Kerodon

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

Example 3.2.2.12. In the special case $n=1$, we can rewrite condition $(b)$ of Theorem 3.2.2.10 as follows:

  • Let $f$, $g$, and $h$ be edges of $X$ which begin and end at the vertex $x$. Then the equality $[h] = [g] [f]$ holds (in the fundamental group $\pi _{1}(X,x)$) if and only if there exists a $2$-simplex $\sigma $ of $X$ which witnesses $h$ as a composition of $f$ and $g$ (in the sense of Definition 1.3.4.1), as indicated in the diagram

    \[ \xymatrix@R =50pt@C=50pt{ & x \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & x. } \]

It follows that the fundamental group $\pi _{1}(X,x)$ can be identified with the automorphism group of $x$ as an object of the fundamental groupoid $\pi _{\leq 1}(X) = \mathrm{h} \mathit{X}$.