Proposition 7.4.3.13 (05RW)

Proposition 7.4.3.13. Let $\kappa $ be an uncountable cardinal, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a diagram, and let $\beta : \mathscr {F} \rightarrow \underline{X}_{\operatorname{\mathcal{C}}}$ be a natural transformation. Suppose we are given a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and a commutative diagram

\[ \xymatrix { & \mathscr {F}|_{ \operatorname{\mathcal{E}}} \ar [dr]^{ \beta |_{\operatorname{\mathcal{E}}} } & \\ \underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}} \ar [ur]^{ \alpha } \ar [rr]^{\gamma } & & \underline{X}_{\operatorname{\mathcal{E}}} } \]

in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Assume that $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small and that $\alpha $ exhibits $\mathscr {F}$ as a covariant transport representation for $U$ (Definition 7.4.1.8). The following conditions are equivalent:

$(1)$: The natural transformation $\beta $ exhibits $X$ as a colimit of $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$.
$(2)$: The natural transformation $\gamma $ determines a weak homotopy equivalence of simplicial sets $\operatorname{\mathcal{E}}\rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}^{< \kappa } }( \Delta ^0, X )$.

Proof. Using Variant 7.4.2.14, we see that $\alpha $ exhibits $\mathscr {F}$ as a left Kan extension of the constant diagram $\underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}}$ along $U$. By virtue of Corollary 7.3.8.20, this is equivalent to the assertion that $\gamma $ exhibits $X$ as a colimit of the diagram $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$. The equivalence of $(1)$ and $(2)$ now follows from Variant 7.1.2.11. $\square$