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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 $\varphi,\varphi':\Lambda\rightarrow \Gamma$ …

I am reading https://arxiv.org/abs/0806.4160 to understand orbifolds as stacks. Definition : Let $D\rightarrow Man$ be a stack over category of manifolds. An atlas for $D$ is a manifold $X$ and a …

A decade ago, Gil Kalai asked the question What precisely Is "Categorification"? After seeing some answers and some online pages, I think one of the meanings of categorification is the following: …

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 …

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

An atlas for a stack $\mathcal{D}\rightarrow Man$ is a smooth manifold $X$ and a map of stacks $p:\underline{X}\rightarrow \mathcal{D}$ such that, given a smooth manifo …

I am reading Differentiable Stacks and Gerbes by Kai Behrend and Ping Xu. They define gerbe over a stack as follows. Let $\mathfrak{X}$ be a differentiable stack. An $\mathfrak{S}$-stack $\mathfr …

Let $\mathcal{D}$ be a stack. An atlas for stack $\mathcal{D}$ is given by a smooth manifold $X$ and a map of stacks $p:\underline{X}\rightarrow \mathcal{D}$ such that, for any manifold $M$ a …

Suppose $F:\mathcal{X}\rightarrow\mathcal{Y}$ is gerbe over stack and $p:X\rightarrow \mathcal{X}$ is an atlas $\mathcal{X}$. Does this imply $F\circ p:X\rightarrow \mathcal{Y}$ is an atlas for $\m …

Let $\mathcal{C}$ be a category. Fixing an object $U$ of $\mathcal{C}$, there are some obvious functors we can associate to it, for example: $h_U:\mathcal{C}^{op}\rightarrow \text{Set}$ given by $V\ …

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 …

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.

Definition I am aware of for Lie groupoid is that (among other things) the source and target maps $s,t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ are submersions. On page 9 of Du Li's thesis Higher Grou …