**3**

votes

**0**answers

198 views

### Homotopy category of groupoids

The nlab Ho(Cat) page says: morphisms in the homotopy category of groupoids $Ho(Gpd)$, have two equivalent description:
iso-classes of functors.
formally invert equivalence functors (i.e. ...

**2**

votes

**2**answers

184 views

### Classifying map of a principal bundle

Given a topological group, say $G$, it is well known that the classifying space $BG$ classifies $G$ principal bundles. Under some mild assumption on $G$, one model of $BG$ is the geometric realization ...

**0**

votes

**1**answer

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}, ...

**1**

vote

**0**answers

37 views

### Map to orbit space of proper Lie groupoid

Recall the definition of a Lie groupoid, and I will assume that all manifolds involved are Hausdorff. A Lie groupoid $G$ is proper if the map $(s,t)\colon G_1 \to G_0\times G_0$ is proper. The orbit ...

**3**

votes

**0**answers

88 views

### Relation between graphs and groupoid $C^*$-algebras

In the paper "Graphs, groupoids and Cuntz-Krieger algebras" by Kumijan, Pask, Raeburn, Renault it was shown (if I understand it correctly) that whenever $G$ is a row-finite directed graph
with no ...

**2**

votes

**3**answers

221 views

### How to detect if a simplicial set is the nerve of a groupoid?

I have the following question.
Suppose I have a simplicial set. Is there a way to detect if it actually is isomorphic to a nerve of a groupoid?
I've seen the fact that if you have a nerve ...

**11**

votes

**4**answers

727 views

### How do you define the strict infinity groupoids in Homotopy Type Theory?

In the setting of Homotopy Type Theory, how would you construct $\mathrm{isStrict} : U \rightarrow U$ which is inhabited exactly when the first type is (equivalent to?) a strict $\infty$-groupoid?
...

**4**

votes

**1**answer

178 views

### What's the link between topological spaces as locales and topological spaces as infinity-groupoids?

I've seen texts that talk about topological spaces being essentially locales, like Topology via Logic by Vickers, and texts related to homotopy theory that talk about topological spaces being ...

**7**

votes

**0**answers

267 views

### Albrecht Fröhlich's text `Groupoids, groupoid spaces and cohomology' (1965)

I am looking for Albrecht Fröhlich's unpublished text `Groupoids, groupoid spaces and cohomology' (1965). In this text Fröhlich defines cohomology of a group with coefficients in a groupoid (this was ...

**2**

votes

**0**answers

52 views

### Is there a Fourier transform for principal r-discrete groupoids?

I have been looking for an analog of the Fourier transform for groupoids coming from tilings (which are generally principal and r-discrete), however all the generalizations I have found assume that ...

**1**

vote

**0**answers

93 views

### Reduced C*-algebras of locally compact etale Hausdorff groupoids

Let $G$ be an étale locally compact Hausdorff groupoid (possibly second-countable) and let $a\in C_{\textrm{red}}^*(G)$. Is it true that for all $\varepsilon>0$ there is $s\in C_c(G)$ satisfying ...

**2**

votes

**2**answers

401 views

### Groupoids vs. action groupoids

Let $A\rightrightarrows X$ be a groupoid, where $X$ is the set of objects and $A$ is the set of arrows.
My favorite example of a groupoid is an action groupoid. If a group $G$ acts on the left on a ...

**2**

votes

**1**answer

186 views

### Equivalence and weak equivalence of groupoids

Let $A\rightrightarrows X$ be a groupoid, where $X$ is the set of objects and $A$ is the set of arrows.
My favorite example of a groupoid is an action groupoid. If a group $G$ acts on the left on a ...

**2**

votes

**0**answers

78 views

### Bisections in Kan Complexes

Kan Complexes can be seen as a generalization of groupoids, mostly called (weak)
infinity groupoids in this context.
On groupoids we can define the \textbf{group of bisections} the following way:
...

**3**

votes

**5**answers

368 views

### Connected groupoids and action groupoids

It is written in Wikipedia http://en.wikipedia.org/wiki/Groupoid, that any connected groupoid $A\rightrightarrows X$ is isomorphic to an action groupoid $G\ltimes X$ coming from a transitive action ...

**2**

votes

**2**answers

250 views

### Constructing a stack (gerbe) from a connected groupoid

Let $\mathcal{G}=(A\rightrightarrows X)$ be a groupoid.
Here $X={\rm Ob}(\mathcal{G})$, $A={\rm Ar}(\mathcal{G})$,
and we have 5 maps:
$s,t\colon A\to X$ (the source and the target, surjective),
...

**4**

votes

**2**answers

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 ...

**1**

vote

**2**answers

306 views

### Semidirect product of groupoids

I am trying to understand the semidirect product of groupoids, as defined in this answer by Theo Johnson-Freyd. Part of my difficulty is that although the definition of a 2-group makes sense, I am ...

