## Homotopy Groups

Let $X$ be a pointed fibrant simplicial set. The homotopy group $\pi_n(X)$, as a set, is defined by $\pi_n(X)=[S^n,X]$, i.e.  the set of the pointed homotopy classes of all pointed simplicial maps from $S^n$ to $X$. $\pi_n(X)=\pi_n(|X|)$ as sets.

An element $x\in X_n$ is said to be spherical if $d_i x=*$ for all $0\leq i\leq n$.

Given a spherical element $x\in X_n$, then its representing map $f_x:\Delta[n]\to X$ factors through the simplicial quotient set $S^n=\Delta[n]/\partial\Delta[n]$. Conversely, any simplicial map $f:S^n\to X$ gives a spherical element $f(\sigma_n)\in X_n$, where $\sigma_n$ is the nondegenerate element in $S^n_n$. This gives a one-to-one correspondence from the set of spherical elements in $X_n$ to the set of simplicial maps $S^n\to X$.

Path product and fundamental groupoids
Let $\sigma_1=(0,1)\in\Delta[1]_1$. A path is a simplicial map $\lambda:\Delta[1]\to X$.