Higher Homotopy Groups

We can generalise the idea to higher homotopy groups as follows.

Let $X$ be a pointed fibrant simplicial set. The fundamental group $\pi_n(X)$ is the quotient set of the spherical elements in $X_n$ subject to the relation generated by $x\sim x'$ if there exists $w\in X_{n+1}$ such that $d_0w=x$ , $d_1w=x'$ and $d_jw=*$ for $j>1$ .

The product structure in $\pi_n(X)$ is given by: $[x]+[x']=[d_1w]$ , where $w\in X_{n+1}$ such that $d_0w=x'$ , $d_2w=x$ and $d_jw=*$ for $j>2$ . Furthermore, the map $|\cdot|:\pi_n(X)\to\pi_n(|X|)$ preserves the product structure.