Definition 9.5.1.1. Let $\kappa $ be a small regular cardinal. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-presentable if it is both cocomplete and $\kappa $-accessible (Definition 9.4.6.1).
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9.5.1 Presentable $\infty $-Categories
It will be convenient to work with a slightly more quantitative version of Definition 9.5.0.1.
Remark 9.5.1.2. An $\infty $-category $\operatorname{\mathcal{C}}$ is presentable (in the sense of Definition 9.5.0.1) if and only if it is $\kappa $-presentable (in the sense of Definition 9.5.1.1) for some small regular cardinal $\kappa $.
Example 9.5.1.3. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1). Then $\operatorname{\mathcal{S}}$ is presentable: in fact, it is $\aleph _0$-presentable (see Examples 7.6.6.2 and 9.4.6.13).
Example 9.5.1.4. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1). Then $\operatorname{\mathcal{QC}}$ is presentable: in fact, it is $\aleph _0$-presentable (see Examples 7.6.6.3 and 9.4.6.13).
Example 9.5.1.5. Let $(Q, \leq )$ be a partially ordered set. Then $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(Q)$ is a presentable $\infty $-category if and only if the set $Q$ is small and every subset of $Q$ has a least upper bound. See Corollary 9.4.6.22 and Example 7.6.6.10.
Remark 9.5.1.6 (Sizes of Presentable $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. Then we have two possibilities:
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to the nerve of a partially ordered set $Q$. In this case, Example 9.5.1.5 shows that $Q$ is small, so that $\operatorname{\mathcal{C}}$ is essentially small.
The $\infty $-category $\operatorname{\mathcal{C}}$ is not equivalent to the nerve of a partially ordered set. In this case, Proposition 7.1.2.15 shows that $\operatorname{\mathcal{C}}$ is not essentially small.
Proposition 9.5.1.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be a small regular cardinal. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-presentable.
The $\infty $-category $\operatorname{\mathcal{C}}$ is an $\operatorname{Ind}_{\kappa }$-completion of an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$ which is $\kappa $-cocomplete.
Moreover, if these conditions are satisfied, then we can take $\operatorname{\mathcal{C}}_0$ to be the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects.
Remark 9.5.1.8. In the situation of Proposition 9.5.1.7, the $\infty $-category $\operatorname{\mathcal{C}}_0$ is unique up to Morita equivalence (see Proposition 9.4.1.19). If $\kappa $ is uncountable, the assumption that $\operatorname{\mathcal{C}}_0$ is $\kappa $-cocomplete guarantees that it is idempotent-complete (Corollary 8.5.4.19), so $\operatorname{\mathcal{C}}_0$ is unique up to equivalence. In the case $\kappa = \aleph _0$, this is not necessarily true. For example, if $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ is the $\infty $-category of spaces, then we can take $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{S}}_{\mathrm{fin}}$ to be the full subcategory spanned by the essentially finite spaces (Proposition 9.3.2.16), which is finitely cocomplete but not idempotent-complete (Proposition 9.2.6.3 and Warning 9.2.6.8).
Our proof of Proposition 9.5.1.7 will require some preliminaries. We begin with a variant of Theorem 9.3.6.4.
Proposition 9.5.1.9. Let $\kappa \trianglelefteq \lambda $ be regular cardinals and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{C}}$. Then the composite functor exhibits $\operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$ as a $\lambda $-cocompletion of $\operatorname{\mathcal{C}}$.
