**5**

votes

**5**answers

1k views

### Groupoid actions on spaces

The action of a group $G$ on a topological space $X$ can be viewed as a functor $F: G \to \mathcal{Top}$ with $F(*)=X$. (Here I'm viewing a group as a category with one object, $ * $, and the ...

**2**

votes

**1**answer

396 views

### About properties of groupoid C*-algebras

I'm interested in the following kind of questions about groupoid $C^*$-algebras.
1) If $G_1 \times_{H} \ G_2$ is a fibre product of (nice) groupoids do we have something like $$C^\star(G_1 \times_{H} ...

**6**

votes

**1**answer

313 views

### How equivalent are the theories of reduced and groupal $\infty$-groupoids?

I hope that my question is sufficiently trivial that someone will be able to give me a pedantic answer, and not so trivial that no one takes the time to give an answer. My motivation for asking this ...

**7**

votes

**2**answers

256 views

### How to interpret topologically that the equalizer in Groupoids of ${\rm id}, {\rm id}: BG \rightrightarrows BG$ is $G/G$ (adjoint action)?

Let $G$ be a (discrete) group, and $1/G$ the corresponding groupoid with one object. Consider the diagram in (the 2-category) Groupoids:
(This diagram is equivalent to the pair of parallel arrows ...

**5**

votes

**2**answers

477 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, ...

**8**

votes

**0**answers

355 views

### Quotients of topological groupoids

The issues that arise when moving from topological groups to topological groupoids are (at least to me) both subtle and interesting. Recently, I was reading a paper of R. Brown and J.P.L. Hardy from ...

**4**

votes

**2**answers

375 views

### $\infty-$groupoid of $A_{\infty}$ algebras

Hello,
Consider first the following $2-$groupoid of Algebras over $\mathbb{C}$. Objects are Algebras, $1-$morphisms are isomorphisms, and a $2-$morphism between the isoms $f$ and $g$ from $A$ to $B$ ...

**5**

votes

**2**answers

231 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 ...

**4**

votes

**1**answer

420 views

### Colimits of topological groupoids

Let $G$ and $H$ be two topological groupoids. Suppose that I have two morphisms $G \rightrightarrows H$ and I want to take the 2-coequalizer of these maps. I'd like an explicit description of (a ...

**2**

votes

**1**answer

306 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 ...

**2**

votes

**2**answers

227 views

### Is the cohomology of the corresponding Lie algebroid an invariant under equivalence of source-simply-connected Lie groupoids?

Recall the related notions of Lie groupoid, Lie algebroid, generalized morphism of Lie groupoids, and cohomology of Lie algebroid. Henceforth, I will drop the word "Lie" for all those things listed ...

**14**

votes

**4**answers

617 views

### What is the 2-category whose 0-objects are Lie algebroids?

Recall the notion of Lie algebroid (n Lab, Wikipedia). One motivation for studying Lie algebroids is that they are infinitesimal versions of Lie groupoids, and Lie groupoids present stacks. In ...

**7**

votes

**2**answers

462 views

### Is a groupoid determined by its Hopfish algebra?

This is a follow up to my question What is the precise relationship between groupoid language and noncommutative algebra language?. I will briefly review some definitions; for details, a good place ...

**16**

votes

**3**answers

942 views

### What is the precise relationship between groupoid language and noncommutative algebra language?

I have sitting in front of me two 2-categories. On the left, I have the 2-category GPOID, whose:
objects are groupoids;
1-morphisms are (left-principal?) bibundles;
2-morphisms are bibundle ...

**5**

votes

**4**answers

2k views

### Geometric interpretation of the fundamental groupoid

Motivation
The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy ...

**9**

votes

**1**answer

401 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 ...

**1**

vote

**3**answers

154 views

### Given an object in a Lie groupoid, does there exist a subgroupoid for which the object has no automorphisms but retains its equivalence class?

For this question, I am in the smooth finite-dimensional category. All objects are $\mathcal C^\infty$ manifolds and all maps are smooth. Recall the notion of groupoid, and let's restrict our ...

**4**

votes

**1**answer

390 views

### Do Lie algebroids pull back (along submersions)?

There are more general definitions, but for my purposes a Lie algebroid on a smooth manifold $X$ is a vector bundle $A \to X$, a map $\rho: A \to {\rm T}X$ of vector bundles over $X$, and a bracket ...

**3**

votes

**3**answers

327 views

### What is an obviously coordinate-independent description of the Chevellay-Eilenberg complex for a Lie algebroid?

I've read in many places, including the n-Lab page, that a Lie algebroid (which I think of as in the first definition on the n-Lab page) is the same as a vector bundle $A \to X$ and a (properties?) ...

**4**

votes

**2**answers

900 views

### How can I understand the “groupoid” quotient of a group action as some sort of “product”?

Recall the notion of groupoid (Wikipedia, nLab). An important construction of groupoids is as "action groupoids" for group actions. Namely, let $X$ be a groupoid and $G$ a group, and suppose that ...

**5**

votes

**1**answer

327 views

### Is there a “groupoid integral” with values in a groupoid?

Let $G = \{G_1 \rightrightarrows G_0\}$ be a finite groupoid, i.e. $G_1,G_0$ are both finite sets, and let $A$ be $\mathbb Q$-module. Regard $A$ as a discrete groupoid $A \rightrightarrows A$, and ...

**5**

votes

**1**answer

265 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 ...

**5**

votes

**5**answers

912 views

### Category = Groupoid x Poset?

Is it possible to split a given category $C$ up into its groupoid of isomorphisms and a category that resembles a poset?
"Splitting up" should be that $C$ can be expressed as some kind of extension ...

**9**

votes

**2**answers

1k views

### What is the infinite-dimensional-manifold structure on the space of smooth paths mod thin homotopy?

This question is motivated by the recent paper An invitation to higher gauge theory by Baez and Huerta, and the 2007 paper Parallel Transport and Functors by Schreiber and Waldorf.
Let $M$ be a ...

**6**

votes

**4**answers

546 views

### Groupoid structure on G/H?

Let $G$ be a group and let $H$ be a subgroup. If $H$ is normal in $G$, then $G/H$ has a group structure. But in general, can there be a groupoid structure on $G/H$(left cosets or right cosets) that ...

**8**

votes

**6**answers

2k 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? ...

**12**

votes

**1**answer

755 views

### How difficult is Morse theory on stacks?

The title is a little tongue-in-cheek, since I have a very particular question, but I don't know how to condense it into a pithy title. If you have suggestions, let me know.
Suppose I have a Lie ...

**5**

votes

**1**answer

655 views

### Normal coordinates for a manifold with volume form

I'm hoping that the following are true. In fact, they are probably easy, but I'm not seeing the answers immediately.
Let $M$ be a smooth $m$-dimensional manifold with chosen positive smooth density ...

**6**

votes

**1**answer

297 views

### Is there a theory of differential equations for smooth correspondences?

This question is very closely related to another one I just asked. The general question is to what extent there is a theory of differential equations for smooth correspondences (between a smooth ...

**9**

votes

**3**answers

334 views

### Groupoid of moves on trivalent fatgraph

Let $T$ be a finite trivalent fatgraph - i.e. a graph with a cyclic order of the edges at each vertex. Then there are certain basic "moves" we can perform on $T$: an embedded edge can be collapsed and ...

**40**

votes

**21**answers

6k views

### What's a groupoid? What's a good example of a groupoid?

Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?