Kerodon

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

Proposition 8.2.4.9. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories, let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor, and suppose we are given a morphism of couplings

8.49
\begin{equation} \begin{gathered}\label{equation:representability-as-left-cofinality} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [r]^-{ \widetilde{G} } \ar [d]^{\mu } & \operatorname{\mathcal{C}}\ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{+} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} } \end{gathered} \end{equation}

where $\mu = (\mu _{-}, \mu _{+})$ is the twisted arrow coupling of Example 8.2.0.2. The following conditions are equivalent:

$(1)$

The diagram (8.49) exhibits $\lambda $ as represented by $G$, in the sense of Definition 8.2.4.1. That is, the functor $\widetilde{G}$ carries isomorphisms of $\operatorname{\mathcal{C}}_{+}$ to universal objects of $\operatorname{\mathcal{C}}$.

$(2)$

The functor $\widetilde{G}$ is left cofinal.

Proof. By virtue of Proposition 8.2.1.7, the functors $\lambda _{+}$ and $\mu _{+}$ are cocartesian fibrations, and the functor $\widetilde{G}$ carries $\mu _{+}$-cocartesian morphisms of $\operatorname{Tw}(\operatorname{\mathcal{C}}_{+} )$ to $\lambda _{+}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$. By virtue of Corollary 7.2.3.15, the functor $\widetilde{G}$ is left cofinal if and only if, for every object $C \in \operatorname{\mathcal{C}}_{+}$, the induced map $\widetilde{G}_{C}: \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \times _{ \operatorname{\mathcal{C}}_{+} } \{ C \} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ C\} $ is left cofinal. It will therefore suffice to show that the following conditions are equivalent, for each object $C \in \operatorname{\mathcal{C}}_{+}$.

$(1_ C)$

The image $\widetilde{G}( \operatorname{id}_ C )$ is a universal object of $\operatorname{\mathcal{C}}$: that is, it is initial when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ C\} $.

$(2_ C)$

The functor $\widetilde{G}_{C}$ is left cofinal.

The equivalence $(1_ C) \Leftrightarrow (2_ C)$ is a special case of Corollary 7.2.1.9, since $\operatorname{id}_{C}$ is initial when viewed as an object of the $\infty $-category $\operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \times _{ \operatorname{\mathcal{C}}_{+} } \{ C \} $ (see Proposition 8.1.2.1). $\square$