\[ \operatorname{\mathcal{C}}\xrightarrow {h} \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} ) \]
Proof. Using Proposition 8.4.5.3, we can choose a functor $h': \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}'$ which exhibits $\widehat{\operatorname{\mathcal{C}}}'$ as a $\lambda $-cocompletion of $\operatorname{\mathcal{C}}$. Without loss of generality, we may assume that $\widehat{\operatorname{\mathcal{C}}}$ is the smallest full subcategory of $\widehat{\operatorname{\mathcal{C}}}'$ which contains the essential image of $h'$ and is closed under $\kappa $-small colimits. Let $T: \operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} ) \rightarrow \widehat{\operatorname{\mathcal{C}}}'$ be an $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of the inclusion functor $\widehat{\operatorname{\mathcal{C}}} \hookrightarrow \widehat{\operatorname{\mathcal{C}}}'$ (Definition 9.3.1.12). It follows from Proposition 9.2.5.24 that every object of $\widehat{\operatorname{\mathcal{C}}}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\widehat{\operatorname{\mathcal{C}}}'$, so the functor $T$ is fully faithful (Proposition 9.3.2.1). To complete the proof, it will suffice to show that $T$ is essentially surjective. Without loss of generality, we may assume that $\lambda $ is uncountable (otherwise, the result is immediate from Example 9.3.1.10). In this case, Variant 9.3.4.18 guarantees that every object $X \in \widehat{\operatorname{\mathcal{C}}}'$ can be realized as the colimit of a diagram $K \xrightarrow {F} \operatorname{\mathcal{C}}\xrightarrow {h'} \widehat{\operatorname{\mathcal{C}}}'$, where $K$ is a $\lambda $-small simplicial set. Applying Lemma 9.1.7.18 we can realize $K$ as the colimit of a diagram
\[ A \rightarrow \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto K_{\alpha }, \]
where $A$ is a $\lambda $-small $\kappa $-directed partially ordered set and each $K_{\alpha }$ is a $\kappa $-small simplicial set. For each $\alpha \in A$, the diagram $(h' \circ F)|_{ K_{\alpha } }$ has some colimit $X_{\alpha } \in \widehat{\operatorname{\mathcal{C}}}$. Since $K$ is also a categorical colimit of the diagram $\{ K_{\alpha } \} _{\alpha \in A}$ (Proposition 9.1.6.1), we can promote the construction $\alpha \mapsto X_{\alpha }$ to a $\lambda $-small $\kappa $-filtered diagram $\operatorname{N}_{\bullet }(A) \rightarrow \widehat{\operatorname{\mathcal{C}}}$ having colimit $X$ (see Proposition 7.5.8.12), so that $X$ belongs to the essential image of $T$. $\square$
Corollary 9.5.1.10. Let $\kappa \trianglelefteq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete. Then $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $\lambda $-cocomplete.
Proof. Using Proposition 8.4.5.3, we can choose a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{C}}$. Moreover, $h$ is fully faithful, so the functor $H = \operatorname{Ind}_{\kappa }^{\lambda }(h): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$ is also fully faithful (Corollary 9.3.2.2). If $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete, then the functor $h$ admits a left adjoint (Proposition 8.4.5.13). It follows that $H$ also admits a left adjoint (Exercise 9.3.3.11), and therefore induces an equivalence from $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ to a reflective localization of $\operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$. By virtue of Corollary 7.1.4.24, it will suffice to show that the $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$ is $\lambda $-cocomplete, which follows from Proposition 9.5.1.9. $\square$
Corollary 9.5.1.11. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ denote the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda $-cocomplete.
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete.
The $\infty $-category $\operatorname{\mathcal{C}}_{< \kappa }$ is $\kappa $-cocomplete.
Proof. The implication $(1) \Rightarrow (2)$ is immediate, the implication $(2) \Rightarrow (3)$ follows from Proposition 9.2.5.24, and the implication $(3) \Rightarrow (1)$ follows from Corollary 9.5.1.10. $\square$
Corollary 9.5.1.12. Let $\kappa $ be a small regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-compactly generated, and let $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ denote the full subcategory spanned by the $\kappa $-compact objects. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete.
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete.
The $\infty $-category $\operatorname{\mathcal{C}}_{< \kappa }$ is $\kappa $-cocomplete.
Proof. Apply Corollary 9.5.1.11 in the special case where $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal. $\square$
Proof of Proposition 9.5.1.7. Combine Proposition 9.4.6.2 with Corollaries 9.5.1.11 and 9.5.1.12. $\square$