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Let $M$ be a manifold. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Let $P(M,G)$ be a principal bundle. Recall that, a connection on $P(M,G)$ is a distribution $\mathcal{H}\subseteq …

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 …

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$Let $G$ be a Lie group and $\pi:P\rightarrow M$ be a principal $G$ bundle. The notion of reduction of structure group is standard but I will rec …

The background of this question is the talk given by Kevin Buzzard. I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here. …

Let $(\mathcal{C},\mathcal{I})$ and $(\mathcal{D},\mathcal{J})$ be sites where $\mathcal{C}, \mathcal{D}$ are categories and $\mathcal{I}$ and $\mathcal{J}$ are Grothendieck topologies on $\mathcal{C …

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 …

What are the occurrences of the notion of a stack outside algebraic geometry, differential geometry, and general topology? In most of the references, the introduction of the notion of a stack takes …

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.

Question is the following: Is the functor $H^n_{dR}:\text{Man}\rightarrow \text{Set}$ a sheaf with respect to open cover topology on $\text{Man}$? More generally, are cohomology functors sheaves in …

Consider the category of smooth manifolds $\text{Man}$. I quote from n-lab page: Manifolds are fantastic spaces. It’s a pity that there aren’t more of them. I understand that this category $\text{Ma …

I have made the following observation (hopefully a correct one) when reading the paper Orbifolds as stacks: They start with the strict $2$-category category of Lie groupoids, functors, natural transfo …

Suppose a term(inology) is recently (in last 20 years) introduced in research mathematics. It might happen that some one who wish to use it, in the same area of research, for different purposes or se …

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. …

The word “topological stack” has at least three usages: A stack $\mathcal{D}\rightarrow \text{Top}$ is said to be a topological stack if there is a a morphism of stacks $p: \underline{M}\rightarrow …

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 …