Kerodon

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

Warning 5.3.4.3. Let $S$ be a simplicial set. We have now assigned two different meanings to the term “last vertex map”:

  • The last vertex map $\lambda _{S}$ of Construction 3.3.4.3, which is a morphism of simplicial sets from the subdivision $\operatorname{Sd}(S)$ to $S$.

  • The last vertex map $\lambda _{S}^+$ of Construction 5.3.4.2, which is a morphism of simplicial sets from the nerve $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ to $S$.

These meanings are closely related. If the simplicial set $S$ is braced (Definition 3.3.1.1), then we can identify the subdivision $\operatorname{Sd}(S)$ with the nerve of the full subcategory $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} \subseteq \operatorname{{\bf \Delta }}_{S}$ spanned by the nondegenerate simplices of $S$ (Proposition 3.3.3.15). Under this identification, the last vertex map $\lambda _{S}$ of Construction 3.3.4.3 is given by the composition

\[ \operatorname{Sd}(S) \simeq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S}^{\mathrm{nd} }) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S}) \xrightarrow { \lambda ^{+}_{S} } S, \]

where $\lambda ^{+}_{S}$ is the last vertex map of Construction 5.3.4.2 (Example 3.3.4.7).