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Questions tagged [groupoids]

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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Locally compact groupoid with a Haar system such that the range map restricted to isotropy groupoid is open

Can somebody provide an example of a locally compact groupoid $G$ with a Haar system such that the range map restricted to isotropy groupoid of $G$ is open? I could not find any specific example for ...
K N Sridharan's user avatar
15 votes
0 answers
185 views

Are Lie groupoids just groupoids internal to smooth manifolds?

It seems to be common to say "no" - but is this true? Two weeks ago I asked for a counterexample, but received no replies. To give some background, let's recall that the difference between ...
Konrad Waldorf's user avatar
7 votes
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114 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, $G_1 \;{}_s\!\times_t G_1$, ...
Konrad Waldorf's user avatar
9 votes
0 answers
300 views

Is $\mathcal{B}(\mathcal{H})$ a groupoid $C^*$-algebra?

Let $\mathcal{H}$ be a complex Hilbert space, and $\mathcal{B}(\mathcal{H})$ be the $C^{\ast}$-algebra of bounded operators on $\mathcal{H}$. Is there an étale groupoid $\mathcal{G}$ such that its $C^{...
Luiz Felipe Garcia's user avatar
1 vote
0 answers
66 views

Root systems of Weyl groupoids

I am working with the notion of Weyl Groupoids as introduced in "A generalization of Coxeter groups, root systems, and Matsumoto’s theorem" by Heckenberger and Yamane. The authors generalize ...
Tim's user avatar
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0 votes
1 answer
295 views

Can we generalise groupoids to monoid-oids? [closed]

Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories. Groupoids correspond to small categories where every morphism is an ...
Diego de la Paz's user avatar
6 votes
1 answer
392 views

A possible alternative model for $\infty$-groupoids

I've been studying homotopy theory on myself for quite some time now, and it is to my understanding that there's still no generally accepted definition for $\infty$-groupoids. The closest to this is ...
XiaohuWang's user avatar
6 votes
0 answers
190 views

What is the standard groupoid model of the Cuntz algebra?

I know that the Cuntz algebras $\mathcal{O}_n$, $n=1,2,...,\infty$, have groupoid models. I.e. they can be realised as groupoid C*-algebras. Can you describe the standard groupoid model for $\mathcal{...
Panini's user avatar
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1 vote
0 answers
320 views

Two different definitions of condensed groupoid

I am searching for a condensed version of a topological groupoid and I found two possible definitions. $\textbf{Definition 0:}$ A condensed groupoid(0) is a functor $X: \mathrm{Extr}^{\mathrm{op}} \...
Luiz Felipe Garcia's user avatar
6 votes
0 answers
139 views

Mapping space between $n$-groupoids is an $n$-groupoid

Consider two simplicial sets $K$ and $L$. Their mapping space (or mapping complex) is the internal hom of simplicial sets, i.e. $\underline{\mathrm{Hom}}(K,L)$, where $$ \underline{\mathrm{Hom}}(K,L)...
SetR's user avatar
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15 votes
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Is there a higher analog of "category with all same side inverses is a groupoid"?

There is a (most likely folklore) theorem - if in a category every morphism has a right inverse then that category is a groupoid. The proof is an honest oneliner: for $x:A\to B$ find $x':B\to A$ with $...
მამუკა ჯიბლაძე's user avatar
3 votes
0 answers
186 views

Transitive action on domino tilings

Fix a $n \times m$ rectangle and consider the set $S_{n,m}$ of all its dominos tilings. Here are examples with $n=m=8$. The set $S_{n,m}$ is empty if and only if $nm$ is odd, and for small $nm$, its ...
Sebastien Palcoux's user avatar
3 votes
1 answer
318 views

Can graphs of groups be thought of as "graph objects" in the category of groupoids?

An undirected graph is sometimes defined as a pair of sets $V$ and $E$ (vertices and oriented edges), together with two maps $i,f: E\to V$ (sending a directed edge its initial/final vertex) and a map $...
Antoine Labelle's user avatar
3 votes
0 answers
119 views

Abelianisation of Groupoids

I was wondering what a good source for the properties (or even the existence) of the abelianisation of a (2-) groupoid would be? A naive construction would certainly be to abelianise the automorphisms ...
curious math guy's user avatar
5 votes
0 answers
339 views

Does $\mathit{Suz}$ contain $M_{13}$?

$\newcommand\Suz{\mathit{Suz}}$I recently noticed that the Suzuki group $\Suz$ has as subgroups classes of both $L_3(3)$ and $M_{12}$, both of which are also subgroups of the Mathieu groupoid $M_{13}$....
Daniel Sebald's user avatar
3 votes
1 answer
153 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 ...
Rylee Lyman's user avatar
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8 votes
1 answer
413 views

Connection between Grothendieck's homotopy hypothesis and Lie's second and third theorems?

