Definition 9.2.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small filtered colimits. We say that an object $C \in \operatorname{\mathcal{C}}$ is compact if the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet ): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is finitary: that is, it preserves small filtered colimits.
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9.2.2 Compact Objects
Definition 9.2.0.2 has a counterpart in the setting of $\infty $-categories.
Warning 9.2.2.2. In the formulation of Definition 9.2.2.1, we have implicitly assumed that the $\infty $-category $\operatorname{\mathcal{C}}$ is locally small (so that the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet ): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is well-defined). More generally, let $\mu $ be a regular cardinal which is not small such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small. In this case, we say that $C \in \operatorname{\mathcal{C}}$ is compact if the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet ): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is finitary. It follows from Corollary 7.4.3.8 that this condition does not depend on the choice of $\mu $.
Example 9.2.2.3. Let $\operatorname{\mathcal{C}}$ be a category which admits small filtered colimits. Then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an $\infty $-category which admits small filtered colimits (Corollary 9.1.9.13). Moreover, an object $C \in \operatorname{\mathcal{C}}$ is compact (in the sense of Definition 9.2.0.2) if and only if it is compact when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 9.2.2.1). This follows from Proposition 9.1.9.10, since the inclusion functor $\operatorname{N}_{\bullet }(\operatorname{Set}) \hookrightarrow \operatorname{\mathcal{S}}$ is finitary (see Variant 9.1.6.4).
Example 9.2.2.4. The Kan complex $\Delta ^0$ corepresents the identity functor $\operatorname{id}: \operatorname{\mathcal{S}}\rightarrow \operatorname{\mathcal{S}}$ (see Proposition 5.6.6.17), and is therefore compact when viewed as an object of the $\infty $-category of spaces $\operatorname{\mathcal{S}}$. See Corollary 9.2.7.5 for a more general statement.
Example 9.2.2.5. Let $X$ be a Kan complex, which we view as an $\infty $-category in which every morphism is an isomorphism. Then $X$ admits small filtered colimits which are preserved by any functor of $\infty $-categories $X \rightarrow \operatorname{\mathcal{C}}$ (see Example 7.3.9.5). It follows that every vertex $x \in X$ is a compact object of $X$.
It will sometimes be useful to consider an infinitary generalization of Definition 9.2.2.1.
Definition 9.2.2.6. Let $\kappa $ be a small regular cardinal, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small $\kappa $-filtered colimits. We say that an object $C \in \operatorname{\mathcal{C}}$ is $\kappa $-compact if the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, \bullet )$ is $\kappa $-finitary: that is, it preserves small $\kappa $-filtered colimits. We will sometimes write $\operatorname{\mathcal{C}}_{< \kappa }$ for the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects.
Warning 9.2.2.7. Let $\kappa $ be an uncountable regular cardinal. We have now assigned two meanings to the notation $\operatorname{\mathcal{C}}_{< \kappa }$ in the cases $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{C}}= \operatorname{\mathcal{QC}}$:
Following the convention of Remark 5.5.4.12 and Variant 5.5.4.13, $\operatorname{\mathcal{C}}_{< \kappa }$ denotes the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects which are $\kappa $-small when viewed as simplicial sets.
Following the convention of Definition 9.2.2.6, $\operatorname{\mathcal{C}}_{< \kappa }$ denotes the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects which are $\kappa $-compact.
However, these conventions are almost compatible: we will see later that a Kan complex is $\kappa $-compact (as an object of $\operatorname{\mathcal{S}}$) if and only if it is essentially $\kappa $-small (Proposition 9.2.7.11). Similarly, a small $\infty $-category is $\kappa $-compact (as an object of $\operatorname{\mathcal{QC}}$) if and only if it is essentially $\kappa $-small (Proposition 9.2.8.15).
Example 9.2.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small filtered colimits. Then an object $C \in \operatorname{\mathcal{C}}$ is compact (in the sense of Definition 9.2.2.1) if and only if it is $\aleph _0$-compact (in the sense of Definition 9.2.2.6).
Example 9.2.2.9. Let $\kappa $ be a (small) regular cardinal. For every (small) set $S$, the following conditions are equivalent:
The set $S$ is $\kappa $-small: that is, it has cardinality $< \kappa $.
The set $S$ is $\kappa $-compact when viewed as an object of (the nerve of) the category of sets.
