**11**

votes

**4**answers

673 views

### Brandt's definition of groupoids (1926)

The definition of a category is usually attributed to Mac Lane and Eilenberg (1945). What seems to be less known is that the german mathematician Heinrich Brandt has developed the notion of a groupoid ...

**10**

votes

**4**answers

472 views

### The groupoid of algebraic expressions and proofs

Fix a set of variables $V$, and suppose we're given a presentation of a monosorted algebraic theory, with variable symbols taken from $V$. For the sake of example, suppose the presentation consists of ...

**1**

vote

**0**answers

42 views

### Local section of Lie Groupoids

Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...

**2**

votes

**1**answer

248 views

### Reference request for generalization of groups with out identity element?

In other words what do we call a magma which is associative and has divisibility property but not existence of identity? Or a groupoid when it loses the identity property?
A reference on such ...

**1**

vote

**0**answers

66 views

### Lower periodic subsets of groups and semigroups

Suppose that $A$ and $B$ are subsets of a group or semigroup. We call $A$ left
upper [resp. lower] $B$-periodic if $BA\subseteq A$
[resp. $A\subseteq BA$]. If $A$ is both left upper and
lower ...

**0**

votes

**0**answers

46 views

### One dimensional foliation of surfaces with prescribed graph of foliation

According to the definition of the graph of a foliation by Winkelnkemper we ask the following questions:
Let $G$ be one of the following non hausdorff 3 dim manifold
1) $G$ is a ...

**2**

votes

**0**answers

94 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 ...

**14**

votes

**0**answers

529 views

### What's the big deal about $M_{13}$?

$M_{13}$ is the Mathieu groupoid defined by Conway in
Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11,
London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...

**2**

votes

**3**answers

333 views

### Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?

Let $\pi _1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi _1$ is left adjoint to $N.$
On ...

**3**

votes

**1**answer

139 views

### How to construct a free 2-group on a groupoid?

Let G
be a groupoid. I'm wondering how to construct the free 2-group on G.
By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$
equipped with a functor ...

**5**

votes

**0**answers

218 views

### Is the crossed product $\mathcal{K} \rtimes G$ a groupoid algebra?

Suppose G, a discrete group acting on the compact operators $\mathcal{K}$ by automorphism of C*-algebra $\mathcal{K}$. Can we view the crossed product as a groupoid C*-algebra of some groupoid?
This ...

**3**

votes

**0**answers

53 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 ...

**14**

votes

**4**answers

611 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 ...

**2**

votes

**0**answers

37 views

### Connection on 3-bundle given as triplet of forms

A connection on a bundle is given locally by a Lie algebra-valued 1-form.
Gauge transformations act in the usual way on the forms, and form a groupoid.
A connection on a 2-bundle is given locally by ...

**11**

votes

**4**answers

786 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?
...

**3**

votes

**0**answers

248 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

218 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

76 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}, ...

**2**

votes

**3**answers

309 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 ...

**3**

votes

**0**answers

104 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 ...

**39**

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)?

**5**

votes

**1**answer

213 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 ...

**1**

vote

**2**answers

366 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 ...

**7**

votes

**0**answers

277 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

57 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 ...

**9**

votes

**3**answers

330 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 ...

**6**

votes

**1**answer

329 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 ...

**4**

votes

**2**answers

886 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 ...

**1**

vote

**0**answers

102 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

433 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 ...

**3**

votes

**5**answers

407 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 ...

**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

229 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

79 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:
...

**2**

votes

**2**answers

269 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),
...

**17**

votes

**5**answers

2k 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 ...

**4**

votes

**2**answers

341 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 ...

**3**

votes

**3**answers

300 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 $H$ ...

**1**

vote

**1**answer

302 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 ...

**15**

votes

**1**answer

847 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 ...

**24**

votes

**5**answers

2k 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 ...

**3**

votes

**1**answer

221 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 ...

**3**

votes

**1**answer

162 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 ...

**7**

votes

**1**answer

304 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 ...

**8**

votes

**5**answers

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? ...

**3**

votes

**2**answers

326 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

109 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 ...

**6**

votes

**4**answers

545 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 ...

**3**

votes

**2**answers

579 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

523 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 ...