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Definition 5.5.3.1 (The Weighted Nerve). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. For every integer $n \geq 0$, we let $(\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F})_{n}$ denote the collection of all pairs $( \overrightarrow {C}, \overrightarrow {\sigma })$, where $\overrightarrow {C}: [n] \rightarrow \operatorname{\mathcal{C}}$ is a functor which we view as a diagram

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

and $\overrightarrow {\sigma }$ is a collection of simplices $\{ \sigma _{j}: \Delta ^{j} \rightarrow \mathscr {F}(C_ j) \} _{0 \leq j \leq n}$ which fit into a commutative diagram of simplicial sets

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

For every nondecreasing function $\alpha : [m] \rightarrow [n]$, we define a map $\alpha ^{\ast }: (\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F})_{n} \rightarrow (\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F})_{m}$ by the formula $\alpha ^{\ast }( \overrightarrow {C}, \overrightarrow {\sigma } ) = (\overrightarrow {C} \circ \alpha , \overrightarrow {\sigma }')$, where $\overrightarrow {\sigma }'$ associates to each $0 \leq i \leq m$ the $i$-simplex $\sigma '_{i}: \Delta ^{i} \rightarrow \mathscr {F}( \alpha (i) )$ given by the composition

\[ \Delta ^{i} \xrightarrow { \alpha |_{ \{ 0 < 1 < \cdots < i \} } } \Delta ^{ \alpha (i) } \xrightarrow { \sigma _{ \alpha (i) } } \mathscr {F}( \alpha (i) ). \]

By means of these restriction maps, we regard the construction $[n] \mapsto (\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F})_{n}$ as a simplicial set. We will denote this simplicial set by $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} $ and refer to it as the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$. Note that there is an evident projection map $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, given on simplices by the construction $( \overrightarrow {C}, \overrightarrow {\sigma } ) \mapsto \overrightarrow {C}$.