To prove the implication $(1) \Rightarrow (2)$, we can use the decomposition $S \simeq \coprod _{s \in S} \{ s\} $ to reduce to the case where $S$ consists of a single element (Corollary 9.2.2.22). In this case, $S$ corepresents the identity functor $\operatorname{id}: \operatorname{Set}\rightarrow \operatorname{Set}$, which preserves all colimits. For the converse, we observe that $S$ can be realized as a $\kappa $-filtered colimit of $\kappa $-small subsets of itself. Consequently, if $S$ is $\kappa $-compact, then the identity function $\operatorname{id}: S \rightarrow S$ factors through some $\kappa $-small subset $S_0 \subseteq S$. It follows that $S = S_0$ is $\kappa $-small.
Variant 9.2.2.10. Let $\kappa $ be a (small) regular cardinal and let $X$ be a (small) simplicial set. Then $X$ is $\kappa $-compact (as an object of the ordinary category of simplicial sets) if and only if is $\kappa $-small (in the sense of Definition 4.7.4.1).
Proof. If $\kappa = \aleph _0$, this follows from Variant 9.2.0.5. We may therefore assume without loss of generality that $\kappa $ is uncountable. Assume first that $X$ is a $\kappa $-small simplicial set, and let $\operatorname{{\bf \Delta }}_{X}$ be the category of simplices of $X$ (Construction 1.1.3.9). Then $X$ can be realized as a colimit $\varinjlim _{ ([n],\sigma ) \in \operatorname{{\bf \Delta }}_{X} } \Delta ^ n$ (see the proof of Proposition 1.1.3.11, or Theorem 8.4.2.1). Each of the standard simplices $\Delta ^ n$ is a $\kappa $-compact object of the category $\operatorname{Set_{\Delta }}$, since the evaluation functor $(Y_{\bullet } \in \operatorname{Set_{\Delta }}) \mapsto Y_{n}$ preserves colimits (Remark 1.1.0.8). Our assumption that $X$ is $\kappa $-small guarantees that the category $\operatorname{{\bf \Delta }}_{X}$ is $\kappa $-small (Proposition 4.7.4.10), so that $X$ is $\kappa $-compact by virtue of Proposition 9.2.2.21.
We now prove the converse. As in Example 9.2.2.9, we can realize $X$ as a $\kappa $-filtered colimit $\varinjlim _{\alpha \in A} X_{\alpha }$, where $\{ X_{\alpha } \} _{\alpha \in A}$ is the collection of all $\kappa $-small simplicial subsets of $X$. If $X$ is $\kappa $-compact, then the identity map $\operatorname{id}_{X}$ factors through some $X_{\alpha }$, so that $X = X_{\alpha }$ is $\kappa $-small. $\square$
Variant 9.2.2.11. Let $\kappa $ be a regular cardinal, let $Y$ be a set, and let $P(Y)$ be the partially ordered collection of all subsets of $Y$. Then a subset $X \subseteq Y$ is $\kappa $-small if and only if it is $\kappa $-compact when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }( P(Y) )$.
Definition 9.2.2.12. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. We say that an object $C \in \operatorname{\mathcal{C}}$ is $(\kappa , \lambda )$-compact if the corepresentable functor is $(\kappa ,\lambda )$-finitary: that is, it preserves $\lambda $-small $\kappa $-filtered colimits. Here $\mu $ is any cardinal of cofinality $\geq \lambda $ such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small (it follows from Corollary 7.4.3.8 that this condition does not depend on the choice of $\mu $).
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, \bullet ): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu } \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \]
Remark 9.2.2.13. In the situation of Definition 9.2.2.12, the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet )$ is only well-defined up to isomorphism. However, the condition that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet )$ is $(\kappa ,\lambda )$-finitary depends only on its isomorphism class.
Remark 9.2.2.14. 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 $\operatorname{\textnormal{\cjRL {t}}}$. If $\operatorname{\mathcal{C}}$ is an $\infty $-category which admits small $\kappa $-filtered colimits, then an object $C \in \operatorname{\mathcal{C}}$ is $\kappa $-compact (in the sense of Definition 9.2.2.6) if and only if it is $(\kappa , \operatorname{\textnormal{\cjRL {t}}})$-compact (in the sense of Definition 9.2.2.12. In particular, if $\operatorname{\mathcal{C}}$ admits small filtered colimits, then an object $C \in \operatorname{\mathcal{C}}$ is compact if and only if it is $(\aleph _0, \operatorname{\textnormal{\cjRL {t}}})$-compact.
