We now study $\infty $-categories $\operatorname{\mathcal{C}}$ which contain “enough” compact objects.
Definition 9.2.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is compactly generated if it satisfies the following conditions:
The $\infty $-category $\operatorname{\mathcal{C}}$ admits small filtered colimits.
Every object $C \in \operatorname{\mathcal{C}}$ can be realized as the colimit of a small filtered diagram $\{ C_{\alpha } \} $, where each $C_{\alpha }$ is a compact object of $\operatorname{\mathcal{C}}$.
Example 9.2.6.2. Every Kan complex $X$ is compactly generated when viewed as an $\infty $-category: see Examples 9.2.2.5 and 7.3.9.5.
Definition 9.2.6.3. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated if it satisfies the following conditions:
The $\infty $-category $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits.
Every object $C \in \operatorname{\mathcal{C}}$ can be realized as the colimit of a small $\kappa $-filtered diagram $\{ C_{\alpha } \} $, where each $C_{\alpha }$ is a $\kappa $-compact object of $\operatorname{\mathcal{C}}$.
Example 9.2.6.4. An $\infty $-category $\operatorname{\mathcal{C}}$ is compactly generated (in the sense of Definition 9.2.6.1) if and only if it is $\aleph _0$-compactly generated (in the sense of Definition 9.2.6.3). See Example 9.2.2.8.
Variant 9.2.6.5. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated if it satisfies the following conditions:
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\lambda $-small $\kappa $-filtered colimits.
Every object $C \in \operatorname{\mathcal{C}}$ can be realized as the colimit of a $\lambda $-small $\kappa $-filtered diagram $\{ C_{\alpha } \} $, where each $C_{\alpha }$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$.
Remark 9.2.6.6. Following the convention of Remark 4.7.0.5, a regular cardinal $\kappa $ is small if it satisfies $\kappa < \operatorname{\textnormal{\cjRL {t}}}$, for some fixed strongly inaccessible cardinal $\operatorname{\textnormal{\cjRL {t}}}$. In this case, an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated (in the sense of Definition 9.2.6.3) if and only if it is $(\kappa ,\operatorname{\textnormal{\cjRL {t}}})$-compactly generated (in the sense of Variant 9.2.6.5). In particular, $\operatorname{\mathcal{C}}$ is compactly generated if and only if it is $(\aleph _0, \operatorname{\textnormal{\cjRL {t}}})$-compactly generated. See Remark 9.2.2.14.
Example 9.2.6.7. Let $\kappa $ be an uncountable regular cardinal. An $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\kappa )$-compactly generated if and only if it is idempotent complete (see Example 9.2.2.15). If $\kappa = \aleph _0$, every $\infty $-category is $(\kappa ,\kappa )$-compactly generated.
Proposition 9.2.6.8. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated.
There exists a functor of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ which exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}'$ (Variant 9.2.1.7).
There exists a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ for which the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}_0$.
Moreover, if these conditions are satisfied, then we can take $\operatorname{\mathcal{C}}_0$ to be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $(\kappa ,\lambda )$-compact objects.
Proof. The implication $(3) \Rightarrow (2)$ is immediate, and the implication $(2) \Rightarrow (1)$ follows from Proposition 9.2.3.3 and Corollary 9.2.4.23. To complete the proof, assume that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects. Applying the criterion of Proposition 9.2.3.3, we conclude that the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}_0$. $\square$
Corollary 9.2.6.9. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated.
There exists a functor of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ which exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}'$ (Definition 9.2.1.4).
There exists a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ for which the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}_0$.
Moreover, if these conditions are satisfied, then we can take $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{C}}_{< \kappa }$ to be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects.
Proof. Apply Proposition 9.2.6.8 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a strongly inaccessible cardinal (Remark 9.2.6.6). $\square$
Corollary 9.2.6.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is compactly generated.
There exists a functor of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ which exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}'$ (Definition 9.2.1.1).
There exists a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ for which the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}_0$.
Moreover, if these conditions are satisfied, then we can take $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{C}}_{< \aleph _0}$ to be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the compact objects.
Proof. Apply Corollary 9.2.6.9 in the special case $\kappa = \aleph _0$ (see Example 9.2.6.4). $\square$
Remark 9.2.6.11. Let $\kappa \leq \lambda $ be regular cardinals, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. To show that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, it suffices to verify the following (which is a priori weaker than condition $(b_{\kappa ,\lambda })$ of Variant 9.2.6.5) :
Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects. Then $\operatorname{\mathcal{C}}_0$ generated $\operatorname{\mathcal{C}}$ under $\lambda $-small $\kappa $-filtered coliits That is, if $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is a replete full subcategory which contains $\operatorname{\mathcal{C}}_0$ and is closed under $\lambda $-small $\kappa $-filtered colimits, then $\operatorname{\mathcal{C}}' = \operatorname{\mathcal{C}}$.
