# 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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### Automorphism group of a torsor

Given a site $C$ and an object $U$, let $G$ be a sheaf of groups on this site and let $F$ be $G$-torsor, see the Stacks Project for the general definition.
By restriction on the over category $C/U$ (...

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### What is the category of algebras for the finitely supported measures monad?

In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad.
I have three ...

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**1**answer

102 views

### Continuity of functions on étale groupoids

Let $\mathcal G$ be an étale groupoid with a locally compact, Hausdorff unit space $\mathcal G^{(0)}$. If $U⊆\mathcal G$ is an open subset, which is Hausdorff in the induced topology, and if $f$ is ...

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### Cencov's “categories of figures”

In his 1982 book Statistical Decision Rules and Optimal Inference, N. N. Cencov studies statistical models (parametrized families of probability distributions) from an unconventional category-...

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### Convolution product in an étale groupoid

I am going through Sims - Étale groupoids and their $C^*$ algebras and at Lemma 3.1.4. the author says that
$f^**f\in C_c(G^{(0)})$ is supported on $s(supp(f))$ and $(f^**f)(s(\gamma))=|f(\gamma)|^2$ ...

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### Fundamental groupoid and fibration

In this post, it is said that a functor from the fundamental groupoid of a space $X$ (denoted by $\Pi(X)$) to the category $\mathrm{Vect}$ of vector spaces gives a flat vector bundle over $X$. But I ...

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### Why did Voevodsky consider categories “posets in the next dimension”, and groupoids the correct generalisation of sets?

Earlier today, I stumbled upon this article written by V. Voevodsky about the "philosophy" behind the Univalent Foundations program. I had read it before around the time of his passing, and one ...

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### Does the 2 category of Groupoids Admit the Vector Space Monad?

We can see here in Jacob's 2013 paper, that he seems to state that a particular kind of multiset monad is actually a vector space monad.
3.2. Vector spaces. For a semiring S one can define the ...

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**1**answer

217 views

### Is Quillen's bracket a “universal enveloping” something?

$\newcommand{\G}{\mathcal{G}}$
In K-theory, there is a construction due to Quillen as follows. Let $(\G, \oplus, 0)$ be a monoidal groupoid. Then the bracket $\langle \G, \G \rangle$, sometimes also ...

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207 views

### Slice theorem for proper groupoids

Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$.
Assume $G$ is étale, i.e., the source and range maps of $G$ are local ...

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### Frobenius monads and groupoids

For a while, I was looking for a Frobenius monad on Set. It doesn't exist as pointed out here. I am now looking at the 2-category of groupoids. Does the 2-category of groupoids admit a Frobenius ...

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### The étale topos of a scheme is the classifying topos of which groupoid?

[Sent here from Math.StackExchange by suggestion of an user.]
By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. ...

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**2**answers

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### Why study orbifolds? [closed]

Question is as in the title.
Why study orbifolds?
I study orbifolds as locally compact Hausdorff spaces $X$ having an orbifold structure, i.e., there exists an orbifold groupoid (proper foliatio. ...

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175 views

### Isotropy group of a Lie groupoid is a Lie group

I am trying to see that Isotropy group/object group/vertex group of a Lie groupoid is a Lie group.
Let $\mathcal{G}$ be a Lie groupoid and $x$ be an object in $\mathcal{G}$ i.e., $x\in \mathcal{G}_0$...

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### Some generalizations of the graph of a foliation

The graph of a foliation $\mathcal{F}$ of a manifold $M$ is the space of all triple $(x,y,[\gamma])$ where $x,y$ lies on the same leaf $F$ and $[\gamma]$ is the equivalent class of a ...

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### The groupoid $C^*$ algebra associated to a certain groupoid

Let $\mathbb{N}$ be the set of all natural numbers. We define a groupoid structure on $\mathbb{N}^{\mathbb{N}}$ as follows:
We put $G^1=\mathbb{N}^{\mathbb{N}},\;G^0 =\{(a_n)\in G^1\mid a_{2n-1}=...

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### What are Lie groupoids intuitively?

I am trying to understand about Lie groupoids but not able to get feeling for what it actually is.
So, question here is,
What are Lie groupoids? How similar are they to Lie groups, Groupoids and ...

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**1**answer

207 views

### moduli stack of double covers of $\mathbb{P}^1$ with one marked point

I am trying to improve my moderate knowledge of moduli spaces/stacks by examining the moduli stack of stable double covers of $\mathbb{P}^1$ with one marked point.
My idea is to ignore the stack ...

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### Group objects in $\infty$-categories

A groupoid object in an $(\infty,1)$ category $\mathcal{C}$ is a functor $G:N(FinSet)^{op} \to \mathcal{C}$ such that for any partition $[n]=S \cup S'$ intersecting in $s$, the object $G([n]$ is the ...

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### Relation between the Hochschild cohomology of group algebras and groupoids

Is there a known relation between the Hochschild cohomology of group algebras and cohomology of groupoids?
Clarification: It is known that 1-dimensional Hochschild cohomology of the Group algebra C[...

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**2**answers

343 views

### Coverings of a space and coverings of a groupoid

In algebraic topology, there is the well-known notion of coverings of a space. It is very nice, it has many properties, but I find it frustrating that:
1) some hypotheses are needed for them to work ...

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### K-theory of a discrete groupoid crossed product

Does there exist a method to compute the K-theory
$$K(A \rtimes G)$$
for a discrete, countable groupoid $G$ and $G$-$C^*$-algebra $A$? In good cases, say $G$ is ameanable.
Say, via Baum--Connes and a ...

