Proposition 9.5.3.1 (Presentability of Slice and Coslice $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a small diagram. Then the slice and coslice $\infty $-categories $\operatorname{\mathcal{C}}_{/f}$ and $\operatorname{\mathcal{C}}_{f/}$ are also presentable.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.5.3 Stability Properties of Presentable $\infty $-Categories
We now show that the condition of presentability is preserved by various categorical constructions.
Proof. It follows from Corollary 9.4.8.5 that $\operatorname{\mathcal{C}}_{/f}$ and $\operatorname{\mathcal{C}}_{f/}$ are accessible. It will therefore suffice to show that $\operatorname{\mathcal{C}}_{/f}$ and $\operatorname{\mathcal{C}}_{f/}$ are cocomplete, which follows from Corollary 7.1.4.22 and Remark 7.1.3.12, respectively. $\square$
Proposition 9.5.3.2 (Presentability of Diagram $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $K$ be a small simplicial set. Then the diagram $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is presentable.
Proof. It follows from Proposition 9.4.8.1 that $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is accessible and from Proposition 7.1.8.2 that it is cocomplete. $\square$
Proposition 9.5.3.3 (Presentability of Oriented Fiber Products). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be presentable $\infty $-categories and let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category. Suppose we are given a pair of functors $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$. If $F_{-}$ is cocontinuous and $F_{+}$ is accessible, then the oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is presentable.
Proof. Proposition 9.4.8.3 guarantees that $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is accessible and Corollary 7.1.9.5 guarantees that it is cocomplete. $\square$
Remark 9.5.3.4. In the situation of Proposition 9.5.3.3, the homotopy fiber product $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is accessible (Proposition 9.4.8.8). If the functors $F_{-}$ and $F_{+}$ are both cocontinuous, then $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is closed under small colimits in the presentable $\infty $-category $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ (Remark 7.1.9.6), and is therefore also presentable.
Corollary 9.5.3.5. Suppose we are given a categorical pullback diagram of $\infty $-categories where $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ are presentable and the functors $F_{-}$ and $F_{+}$ are cocontinuous. Then the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is presentable.
9.15
\begin{equation} \begin{gathered}\label{equation:presentable-categorical-pullback} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm } \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{-} \ar [d]^{F_{-} } \\ \operatorname{\mathcal{C}}_{+} \ar [r]^-{F_{+} } & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}
Proof. The assumption that (9.15) is a categorical pullback square guarantees that $\operatorname{\mathcal{C}}_{\pm }$ is equivalent to the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$, which is presentable by Remark 9.5.3.4. $\square$
We now formulate a more general version of Corollary 9.5.3.5. Recall that every accessible $\infty $-category is $\operatorname{\Omega }^{+}$-small, where $\operatorname{\Omega }$ denotes the fixed strongly inaccessible cardinal of Remark 4.7.0.5 (see Remark 9.4.6.14).
Proposition 9.5.3.6 (Limits of Presentable $\infty $-Categories). Suppose we are given a small diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ satisfying the following conditions:
For every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is presentable.
For every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}(e): \mathscr {F}(C) \rightarrow \mathscr {F}(C')$ is cocontinuous.
Then the limit $\varprojlim ( \mathscr {F} )$ is also a presentable $\infty $-category.
Proof. It follows from Corollary 9.4.8.11 that the limit $\varprojlim (\mathscr {F})$ is an accessible $\infty $-category, and from Proposition 7.6.6.24 that it is cocomplete. $\square$
Example 9.5.3.7 (Products of Presentable $\infty $-Categories). Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a small collection of presentable $\infty $-categories. Then the product $\operatorname{\mathcal{C}}= \prod _{i \in I} \operatorname{\mathcal{C}}_ i$ is presentable. This can be regarded as a special case of Proposition 9.5.3.6 (see Example 7.6.1.17). Alternatively, it follows from Corollary 9.4.6.19 (which guarantees that $\operatorname{\mathcal{C}}$ is accessible) and Example 7.1.3.11 (which guarantees that it is cocomplete).
Remark 9.5.3.8. In the situation of Proposition 9.5.3.6, suppose that $\operatorname{\mathcal{D}}$ is a cocomplete $\infty $-category. Then a functor $T: \operatorname{\mathcal{D}}\rightarrow \varprojlim (\mathscr {F})$ is cocontinuous if and only if, for each $C \in \operatorname{\mathcal{C}}$, the composite functor $\operatorname{\mathcal{D}}\xrightarrow {T} \varprojlim (\mathscr {F}) \rightarrow \mathscr {F}(C)$ is cocontinuous. See Proposition 7.6.6.24.