Example 9.2.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be a regular cardinal. Assume either that $\kappa = \aleph _0$ or that $\operatorname{\mathcal{C}}$ is idempotent complete, so that $\operatorname{\mathcal{C}}$ admits $\kappa $-small $\kappa $-filtered colimits. Then any object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\kappa )$-compact. See Proposition 9.1.9.17.
Example 9.2.2.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits sequential colimits. Then an object $C \in \operatorname{\mathcal{C}}$ is $(\aleph _0, \aleph _1)$-compact if and only if the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, \bullet )$ commutes with sequential colimits. See Variant 9.1.9.11.
Remark 9.2.2.17 (Monotonicity). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits, and let $C$ be an object which is $(\kappa ,\lambda )$-compact. Then $C$ is also $(\kappa ',\lambda ')$-compact for all regular cardinals $\kappa '$ and $\lambda '$ satisfying $\kappa \leq \kappa ' \leq \lambda ' \leq \lambda $. See Remark 9.1.9.18.
Definition 9.2.2.12 makes sense for every pair of regular cardinals $\kappa \leq \lambda $. However, it is primarily of interest in the case $\kappa \trianglelefteq \lambda $, in the sense of Definition 9.1.7.5 (see Warning 9.1.9.7).
Proposition 9.2.2.18. Let $\kappa $, $\lambda $, and $\mu $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda \trianglelefteq \mu $, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\mu $-small $\kappa $-filtered colimits. Then an object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\mu )$-compact if and only if it is both $(\kappa ,\lambda )$-compact and $(\lambda ,\mu )$-compact.
Proof. Apply Proposition 9.1.9.19 to the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, \bullet )$. $\square$
Corollary 9.2.2.19. Let $\kappa $ and $\lambda $ be small regular cardinals satisfying $\kappa \trianglelefteq \lambda $, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small $\kappa $-filtered colimits. Then an object $C \in \operatorname{\mathcal{C}}$ is $\kappa $-compact if and only if it is both $(\kappa ,\lambda )$-compact and $\lambda $-compact.
Corollary 9.2.2.20. Let $\lambda $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small filtered colimits. Then an object $C \in \operatorname{\mathcal{C}}$ is compact if and only if it both $(\aleph _0, \lambda )$-compact and $\lambda $-compact.
Proposition 9.2.2.21. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. Then the collection of $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{C}}$ is closed under $\kappa $-small colimits.
Proof. Fix an uncountable cardinal $\mu $ having cofinality $\geq \lambda $ and exponential cofinality $\geq \kappa $, and such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small. Let $\operatorname{Fun}^{( \kappa , \lambda )-\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ spanned by the $(\kappa , \lambda )$-finitary functors. Then the subcategory $\operatorname{Fun}^{( \kappa , \lambda )-\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ is closed under $\kappa $-small limits (Remark 9.1.9.9). By definition, an object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact if and only if the contravariant Yoneda embedding $h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ carries $C$ to an object of $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$. The desired result now follows from the observation that $h^{\bullet }$ preserves $\kappa $-small limits (see Proposition 7.4.1.18). $\square$
Corollary 9.2.2.22. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small $\kappa $-filtered colimits. Then the collection of $\kappa $-compact objects of $\operatorname{\mathcal{C}}$ is closed under $\kappa $-small colimits.
Corollary 9.2.2.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small filtered colimits. Then the collection of compact objects of $\operatorname{\mathcal{C}}$ is closed under finite colimits.
Remark 9.2.2.24 (Retracts). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits, and let $C \in \operatorname{\mathcal{C}}$ be an object. If $C$ is $(\kappa ,\lambda )$-compact, then any retract of $C$ is also $(\kappa ,\lambda )$-compact. This follows from Corollary 8.5.1.14. If $\kappa $ is uncountable, then it can also be deduced from Proposition 9.2.2.21, since any retract of $C$ can be realized as the colimit of a diagram $C \xrightarrow {e} C \xrightarrow {e} C \xrightarrow {e} C \rightarrow \cdots $; see Proposition 8.5.4.17.