9.4.4 Compact Generation of Homotopy Fiber Products
Let $\kappa $ be a small regular cardinal. Suppose we are given $\kappa $-compactly generated $\infty $-categories $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$, together with $\kappa $-compact functors $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$. Then Corollary 9.4.3.6 asserts that the oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is also $\kappa $-compactly generated. Beware that the analogous statement need not hold for the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$.
Exercise 9.4.4.1. Let $\operatorname{\mathcal{C}}$ be (the nerve of) the category whose objects are linearly ordered sets and whose morphisms are strictly increasing functions. Let $\operatorname{\mathcal{C}}_{-} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those linearly ordered sets $(Q, \leq )$ where $Q$ is either infinite or of even cardinality, and let $\operatorname{\mathcal{C}}_{+} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those linearly ordered sets $(Q, \leq )$ where $Q$ is either infinite or of odd cardinality. Show that:
The $\infty $-categories $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ are compactly generated.
The inclusion functors $\operatorname{\mathcal{C}}_{-} \hookrightarrow \operatorname{\mathcal{C}}\hookleftarrow \operatorname{\mathcal{C}}_{+}$ are compact.
The homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is nonempty but does not contain any compact objects. In particular, it is not compactly generated.
Our goal in this section is to show that, under some additional assumptions, we can avoid the phenomenon illustrated in Exercise 9.4.4.1. We can state a preliminary version of our main result as follows:
Proposition 9.4.4.2. Let $\kappa $ be a small uncountable regular cardinal and suppose we are given a categorical pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm } \ar [r]^-{G_{+}} \ar [d]^{G_{-}} & \operatorname{\mathcal{C}}_{+} \ar [d]^{ F_{+} } \\ \operatorname{\mathcal{C}}_{-} \ar [r]^-{ F_{-} } & \operatorname{\mathcal{C}}} \]
which satisfies the following conditions:
The $\infty $-categories $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ admit small filtered colimits, and the functors $F_{-}$ and $F_{+}$ are finitary.
The $\infty $-categories $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $\kappa $-compactly generated, and the functors $F_{-}$ and $F_{+}$ are $\kappa $-compact.
Then the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $\kappa $-compactly generated. Moreover, an object $X \in \operatorname{\mathcal{C}}_{\pm }$ is $\kappa $-compact if and only if $G_{-}( X)$ and $G_{+}(X)$ are $\kappa $-compact objects of the $\infty $-categories $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_+$, respectively.
Warning 9.4.4.3. In the statement of Proposition 9.4.4.2, the assumption that $\kappa $ is uncountable cannot be omitted. In particular, Proposition 9.4.4.2 cannot be used to show that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is compactly generated (or to characterize the compact objects of $\operatorname{\mathcal{C}}_{\pm }$). See Exercise 9.4.4.1.
Proposition 9.4.4.2 is a consequence of the following more general result:
Theorem 9.4.4.4. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories which are $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Suppose we are given a categorical pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm } \ar [r]^-{G_{+}} \ar [d]^{G_{-}} & \operatorname{\mathcal{C}}_{+} \ar [d]^{ F_{+} } \\ \operatorname{\mathcal{C}}_{-} \ar [r]^-{ F_{-} } & \operatorname{\mathcal{C}}, } \]
where the functors $F_{-}$ and $F_{+}$ are $(\kappa ,\lambda )$-compact. Assume that the following additional condition is satisfied:
- $(\star )$
Let $\operatorname{\mathcal{C}}_{-}^{0}$ and $\operatorname{\mathcal{C}}_{+}^{0}$ be the full subcategories of $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ spanned by the $(\kappa ,\lambda )$-compact objects. Then there exists a regular cardinal $\kappa _0 < \kappa $ such that $\operatorname{\mathcal{C}}_{-}^{0}$ 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.
The first assertion is a special case of Proposition 7.1.9.1, and the second follows by combining Proposition 7.1.9.1 with the characterization of $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{C}}_{\pm }$ supplied by Theorem 9.4.4.4.
Applying this observation in the special case where $H$ is the identity functor $\operatorname{id}_{ \operatorname{\mathcal{C}}_{\pm } }$, we see that the functors $G_{-}: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{-}$ and $G_{+}: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{+}$ are $(\kappa ,\lambda )$-compact.
Proof of Proposition 9.4.4.2.
Apply Theorem 9.4.4.4 in the special case where $\kappa _0 = \aleph _0$ and $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal.
