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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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 ...
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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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What is the local structure of a Lie groupoid?
A manifold is locally $\mathbb R^n$. An orbifold is locally $\mathbb R^n/\{\text{finite group}\}$. Is there a similar way to think about the local structure of a Lie groupoid $G_1 \rightrightarrows ...
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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, ...
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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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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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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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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 ...
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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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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 ...
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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 ...
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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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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 $\...
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Is the crossed product $\mathcal{K} \rtimes G$ a groupoid algebra?
Suppose G, a discrete group acting on the compact operators $\mathcal{K}$ by automorphism of C*-algebra $\mathcal{K}$. Can we view the crossed product as a groupoid C*-algebra of some groupoid?
This ...
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What condition on a "bibundle between categories" generalizes "right-principal bibundle between groupoids"?
My question is long on background and motivation, and almost but not quite answered over at the nLab. I'll write up a bunch before asking my question (feel free to skip to the end or look at 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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Is there a "groupoid integral" with values in a groupoid?
Let $G = \{G_1 \rightrightarrows G_0\}$ be a finite groupoid, i.e. $G_1,G_0$ are both finite sets, and let $A$ be $\mathbb Q$-module. Regard $A$ as a discrete groupoid $A \rightrightarrows A$, and ...
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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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Is there a theory of differential equations for smooth correspondences?
This question is very closely related to another one I just asked. The general question is to what extent there is a theory of differential equations for smooth correspondences (between a smooth ...
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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{...
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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 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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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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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 $\mathcal{...
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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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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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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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Do Lie algebroids pull back (along submersions)?
There are more general definitions, but for my purposes a Lie algebroid on a smooth manifold $X$ is a vector bundle $A \to X$, a map $\rho: A \to {\rm T}X$ of vector bundles over $X$, and a bracket $[,...
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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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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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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 ...
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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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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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$\infty-$groupoid of $A_{\infty}$ algebras
Hello,
Consider first the following $2-$groupoid of Algebras over $\mathbb{C}$. Objects are Algebras, $1-$morphisms are isomorphisms, and a $2-$morphism between the isoms $f$ and $g$ from $A$ to $B$ ...
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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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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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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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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 ...
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Algebraic stacks as (étale) groupoid algebraic 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 ...
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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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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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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 ...
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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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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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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 ...
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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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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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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 (...
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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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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 ...