9.4 Compact Generation and Accessibility
In the development of category theory, it is often important to be mindful of some dichotomy between “large” and “small”. This is for two reasons:
Many of the categories which appear naturally can be fit into a specific mold: their objects are sets equipped with some additional structure, and their morphisms are functions which preserve the structure. Categories of this form (such as the categories of sets, groups, and partially ordered sets) tend not to be small.
From a theoretical perspective, it is technically convenient to work with categories which admit small colimits (or at least some large class of colimits). Such categories are rarely small: a small category which admits small coproducts is equivalent to a partially ordered set (Proposition 7.1.2.15).
These motivations are connected with one another. Large categories $\operatorname{\mathcal{C}}$ which arise in mathematical practice tend to be large in a very specific way: they are obtained from a small subcategory by (freely) adjoining certain colimits. This observation was articulated more precisely in the work of Makkai-Paré ([MR1031717]).
Definition 9.4.0.1 (Makkai-Paré). A category $\operatorname{\mathcal{C}}$ is accessible if there exists a small regular cardinal $\kappa $ satisfying the following conditions:
- $(1_{\kappa })$
The category $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits.
- $(2_{\kappa })$
Every object of $\operatorname{\mathcal{C}}$ can be realized as a small $\kappa $-filtered colimit of $\kappa $-compact objects of $\operatorname{\mathcal{C}}$.
- $(3_{\kappa })$
The full subcategory of $\kappa $-compact objects $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ is essentially small.
Our goal in this section is to study the $\infty $-categorical counterpart of Definition 9.4.0.1. Fix a small regular cardinal $\kappa $. We will say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated if it satisfies conditions $(1_{\kappa })$ and $(2_{\kappa })$ of Definition 9.4.0.1. In §9.4.1 we show that, if this condition is satisfied, then the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}_{< \kappa }$ are essentially interchangeable data. More precisely, Corollary 9.4.1.23 implies that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}_{< \kappa }$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{$\kappa $-compactly generated $\infty $-categories $\operatorname{\mathcal{C}}$} \} / \textnormal{Equivalence} \ar [d]^{\sim } \\ \{ \textnormal{Idempotent-complete $\infty $-categories $\operatorname{\mathcal{C}}_0$} \} / \textnormal{Equivalence}, } \]
where the inverse bijection is given by the construction $\operatorname{\mathcal{C}}_0 \mapsto \operatorname{Ind}_{\kappa }(\operatorname{\mathcal{C}}_0)$ studied in §9.3.
The preceding bijection has a counterpart at the level of functors. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\kappa $-compactly generated $\infty $-categories. We will say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-compact if it is $\kappa $-finitary (that is, it preserves small $\kappa $-filtered colimits) and carries $\kappa $-compact objects of $\operatorname{\mathcal{C}}$ to $\kappa $-compact objects of $\operatorname{\mathcal{D}}$. In §9.4.2, we show that the restriction functor $F \mapsto F|_{ \operatorname{\mathcal{C}}_{< \kappa } }$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{$\kappa $-compact functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} / \textnormal{Isomorphism} \ar [d]^{\sim } \\ \{ \textnormal{Functors $F_0: \operatorname{\mathcal{C}}_{< \kappa } \rightarrow \operatorname{\mathcal{D}}_{< \kappa }$}\} / \textnormal{Isomorphism}, } \]
where the inverse bijection is given by the construction $F_0 \mapsto \operatorname{Ind}_{\kappa }(F_0)$ (see Remark 9.4.2.10 and Example 9.4.2.11).
The bulk of this section is devoted to studying the closure properties of compact generation. Let $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$ be $\kappa $-compactly generated $\infty $-categories, and suppose we are given $\kappa $-compact functors $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$. In §9.4.3, we show that the oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is also $\kappa $-compactly generated (moreover, this is true more generally if the functor $F_{+}$ is only assumed to be $\kappa $-finitary: see Theorem 9.4.3.4). This has several immediate consequences:
If $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated and $X$ is an object of $\operatorname{\mathcal{C}}$, then the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ is $\kappa $-compactly generated (Corollary 9.4.3.2).
