Kerodon

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

Warning 3.2.2.13. Let $(X,x)$ be a pointed Kan complex, so that $\pi _{1}(X,x)$ can be identified with the set $\operatorname{Hom}_{\pi _{\leq 1}(X) }(x,x)$ of homotopy classes of paths from $x$ to itself. We have adopted the convention that the multiplication on $\pi _{1}(X,x)$ is given by composition in the homotopy category $\mathrm{h} \mathit{X}$. In other words, if $f,g: x \rightarrow x$ are edges which begin and end at $x$, then the product $[g] [f] \in \pi _{1}(X,x)$ is the homotopy class of a path which can be described informally as traversing the path $f$ first, followed by the path $g$. Beware that the opposite convention is also common in the literature (note that his issue is irrelevant for the higher homotopy groups $\{ \pi _{n}(X,x) \} _{n \geq 2}$, since they are abelian).