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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 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 …

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 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 …

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 …

Over a large site there is in general no sheafification functor https://ncatlab.org/nlab/show/sheafification. For how to get around this, the keyword is dense subsite https://ncatlab.org/nlab …

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 …

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 …

The term Lie $2$-groupoid is used in the literature in more than one context. Some examples are given below: Ginot and Stiénon's paper $G$-gerbes, principal $2$-group bundles and characteristic clas …

This is not a complete answer, too long for a comment. If we start with an arbitrary site $\mathcal{C}$ and if we want to define the notion of a $G$-torsor over $\mathcal{C}$, then $G$ is not expecte …

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 …