Remark 8.1.2.8 (03JZ)

Remark 8.1.2.8. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $X \in \operatorname{\mathcal{C}}$ be a vertex. Then an $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}_{X/}$ can be identified with an $(n+1)$-simplex of $\operatorname{\mathcal{C}}$, which we represent informally as a diagram

\[ X \xrightarrow {f} Y_0 \xrightarrow {v_1} Y_1 \xrightarrow {v_2} Y_2 \rightarrow \cdots \xrightarrow {v_ n} Y_ n. \]

The morphism $\iota _{X}$ of Construction 8.1.2.7 carries $\sigma $ to a $(2n+1)$-simplex $\tau $ of $\operatorname{\mathcal{C}}$, which can be represented informally by the diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^-{f} & X \ar [l]_-{\operatorname{id}} & X \ar [l]_-{\operatorname{id}} & \cdots \ar [l]_-{\operatorname{id}} & X \ar [l]_-{\operatorname{id}} \\ Y_0 \ar [r]^-{v_1} & Y_1 \ar [r]^-{v_2} & Y_2 \ar [r] & \cdots \ar [r]^-{v_ n} & Y_ n. } \]

Note that $\sigma $ can be recover from $\tau $ (by composing with the inclusion map $\Delta ^{n+1} \hookrightarrow \Delta ^{2n+1}$, given on vertices by $i \mapsto i+n$). It follows that $\iota _{X}$ is a monomorphism of simplicial sets $\operatorname{\mathcal{C}}_{X/} \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ (as suggested by our terminology).