# 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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### 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 ...

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### Integrating the Riemann curvature tensor over a singular 2-disc

There's a classic characterization of the Riemann curvature tensor. Say, take a Riemann metric on an open subset $U$ of $\mathbb R^n$. Given a point $p \in U$ and two vectors $v,w \in T_p U$ you ...

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### 1-connected infinity groupoids, groupoids and 1-connected spaces

I am exploring a bit the world of groupoids. What I have in mind is that infinity groupoids correspond to spaces. So my first question is the following:
Consider the model category $\infty-Grpd$ of ...

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431 views

### Homotopy of functors

Recently I have read two different proposals for a notion of homotopy between functors, and I am curious which contexts each best lend themselves to. The first comes from Ming-Jung Lee's 1972 paper ...

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191 views

### Property of representations of reductive group schemes over characteristic 0 field

I originally posted this on Maths SE, but then I thought it MO might be more fitting.
Let $k$ be a characteristic $0$ field and let $G$ be a linear algebraic group scheme over $k$. Then is it true ...

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### J. F. Adams Proof of Cellular Approximation Theorem

In Ronald Brown's discussion of the proof of The Cellular Approximation Theorem in Topology and Groupoids Sec. 7.6 he writes that, "the elegant formulation of the proof is due to J. F. Adams." Does ...

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### Which $\infty$-groupoids correspond to simplicial abelian groups?

Kan complexes model $\infty$-groupoids, so since every simplicial abelian group is a Kan complex, every simplicial abelian group yields an $\infty$-groupoid. What sort of $\infty$-groupoids do you ...

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### Inverse semigroups and partial symmetries

I recently ran across the idea of inverse semi-groups in the context of partial symmetries, where the symmetry only acts on part of the system and not the entire system (e.g., in quasi-crystals).
My ...

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### Space with semi-locally simply connected open subsets

A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $...

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### Homologous quotient of fundamental groupoid

Let $X$ be a connected space and $\Pi_1(X)$ be its fundamental groupoid. We consider the homologous relation $\mathcal R$ on every morphism space: $f,g\in \Pi_1(X)(p,q)$ are related if the singular ...

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### Does the 1-category construction of a topos of presheaves extend to the 2-Category of Groupoids?

In the case of 1-categories, we know there is a functor category
$PSh(C):=[C^{op},Set]$, where $C$ is a small category,
and this functor category is a topos. I am hoping this will extend to the case ...

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### Is the 2-сategory of groupoids locally presentable?

I am wondering if the 2-сategory of groupoids is locally presentable. Locally presentable means the category is accessible and co-complete.
It has been pointed out that the category of groupoids is ...

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### Generalize $H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R)$ to fundamental Groupoid

Let $X$ be a path-connected smooth manifold, it is known that: $$H^1(X):=H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R).$$ Explicitly, a closed one-form $\alpha$ gives a function on $\pi_1(X)$ by $[\...

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### On fundamental groupoid of fundamental groupoid

Given a topological space $X$, we have the notion of the fundamental groupoid $\Pi_1(X)$.
Here, the fundamental groupoid $\Pi_1(X)$ is made into a topological groupoid giving a topology on the ...

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### Equivalence De Rham and Dolbeault groupoids

I believe there is an error or incompleteness in Goldman's and Xia's proof of the equivalence of the De Rham and Dolbeault groupoids, contained in
Rank One Higgs Bundles and Representations of ...

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### Restricting functions to an isotropy group

Let $\mathcal G$ be a locally compact, étale groupoid and let $x$ be a point in the unit space of $\mathcal G$. Writing $\mathcal G(x)$ for the isotropy group at $x$, consider the map
$$
f∈C_c(\...

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### What category of toposes is monadic over the 2-category of groupoids?

Excuse my lack of understanding of monadicity, but I have been looking at toposes and monads. I see Lambek showed that the category of Toposes are monadic over the category of categories. I see the ...

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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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241 views

### 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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108 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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231 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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229 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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590 views

### 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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241 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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### 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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### 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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472 views

### 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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### 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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332 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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### 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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### 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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### 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 ...