All Questions
17 questions
7
votes
0
answers
120
views
Example of a groupoid internal to the category of smooth manifolds that is not a Lie groupoid
This questions is about the distinction between:
Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\!\times_t G_1$, ...
- 4,519
3
votes
1
answer
156
views
Compact groupoid presentations for closed 2-orbifolds (or finite graphs of finite groups)?
A result of Noohi says that if $\mathsf{X}$ and $\mathsf{Y}$ are topological stacks and $\mathsf{Y}$ admits a groupoid presentation $\mathcal{G}$ in which both $\mathcal{G}_0$ and $\mathcal{G}_1$ are ...
- 1,996
6
votes
0
answers
299
views
Is there a Geometric/Smooth version of Homotopy Hypothesis using the path $\infty$-Groupoid of a Smooth Space?
A version of Homotopy Hypothesis says that the Fundamental $n$-grupoids model Homotopy $n$-types... and if we continue upto $\infty$, then the Fundamental $\infty$- groupoids or Kan Complexes model ...
- 1,215
8
votes
0
answers
205
views
What are the Newton groupoids from Drinfeld's paper on the Grinberg-Kazhdan theorem?
The paper the Grinberg-Kazhdan formal arc theorem and the Newton groupoids by Drinfeld seems to contain many interesting things which are beyond me. For now, I am trying to get some intuition for the ...
- 10.5k
4
votes
1
answer
334
views
Gluing together together differentiable stacks
I am trying to figure out the conditions under which you can glue together a collection of (differentiable) stacks by equivalences, and get a differentiable stack.
More precisely, I have a collection ...
- 43
3
votes
1
answer
234
views
Is the cotangent complexes of groupoids bounded above by degree $1$?
Let $\mathcal{X}$ be a stack given by a groupoid $X_1\rightrightarrows X_0$, where $X_0$ and $X_1$ are smooth $k$-varieties. Let $\mathbb{L}_{\mathcal{X}/k}$ be the cotangent complex of $\mathcal{X}$....
- 9,009
3
votes
0
answers
124
views
Representable torsors on geometric groupoid
Let $(C,\tau,\mathbb P)$ be a geometric context, as defined by Toen and Vezzosi. Let $(X_1\rightrightarrows X_0)$ be a groupoid object in $C$ such that the source and target morphisms are in $\mathbb ...
- 41
0
votes
1
answer
164
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}, \underline{...
- 1,271
5
votes
2
answers
503
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 ...
- 18.7k
4
votes
2
answers
570
views
Algebraic stacks as (étale) groupoid algebraic 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 ...
- 35.4k
6
votes
2
answers
553
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 ...
- 15.5k
7
votes
3
answers
813
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, ...
- 54.6k
6
votes
2
answers
375
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 ...
- 54.6k
2
votes
1
answer
395
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 ...
- 15.5k
9
votes
1
answer
539
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 you'...
- 15.5k
7
votes
1
answer
403
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 ...
- 54.6k
16
votes
6
answers
4k
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? (...
- 5,052