Example 5.3.3.12 (036M): The Weighted Nerve of a Cone

Example 5.3.3.12 (The Weighted Nerve of a Cone). Let $\operatorname{\mathcal{C}}$ be a category and let $\operatorname{\mathcal{C}}^{\triangleright }$ denote the right cone on $\operatorname{\mathcal{C}}$ (Example 4.3.2.5), and let ${\bf 1} \in \operatorname{\mathcal{C}}^{\triangleright }$ denote the final object. Suppose we are given a diagram of simplicial sets $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$ and $Y = \overline{\mathscr {F}}( {\bf 1} )$, so that $\overline{\mathscr {F}}$ determines a natural transformation $\alpha : \mathscr {F} \rightarrow \underline{Y}$ (where $\underline{Y}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denotes the constant functor taking the value $Y$). Combining Remark 5.3.3.10 with Example 5.3.3.9, we obtain morphisms of simplicial sets

\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \xrightarrow {\alpha } \operatorname{N}_{\bullet }^{\underline{Y}}(\operatorname{\mathcal{C}}) \simeq Y \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow Y. \]

Unwinding the definitions, there is a canonical isomorphism of simplicial sets

\[ \operatorname{N}_{\bullet }^{ \overline{\mathscr {F}} }(\operatorname{\mathcal{C}}^{\triangleright }) \simeq \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \star _{Y} Y, \]

where the right hand side denotes the relative join of Construction 5.2.3.1.