# fundamental groupoid

###### Definition 1.

Given a topological space $X$ the fundamental groupoid $\Pi_{1}(X)$ of $X$ is defined as follows:

• The objects of $\Pi_{1}(X)$ are the points of $X$

 $\mathrm{Obj}(\Pi_{1}(X))=X\,,$
• morphisms are homotopy classes of paths “rel endpoints” that is

 $\mathrm{Hom}_{\Pi_{1}(X)}(x,y)=\mathrm{Paths}(x,y)/\sim\,,$

where, $\sim$ 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 groupoid with the inverse of (a morphism represented by) a path being (the homotopy class of) the “reverse” path. Notice that for $x\in X$, the group of automorphisms of $x$ is the fundamental group of $X$ with basepoint $x$,

 $\mathrm{Hom}_{\Pi_{1}(X)}(x,x)=\pi_{1}(X,x)\,.$
###### Definition 2.

Let $f\colon\thinspace X\to Y$ be a continuous function between two topological spaces. Then there is an induced functor

 $\Pi_{1}(f)\colon\thinspace\Pi_{1}(X)\to\Pi_{1}(Y)$

defined as follows

• on objects $\Pi_{1}(f)$ is just $f$,

• on morphisms $\Pi_{1}(f)$ is given by “composing with $f$”, that is if $\alpha\colon\thinspace I\to$ $X$ is a path representing the morphism $[\alpha]\colon\thinspace x\to y$ then a representative of $\Pi_{1}(f)([\alpha])\colon\thinspace f(x)\to f(y)$ is determined by the following commutative diagram