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Definition (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 $\operatorname{N}^{\mathscr {F}}_{n}( \operatorname{\mathcal{C}})$ denote the collection of all pairs $( \sigma , \tau )$, where $\sigma : [n] \rightarrow \operatorname{\mathcal{C}}$ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which we identify with a diagram

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

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

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

For every nondecreasing function $\alpha : [m] \rightarrow [n]$, we define a map $\alpha ^{\ast }: \operatorname{N}^{\mathscr {F}}{n}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\mathscr {F}}{m}( \operatorname{\mathcal{C}})$ by the formula $\alpha ^{\ast }( \sigma , \tau ) = (\sigma \circ \alpha , \tau ')$, where $\tau ' = \{ \tau '_{i}: \Delta ^{i} \rightarrow \mathscr {F}( \alpha (i) ) \} _{0 \leq i \leq m}$ is determined by the requirement that each $\tau '_{i}$ is equal to the composition

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

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