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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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}} \...
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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)...
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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 $...
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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 ...
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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 $...
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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 ...
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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}$....
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 (...
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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 ...
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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 ...
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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 ...
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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:...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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:
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}, \...
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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 ...
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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 ...
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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 $g \cdot (g'K,g''K) = (gg'K,gg''K)$...
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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 $\...
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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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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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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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