Editing Fundamental group (section)

===Fundamental groupoid===
The ''[[fundamental groupoid]]'' is a variant of the fundamental group that is useful in situations where the choice of a base point <math>x_0 \in X</math> is undesirable. It is defined by first considering the [[category (mathematics)|category]] of [[Moore path|path]]s in <math>X,</math> i.e., continuous functions
:<math>\gamma \colon [0, r] \to X</math>,
where ''r'' is an arbitrary non-negative real number. Since the length ''r'' is variable in this approach, such paths can be concatenated as is (i.e., not up to homotopy) and therefore yield a category.<ref>{{harvtxt|Brown|loc=§6.1|2006}}</ref> Two such paths <math>\gamma, \gamma'</math> with the same endpoints and length ''r'', resp. ''r''' are considered equivalent if there exist real numbers <math>u,v \geqslant 0</math> such that <math>r + u = r' + v</math> and <math> \gamma_u, \gamma'_v \colon [0, r + u] \to X</math> are homotopic relative to their end points, where <math> \gamma_u (t) = \begin{cases} \gamma(t), & t \in [0, r] \\ \gamma(r), & t \in [r, r + u]. \end{cases} </math><ref>{{harvtxt|Brown|2006|loc=§6.2}}</ref><ref>{{harvtxt|Crowell|Fox|1963}} use a different definition by reparametrizing the paths to length ''1''.</ref>

The category of paths up to this equivalence relation is denoted <math>\Pi (X).</math> Each morphism in <math>\Pi (X)</math> is an [[isomorphism]], with inverse given by the same path traversed in the opposite direction. Such a category is called a [[groupoid]]. It reproduces the fundamental group since
:<math>\pi_1(X, x_0) = \mathrm{Hom}_{\Pi(X)}(x_0, x_0)</math>.

More generally, one can consider the fundamental groupoid on a set ''A'' of base points, chosen according to the geometry of the situation; for example, in the case of the circle, which can be represented as the [[union (set theory)|union]] of two connected open sets whose intersection has two components, one can choose one base point in each component. The [[Seifert–van Kampen theorem|van Kampen theorem]] admits a version for fundamental groupoids which gives, for example, another way to compute the fundamental group(oid) of <math>S^1.</math><ref>{{harvtxt|Brown|2006|loc=§6.7}}</ref>