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318 views

### A quantity associated with a smooth groupoid

Assume that $(G,G^0,r,s)$ is a smooth groupoid such that $G$ is a compact connected manifold.
The graph of "source" and "range" maps $s, r: G \to G^0$ are compact submanifolds $S$ and $R$ of $G\times ...

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### Category of representations of the path-groupoid

The path-groupoid $\mathcal{P}_1(X)$ of a (smooth) topological space $X$ is a refinement of the fundamental groupoid $\Pi_1(X)$ whose morphisms are given by (piecewise smooth) paths in $X$ up to thin-...

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### What is a “Twisted Groupoid”

Li and Barlak say in here, page 2 that a "twisted groupoid $(G, \Sigma)$ of two topological groupoids $G$, $\Sigma$ is a central groupoid extension of the form $\mathbb{T}\times G^{\left(0\right)}\...

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312 views

### Classification of weak 3-groups

Weak 2-groups can be classified by the data $(\pi_1,\pi_2, t, \omega)$, where $\pi_1$ is a group, $\pi_2$ an Abelian group, $t: \pi_1 \to Aut(\pi_2)$, and $\omega \in H^3(B\pi_1,\pi_2)$.
I wonder do ...

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### Twist of Groupoid: Surjection identity on (common) unit space?

Let $\mathbb{T}\times G^{\left(0\right)}\overset{i}{\rightarrowtail}\Sigma\overset{p}{\twoheadrightarrow}G$ a central groupoid extension where $G$ is an étale, hausdorff, locallycompact, second ...

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### A question on groupoids and measurable fields of Hilbert spaces

Suppose that we have the following data:
$ \mathcal{G} $ is a locally compact Hausdorff groupoid, with its source and
range maps denoted by $ s $ and $ r $ respectively.
$ (\lambda^{x})_{x \in \...

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### Gluing together together differentiable stacks

I am trying to figure out the conditions under which you can glue together a collection of (differentiable) stacks by equivalences, and get a differentiable stack.
More precisely, I have a collection ...

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### Does the $\infty$-groupoid functor $\Pi$ commute with pushouts of nice spaces?

Given a pushout diagram of nice topological spaces (such as CW complexes), does the infinity groupoid functor $\Pi(-)$ commute with the pushout? More precisely, does the pushout diagram get sent to a ...

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### Left adjoint to inclusion of Connected Groupoids into Groupoids

Let $Gpd$ denote the category of groupoids and functors. Let $Gpd_{con}$ denote the subcategory spanned by connected groupoids, i.e for every $x,y\in Ob(Gpd_{con})$, there is at least one morphism $x\...

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### Certain groupoid and its $C^{*}$ algebra

Let $X$ be a finite subset of real numbers. Let $G$ be the collection of all non empty subsets of $X$ and $G_{0}$ be the collection of all singleton subsets of $X$.
We define two maps $r,s:G \...

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### The holonomy groupoid of certain one dimensional foliations of 2 dimensional Euclidean regions

What Is the first fundamental group of each of the following $3$ dimensional Hausdorff manifolds? What about homology groups of these 3-manifolds? Is the first one a contractible manifold?
The ...

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### The pair $(Gl(n,\mathbb{R}), O(n) )$ as a groupoid

"Is there a topological groupoid structure on the pair $(Gl(n,\mathbb{R}), O(n))$, with their standard topologies?"
This is already asked here but this linked question is a very general question, so ...

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377 views

### A topological groupoid structure on a pair $(X,A)$

Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.
Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding ...

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### Reference for the existence of bicolimits in groupoids and categories?

I am looking for a reference of these, I would say, very well known facts. (strangely though finding a reference was bit trick for me).
Let $C$ be a category and $F:C\rightarrow Cat$ a 2-functor in ...

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**1**answer

385 views

### Groupoid cardinality and Egyptian fraction representations of 1

It is well-known that any rational number can be represented using a sum of distinct Egyptian fractions (that is, rational fractions of the form $1/n$ with $n\in\mathbb{N}$). This may be proven by ...

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### Group actions on principal groupoids

Suppose that $\mathcal{G}$ is etale principal groupoid and that $G$ is a discrete (or finite) group acting freely on the locally compact unit space $\mathcal{G}^0$ (or assuming compactness, if ...

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### Existence of enough local sections

Let $\pi: G\to X$ be a continuous open (!) surjection of locally compact Hausdorff spaces. Assume that each fiber $G_x=\pi^{-1}(x)$, $x\in X$ carries a group structure making it a locally compact ...

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### Groupoid isomorphism vs. group isomorphism

Assume that $\Gamma$ is a group with neutral element $e$. We associate to $\Gamma$ the following groupoid $G$:
$G=\Gamma \times \Gamma,\;\;\;G^{(0)}=\Gamma \times \{0\},\;\;s(a,b)=(a,e),\;\;\; r(a,...

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### Representation theory of groupoids

Is there for a groupoid an representation theorem as Representation theorem of Wagner-Preston for inverse semigroups?

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### Are all locally compact anisotropic groupoids etale up to equivalence?

By groupoid I mean "open topological groupoid",i.e. topological groupoids whose source and target maps are open surjections, and the notion of equivalence I'm considering is the isomorphism in the ...

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### 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 ...

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### 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 ...

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### 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. ...

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### 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 $B$-...

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**1**answer

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### 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 ...

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### 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 $\mathcal{X}$....

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### 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, ...

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### 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 ...