The collection of presentable $\infty $-categories can be organized into an $\infty $-category.
Construction 9.5.3.9. We define a (non-full) subcategory $\operatorname{\mathcal{QC}}^{\operatorname{LPr}} \subseteq \operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ as follows:
An object $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ if and only if $\operatorname{\mathcal{C}}$ is a presentable $\infty $-category.
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then a morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ if and only if $F$ is a cocontinuous functor.
We will refer to $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ as the $\infty $-category of presentable $\infty $-categories.
Remark 9.5.3.10. The notation of Construction 9.5.3.9 is intended to remind the reader that morphisms in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ are functors which are left adjoints (see Theorem 9.5.6.1 and compare with Construction 9.5.6.3).
Proposition 9.5.3.11. The $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ admits small limits which are preserved by the inclusion functor $\operatorname{\mathcal{QC}}^{\operatorname{LPr}} \hookrightarrow \operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$.
Proof. Combine Proposition 9.5.3.6 with Remark 9.5.3.8. $\square$
Remark 9.5.3.12. We will later show that the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ admits small colimits (Theorem 
). Beware that these are not computed at the level of the underlying $\infty $-category: that is, the inclusion functor $\operatorname{\mathcal{QC}}^{\operatorname{LPr}} \hookrightarrow \operatorname{\mathcal{QC}}_{ \leq \operatorname{\Omega }}$ does not preserve colimits in general.
Definition 9.5.3.13 (Presentable Fibrations). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We will say that $U$ is a presentable fibration if it satisfies the following conditions:
The morphism $U$ is a locally cocartesian fibration of simplicial sets.
For each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a presentable $\infty $-category.
For every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ is cocontinuous.
Remark 9.5.3.14. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a presentable fibration of simplicial sets (in the sense of Definition 9.5.3.13). Then $U$ is an edgewise accessible inner fibration (in the sense of Definition 9.4.8.12). See Corollary 9.4.8.25.
Example 9.5.3.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then $U$ is presentable (in the sense of Definition 9.5.3.13) if and only if it admits a covariant transport representation taking values in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ of Construction 9.5.3.9.
Let $\operatorname{\mathcal{C}}$ be a small simplicial set and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a presentable cocartesian fibration, so that $U$ admits a covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{\operatorname{LPr}}$. It follows from Proposition 7.4.4.1 that the $\infty $-category of cocartesian sections $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ can be identified with a limit of $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$), and is therefore a presentable $\infty $-category (Proposition 9.5.3.6). This observation has a “lax” counterpart:
Proposition 9.5.3.16. Let $\operatorname{\mathcal{C}}$ be a small simplicial set and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a presentable fibration. Then the $\infty $-category of sections $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is presentable.
Example 9.5.3.17. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $K$ be a small simplicial set. Applying Proposition 9.5.3.16 to the projection map $U: \operatorname{\mathcal{C}}\times K \rightarrow K$, we recover the assertion that the diagram $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is presentable (Proposition 9.5.3.2).
Proposition 9.5.3.16 is a consequence of the following more general assertion:
Lemma 9.5.3.18. Let $\operatorname{\mathcal{C}}$ be a small simplicial set and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration satisfying the following conditions:
For each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is presentable.
For each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{C}$ is accessible.
Then the $\infty $-category of sections $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is presentable.
Proof. It follows from Corollary 9.4.8.23 that the fibration $U$ is edgewise accessible, so that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is accessible (Corollary 9.4.8.29). Corollary 7.1.10.5 guarantees that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is cocomplete. $\square$
Remark 9.5.3.19. In the situation of Lemma 9.5.3.18, small colimits in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ are computed levelwise: that is, they are preserved by the evaluation functor for each vertex $C \in \operatorname{\mathcal{C}}$. See Corollary 7.1.10.5.
\[ \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \{ C\} , \operatorname{\mathcal{E}}) = \operatorname{\mathcal{E}}_{C} \]
Proof of Proposition 9.5.3.16. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a presentable fibration of simplicial sets. It follows from Corollary 9.5.6.2 that $U$ is a locally cartesian fibration whose contravariant transport functors are accessible. Applying Lemma 9.5.3.18, we conclude that the $\infty $-category $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is presentable. $\square$