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.

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

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

  $$\xymatrix{{I}\ar[d]_{{\alpha}}\ar@{-->}[dr]^{{\Pi_{1}(f)(\alpha)}}\\ {X}\ar[r]_{f}&{Y}}$$