Construction 8.6.1.11 (04CH): Conjugacy for Weighted Nerves

Construction 8.6.1.11 (Conjugacy for Weighted Nerves). Let $\operatorname{QCat}$ denote the category of $\infty $-categories (which we regard as a full subcategory of the category of simplicial sets). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, and let $\operatorname{\mathcal{E}}= \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the weighted nerve of Definition 5.3.3.1. We will identify $n$-simplices of $\operatorname{\mathcal{E}}$ with pairs $(\sigma _{+}, \tau _{+})$, where $\sigma _{+} = ( C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n )$ is an $n$-simplex of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$ and $\tau _{+} = ( \tau _0, \tau _1, \cdots , \tau _ n )$ is the datum of a commutative diagram

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

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Corollary 5.3.3.16, given on $n$-simplices by the formula $U(\sigma _{+},\tau _{+}) = \sigma _{+}$.

Let $\mathscr {F}^{\operatorname{op}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ denote the functor given by the formula $\mathscr {F}^{\operatorname{op}}(C) = \mathscr {F}(C)^{\operatorname{op}}$, and let $\operatorname{\mathcal{E}}^{\dagger }$ denote the $\infty $-category ${\operatorname{N}_{\bullet }^{\mathscr {F}^{\operatorname{op}}}(\operatorname{\mathcal{C}})}^{\operatorname{op}}$. Unwinding the definitions, we see that $n$-simplices of $\operatorname{\mathcal{E}}^{\dagger }$ can be identified with pairs $(\sigma _{-}, \tau _{-})$, where $\sigma _{-} = (C'_ n \rightarrow C'_{n-1} \rightarrow \cdots \rightarrow C'_{0})$ is an $n$-simplex of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}}$ and $\tau _{-} = (\tau '_ n, \tau '_{n-1}, \cdots , \tau '_0 )$ is the datum of a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ n\} \ar [d]_{\tau '_{n}} \ar@ {^{(}->}[r] & \operatorname{N}_{\bullet }( \{ n-1 < n \} ) \ar [d]^{\tau '_{n-1}} \ar@ {^{(}->}[r] & \cdots \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]^{ \tau '_0 } \\ \mathscr {F}(C'_ n) \ar [r] & \mathscr {F}(C'_{n-1}) \ar [r] & \cdots \ar [r] & \mathscr {F}(C'_0). } \]

Corollary 5.3.3.16 supplies a cartesian fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}}$, given on $n$-simplices by the formula $U^{\dagger }( \sigma _{-}, \tau _{-} ) = \sigma _{-}$.

Let us identify $n$-simplices of the fiber product $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}} } \operatorname{Tw}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) )$ with quadruples $( \sigma _{-}, \tau _{-}, \sigma _{+}, e)$, where $\sigma _{-} = (C'_ n \rightarrow \cdots \rightarrow C'_0)$ is an $n$-simplex of $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}}$, $\tau _{-} = ( \tau '_{n}, \cdots , \tau '_0 )$ is as above, $\sigma _{+} = ( C_0 \rightarrow \cdots \rightarrow C_ n )$ is an $n$-simplex of $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$, and $e: C'_0 \rightarrow C_0$ is a morphism in the category $\operatorname{\mathcal{C}}$. For $0 \leq i \leq n$, we let $\tau _{i}$ denote the $i$-simplex of $\mathscr {F}( C_ i )$ given by the composition

\[ \Delta ^{i} \hookrightarrow \Delta ^ n \xrightarrow { \tau '_{0} } \mathscr {F}( C'_0 ) \xrightarrow { \mathscr {F}(e) } \mathscr {F}(C_0 ) \rightarrow \mathscr {F}( C_ i ). \]

Set $\tau _{+} = ( \tau _0, \cdots , \tau _ n )$, so that the pair $( \sigma _{+}, \tau _{+} )$ determines an $n$-simplex of the simplicial set $\operatorname{\mathcal{E}}$. The construction $( \sigma _{-}, \tau _{-}, \sigma _{+}, e) \mapsto (\sigma _{+}, \tau _{+} )$ is compatible with face and degeneracy operators, and therefore determines a functor of $\infty $-categories

\[ T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}} } \operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{\mathcal{E}}. \]