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Construction 5.6.4.1 (The Comparison Map). Let $\operatorname{\mathcal{C}}$ be a category and let $\overrightarrow {C}$ be an $n$-simplex of the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, given by a diagram

\[ C_0 \rightarrow C_1 \rightarrow C_2 \rightarrow \cdots \rightarrow C_{n-1} \rightarrow C_ n \]

in the category $\operatorname{\mathcal{C}}$. Let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets and suppose that we are given a collection of simplices $\overrightarrow {\sigma } = \{ \sigma _{j}: \Delta ^{j} \rightarrow \mathscr {F}(C_ j) \} _{0 \leq j \leq n}$ which fit into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{0} \ar [d]_{\sigma _0} \ar@ {^{(}->}[r] & \Delta ^{1} \ar [d]^-{\sigma _1} \ar@ {^{(}->}[r] & \Delta ^{2} \ar@ {^{(}->}[r] \ar [d]_{\sigma _2} & \cdots \ar [d] \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]_{\sigma _ n} \\ \mathscr {F}(C_0) \ar [r] & \mathscr {F}(C_1) \ar [r] & \mathscr {F}(C_2) \ar [r] & \cdots \ar [r] & \mathscr {F}(C_ n). } \]

To this data, we can associate a commutative diagram of simplicial sets

5.58
\begin{equation} \begin{gathered}\label{equation:compare-with-weighted-nerve} \xymatrix@R =50pt@C=50pt{ \Delta ^{n} \ar [r] \ar [d]^{ \overrightarrow {C}} & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{ \Delta ^{0} / } \ar [d] \\ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \ar [r]^-{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F}) } & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }}), } \end{gathered} \end{equation}

where the upper horizontal map is given by the simplicial functor

\[ F: \operatorname{Path}[ \{ x\} \star [n] ]_{\bullet } \rightarrow \operatorname{Set_{\Delta }} \]

described as follows:

  • The functor $F$ carries $x$ to the simplicial set $\Delta ^{0}$ (so that $F$ can be identified with an $n$-simplex of the coslice simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{\Delta ^{0}/}$).

  • The restriction of $F$ to the simplicial path category $\operatorname{Path}[n]_{\bullet }$ is given by the composition

    \[ \operatorname{Path}[n]_{\bullet } \rightarrow [n] \xrightarrow { \overrightarrow {C}} \operatorname{\mathcal{C}}\xrightarrow { \mathscr {F}} \operatorname{Set_{\Delta }} \]

    (as required by the commutativity of the diagram (5.58)).

  • For $0 \leq m \leq n$, the induced map of simplicial sets

    \[ \operatorname{Hom}_{\operatorname{Path}[ \{ x\} \star [n] ]}( x, m)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( F(x), F(m) )_{\bullet } = \mathscr {F}( C_ m ) \]

    is given by the composition $\operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star [n] ] }( x, m )_{\bullet } \xrightarrow {\rho } \Delta ^{m} \xrightarrow { \sigma _ m} \mathscr {F}(C_ m)$, where $\rho $ is induced by the morphism of partially ordered sets

    \[ \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star [n] ]}(x,m) \rightarrow [m] \quad \quad (S \subseteq \{ x\} \star [n] ) \mapsto \min ( S \setminus \{ x\} ). \]

Note that we can identify the diagram (5.58) with an $n$-simplex $\theta ( \overrightarrow {C}, \overrightarrow {\sigma } )$ of the simplicial set $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$. The construction $(\overrightarrow {C}, \overrightarrow {\sigma }) \mapsto \theta ( \overrightarrow {C}, \overrightarrow {\sigma } )$ then determines a morphism of simplicial sets $\theta : \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$, which we will refer to as the comparison map.