$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.4.5.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $f$ is right cofinal (Definition 7.2.1.1).
- $(2)$
For every corepresentable functor $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, the composite map $K \xrightarrow {f} \operatorname{\mathcal{C}}\xrightarrow {h} \operatorname{\mathcal{S}}$ has a contractible colimit.
Proof.
Fix an object $X \in \operatorname{\mathcal{C}}$, and let $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor corepresented by $X$ (Theorem 5.6.6.13). Using Proposition 5.6.6.21, we see that $f \circ h^{X}$ is a covariant transport representation for the left fibration $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/} \rightarrow K$. Using Corollary 7.4.5.4, we can reformulate condition $(2)$ as follows:
- $(2')$
For each object $X \in \operatorname{\mathcal{C}}$, the simplicial set $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ is weakly contractible.
The equivalence $(1) \Leftrightarrow (2')$ follows from Theorem 7.2.3.1.
$\square$