Kerodon

Construction 3.2.2.4. Let $(X,x)$ be a pointed Kan complex and let $n$ be a nonnegative integer. We let $\pi _{n}(X,x)$ denote the set $[ \Delta ^ n / \operatorname{\partial \Delta }^ n, X]_{\ast }$ of pointed homotopy classes of maps from $\Delta ^ n / \operatorname{\partial \Delta }^ n$ to $X$ (Notation 3.2.1.10). For $n > 0$, we will refer to $\pi _{n}(X,x)$ as the $n$th homotopy group of $X$ with respect to the base point $x$ (see Theorem 3.2.2.10 below). In the special case $n = 1$, we refer to $\pi _{1}(X,x)$ as the fundamental group of $X$ with respect to the base point $x$.