A groupoid (in the sense of category theory) is a small category in which every morphism is an isomorphism. Groupoids arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical ...

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coherence of inverses in 2-groupoids

Given a 2-groupoid $G$, two objects $a,b$, and two 1-morphisms $f,g:a\rightarrow b$ and a 2-morphism $\alpha : f \rightarrow g$, is it the case that there always exists a 2-morphism $\beta : f^{-1} ...
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24 views

Ref. Request — Non-Transitive Lie Group Actions, Applications to Orbifolds/Groupoids

I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some ...
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1answer
22 views

Why should the source and the target map of a Lie groupoid be submersions?

In the definition of a Lie groupoid, the source and the target maps are required to be submersions. I want to know the reason for that. I write down definitions below. See also ...
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31 views

van Kampen theorem for fundamental groupoid of $X$ relative to $A$

Let $X$ be a manifold with submanifold $A \subseteq X$. Let $\Pi_{1}(X,A)$ denote the homotopy classes of paths with endpoints lying in $A$. This is a Lie groupoid with set of objects $A$. For ...
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71 views

Fundamental groupoid of a contractible space

I read that the fundamental groupoid of a contractible space is indiscrete. How can one show this? I found this as an exercise here.
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41 views

Is the Groupoid of germs associated to an orbifold a Hausdorff proper Lie groupoid?

I was studying the book Introduction to Foliations and Lie Groupoids by I. Moerdijk and J. Mrcun and I have a doubt. On page 140 they give the following definition for a proper Lie groupoid. A ...
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28 views

Isomorphism of Principal Bundles with structure groupoid

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Suppose that $\pi:P\rightarrow B$ is a $\mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)\rightarrow (P,B); s\in [0,1]$, be a homotopy ...
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39 views

Local bisections 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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41 views

Fundamental groupoids is a embedding of category $Cov(B)\to Cov(\Pi(B))$

It is stated in J.P.May's A Concise Course in Algebraic Topology page 29 that the fundamental groupoid functor induces a bijection $$Cov(E,E')\longleftrightarrow Cov(\Pi(E),\Pi(E')).$$ So does that ...
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117 views

The Seifert-Van Kampen theorem as a push-out

My question concerns the proof of the Seifert-Van Kampen theorem. The version of such a statement that interests me is the following. Let $X$ be a topological space, and $U, V\subseteq X$ two ...
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41 views

groupoid of finite sets.

I'm newbie to the category and groupoid, and I got confused about the definition of groupoids. In the definition of groupoid in the Wikipedia, it says a groupoid is a "small" category such that every ...
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45 views

Can the integral group ring construction be extended to groupoids in such a way that it provides a functor from Groupoids to Rings?

If $G$ is a group, and $R$ is a ring, one can construct the group ring $R[G]$, which is a free $R$-module with basis $G$, and with multiplication coming from the original multiplication of $G$. ...
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40 views

Conjugation in a groupoid

In a group $G$ , the conjugation of any element $g \in G$ by $h$ defines an inner automorphism $\psi_h : g \to hgh^{-1}$. If one considers instead a connected groupoid $\mathcal{G}$, is there a ...
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77 views

Constructing the classifying space for a groupoid G

I am working with the classifying space BG of the groupoid G. One definition is as follows: $$ BG = \bigsqcup _n (G_n \times \Delta ^n) /(d_i(g),x) \sim (g, \delta _i (x)). $$ Where the $d_i$ are ...
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59 views

Proving closure of unit space of a Hausdorff groupoid

For Hausdorff topological groups, the set $\{e\}$ containing only the identity is closed. This is because Hausdorff implies T1 which implies singletons are closed. For topological groupoids, defined ...
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86 views

How do you construct the lifted topology of a groupoid cover?

