Theorem 9.4.7.1 (05NJ)

Theorem 9.4.7.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ be a functor of $\infty $-categories. Then $U$ is flat if and only if, for every morphism $f: X \rightarrow Z$ in $\operatorname{\mathcal{E}}$ satisfying $U(X) = 0$ and $U(Z) = 2$, the $\infty $-category $\{ 1\} \times _{ \Delta ^2 } \operatorname{\mathcal{E}}_{ X/ \, /Z }$ is weakly contractible.

Proof of Theorem 9.4.7.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ be a functor of $\infty $-categories and let $X$ be an object of $\operatorname{\mathcal{E}}$ satisfying $U(X) = 0$. By virtue of Proposition 9.4.7.4, it will suffice to show that the following conditions are equivalent:

$(1)$

The inclusion functor

\[ \iota _{X}: \{ 1\} \times _{ \Delta ^2 } \operatorname{\mathcal{E}}_{X/} \hookrightarrow \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times _{ \Delta ^2 } \operatorname{\mathcal{E}}_{X/ } \]

is left cofinal.

$(2)$

For every morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{E}}$ where $U(Z) = 2$, the $\infty $-category $\{ 1\} \times _{\Delta ^2} \operatorname{\mathcal{E}}_{ X/ \, /Z}$ is weakly contractible.

This follows from the cofinality criterion of Theorem 7.2.3.1. $\square$