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I am reading Orbifolds as stacks? Given Lie groupoids $\mathcal{G}$ and $\mathcal{H}$ there is a notion of what is called a bibundle from $\mathcal{G}$ to $\mathcal{H}$ which is supposed to be a agene …

Kirill Mackenzie has a book on the general theory of Lie groupoids and Lie algebroids. Is there such a reference for the general theory of Lie $\infty$-groupoids and Lie $\infty$-algebroids; that cove …

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. …

The question is as in the title. (What are some of the) are there any applications of Homotopical algebra (in the context of Quillen’s book “Homotopical algebra”) in better understanding (or developi …

I am trying to understand what exactly is the Morita equivalence of Lie groupoids. I am reading Ieke Moerdijk’s notes Orbifolds as groupoids. A homomorphism $\phi:\mathcal{H}\rightarrow \mathcal{G}$ …

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 …

A category $\mathcal{C}$ consists of pair of classes $(\mathcal{C}_0, \mathcal{C}_1)$, along with maps $$\mathcal{C}_1\times_{\mathcal{C}_0}\mathcal{C}_1\rightarrow \mathcal{C}_1\rightrightarrows \mat …

Can someone point me to a reference where the notion of "Lie crossed module" appeared for the first time? I see many papers "recall" the definition of the Lie crossed module but, I do not see any ment …

Given a topological space $X$, we have the notion of the fundamental groupoid $\Pi_1(X)$. Here, the fundamental groupoid $\Pi_1(X)$ is made into a topological groupoid giving a topology on the morph …

Given a category $\mathcal{C}$, by a category over $\mathcal{C}$, I mean a category $\mathcal{D}$ along with a functor $\pi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{C}$. Consider the category $Ca …

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 …

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 …