fundamental groupoid

Definition 1.

Given a topological space $X$ the fundamental groupoid ${\mathrm{\Pi }}_1(X)$ of $X$ is defined as follows:

• The objects of ${\mathrm{\Pi }}_1(X)$ are the points of $X$

  $$\mathrm{Obj}({\mathrm{\Pi }}_1(X))=X,$$

• morphisms are homotopy classes of paths “rel endpoints” that is

  $${\mathrm{Hom}}_{{\mathrm{\Pi }}_1(X)}(x,y)=\mathrm{Paths}(x,y)/\sim ,$$

  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 $x$,

$${\mathrm{Hom}}_{{\mathrm{\Pi }}_1(X)}(x,x)={\pi }_1(X,x).$$

Definition 2.

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

$${\mathrm{\Pi }}_1(f):{\mathrm{\Pi }}_1(X)\to {\mathrm{\Pi }}_1(Y)$$

defined as follows

• on objects ${\mathrm{\Pi }}_1(f)$ is just $f$,

• on morphisms ${\mathrm{\Pi }}_1(f)$ is given by “composing with $f$”, that is if $\alpha :I\to$ $X$ is a path representing the morphism $[\alpha ]:x\to y$ then a representative of ${\mathrm{\Pi }}_1(f)([\alpha ]):f(x)\to f(y)$ is determined by the following commutative diagram