Editing Groupoid (section)

=== Fundamental groupoid ===
{{main|Fundamental groupoid}}
Given a [[topological space]] {{tmath|1= X }}, let <math>G_0</math> be the set {{tmath|1= X }}. The morphisms from the point <math>p</math> to the point <math>q</math> are [[equivalence class]]es of [[continuous function (topology)|continuous]] [[path (topology)|path]]s from <math>p</math> to {{tmath|1= q }}, with two paths being equivalent if they are [[homotopic]].
Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is [[associative]]. This groupoid is called the [[fundamental groupoid]] of {{tmath|1= X }}, denoted <math>\pi_1(X)</math> (or sometimes, {{tmath|1= \Pi_1(X) }}).<ref>{{cite web |url=https://ncatlab.org/nlab/show/fundamental+groupoid |title=fundamental groupoid in nLab |website=ncatlab.org |access-date=2017-09-17 }}</ref> The usual fundamental group <math>\pi_1(X,x)</math> is then the vertex group for the point {{tmath|1= x }}.

The orbits of the fundamental groupoid <math>\pi_1(X)</math> are the path-connected components of {{tmath|1= X }}. Accordingly, the fundamental groupoid of a [[path-connected space]] is transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are [[Equivalence of categories|equivalent]] as categories (see the section [[Groupoid#Relation to groups|below]] for the general theory).

An important extension of this idea is to consider the fundamental groupoid <math>\pi_1(X,A)</math> where <math>A\subset X</math> is a chosen set of "base points". Here <math>\pi_1(X,A)</math> is a (full) subgroupoid of {{tmath|1= \pi_1(X) }}, where one considers only paths whose endpoints belong to {{tmath|1= A }}. The set <math>A</math> may be chosen according to the geometry of the situation at hand.