I'm not an expert on homotopy theory, but I speculated about this in my thesis, so I figured I'd ask about it here. As I understand it, the homotopy hypothesis says that $\infty$-groupoids, with $\...
Josh Lackman's user avatar
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4 votes
1 answer
189 views

Conformal groupoid

I asked this over on Math.SE but it remained completely silent for over a week so I've deleted it and am reposting it here (I'm not really sure which site it fits better). The question itself is ...
J_P's user avatar
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2 votes
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Does the convolution $C^*$-algebra of locally compact Hausdorff groupoids recover back the respective groupoid?

First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known. Let $X$ be a locally compact Hausdorff groupoid (or Lie ...
user40276's user avatar
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3 votes
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Skeletal topological groupoid

Let $G$ be a topological groupoid with the property that any two isomorphic objects are topologically indistinguishable in $\mathrm{Ob}(G)$. Does that imply that $G$ is equivalent to a skeletal ...
user478652's user avatar
4 votes
1 answer
100 views

Extension of an orbifold structure from punctured balls to balls

Let $\hat{D} := D \backslash \{0\}$ be a ball in $R^n$ with the origin $\{0\}$ removed. Assume that $\hat{D}$ has a structure as an orbifold (may be distinct from its standard manifold structure). Is ...
Hao Yu's user avatar
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4 votes
2 answers
466 views

Topological groupoids and equivariant sheaves

Some statements that are true for ordinary groupoids fail for topological groupoids (by which I mean groupoids internal to the category of topological spaces): for instance, every ordinary groupoid is ...
user478652's user avatar
2 votes
0 answers
233 views

Convolution of continuous compactly supported functions on étale groupoid is continuous

Let $G$ be an étale Hausdorff groupoid, i.e. a topological groupoid $G$ such that the source and range maps $s,r: G \to G$ are local homeomorphisms. Consider the complex vector space $C_c(G)$ of ...
Andromeda's user avatar
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4 votes
0 answers
348 views

Is there a name for objects all of whose endomorphisms are automorphisms?

I am looking for a descriptive adjective to describe the following special property that some objects in some categories enjoy: their endomorphism monoids are groups. Of course, one way this could ...
Theo Johnson-Freyd's user avatar
2 votes
0 answers
52 views

A foliation with prescribed graph of foliation

**I have already asked this question on MSE https://math.stackexchange.com/questions/4272279/1-dimensional-foliation-of-surfaces-with-prescribed-graph-of-foliation ** Definition of the graph of a ...
Ali Taghavi's user avatar
4 votes
0 answers
230 views

Graphs with high girth and low diameter

As the title says, I'm interested in graphs with high girth and low diameter. Given a class $\Gamma$ of finite $k$-regular graphs, call a $\Gamma$-graph GD-extremal if every $\Gamma$-graph either has ...
Robin Saunders's user avatar
4 votes
0 answers
103 views

Smash products of pointed groupoids

The category of pointed sets $\mathsf{Sets}_*$ has a symmetric closed monoidal category structure $(\wedge,S^0)$, which, analogously to the symmetric monoidal $\infty$-category of pointed spaces, is (...
Emily's user avatar
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2 votes
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Does this groupoid have a quasi-diagonal reduced $C^*$-algebra?

Let $H$ be a discrete group, and let $X$ be the one-point compactification of $\mathbb{N}$. Consider the étale groupoid $G = H \times \{\infty\} \sqcup \mathbb{N}$, whose unit space is $X$, and with ...
Diego Martinez's user avatar
0 votes
0 answers
157 views

A group acts on a groupoid

Let $G$ be a group. Let $(\Pi,\circ)$ be a groupoid. Suppose I have a $G$-action on every morphism space $\Pi(p,q)$, denoted by $G\times \Pi(p,q)\to \Pi(p,q)$, $(g, \sigma)\mapsto g\cdot \sigma$. (For ...
Hang's user avatar
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7 votes
1 answer
292 views

Filtered 2-colimits commute with finite 2-limits

Is there an explicit proof anywhere in the literature that filtered 2-colimits commute with finite 2-limits (all meant in the weak bicategorical sense) in the 2-category of groupoids? I have only ...
Mike Shulman's user avatar
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2 votes
1 answer
131 views

Is it possible to characterize the elements of the C$^*$-algebra of an open subgroupoid?

$\newcommand{\Cstar}{C^*_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$Let $\G$ be an etale groupoid, let $U$ be an open subset of $\G^{(0)}$, and let $$ \H = \{\gamma \in \G:...
Ruy's user avatar
  • 2,233
4 votes
0 answers
123 views

Restricting a function defined on an étale groupoid to an isotropy group

Let $\mathcal G$ be an étale groupoid, let $x$ be a point in the unit space of $\mathcal G$, and let $\mathcal G(x)$ be the isotropy group of $x$. If $f$ is a continuous, complex valued, compactly ...
Ruy's user avatar
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4 votes
1 answer
257 views

Hopf "algebroid" structure of a groupoid convolution algebra?

This question is already posted in math.stackexchange, but didn't receive any answer. I'm not sure if this question fits in here, but surely someone in here can guide me to the correct answer. To make ...
Bumblebee's user avatar
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8 votes
1 answer
331 views

Amenable groupoid C*-algebras satisfy the UCT in English?

