifolds to be Hausdorff and/or second countable. …