homotopy invariance

Let $ℱ$ be a functor from the category of topological spaces to some category $𝒞$. Then $ℱ$ is called homotopy invariant if for any two homotopic maps $f,g:X\to Y$ between topological spaces $X$ and $Y$ the morphisms $ℱf$ and $ℱg$ in $𝒞$ induced by $ℱ$ are identical.

Suppose $ℱ$ is a homotopy invariant functor, and $X$ and $Y$ are homotopy equivalent topological spaces. Then there are continuous maps $f:X\to Y$ and $g:Y\to X$ such that $g\circ f\simeq {\mathrm{id}}_X$ and $f\circ g\simeq {\mathrm{id}}_Y$ (i.e. $g\circ f$ and $f\circ g$ are homotopic to the identity maps on $X$ and $Y$, respectively). Assume that $ℱ$ is a covariant functor. Then the homotopy invariance of $ℱ$ implies

$$ℱg\circ ℱf=ℱ(g\circ f)={\mathrm{id}}_{ℱX}$$

and

$$ℱf\circ ℱg=ℱ(f\circ g)={\mathrm{id}}_{ℱY}.$$

From this we see that $ℱX$ and $ℱY$ are isomorphic in $𝒞$. (The same argument clearly holds if $ℱ$ is contravariant instead of covariant.)

An important example of a homotopy invariant functor is the fundamental group ${\pi }_1$; here $𝒞$ is the category of groups.