0
votes
1answer
71 views

Groupoid as a 2-coequaliser

Let $G=(G_1, G_0, s, t, u, i,\circ)$ be a groupoid, where $s, t$ are source and target maps, $i$ is the inverse, $u$ is the unit, and $\circ$ is the composition. Denote $\underline{G_1}, ...
4
votes
2answers
310 views

How to specify a finite group up to inner automorphism?

I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...
0
votes
0answers
141 views

Terminology for a partition of unity for an étale groupoid

I would like to ask about terminology for a partition of unity for an ├ętale groupoid. I am reading the lecture notes "Cohomology of Stacks" by Prof. Behrend. A partition of unity is defined in ...
3
votes
2answers
359 views

Algebraic stacks as (etale) groupoid alg.spaces/schemes

Assume given an algebraic stack() $\mathcal{X}$ with presentation $X_0 \to \mathcal{X}$, and the corresponding groupoid $X = (X_0\times_\mathcal{X} X_0 \rightrightarrows X_0)$ in algebraic spaces (or ...
5
votes
2answers
453 views

Automorphism groups and etale topological stacks

Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a ...
4
votes
2answers
454 views

Is there a “geometric” language that describes the equivalence groupoid of a foliated manifold?

Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions. Here, ...
5
votes
2answers
219 views

What condition on a “bibundle between categories” generalizes “right-principal bibundle between groupoids”?

My question is long on background and motivation, and almost but not quite answered over at the nLab. I'll write up a bunch before asking my question (feel free to skip to the end or look at the ...
2
votes
1answer
301 views

Weak colimits of weak and strict presheaves in groupoids

Let $C$ be a small category, and for this question, let groupoid mean an (essentially small) groupoid. There are two 2-categories in question: the 2-category of strict presheaves in groupoids and ...
9
votes
1answer
376 views

Double Category of Topological Stacks

There are two equivalent ways of describing topological stacks. One is the "stacky" definition, that is, a topological stack is a stack $\mathbb{X}$ on $Top$ (a Grothendieck universe thereof, if ...
5
votes
1answer
247 views

What is the local structure of a Lie groupoid?

A manifold is locally $\mathbb R^n$. An orbifold is locally $\mathbb R^n/\{\text{finite group}\}$. Is there a similar way to think about the local structure of a Lie groupoid $G_1 \rightrightarrows ...
7
votes
5answers
1k views

Representation of Groupoids

The title is vague, my actuall question is the following: Has the representations of groupoids been systematically studied? Is there any new phenomenon, compare with the representation of groups? ...