Editing Path (topology) (section)

== Fundamental groupoid ==

There is a [[Category theory|categorical]] picture of paths which is sometimes useful. Any topological space <math>X</math> gives rise to a [[Category (mathematics)|category]] where the objects are the points of <math>X</math> and the [[morphism]]s are the homotopy classes of paths. Since any morphism in this category is an [[isomorphism]], this category is a [[groupoid]] called the [[fundamental groupoid]] of <math>X.</math> Loops in this category are the [[endomorphism]]s (all of which are actually [[automorphism]]s). The [[automorphism group]] of a point <math>x_0</math> in <math>X</math> is just the fundamental group based at <math>x_0</math>. More generally, one can define the fundamental groupoid on any subset <math>A</math> of <math>X,</math> using homotopy classes of paths joining points of <math>A.</math> This is convenient for [[Van Kampen's Theorem]].