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Consider a manifold $M$. Then, we have the notion of differential forms on $M$ and complex associated to that, denoted by $$\cdots\rightarrow \Omega^{k-1}(M)\rightarrow \Omega^k(M)\rightarrow \Omega^{ …

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Let $ M$ be a manifold and $\mathfrak{X}(M)$ be its Lie algebra of vector fields on $M$. Let $G\times M\rightarrow M$ be an action of t …

Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fiber bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G …

I am trying to see that Isotropy group/object group/vertex group of a Lie groupoid is a Lie group. Let $\mathcal{G}$ be a Lie groupoid and $x$ be an object in $\mathcal{G}$ i.e., $x\in \mathcal{G}_0 …

Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions. When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some …

Let $\theta:G\rightarrow H$ be a morphism of Lie groups. Given $G$ we have classifying space $BG$ and given $H$ we have classifying space $BH$. This $\theta:G\rightarrow H$ gives a map $B\theta:BG\ri …

Given a morphism of Lie groups $ \theta:G\rightarrow H$  and a principal $G$ bundle $ \pi:P\rightarrow M$ there are (at least) two ways to assign a principal $ H$ bundle. See that the morphism of Li …

Lie's third theorem : Given any finite dimensional Lie algebra $\mathfrak{g}$, there exists a Lie group $G$ whose Lie algebra is equal to $\mathfrak{g}$. In one of the talks, speaker mentions that thi …

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 f …

I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle. I am not comfortable with these notions and google …

I am trying to understand about Lie groupoids but not able to get feeling for what it actually is. So, question here is, What are Lie groupoids? How similar are they to Lie groups, Groupoids and …