fundamental groupoid

Definition 1.

Given a topological spaceMathworldPlanetmath X the fundamental groupoidMathworldPlanetmathPlanetmathPlanetmath Π1(X) of X is defined as follows:

It is easily checked that the above defined category is indeed a groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath with the inverseMathworldPlanetmathPlanetmathPlanetmath of (a morphism represented by) a path being (the homotopy class of) the “reverse” path. Notice that for xX, the group of automorphismsPlanetmathPlanetmathPlanetmathPlanetmath of x is the fundamental groupMathworldPlanetmathPlanetmath of X with basepoint x,

Definition 2.

Let f:XY be a continuous function between two topological spaces. Then there is an induced functorMathworldPlanetmath


defined as follows

  • on objects Π1(f) is just f,

  • on morphisms Π1(f) is given by “composing with f”, that is if α:I X is a path representing the morphism [α]:xy then a representative of Π1(f)([α]):f(x)f(y) is determined by the following commutative diagramMathworldPlanetmath