**3**

votes

**3**answers

291 views

### What are the symmetries of a principal homogeneous bundle?

Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where:
$G$ is a Lie group, and $H$ ...

**24**

votes

**5**answers

1k views

### How many binary operations are associative?

Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be ...

**1**

vote

**1**answer

292 views

### When are $k$-sectors of a Lie groupoid a manifold?

Let ${\mathcal{G} = \lbrace s,t:G_1 \to G_0 \rbrace}$ be a Lie groupoid. Define
$$(\mathcal{G}^k)_0:=\lbrace (a_1,\dots,a_k) \in G_1^k\mid s(a_1)=t(a_1)=\dots=s(a_k)=t(a_k) \rbrace$$
(This is the ...

**3**

votes

**1**answer

157 views

### reference request for essential equivalence of top. groupoids

Let $G$ and $H$ be two topological groupoids. Recall that a morphism $F \colon H \to G$ is called an essential equivalence, if
the map $t \circ \pi_1 \colon G_1 \times_{G_0} H_0 \to G_0$ is an open ...

**3**

votes

**1**answer

198 views

### Simplicity of reduced C*-algebras for non-Hausdorff etale groupoids

It is known that for a Hausdorff locally compact etale groupoid, the reduced C*-algebra is simple iff the groupoid is minimal (meaning the orbit of each unit is dense) and topologically principal ...

**17**

votes

**5**answers

1k views

### How should one understand orbifold fundamental groups?

I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? In what follows I give definitions and more precise ...

**6**

votes

**1**answer

287 views

### Groupoids and hypergroups

There are two generalizations of usual groups: groupoids, where the multiplication operation becomes "partial", and hypergroups, for which the result of multiplying two elements is a probability ...

**3**

votes

**2**answers

306 views

### Cartesian cubes and groupoids

Given a groupoid $G,$ one can consider the canonical epimorphism $$G_0 \to G.$$ Since it is an epimorphism in the $2$-topos of groupoids, $G$ is the weak colimit of the corresponding Cech diagram ...

**2**

votes

**0**answers

105 views

### spaces of projections

Let $\mathbb{K}$ be the compact operators on a separable infinite dimensional Hilbert space. Denote by $\mathcal{P}(\mathbb{K})$ the space of projections in $\mathbb{K}$. If I am not terribly wrong ...

**5**

votes

**1**answer

286 views

### Continuous and smooth Lie groupoid cohomology

In the paper by Weinstein and Xu: Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), there are two versions of Lie groupoid cohomology. The same differential ...

**15**

votes

**1**answer

786 views

### Convolution algebras for double groupoids?

There is a lot of work of course on convolution algebras of measured groupoids, and this gives "Noncommutative geometry". However there is a lot of interest in algebraically structured groupoids, for ...

**3**

votes

**2**answers

536 views

### Examples of Simplicial Groupoids in Nature

For me, a simplicial groupoid is a simplicial object in ${\mathbf{Grpd}}$. I am more general than Goerss-Jardine in this definition.
Do you have examples simplicial groupoids that occur in nature? ...

**15**

votes

**1**answer

467 views

### Toposes (topoi) as classifying toposes of groupoids

A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Buntz and Moerdijk have shown that if the topos has ...

**0**

votes

**0**answers

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 étale groupoid.
I am reading the lecture notes "Cohomology of Stacks" by Prof. Behrend. A partition of unity is defined in ...

**2**

votes

**1**answer

354 views

### A category being self-dual vs. it being a groupoid

What is the relationship between self-duality and groupoid-ness? Does any condition imply the other? Is there an example which helps understand the difference?
To go from a self-duality $F$ on a ...

**15**

votes

**3**answers

393 views

### Characterizing Groupoids via Quotients?

A groupoid is a category in which all morphisms are invertible.(*) The groupoids form a very nice subclass of categories. The inclusion of the groupoids into the 2-category of small categories admits ...

**3**

votes

**2**answers

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 ...

**9**

votes

**3**answers

911 views

### Groupoids vs Pseudogroups

(Warning: I'm not an expert in the topic) Let's work in a "geometric" category, for example the category $\mathfrak{Diff}$ of "manifolds" (without the requirements of connectedness and second ...

**7**

votes

**0**answers

411 views

### 2 questions on the groupoid algebra

Dear All:
I would like some refs and/or thoughts on the following two related
questions:
1) If I am not mistaken, there is a " Groupoid Convolution Algebra"
(GCA)
contravariant functor from the ...

**5**

votes

**2**answers

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 ...

**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

334 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

302 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

235 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 ...

**4**

votes

**2**answers

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, ...

**8**

votes

**0**answers

343 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

360 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

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 ...

**4**

votes

**1**answer

384 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

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 ...

**2**

votes

**2**answers

218 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

**3**answers

535 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 ...