Construction 8.1.7.1 (049F)

Construction 8.1.7.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection map of Notation 8.1.1.6, carrying each vertex $(f: X \rightarrow Y)$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to the vertex $Y \in \operatorname{\mathcal{C}}$. Under the bijection supplied by Proposition 8.1.3.7, we can identify $\lambda _{+}$ with a morphism of simplicial sets $\rho _{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. If $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$, which we display informally as a diagram

\[ X_0 \xrightarrow {f_1} X_1 \xrightarrow {f_2} X_2 \rightarrow \cdots \xrightarrow {f_ n} X_ n, \]

then $\rho _{+}(\sigma )$ is an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ which can be depicted informally as a diagram

\[ \xymatrix@R =20pt@C=20pt{ X_0 \ar [dr]^-{f_1} & & X_1 \ar [dr]^{f_2} \ar [dl]^{\operatorname{id}} & & \cdots \ar [dl] \ar [dr] & & X_{n-1} \ar [dr]^{f_ n} \ar [dl]^{\operatorname{id}} & & X_ n \ar [dl]^{\operatorname{id}} \\ & X_1 \ar [dr]^{f_2} & & X_2 \ar [dl]^{\operatorname{id}} \ar [dr]^{f_3} & \cdots & X_{n-1} \ar [dl]^{\operatorname{id}} \ar [dr]^{f_ n} & & X_{n} \ar [dl]^{\operatorname{id}} & \\ & & \cdots \ar [dr]^{f_{n-1}} & & \cdots \ar [dl]^{\operatorname{id}} \ar [dr]^{f_ n} & & \cdots \ar [dl]^{\operatorname{id}} & & \\ & & & X_{n-1} \ar [dr]^{f_ n} & & X_{n} \ar [dl]^{\operatorname{id}} & & & \\ & & & & X_ n. & & & & } \]

Note that $\rho _{+}$ is a monomorphism of simplicial sets.