Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 3.2.2.16. Let $X$ be a Kan complex and let $x$ be a vertex of $X$. Then $x$ can also be regarded as a vertex of the opposite simplicial set $X^{\operatorname{op}}$, which is also a Kan complex. For $n \geq 1$, we have an evident bijection $\varphi : \pi _{n}(X,x) \simeq \pi _{n}(X^{\operatorname{op}}, x)$. If $n \geq 2$, then this bijection is an isomorphism of abelian groups. Beware that, in the case $n=1$, it is generally not an isomorphism of groups: instead, it is an anti-isomorphism (that is, it satisfies the identity $\varphi ( \xi \xi ' ) = \varphi (\xi ') \varphi (\xi )$ for $\xi , \xi ' \in \pi _{1}(X,x)$; see Warning 3.2.2.13 above).