Remark 3.2.2.15. Let $(X,x)$ be a pointed Kan complex. For $n \geq 2$, the homotopy group $\pi _{n}(X,x)$ is abelian. We will generally emphasize this by using additive notation for the group structure on $\pi _{n}(X,x)$: that is, we denote the group law by
\[ +: \pi _{n}(X,x) \times \pi _{n}(X,x) \rightarrow \pi _{n}(X,x) \quad \quad (\xi , \xi ') \mapsto \xi + \xi '. \]
With this convention, we can restate property $(b)$ of Theorem 3.2.2.10 as follows:
- $(b)$
Let $f: \operatorname{\partial \Delta }^{n+1} \rightarrow X$ be a morphism of simplicial sets, corresponding to a tuple $(\sigma _0, \sigma _1, \ldots , \sigma _{n+1})$ of $n$-simplices of $X$. Then $f$ extends to an $(n+1)$-simplex of $X$ if and only if the sum $\sum _{i = 0}^{n+1} (-1)^{i} [ \sigma _ i ]$ vanishes in $\pi _{n}(X,x)$.