Lemma 9.4.7.2 (05NK)

Lemma 9.4.7.2. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{E}}$ be an $\infty $-category which is essentially $\kappa $-small, let $X$ be an object of $\operatorname{\mathcal{E}}$, and let $h_{X}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor corepresented by $X$. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ be an inner fibration of $\infty $-categories, let $\operatorname{\mathcal{E}}_{\geq 1}$ be the full subcategory of $\operatorname{\mathcal{E}}$ spanned by those objects $E$ satisfying $U(E) \geq 1$, and let $\operatorname{\mathcal{E}}_{1}$ denote the full subcategory spanned by those objects satisfying $U(E) = 1$. If $U$ is flat, then the functor $h_{X}|_{ \operatorname{\mathcal{E}}_{\geq 1} }$ is left Kan extended from $\operatorname{\mathcal{E}}_{1}$.

Proof. Choose a functor $\mathscr {F}_{\geq 1}: \operatorname{\mathcal{E}}_{\geq 1} \rightarrow \operatorname{\mathcal{S}}$, and set $\mathscr {F}_{1} = \mathscr {F}_{\geq 1}|_{ \operatorname{\mathcal{E}}_{1} }$. By virtue of Corollary 7.3.6.13, it will suffice to show that the restriction map

\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{\geq 1}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{\geq 1} }, \mathscr {F}_{\geq 1} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{1}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{1} }, \mathscr {F}_{1} ) \]

is a homotopy equivalence of Kan complexes. Let $\operatorname{\mathcal{E}}_{\leq 1}$ be the full subcategory of $\operatorname{\mathcal{E}}$ spanned by those objects $E$ satisfying $U(E) \leq 1$, and choose a regular cardinal $\lambda $ of exponential cofinality $\geq \kappa $. Then the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ is $\kappa $-complete (Example 7.6.6.4). Since $\operatorname{\mathcal{E}}_{\leq 1}$ is essentially $\kappa $-small, the functor $\mathscr {F}_{1}$ admits a right Kan extension $\mathscr {F}_{\leq 1}: \operatorname{\mathcal{E}}_{\leq 1} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$. Our assumption that $U$ is flat guarantees that the restriction functor

\[ T: \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{\leq 1}, \operatorname{\mathcal{S}}^{< \lambda } ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{1}, \operatorname{\mathcal{S}}^{< \lambda } ) } \operatorname{Fun}( \operatorname{\mathcal{E}}_{\geq 1}, \operatorname{\mathcal{S}}^{< \lambda } ) \]

is a trivial Kan fibration, so that we can choose a functor $\mathscr {F}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ satisfying $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{\leq 1} } = \mathscr {F}_{\leq 1}$ and $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{\geq 1} } = \mathscr {F}_{\geq 1}$. Since $T$ is a trivial Kan fibration, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}}^{< \lambda } ) }( h_{X}, \mathscr {F} ) \ar [r]^-{\theta '} \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{\leq 1}, \operatorname{\mathcal{S}}^{< \lambda } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{\leq 1} }, \mathscr {F}_{\leq 1} ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{\geq 1}, \operatorname{\mathcal{S}}^{< \lambda } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{\geq 1} }, \mathscr {F}_{\geq 1} ) \ar [r]^-{\theta } & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{1}, \operatorname{\mathcal{S}}^{< \lambda } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{1} }, \mathscr {F}_{1} ). } \]

Our assumption that $\mathscr {F}_{\leq 1}$ is right Kan extended from $\mathscr {F}_{1}$ guarantees that the right vertical map is a homotopy equivalence (Corollary 7.3.6.13). Consequently, to show that $\theta $ is a homotopy equivalence, it will suffice to show that $\theta '$ is a homotopy equivalence. This follows from the $\infty $-categorical version of Yoneda's lemma (Proposition 8.3.1.3): the source and target of $\theta '$ can both be identified with the Kan complex $\mathscr {F}(X)$. $\square$