If I have a particularly nice space $X$ (Hausdorf, locally path connected, semi-locally 1-connected, I think), then then there is an equivalence of categories between the cover category over $X$ and ...
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95 views

Groupoid $C^*$ algebra of product groupoid

Let $G$ and $H$ be locally compact (Hausdorff, second countable) groupoids with Haar systems $\mu$ and $\nu$, respectively. Is it true then that the (full) groupoid $C^*$-algebras satisfy $$ ...
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25 views

A certain product of C*-algebras

So, I am looking for some kind of 'product' $\bullet$ on the category of (unital?) $C^*$-algebras satisfying that $M_n(\mathbb{C})\bullet M_m(\mathbb{C}) = M_{m+n}(\mathbb{C})$ where ...
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Is there a “natural” / “categorical” definition of the “parity” of a permutation?

Given a permutation $\sigma$ on $n$ elements (i.e. $\sigma \in S_n$), there is a notion of "parity" (or "sign" or "signature") of $\sigma$, which can be defined in several equivalent ways (look here). ...
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42 views

Relationship between groupoid morphisms and the induced functor on their categories of actions?

This is similar to a question I asked recently, but this time specifically for groupoids. Suppose $f: A \rightarrow B$ is a groupoid morphism. Let $f^\ast: [B, \text{Set}] \rightarrow [A, ...
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2answers
89 views

Which constructions on a category are still interesting for a groupoid?

By a groupoid, I mean a (small) category in which every morphism is an isomorphism. It looks to me that constructions on a category like the opposite ("dual") category or products and (co-)limits ...
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3answers
115 views

From groups to groupoids.

Let $\mathcal{G}$ be a groupoid and $p$ an object in $\mathcal{G}.$ It is well known that the set ${\rm Mor}_{\mathcal{G}}(p,p)$ is a group. I would like to know if there is a way to recognize a ...
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42 views

Determination of a connected groupoid by its objects and by a set of automorphisms. [duplicate]

One may readily show that a connected groupoid $G$ is determined up to isomorphism by a group (one of the groups $\hom_G(x,x)$) and by a set (the set of all objects). This is the nature of the problem ...
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47 views

Groupoid-valued presheaf as a colimit of representables

Is there a specific way to see a presheaf of groupoids as a colimit of representables ? As you can understand I'm looking for a similar result to the well-known fact that presheaves of sets are ...
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69 views

Torsors for 2-groups

Let $\mathbb{G}$ be a 2-group, by which I mean a strict monoidal category in which all objects are invertible (up to coherent isomorphisms) and all morphisms are invertible (strictly). What is the ...
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175 views

Which of these constructions are left adjoints?

A groupoid can be regarded as a small category in which every arrow is an isomorphism A monoid can be regarded as a small category with only a single object A preorder can be regarded as a small ...
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1answer
362 views

Where to get help with Homotopy type theory?

I'm trying to understand the Homotopy Type Theory book. I find myself completely lost in Chapter 2, especially when it starts using higher groupoids. What is the recommended background for this book? ...
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1answer
84 views

Should the first be the last by composition of paths?

Given two paths $f,g:\mathbb{I}\rightarrow X$ with $f\left(1\right)=g\left(0\right)$ there is a composite $f.g$ defined by $t\mapsto f\left(2t\right)$ if $2t\leq1$ and $t\mapsto g\left(2t-1\right)$ ...
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448 views

Statement about Homotopy in Brown's “Topology & Groupoids”

I am trying to understand a statement in Brown's Topology and Groupoids, 7.2.5 (Corollary 1), page 270. Let's first have some preliminary remarks Let $X,Y$ be topological spaces. The track groupoid ...
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82 views

Is a groupoid a universal algebra?

I was trying to use the first isomorphism theorem on groupoids. From the wikipedia page I know that it holds for groups, rings and algebras. So are groupoids algebras? Or, in other words, does the ...
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153 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
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407 views

A comparison between the fundamental groupoid and the fundamental group

Are there two path connected topological spaces $X,Y$ such that the fundamental groupoid of $X$ is not isomorphic to the fundamental groupoid of $Y$ but the fundamental group of $X$ is isomorphic to ...
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352 views

Existence of a certain functor $F:\mathrm{Grpd}\rightarrow\mathrm{Grp}$

Let $\mathrm{Grpd}$ denote the category of all groupoids. Let $\mathrm{Grp}$ denote the category of all groups. Are there functors $F\colon\mathrm{Grpd}\rightarrow \mathrm{Grp}, ...
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2answers
312 views

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 the action groupoid coming from a transitive action of some group ...
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2answers
146 views

In a groupoid, do any two objects have a morphism between them?

