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.


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