Kerodon

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

Construction 3.3.4.3. Let $X$ be a simplicial set. For every $n$-simplex $\sigma : \Delta ^ n \rightarrow X$, we let $\rho _{X}(\sigma )$ denote the composite map

\[ \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) \xrightarrow { \mathrm{Max} } \Delta ^ n \xrightarrow {\sigma } X, \]

which we regard as an $n$-simplex of the simplicial set $\operatorname{Ex}(X)$ of Construction 3.3.2.5. It follows from Remark 3.3.4.2 that the construction $\sigma \mapsto \rho _{X}(\sigma )$ determines a map of simplicial sets $\rho _{X}: X \rightarrow \operatorname{Ex}(X)$.

Let $u: X \rightarrow \operatorname{Ex}( \operatorname{Sd}(X) )$ be a map of simplicial sets which exhibits $\operatorname{Sd}(X)$ as a subdivision of $X$ (Definition 3.3.3.1). Then there is a unique map of simplicial sets $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$ for which the composition $X \rightarrow \operatorname{Ex}( \operatorname{Sd}(X) ) \xrightarrow { \operatorname{Ex}( \lambda _ X ) } \operatorname{Ex}(X)$ is equal to $\rho _{X}$. We will refer to $\lambda _{X}$ as the last vertex map of $X$.