This is a question about the definition of a groupoid (in category theory). I've read the Wikipedia article, and it says that a groupoid is a small category in which every morphism is an isomorphism. ...
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2answers
82 views

Groupoids isomorphism

Let $G, G'$ be two groups and $X=\{x,y\}$ be a set of two elements. Consider a groupoid $\mathcal{G}$ with objects from $X$ such that Hom$(x,x)=G$ and Hom$(y,y)=G'$. Suppose Hom$(x,y) \neq ...
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124 views

Properties of the Category of topological spaces with $n$ basepoints.

I've recently encounted a problem in my reading which would seem to be more naturally phrased if the category we work in shifted from the category $\textbf{Top}^*$ of pointed topological spaces, to ...
5
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2answers
181 views

Understanding an Example of a Lie Groupoid

For the definition of Lie groupoid see https://en.wikipedia.org/wiki/Lie_groupoid . In this question I want to understand Example 1.1.17 in "General theory of Lie groupoid and Lie algebroids" by ...
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1answer
248 views

Group of Isomorphisms of a Groupoid

Write a careful proof that every group is the group of isomorphisms of a groupoid. In particular, every group is the group of automorphisms of some object in some category. First can someone tell me ...
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65 views

2-morphisms from spans of spans

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton . Suppose we have a span of span of groupoids as follows and ...
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1answer
906 views

Does May's version of groupoid Seifert-van Kampen need path connectivity as a hypothesis?

May's A Concise Course in Algebraic Topology gives the following statement of the Seifert-van Kampen theorem for fundamental groupoids $\Pi(X)$ (section 2.7): Theorem (van Kampen). Let ...
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1answer
132 views

Etale groupoid and Morita equivalence

Let $\mathcal{G}=(G_{0},G_{1})$ be a groupoid, where $G_{0}$ is the space of objects and $G_{1}$ is the space of morphisms. $\mathcal{G}$ is called etale if both the source and target maps $$ ...
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1answer
206 views

The axiom of choice and connected groupoids

Recall the two definitions of equivalence of categories: Definition 1. An equivalence of categories is a quadruplet $(F, G, \alpha, \beta)$, where $F : \mathbb{C} \to \mathbb{D}$ and $G : \mathbb{D} ...
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325 views

can the statement “a simplicial set is the nerve of a category if and only if it satisfies a horn-filling condition” be tweaked for groupoids?

For some reason I convinced myself that a simplicial set (or maybe I mean directly Kan complex) is homotopy equivalent to the nerve of a groupoid if and only if it has no higher homotopy groups. Is ...
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85 views

When is a morphism of $S$-groupoids a monomorphism?

According to "Champs algébriques" by Laumon and Moret-Bailley, and $S$-groupoid is a category $\mathscr{X}$ and a functor $a: \mathscr{X} \to (\mathrm{Aff}/S)$, where $(\mathrm{Aff}/S)$ is the ...
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1answer
113 views

Relationship between functors

You will have to forgive me as I am very new to category theory - fifth of the way through Categories for a working mathematician. I'm interested in the following; Let $F:A \to B$ and $G:A \to C$ be ...
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1answer
527 views

covering spaces and the fundamental groupoid

Briefly, my question is whether there is a basepoint-free statement of the basic theorem on covering spaces. For a nice space $X$, I would hope that there is an equivalence of categories between ...
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118 views

An example of functions on a groupoid

There are two $C^\ast$-algebras associated to the $\ast$-algebra (under a convolution and the usual involution) $$C_c(G) := \{ f:G\longrightarrow \mathbb C :\:f \text{ has compact support}\}$$ of ...
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1answer
264 views

Topological Space as an $(\infty,0)$-category

Given a topological space $X$, we may wish to consider it as an $(\infty,0)$-category, where the objects are the points of the space, the 1-morphisms are continuous paths between points, the ...
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93 views

Definition of a groupoid of fractions

The title sums it up : I am looking for a definition of "a groupoid of fractions for a category". I would be interested in any example someone might have as well...