Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 8.1.3.2. We will be primarily interested in the special case of Theorem 8.1.3.1 where $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, we can reformulate conditions $(1)$ and $(2)$ as follows:

$(1')$

For every object $C \in \operatorname{\mathcal{C}}$, the induced functor

\[ F_{C}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ C\} \rightarrow \operatorname{\mathcal{E}}_{C} \]

carries initial objects of $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ C\} $ to initial objects of $\operatorname{\mathcal{E}}_{C}$.

$(2')$

The functor $F$ carries $\lambda _{+}$-cocartesian morphisms of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$.

The equivalence $(1) \Leftrightarrow (1')$ follows from Corollary 4.6.6.16, since $\operatorname{id}_{C}$ is an initial object of $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ C\} $ (Proposition 8.1.2.1). Note that a morphism $e$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is $\lambda _{+}$-cocartesian if and only if $\lambda _{-}(e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (Corollary 8.1.1.13). This condition is automatically satisfied when $\lambda _{-}(e)$ is a degenerate edge of $\operatorname{\mathcal{C}}^{\operatorname{op}}$, which shows that $(2') \Rightarrow (2)$. To prove the converse, we can factor $e$ as a composition $e'' \circ e'$, where $e'$ is $\lambda _{-}$-cocartesian and $\lambda _{-}(e'')$ is an identity morphism of $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (see Remark 5.1.3.8). If condition $(2)$ is satisfied, then $F(e'')$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$. If $\lambda _{-}(e)$ is an isomorphism in $\operatorname{\mathcal{C}}^{\operatorname{op}}$, then $\lambda _{-}(e')$ is also an isomorphism in $\operatorname{\mathcal{C}}^{\operatorname{op}}$, so that $e'$ is an isomorphism in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ (Proposition 5.1.1.8). It follows that $U(e')$ is an isomorphism in $\operatorname{\mathcal{E}}$, so that $U(e)$ is also $U$-cocartesian Corollary 5.1.2.4.