If $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated and $X$ is a $\kappa $-compact object of $\operatorname{\mathcal{C}}$, then the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X/}$ is $\kappa $-compactly generated (Corollary 9.4.3.7).
If $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated and $K$ is the nerve of a finite partially ordered set (for example, a standard simplex), then the diagram $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is also $\kappa $-compactly generated (Corollary 9.4.3.9).
The situation for homotopy fiber products is more subtle. Beware that, if $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ are $\kappa $-compact functors between $\kappa $-compactly generated $\infty $-categories, then the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_{+}$ need not be $\kappa $-compactly generated: in fact, it might not contain any $\kappa $-compact objects at all (Exercise 9.4.4.1). However, we will prove in §9.4.4 that it is $\kappa $-compactly generated provided that there exists some regular cardinal $\kappa _0 < \kappa $ such that $F_{-}$ and $F_{+}$ are $\kappa _0$-finitary (for a more general statement, see Theorem 9.4.4.4). In §9.4.5, we apply this result to establish many other stability properties of the collection of $\kappa $-compactly generated $\infty $-categories.
We will say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-accessible if it satisfies the $\infty $-categorical counterpart of Definition 9.4.0.1: that is, if it is $\kappa $-compactly generated and the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ is essentially small. We will say that an $\infty $-category $\operatorname{\mathcal{C}}$ is accessible if it is $\kappa $-accessible for some small regular cardinal $\kappa $. In §9.4.6, we collect some basic facts about accessible $\infty $-categories:
If an $\infty $-category $\operatorname{\mathcal{C}}$ is accessible, then it is $\lambda $-accessible for arbitrarily large values of $\lambda $ (Proposition 9.4.6.16). In particular, each of the full subcategories $\operatorname{\mathcal{C}}_{< \lambda }$ is essentially small (Corollary 9.4.6.7).
If $\operatorname{\mathcal{C}}$ is accessible, then every object $C \in \operatorname{\mathcal{C}}$ is $\lambda $-compact for some $\lambda $: that is, $\operatorname{\mathcal{C}}$ can be realized as the directed union $\bigcup _{\lambda } \operatorname{\mathcal{C}}_{< \lambda }$ (Remark 9.4.6.8). In particular, $\operatorname{\mathcal{C}}$ is locally small.
An essentially small $\infty $-category $\operatorname{\mathcal{C}}$ is accessible if and only if it is idempotent-complete (Proposition 9.4.6.21).
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between accessible $\infty $-categories. We say that the functor $F$ is accessible if it preserves small $\kappa $-filtered colimits for some small regular cardinal $\kappa $ (Definition 9.4.7.1). In §9.4.7 we show that, if this condition is satisfied, then we can choose a small regular cardinal $\lambda \geq \kappa $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\lambda $-accessible and $F$ is $\lambda $-compact: that is, $F$ can be realized as the $\operatorname{Ind}_{\lambda }$-extension of its restriction $F_{< \lambda }: \operatorname{\mathcal{C}}_{< \lambda } \rightarrow \operatorname{\mathcal{D}}_{< \lambda }$ (Proposition 9.4.7.9). In §9.4.8, we combine this observation with the results of §9.4.5 to show that the collection of accessible $\infty $-categories has robust closure properties. For example, if $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ is a small diagram of accessible $\infty $-categories in which the transition maps are accessible functors, then the limit $\varprojlim ( \operatorname{\mathcal{C}}_{\alpha } )$ is also accessible (Corollary 9.4.8.11).
Structure
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Subsection 9.4.1: Compactly Generated $\infty $-Categories
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Subsection 9.4.2: Compact Functors
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Subsection 9.4.3: Compact Generation of Oriented Fiber Products
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Subsection 9.4.4: Compact Generation of Homotopy Fiber Products
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Subsection 9.4.5: Compactly Generated Fibrations
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Subsection 9.4.6: Accessible $\infty $-Categories
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Subsection 9.4.7: Accessible Functors
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Subsection 9.4.8: Stability Properties of Accessible $\infty $-Categories