Warning 11.9.6.3. 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 11.9.6.2, 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 $\operatorname{{\bf \Delta }}^{\mathrm{nd}}_{S} \subseteq \operatorname{{\bf \Delta }}_{S}$ spanned by the nondegenerate simplices of $S$ (Proposition 3.3.3.16). Under this identification, the last vertex map $\lambda _{S}$ of Construction 3.3.4.3 is given by the composition
where $\lambda ^{+}_{S}$ is the last vertex map of Construction 11.9.6.2 (Example 3.3.4.7).