$\square$
Corollary 9.4.4.7 (Compact Generation of Inverse Limits). Let $\kappa \trianglelefteq \lambda $ be regular cardinals and suppose we are given a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_{3} \xrightarrow { U_2} \operatorname{\mathcal{C}}_2 \xrightarrow { U_1 } \operatorname{\mathcal{C}}_1 \xrightarrow {U_0} \operatorname{\mathcal{C}}_0. \]
Assume that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{n}$ is $(\kappa ,\lambda )$-compactly generated, and that each of the functors $U_{n}$ is a $(\kappa ,\lambda )$-compact isofibration, and that the following additional condition is satisfied:
- $(\star )$
There exists a regular cardinal $\kappa _0 < \kappa $ such that, for each $n \geq 0$, the full subcategory $\operatorname{\mathcal{C}}_{n}^{0} \subseteq \operatorname{\mathcal{C}}_ n$ spanned by the $(\kappa ,\lambda )$-compact objects admits $\kappa _0$-sequential colimits, which are preserved by functors $U_{n}|_{ \operatorname{\mathcal{C}}_{n+1}}: \operatorname{\mathcal{C}}_{n+1}^{0} \rightarrow \operatorname{\mathcal{C}}_{n}^{0}$.
Then the inverse limit $\varprojlim ( \operatorname{\mathcal{C}}_{n} )$ is $(\kappa ,\lambda )$-compactly generated. Moreover, an object $X = \{ X_ n \} _{n \geq 0} \in \varprojlim ( \operatorname{\mathcal{C}}_{n} )$ is $(\kappa ,\lambda )$-compact if and only if each $X_ n$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{n}$.
Proof.
Set $\operatorname{\mathcal{C}}= \prod _{n} \operatorname{\mathcal{C}}_{n}$. Since $\kappa $ is uncountable, Proposition 9.4.3.13 implies that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, and that an object $\{ X_ n \} _{n \geq 0} \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact if and only if each $X_{n}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{n}$. Moreover, an analogous statement holds for the product $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}$. Let $S: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the shift functor (given on objects by the construction $\{ X_ n \} _{n \geq 0} \mapsto \{ U_{n}( X_{n+1} ) \} _{n \geq 0}$). The desired result now follows by applying Theorem 9.4.4.4 to the categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \varprojlim (\operatorname{\mathcal{C}}_{n}) \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ (\operatorname{id}, \operatorname{id}) } \\ \operatorname{\mathcal{C}}\ar [r]^-{ (\operatorname{id},S) } & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}} \]
supplied by Corollary 4.5.6.19.
$\square$
Corollary 9.4.4.8. Let $\kappa $ be a small uncountable regular cardinal and suppose we are given a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_{3} \xrightarrow { U_2} \operatorname{\mathcal{C}}_2 \xrightarrow { U_1 } \operatorname{\mathcal{C}}_1 \xrightarrow {U_0} \operatorname{\mathcal{C}}_0. \]
Assume that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{n}$ is $\kappa $-compactly generated and admits small filtered colimits and that each of the functors $U_{n}: \operatorname{\mathcal{C}}_{n+1} \rightarrow \operatorname{\mathcal{C}}_{n}$ is a finitary isofibration which preserves $\kappa $-compact objects. Then the limit $\varprojlim ( \operatorname{\mathcal{C}}_{n} )$ is $\kappa $-compactly generated. Moreover, an object $X = \{ X_ n \} _{n \geq 0} \in \varprojlim ( \operatorname{\mathcal{C}}_{n} )$ is $\kappa $-compact if and only if each $X_ n$ is a $\kappa $-compact object of $\operatorname{\mathcal{C}}_{n}$.
Proof.
Apply Corollary 9.4.4.7 in the special case where $\kappa _0 = \aleph _0$ and $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal.
$\square$
We begin by proving a slight variant of Theorem 9.4.4.4, where we replace the hypothesis $\kappa \trianglelefteq \lambda $ by set-theoretic assumptions on the $\infty $-categories themselves.
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 4.7.5.17, 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 7.1.4.22, 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.
$\square$
The following is a restatement of Theorem 9.4.4.9:
Corollary 9.4.4.10. We Let $\kappa \leq \lambda $ be regular cardinals and let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories satisfying conditions $(a)$, $(b)$, and $(c)$ of Theorem 9.4.4.9. Then the homotopy fiber product Then the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda )$-compactly generated. Moreover, an object of $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda )$-compact if and only if its images in $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $(\kappa ,\lambda ))$-compact.
Proof.
Using Corollary 4.5.2.24 (and Corollary 4.5.2.21), we can reduce to the case where $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ is an isofibration. In this case, the inclusion map $\operatorname{\mathcal{C}}_{-} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \hookrightarrow \operatorname{\mathcal{C}}_{-} \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_{+}$ is an equivalence of $\infty $-categories (Corollary 4.5.2.29). The desired result now follows by applying Theorem 9.4.4.9 to the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{-} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{+} \ar [d]^{ F_{+} } \\ \operatorname{\mathcal{C}}_{-} \ar [r]^-{ F_{-} } & \operatorname{\mathcal{C}}, } \]
which is a categorical pullback square by virtue of Corollary 4.5.2.28.
