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 ...

learn more… | top users | synonyms

40
votes
5answers
2k views

What structure does the alternating group preserve?

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other ...
26
votes
1answer
948 views

Purely combinatorial proof that$ (e^x)' = e^x$

At the beginning of Week 300 of John Baez's blog, Baez gives a proof that the "number" of finite sets (more specifically, the cardinality of the groupoid of all finite sets, where an object in the ...
23
votes
1answer
404 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}, ...
18
votes
4answers
396 views

What are the ramifications of the fact that the first homotopy group can be non-commutative, whilst the higher homotopy groups can't be?

Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to ...
16
votes
1answer
714 views

Categorification of $\pi$?

Is there a categorification of $\pi$? I have to admit that this is a very vague question. Somehow it is motivated by this recent MO question, which made me stare at some digits and somehow forgot my ...
12
votes
2answers
460 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 ...
12
votes
1answer
1k 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 ...
11
votes
2answers
209 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 ...
10
votes
3answers
482 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 ...
9
votes
4answers
247 views

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). ...
8
votes
4answers
2k views

definition of a groupoid

Notation: Underlining $\underline{G}$ denotes a category and $\underline{G}(x,y)$ the class of morphisms from $x$ to $y$. On the Wiki page about groupoids, it is written (I write here my own more ...
8
votes
1answer
103 views

How are groupoids richer structures than sets of groups?

This has been bugging me for quite some time: My intuition with categories is, that I can simply identify isomorphic objects. It does for example not matter, whether the entries in a sudoku are the ...
6
votes
2answers
165 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 ...
6
votes
1answer
405 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? ...
5
votes
2answers
370 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 ...
5
votes
2answers
192 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 ...
5
votes
1answer
121 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 ...
5
votes
1answer
121 views

Universal covering and double cover functors

Cross-posted on MO Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to know ...
5
votes
0answers
45 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 ...
5
votes
0answers
176 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 ...
4
votes
1answer
639 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 ...
4
votes
1answer
91 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 ...
4
votes
1answer
97 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...
4
votes
0answers
44 views

What natural monoidal structure and braiding exists on the category of modules of the convolution algebra of an action groupoid?

Let $S$ be a set with an action $\triangleright$ of a finite group $G$. The action groupoid $S // G$ has as objects the set $S$, and the morphisms from $s_1$ to $s_2$ are just the $g \in G$ that ...
4
votes
0answers
36 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 ...
4
votes
0answers
159 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 ...
3
votes
3answers
136 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 ...
3
votes
1answer
43 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 ...
3
votes
2answers
99 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 ...
3
votes
2answers
92 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 ...
3
votes
1answer
81 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.
3
votes
1answer
91 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 ...
3
votes
0answers
35 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 ...
2
votes
3answers
400 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 ...
2
votes
1answer
273 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 ...
2
votes
1answer
40 views

If functions compose both ways to make automorphisms, are they isomorphisms?

Let's say that we have morphisms $f:A \to B$ and $g : B \to A$ such that $f \circ g$ and $g \circ f$ are both automorphisms (an automorphism is a morphism that is both iso and endo). Are $f$ and $g$ ...
2
votes
1answer
275 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 ...
2
votes
1answer
70 views

Pushforward of a representation?

Suppose that $G$ is a finite group, and $G/N$ is a quotient. Given a representation of $G$, is there a "natural" way to construct a representation on $G/N$? (I.e. a pushforward representation, ...
2
votes
1answer
33 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 ...
2
votes
2answers
157 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. ...
2
votes
1answer
165 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 $$ ...
2
votes
1answer
217 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} ...
2
votes
1answer
115 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 ...
2
votes
1answer
63 views

Proper effective étale groupoid gives an effective orbifold

I have a question about the translation groupoid mentioned in the last paragraph. I don't understand why $N_x/G_x$ is an open embedding. I think because $X$ is the orbit space, we need to mod out ...
2
votes
0answers
37 views

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} ...
2
votes
1answer
54 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 ...
2
votes
0answers
101 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 $$ ...
2
votes
1answer
91 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 $t\mapsto g\left(2t-1\right)$ ...
2
votes
2answers
88 views

groupoids and localization in categories

I've been learning homology (and hence Category Theory) and have been interested in groupoids for some time. I've gotten to localization in categories, and that has sparked the following question: ...
1
vote
2answers
112 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 ...