In this case, the recognition principle of Proposition 9.2.3.3 guarantees that $\operatorname{\mathcal{C}}$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of the full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$.
Beware that, in the setting of Proposition 9.2.6.8, the subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is generally not unique. However, it is unique up to Morita equivalence.
Lemma 9.2.6.12. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Assume that $\operatorname{\mathcal{C}}_0$ generates $\operatorname{\mathcal{C}}$ under $\lambda $-small $\kappa $-filtered colimits (in the sense of Remark 9.2.6.11). Then every $(\kappa ,\lambda )$-compact object $C \in \operatorname{\mathcal{C}}$ is a retract of an object of $\operatorname{\mathcal{C}}_0$.
Proof. Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects $D$ which satisfy the following condition:
Every morphism $f: C \rightarrow D$ factors (up to homotopy) through an object of $\operatorname{\mathcal{C}}_0$.
Then $\operatorname{\mathcal{C}}'$ contains $\operatorname{\mathcal{C}}_0$, and our assumption that $C$ is $(\kappa ,\lambda )$-compact guarantees that $\operatorname{\mathcal{C}}'$ is closed under the formation of $\lambda $-small $\kappa $-filtered colimits. Since $\operatorname{\mathcal{C}}_0$ generates $\operatorname{\mathcal{C}}$ under $\lambda $-small $\kappa $-filtered colimits, we must have $\operatorname{\mathcal{C}}' = \operatorname{\mathcal{C}}$. In particular, $\operatorname{\mathcal{C}}'$ contains the object $C$, so the identity morphism $\operatorname{id}_{C}: C \rightarrow C$ factors through an object of $\operatorname{\mathcal{C}}_0$. $\square$
Proposition 9.2.6.13. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$, and let $\widehat{\operatorname{\mathcal{C}}}_0 \subseteq \widehat{\operatorname{\mathcal{C}}}$ be the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects. Then $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}_0$ as an idempotent completion of $\operatorname{\mathcal{C}}$.
Proof. It follows from Proposition 9.2.3.3 that $h$ induces an equivalence from $\operatorname{\mathcal{C}}$ to a full subcategory of $\widehat{\operatorname{\mathcal{C}}}_{1} \subseteq \widehat{\operatorname{\mathcal{C}}}_0$ which generates $\widehat{\operatorname{\mathcal{C}}}$ under $\lambda $-small $\kappa $-filtered colimits. Applying Lemma 9.2.6.12, we see that every object of $\widehat{\operatorname{\mathcal{C}}}_0$ is a retract of an object of $\widehat{\operatorname{\mathcal{C}}}_{1}$. It will therefore suffice to show that $\widehat{\operatorname{\mathcal{C}}}_0$ is idempotent complete. This follows from Proposition 8.5.4.8, since the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is idempotent complete (Proposition 9.1.9.17) and the full subcategory $\widehat{\operatorname{\mathcal{C}}}_0$ is closed under retracts (Remark 9.2.2.24). $\square$
Exercise 9.2.6.14. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Show that $F$ is a Morita equivalence if and only if $\operatorname{Ind}^{\lambda }_{\kappa }(F): \operatorname{Ind}^{\lambda }_{\kappa }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}^{\lambda }_{\kappa }(\operatorname{\mathcal{D}})$ is an equivalence of $\infty $-categories.
Corollary 9.2.6.15. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is idempotent complete. Then the tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ induces an equivalence from $\operatorname{\mathcal{C}}$ to the full subcategory of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ spanned by the $(\kappa ,\lambda )$-compact objects.
Proposition 9.2.6.16. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable. Then the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Idempotent complete $\infty $-categories} \} / \textnormal{Equivalence} \ar [d]^{\sim } \\ \{ \textnormal{$(\kappa ,\lambda )$-compactly generated $\infty $-categories} \} / \textnormal{Equivalence}. } \]
Proof. Surjectivity follows from Proposition 9.2.6.8, and injectivity from Proposition 9.2.6.13. $\square$
Corollary 9.2.6.17. Let $\kappa $ be a small regular cardinal. Then the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}_{\kappa }(\operatorname{\mathcal{C}})$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Idempotent complete $\infty $-categories} \} / \textnormal{Equivalence} \ar [d]^{\sim } \\ \{ \textnormal{$\kappa $-compactly generated $\infty $-categories} \} / \textnormal{Equivalence}. } \]
Proof. Apply Proposition 9.2.6.16 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a strongly inaccessible cardinal (Remark 9.2.6.6). $\square$
Corollary 9.2.6.18. The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}(\operatorname{\mathcal{C}})$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Idempotent complete $\infty $-categories} \} / \textnormal{Equivalence} \ar [d]^{\sim } \\ \{ \textnormal{Compactly generated $\infty $-categories} \} / \textnormal{Equivalence}. } \]
Proof. Apply Corollary 9.2.6.17 in the special case $\kappa = \aleph _0$ (Example 9.2.6.4). $\square$