Kerodon

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

Proposition 1.1.6.7. Let $X =X_{\bullet }$ and $Y = Y_{\bullet }$ be simplicial sets, and let $f: X \rightarrow Y$ be a morphism of simplicial sets. Then the induced map

\[ \operatorname{Vert}( \mathrm{Gr}(X) ) \amalg \operatorname{Edge}( \mathrm{Gr}(X) ) \simeq X_1 \xrightarrow {f} Y_1 \simeq \operatorname{Vert}( \mathrm{Gr}(Y) ) \amalg \operatorname{Edge}( \mathrm{Gr}(Y) ) \]

is a morphism of directed graphs from $\mathrm{Gr}( X )$ to $\mathrm{Gr}( Y)$, in the sense of Definition 1.1.6.5.

Proof. Since $f$ commutes with the degeneracy operator $s^{0}_0$, it carries degenerate $1$-simplices of $X$ to degenerate $1$-simplices of $Y$, and therefore satisfies requirement $(a)$ of Definition 1.1.6.5. Requirement $(b)$ follows from the fact that $f$ commutes with the face operators $d^{1}_0$ and $d^{1}_1$. $\square$