Construction 11.9.6.2 (01NA): The Last Vertex Map

Construction 11.9.6.2 (The Last Vertex Map). Let $S$ be a simplicial set, let $\operatorname{{\bf \Delta }}_{S}$ denote the category of simplices of $S$ (Construction 1.1.3.9), and let $\tau $ be a $k$-simplex of the nerve $\operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S})$, which we identify with a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{n_0} \ar [r] \ar [drr]^-{ \sigma _0 } & \Delta ^{n_1} \ar [r] \ar [dr]^-{ \sigma _1} & \cdots \ar [r] & \Delta ^{n_{k-1}} \ar [dl]_{ \sigma _{k-1} } \ar [r] & \Delta ^{n_ k} \ar [dll]_{\sigma _{k}} \\ & & S. & & } \]

We let $\lambda ^{+}_{S}(\tau ): \Delta ^{k} \rightarrow S$ denote the map given by the composition

\[ \Delta ^{k} \xrightarrow { f } \Delta ^{n_ k} \xrightarrow { \sigma _ k} X, \]

where $f$ carries each vertex $\{ i\} \subseteq \Delta ^{k}$ to the image of the last vertex $\{ n_ i \} \subseteq \Delta ^{n_ i}$ under the composite map $\Delta ^{n_ i} \rightarrow \Delta ^{n_{i+1}} \rightarrow \cdots \rightarrow \Delta ^{n_ k}$. The construction $\tau \mapsto \lambda ^{+}_{S}(\tau )$ determines a morphism of simplicial sets $\lambda ^{+}_{S}: \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S} ) \rightarrow S$, which we will refer to as the last vertex map. By construction, it carries each object $([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S}$ to the vertex $\sigma (n) \in S$.