As is by now well known, Tu proved in 1998 that the C*-algebras coming from amenable groupoids satisfy the so-called UCT (universal coefficient theorem). Unfortunately, I don't speak french and I've ...
Diego Martinez's user avatar
13 votes
3 answers
1k views

Elementary theory of the category of groupoids?

One axiomatisation of set theory, the Elementary Theory of the Category of Sets, or ETCS for short, comes from category theory and states that sets and functions form a locally cartesian-closed, ...
user avatar
2 votes
0 answers
119 views

Classifying space induces a equivalence of categories between $\operatorname{PBun}_G(M)$ and $\Pi(M,BG)$ for finite groups $G$

Let $G$ be a finite group, $BG$ its classifying space and M a manifold. Then it is mentioned in Schweigert and Woike - Orbifold construction for topological field theories (Remark 2.3 d) that there is ...
Flo's user avatar
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2 votes
0 answers
109 views

A quantity associated to a foliated manifold and its non-commutative interpretation

Let $M$ be a compact $n$-dimensional manifold. Assume that $F$ is a $k$-dimensional foliation of $M$. The graph $G(M,F)$ of this foliation is a $(n+k)$-dimensional manifold. We recall its definition: ...
Ali Taghavi's user avatar
1 vote
0 answers
99 views

Classification of all groupoids $G$ whose automorphism group is in bijective correspondence the automorphism group of $C^*_\text{red}(G)$

Is there a terminology (and a classification) for all groupoids $G$ for which all automorphisms of $C^*_\text{red}G$ are induced from a groupoid automorphism of $G$. (A groupoid automorphism has ...
Ali Taghavi's user avatar
3 votes
0 answers
185 views

Skeleton of $\mathcal{G}$-simplicial complex

I'm trying to prove a result with simplicial complex endowed with an action of a groupoid. Let start with a formal definition : $\textbf{Definition.}$ [$\mathcal{G}$-simplicial complex] Let $\mathcal{...
MacFly's user avatar
  • 53
5 votes
1 answer
515 views

Delooping of a group object as a one object groupoid

According to https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid we can think of delooping of a group as the one object groupoid $BG$ consisting of a single object and whose ...
Adittya Chaudhuri's user avatar
3 votes
0 answers
163 views

Non-commutative duality II: The two algebras of a groupoid

This question is in a sense the continuation of my previous one, Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?. Reading the great answers and the equally good ...
Mirco A. Mannucci's user avatar
15 votes
3 answers
1k views

Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?

Many interesting C*-algebras can be realized as convolution algebras over a groupoid, an idea introduced in 1980 by Jean Renault (this entry in nLab provides plenty of context to the general approach ...
Mirco A. Mannucci's user avatar
7 votes
0 answers
268 views

Is there a practically useful or concrete representation theory/Fourier analysis on finite groupoids?

Fourier analysis on finite groups is well known to be useful for probability theory and combinatorics — consider for example the Fourier analysis on $(\mathbb Z/2\mathbb Z)^n$ which can be used to get ...
Vilhelm Agdur's user avatar
3 votes
1 answer
199 views

Morphisms of $\infty$-groupoids

As far as I understand, there are several ways of defining $\infty$-categories. One of them is to think of $\infty$-cateogries as $top$-enriched categories. Hence we can think of $\infty$-groupoids as ...
curious math guy's user avatar
7 votes
1 answer
243 views

Representation of fundamental groupoid as $2$-sheaf

By https://arxiv.org/abs/1406.4419 (The fundamental groupoid as a terminal costack, Ilia Pirashvili), we know that for a topological space $X$, the $2$-functor $$\text{Top}(X)\rightarrow \text{Gpd}, \...
curious math guy's user avatar
7 votes
0 answers
215 views

Representations of 2-groups and quantum double constructions

Let $G$ be a finite group. The category of its representations (complex linear, finite dimensional, throughout this whole question) is equivalent to $\mathbb{C}[G]$-modules. V. Drinfeld constructed a ...
Student's user avatar
  • 5,028
6 votes
0 answers
293 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 ...
Adittya Chaudhuri's user avatar
1 vote
0 answers
52 views

Promoting representation of a subgroup to representation of an action groupoid

Suppose I have a group $G$, a subgroup $K \leq G$, and a representation $(\sigma, V)$ of $K$. There is a natural left action of $G$ on $X := G/K \times G/K$ given by $g \cdot (g'K,g''K) = (gg'K,gg''K)$...
Ashwin Iyengar's user avatar
19 votes
1 answer
677 views

What is the groupoid cardinality of the category of vector spaces over a finite field?

For any groupoid, it's groupoid cardinality is the sum of the reciprocals of the automorphism groups over the isomorphism classes. Let us consider the category of vector spaces over a finite field $\...
Asvin's user avatar
  • 7,716
8 votes
0 answers
202 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 ...
Arrow's user avatar
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