fundamental groupoid
Definition 1.
Given a topological space the fundamental groupoid
of is defined asfollows:
- •
The objects of are the points of
- •
morphisms
are homotopy classes of paths “rel endpoints” that is
where, denotes homotopy
rel endpoints, and,
- •
composition
of morphisms is defined via concatenation of paths.
It is easily checked that the above defined category is indeed a groupoidwith the inverse
of (a morphism represented by) a path being (the homotopyclass of) the “reverse” path.Notice that for , the group of automorphisms
of is thefundamental group
of with basepoint ,
Definition 2.
Let be a continuous function between two topological spaces.Then there is an induced functor
defined as follows
- •
on objects is just ,
- •
on morphisms is given by “composing with ”, that isif is a path representing the morphism then a representative ofis determined by the following commutative diagram