Proposition 5.3.8.7. Let $\mathscr {F}: A \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by a partially ordered set $(A, \leq )$. Suppose that, for every element $a \in A$, the simplicial set $\mathscr {F}(a)$ is a minimal $\infty $-category. Then the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(A)$ is a minimal $\infty $-category.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\sigma $ and $\sigma '$ be $n$-simplices of $\operatorname{N}_{\bullet }^{\mathscr {F}}(A)$ which are isomorphic relative to $\operatorname{\partial \Delta }^ n$; we wish to show that $\sigma = \sigma '$. Note that $\sigma $ and $\sigma '$ have the same image in $\operatorname{N}_{\bullet }(A)$, which we can identify with a nondecreasing sequence $a_0 \leq a_1 \leq \cdots \leq a_ n$ of elements of $A$. Let us identify $\sigma $ with a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{0} \ar [d]_{\tau _0} \ar@ {^{(}->}[r] & \Delta ^{1} \ar [d]^-{\tau _1} \ar@ {^{(}->}[r] & \Delta ^{2} \ar@ {^{(}->}[r] \ar [d]_{\tau _2} & \cdots \ar [d] \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]_{\tau _ n} \\ \mathscr {F}(a_0) \ar [r] & \mathscr {F}(a_1) \ar [r] & \mathscr {F}(a_2) \ar [r] & \cdots \ar [r] & \mathscr {F}(a_ n) } \]
and $\sigma '$ with a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{0} \ar [d]_{\tau '_0} \ar@ {^{(}->}[r] & \Delta ^{1} \ar [d]^-{\tau '_1} \ar@ {^{(}->}[r] & \Delta ^{2} \ar@ {^{(}->}[r] \ar [d]_{\tau '_2} & \cdots \ar [d] \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]_{\tau '_ n} \\ \mathscr {F}(a_0) \ar [r] & \mathscr {F}(a_1) \ar [r] & \mathscr {F}(a_2) \ar [r] & \cdots \ar [r] & \mathscr {F}(a_ n). } \]
Since $\sigma $ and $\sigma '$ have the same restriction to $\operatorname{\partial \Delta }^ n$, we automatically have $\tau _{i} = \tau '_{i}$ for $i < n$. To complete the proof, it will suffice to show that $\tau _{n} = \tau '_{n}$. Replacing $A$ by the subset $\{ a \in A: a \leq a_ n \} $, we may assume that $a_ n$ is the largest element of $A$. Let $\mathscr {F}': A \rightarrow \operatorname{Set_{\Delta }}$ be the constant functor taking the value $\mathscr {F}(a_ n)$. Then there is a unique natural transformation of functors $\gamma : \mathscr {F} \rightarrow \mathscr {F}'$ which is the identity when evaluated at $a_ n$. Passing to weighted nerves, we obtain a functor of $\infty $-categories
\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(A) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}'}(A) \simeq \operatorname{N}_{\bullet }(A) \times \mathscr {F}(a_ n) \rightarrow \mathscr {F}(a_ n), \]
carrying $\sigma $ and $\sigma '$ to $\tau _ n$ and $\tau '_ n$, respectively. Since $\sigma $ and $\sigma '$ are isomorphic relative to $\operatorname{\partial \Delta }^{n}$, the simplices $\tau _ n$ and $\tau '_ n$ are isomorphic relative to $\operatorname{\partial \Delta }^ n$ as $n$-simplices of $\mathscr {F}(a_ n)$. Our assumption that $\mathscr {F}(a_ n)$ is a minimal $\infty $-category then guarantees that $\tau _ n = \tau '_ n$, as desired. $\square$