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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67 views

Correspondence $\{$principal $G$-bundles on $M\}\leftrightarrow\{$conjugacy classes of homomorphisms $\pi_1(M)\to G\}$

Context. I'm reading Qiaochu's short note Surfaces and the representation theory of finite groups which aims to prove Mednykh's formula inspired by ideas from topological quantum field theory. On page ...
3
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1answer
51 views

Characterization of internal groupoids via pullbacks

The most intuitive way (for me) to define an internal groupoid is as an internal category with extra structure, namely an involution on the object of morphisms which "produces inverses". In Borceux ...
2
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1answer
16 views

The Image of a Groupoid Morphism is Not a Subgroupoid

I am reading Brown's "Topology and Groupoids" book. In it he defines a subgroupoid as a subcategory of a groupoid that contains all the inverses for all the morphisms. A morphism of groupoids is a ...
0
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1answer
35 views

Action groupoid as $G\rightrightarrows \textrm{Bij}(X)$?

Let $G$ be a group and $X$ a set. A left action of $G$ on $X$ can be thought either as a map $G\times X\longrightarrow X$, $(g, x)\longmapsto g\cdot x$, satisfying: $(i)$ $g\cdot (h\cdot x)=(gh)\cdot ...
2
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1answer
83 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, ...
5
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1answer
124 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 ...
6
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2answers
173 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
votes
1answer
123 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 ...
4
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1answer
662 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 ...
12
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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 $\mathcal{O}...
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0answers
28 views

Differentiable stacks and morita morphism

I heard that if $[X_0/G_0]$ and $[X_1/G_1]$ are differentiable stacks, then any morphism between them is naturally equivalent to $$(G_0 \rightrightarrows X_0) \xleftarrow{\simeq} (G_2 \...
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4answers
255 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). ...
9
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1answer
114 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 ...
2
votes
1answer
41 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 $g$ ...
5
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0answers
48 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 ...
2
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1answer
68 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 by ...
2
votes
2answers
91 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: ...
23
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1answer
407 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}, G\colon\mathrm{Grp}\...
2
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0answers
22 views

Morphism from a groupoid to the action groupoid?

Let $\mathcal{G}\rightrightarrows M$ be a groupoid and $G$ be a group acting on $M$. Then we might associate the action groupoid $G\ltimes M\rightrightarrows M$. What are the groupoid morphisms from ...
0
votes
1answer
34 views

Groupoid element with multiple inverse elements?

Simply put, is there a groupoid whose element can have multiple inverse elements? I know how to prove that elements of a semigroup have unique inverses, but this is a bit diferent... If there is such ...
1
vote
1answer
51 views

Conjugacy Class of Isomorphisms Between Two Isomorphic Groups Definition

In Spanier's algebraic topology book, in section 1.7 about the fundamental groupoid, he claims that if $A$ and $B$ are objects in the same component of a groupoid $\mathcal{G}$ (meaning that $hom_{\...
2
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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} \...
3
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0answers
37 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
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1answer
38 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 https://en.wikipedia....
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1answer
40 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 example,...
3
votes
1answer
86 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.
40
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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 words,...
5
votes
0answers
51 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 ...
4
votes
0answers
39 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 of $\...
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0answers
56 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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1answer
48 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 ...
4
votes
0answers
169 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 path-...
2
votes
1answer
55 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 ...
0
votes
1answer
52 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 $G$. http:/...
3
votes
1answer
47 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
1answer
96 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 ...
0
votes
1answer
76 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 ...
2
votes
0answers
105 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 $$ C^*(G\...
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vote
0answers
28 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 $M_n(\mathbb{C}...
1
vote
0answers
54 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, \text{Set}]...
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
3answers
140 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 ...
0
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0answers
43 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 ...
0
votes
1answer
54 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 ...
1
vote
0answers
73 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 ...
11
votes
2answers
220 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 ...
7
votes
1answer
425 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? ...
2
votes
2answers
160 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. ...
3
votes
2answers
98 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 ...
10
votes
3answers
492 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 ...