A groupoid is a category where all morphisms are invertible. This notion can also be seen as an extension of the notion of group. A motivating example is the fundamental groupoid of a topological space with respect to several base points, compared to the usual fundamental group.

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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
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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
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2answers
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
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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}, ...
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0answers
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 ...
2
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3answers
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 ...
3
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0answers
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 ...
36
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22answers
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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)?
4
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1answer
179 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
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2answers
307 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
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0answers
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
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0answers
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 ...
9
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3answers
311 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 ...
5
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1answer
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 ...
4
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2answers
805 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
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0answers
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
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2answers
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 ...
3
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5answers
369 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 ...
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5answers
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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
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1answer
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
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0answers
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: ...
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2answers
251 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), ...
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5answers
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 ...
4
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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 ...
3
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3answers
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 $H$ ...
1
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1answer
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 ...
15
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1answer
788 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
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5answers
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 ...
3
votes
1answer
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 ...
3
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1answer
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 ...
6
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1answer
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 ...
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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? ...
3
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2answers
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
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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 ...
6
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4answers
528 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
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2answers
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
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1answer
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
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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 étale groupoid. I am reading the lecture notes "Cohomology of Stacks" by Prof. Behrend. A partition of unity is defined in ...
4
votes
1answer
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
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1answer
355 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 ...
9
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3answers
913 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 ...
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, ...
15
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3answers
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 ...
4
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4answers
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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 ...
3
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2answers
361 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 ...
7
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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
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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 ...
2
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1answer
335 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
1answer
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
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2answers
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 ...