Theorem 9.4.4.9. Let $\kappa \leq \lambda $ be regular cardinals and suppose we are given a categorical pullback diagram of $\infty $-categories
9.14
\begin{equation} \label{equation:categorical-pullback-lm-accessible} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm } \ar [r]^-{G_{+}} \ar [d]^{G_{-}} & \operatorname{\mathcal{C}}_{+} \ar [d]^{ F_{+} } \\ \operatorname{\mathcal{C}}_{-} \ar [r]^-{ F_{-} } & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}
Assume that:
- $(a)$
The $\infty $-categories $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ are $(\kappa ,\lambda )$-compactly generated, and the functors $F_{-}$ and $F_{+}$ are $(\kappa ,\lambda )$-compact.
- $(b)$
The full subcategories $\operatorname{\mathcal{C}}_{-}^{0} \subseteq \operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}^{0} \subseteq \operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ spanned by the $(\kappa ,\lambda )$-compact objects are essentially $\lambda $-small.
- $(c)$
There exists a regular cardinal $\kappa _0 < \kappa $ for which the $\infty $-categories $\operatorname{\mathcal{C}}_{-}^{0}$ and $\operatorname{\mathcal{C}}_{+}^{0}$ admit $\kappa _0$-sequential colimits, which are preserved by the functors $F_{-}$ and $F_{+}$.
Then the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-compactly generated. Moreover, an object $X \in \operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-compact if and only if $G_{-}( X)$ and $G_{+}(X)$ are $(\kappa ,\lambda )$-compact objects of the $\infty $-categories $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_+$, respectively.
Proof.
Let $\operatorname{\mathcal{C}}^{0}_{\pm }$ denote the full subcategory of $\operatorname{\mathcal{C}}_{\pm }$ spanned by those objects $X$ such that $G_{-}(X)$ and $G_{+}(X)$ are $(\kappa ,\lambda )$-compact, so that (9.14) restricts to a categorical pullback sqaure
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm }^{0} \ar [r]^-{G^{0}_{+}} \ar [d]^{G^{0}_{-}} & \operatorname{\mathcal{C}}^{0}_{+} \ar [d]^{ F^{0}_{+} } \\ \operatorname{\mathcal{C}}^{0}_{-} \ar [r]^-{ F^{0}_{-} } & \operatorname{\mathcal{C}}^{0}. } \]
It follows from Proposition 7.1.9.1 that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda ))$-cocomplete. Moreover Corollary 9.2.8.5 guarantees that every object of $\operatorname{\mathcal{C}}^{0}_{\pm }$ is $(\kappa ,\lambda )$-compact when regarded as an object of $\operatorname{\mathcal{C}}_{\pm }$. To show that $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda ))$-accessible, it will suffice to show that every object $X_{\pm } \in \operatorname{\mathcal{C}}_{\pm }$ can be realized as the colimit of an essentially $\lambda $-small $\kappa $-filtered diagram in $\operatorname{\mathcal{C}}^{0}_{\pm }$. Lemma 9.4.1.18 then guarantees that every $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{\pm }$ is a retract of an object of $\operatorname{\mathcal{C}}^{0}_{\pm }$. Since $\operatorname{\mathcal{C}}_{\pm }^{0}$ is closed under retracts (Remark 9.2.5.18), we conclude that an object of $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-compact if and only if it belongs to $\operatorname{\mathcal{C}}^{0}_{\pm }$.
Fix an object $X_{\pm } \in \operatorname{\mathcal{C}}_{\pm }$ as above. Set $X_{-} = G_{-}( X_{\pm } )$, $X_{+} = G_{+}( X_{\pm } )$, and $X = F_{+}(X_{+} ) = F_{-}( X_{-} )$. Let $\operatorname{\mathcal{K}}$ denote the fiber product $\operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X}$, and define $\infty $-categories $\operatorname{\mathcal{K}}_{-}$, $\operatorname{\mathcal{K}}_{+}$, and $\operatorname{\mathcal{K}}_{\pm }$ similarly, so that Corollary 4.6.4.22 supplies a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}_{\pm } \ar [r]^-{g_{+}} \ar [d]^{g_{-}} & \operatorname{\mathcal{K}}_{+} \ar [d]^{f_{+}} \\ \operatorname{\mathcal{K}}_{-} \ar [r]^-{f_{-}} & \operatorname{\mathcal{K}}. } \]
Applying Corollary 9.3.4.15, we see that the $\infty $-categories $\operatorname{\mathcal{K}}_{-}$, $\operatorname{\mathcal{K}}_{+}$, and $\operatorname{\mathcal{K}}$ are $\kappa $-filtered and essentially $\lambda $-small. Using Corollary 4.9.5.19, we see that $\operatorname{\mathcal{K}}_{\pm }$ is also essentially $\lambda $-small. Note that $\operatorname{Ind}_{\kappa }^{\lambda }( f_{-} )$ can be identified with the functor $(\operatorname{\mathcal{C}}_{-})_{/X_{-}} \rightarrow \operatorname{\mathcal{C}}_{/X}$ induced by $F_{-}$. In particular, $\operatorname{Ind}_{\kappa }^{\lambda }( f_{-} )$ preserves final objects, so $f_{-}$ is right cofinal (Proposition 9.3.7.11). Similarly, the functor $f_{+}: \operatorname{\mathcal{K}}_{+} \rightarrow \operatorname{\mathcal{K}}$ is right cofinal. Combining assumption $(c)$ with Proposition 9.2.5.24 and Corollary 7.1.4.27, we deduce that the $\infty $-categories $\operatorname{\mathcal{K}}_{-}$, $\operatorname{\mathcal{K}}_{+}$ and $\operatorname{\mathcal{K}}$ are $\kappa _0$-sequentially cocomplete, and that the functors $f_{-}$ and $f_{+}$ preserve $\kappa _0$-sequential colimits. Invoking Theorem 9.2.4.3, we conclude that the $\infty $-category $\operatorname{\mathcal{K}}_{\pm }$ is $\kappa $-filtered and that the functors $g_{-}$ and $g_{+}$ are right cofinal. We will complete the proof by showing that the composite map
\[ \overline{q}: \operatorname{\mathcal{K}}_{\pm }^{\triangleright } \rightarrow (\operatorname{\mathcal{C}}_{\pm })_{ / X_{\pm } }^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\pm } \]
is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$. Note that $G_{-} \circ \overline{q}$ can be written as a composition
\[ \operatorname{\mathcal{K}}_{\pm }^{\triangleright } \xrightarrow { g_{-}^{\triangleright } } \operatorname{\mathcal{K}}_{-}^{\triangleright } \rightarrow (\operatorname{\mathcal{C}}_{-})^{\triangleright }_{/X_{-}} \rightarrow \operatorname{\mathcal{C}}_{-}, \]
which is a colimit diagram by virtue Corollary 9.3.4.15 together with the right cofinality of $g_{-}$ (Corollary 7.2.2.3). A similar argument shows that $G_{+} \circ \overline{q}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{+}$, so that $\overline{q}$ is a colimit diagram by Proposition 7.1.9.8.
$\square$