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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2
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1answer
81 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 $t\mapsto g\left(2t-1\right)$ ...
1
vote
1answer
37 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 ...
20
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0answers
304 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}, ...
4
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0answers
23 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
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0answers
145 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 ...
3
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0answers
92 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 ...
2
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0answers
25 views

Proper Lie Groupoid confusion

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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0answers
32 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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0answers
90 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 $$ ...
1
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0answers
23 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 ...
1
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0answers
65 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 ...
1
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0answers
63 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 ...
1
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0answers
114 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 ...
0
votes
0answers
38 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, ...