$\square$
We now return to Theorem 9.4.4.4. Recall that a diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm } \ar [r]^-{G_{+}} \ar [d]^{G_{-}} & \operatorname{\mathcal{C}}_{+} \ar [d]^{ F_{+} } \\ \operatorname{\mathcal{C}}_{-} \ar [r]^-{ F_{-} } & \operatorname{\mathcal{C}}} \]
is a categorical pullback square if and only if the functors $G_{-}$ and $G_{+}$ induce an equivalence from $\operatorname{\mathcal{C}}_{\pm }$ to the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$. It will therefore suffice to prove the following:
Proposition 9.4.4.11. Let $\kappa \trianglelefteq \lambda $ be regular cardinals and let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories satisfying the following conditions:
The $\infty $-categories $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $(\kappa ,\lambda )$-compactly generated and the $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete.
The functor $F_{-}$ and $F_{+}$ are $(\kappa ,\lambda )$-compact.
There exists a regular cardinal $\kappa _0 < \kappa $ for which the full subcategories of $(\kappa ,\lambda )$-compact objects $\operatorname{\mathcal{C}}^{0}_{-} \subseteq \operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}^{0}_{+} \subseteq \operatorname{\mathcal{C}}_{+}$ admits $\kappa _0$-sequential colimits, which are preserved by the functors $F_{-}$ and $F_{+}$.
Then the homotopy fiber product $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda ))$-compactly generated. Moreover, an object of $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda ))$-compact if and only if its images in $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $(\kappa ,\lambda ))$-compact.
Proof.
Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the smallest full subcategory which contains $\operatorname{\mathcal{C}}^0$ and is closed under $\lambda $-small $\kappa $-filtered colimits. Then $\operatorname{\mathcal{C}}'$ is $(\kappa ,\lambda )$-compactly generated and our hypotheses guarantee that the functors $F_{-}$ and $F_{+}$ factor through $\operatorname{\mathcal{C}}'$. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is also $(\kappa ,\lambda )$-compactly generated.
Fix a regular cardinal $\mu $ satisfying $\lambda \trianglelefteq \mu $ such that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ are essentially $\mu $-small. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$ are given as full subcategories of their $\operatorname{Ind}_{\lambda }^{\mu }$-completions $\widehat{\operatorname{\mathcal{C}}}_{-} = \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{C}}_{-} )$, $\widehat{\operatorname{\mathcal{C}}}_{+} = \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{C}}_{+}$, and $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{C}})$, and that $F_{-}$ and $F_{+}$ are the restrictions of $(\lambda ,\mu )$-compact functors $\widehat{F}_{-}: \widehat{\operatorname{\mathcal{C}}}_{-} \rightarrow \widehat{\operatorname{\mathcal{C}}}$ and $\widehat{F}_{+}: \widehat{\operatorname{\mathcal{C}}}_{+} \rightarrow \widehat{\operatorname{\mathcal{C}}}$. Applying Propositions 9.4.1.25, we see that the $\infty $-categories $\widehat{\operatorname{\mathcal{C}}}_{-}$, $\widehat{\operatorname{\mathcal{C}}}_{+}$, and $\widehat{\operatorname{\mathcal{C}}}$ are $(\kappa ,\mu )$-compactly generated, and that their full subcategories of $(\kappa ,\mu )$-compact objects can be identified with $\operatorname{\mathcal{C}}_{-}^{0}$, $\operatorname{\mathcal{C}}_{+}^{0}$ and $\operatorname{\mathcal{C}}^{0}$, respectively. Applying Corollary 9.4.4.10 for the pairs $(\lambda \leq \mu )$ and $(\kappa \leq \mu )$, we obtain the following:
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}_{\pm } = \widehat{\operatorname{\mathcal{C}}}_{-} \times ^{\mathrm{h}}_{ \widehat{\operatorname{\mathcal{C}}} } \widehat{\operatorname{\mathcal{C}}}_{+}$ is $(\lambda ,\mu )$-compactly generated. Moreover, an object of $\widehat{\operatorname{\mathcal{C}}}_{\pm }$ is $(\lambda ,\mu )$-compact if and only if it belongs to the full subcategory $\operatorname{\mathcal{C}}_{\pm }$.
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}_{\pm }$ is $(\kappa ,\mu )$-compactly generated. Moreover, an object of $\widehat{\operatorname{\mathcal{C}}}_{\pm }$ is $(\kappa ,\mu )$-compact if and only if it belongs to the full subcategory $\operatorname{\mathcal{C}}^{0}_{\pm } = \operatorname{\mathcal{C}}^{0}_{-} \times _{\operatorname{\mathcal{C}}^{0}}^{\mathrm{h}} \operatorname{\mathcal{C}}^{0}_{+}$.
Applying Corollary 9.4.1.26, we see that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-compactly generated, and that an object of $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-compact if and only if it belongs to the full subcategory $\operatorname{\mathcal{C}}^{0}_{\pm }$.
$\square$