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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. …

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 …

I am looking for some references related to gerbes and their differential geometry. Almost every article I have seen that is related to gerbes there is a common reference that is Giraud's book Cohomo …

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 …

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 …

The author means there is a zigzag of natural transformations. That is, "a natural transformation between $\varphi_i$ and $\varphi_{i+1}$" is intended to be nonspecific as to the direction of the tr …

In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following. Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\righ …

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 …

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 …

I am reading this paper on Homotopy for functors by Ming-Jung Lee. The author gives a definition (at the beginning of section $3$) as follows : Let $\varphi,\varphi':\Lambda\rightarrow \Gamma$ …

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 …

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 $\phi:\mathcal{H}\rightarrow \mathcal{G}$ …

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 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 …

Let $\phi:\mathcal{G}\rightarrow \mathcal{K}$ and $\psi:\mathcal{H}\rightarrow \mathcal{K}$ be morphisms of Lie groupoids. We define weak pullback/2-fibre product corresponding to $\phi:\mathcal{G}\ …