Example 3.3.4.7 (00YY)

Example 3.3.4.7. Let $X$ be a braced simplicial set, so that the subdivision $\operatorname{Sd}(X)$ can be identified with the nerve of the category $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ of nondegenerate simplices of $X$ (Proposition 3.3.3.17). Under this identification, the last vertex map $\lambda _{X}$ corresponds to a morphism of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} ) \rightarrow X$. Concretely, if $\tau $ is a $k$-simplex of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}})$ corresponding to a diagram

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

then $\lambda _{X}(\tau )$ is the $k$-simplex of $X$ 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 map $\Delta ^{n_ i} \rightarrow \Delta ^{n_ k}$ given by horizontal composition in the diagram.