Definition 9.5.0.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is presentable if it is both cocomplete and accessible (Definition 9.4.6.10).
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9.5 Presentable $\infty $-Categories
Recall that an $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete if every small diagram $K \rightarrow \operatorname{\mathcal{C}}$ admits a colimit (Definition 7.6.6.1). Such $\infty $-categories are rarely small: any small $\infty $-category which is cocomplete is equivalent to the nerve of a partially ordered set (Proposition 7.1.2.15). In this section, we study cocomplete $\infty $-categories $\operatorname{\mathcal{C}}$ which are nevertheless controlled by a small subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$.
The classical counterpart of Definition 9.5.0.1 was introduced by Gabriel and Ulmer in [MR327863] (see [MR1294136] for a more expository account). Beware that our terminology is nonstandard: most authors use the term locally presentable in place of presentable.
Recall that an $\infty $-category $\operatorname{\mathcal{C}}$ is accessible if and only if it is an $\operatorname{Ind}_{\kappa }$-completion of a small idempotent-complete $\infty $-category $\operatorname{\mathcal{C}}_{0}$, for some small regular cardinal $\kappa $ (in which case we can identify $\operatorname{\mathcal{C}}_0$ with the full subcategory of $\operatorname{\mathcal{C}}$ spanned by its $\kappa $-compact objects). Our first goal in this section is to show that, in this case, properties of $\operatorname{\mathcal{C}}$ can be translated to properties of $\operatorname{\mathcal{C}}_0$:
The $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete (and therefore presentable) if and only if the $\infty $-category $\operatorname{\mathcal{C}}_0$ is $\kappa $-cocomplete (Proposition 9.5.1.7).
Assume that $\operatorname{\mathcal{C}}$ is cocomplete, and let $\operatorname{\mathcal{D}}$ be another cocomplete $\infty $-category. Then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is cocontinuous if and only if arises as the $\operatorname{Ind}_{\kappa }$-extension of a $\kappa $-cocontinuous functor $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$ (Corollary 9.5.2.6).
Both of these results are special cases of more general facts concerning $(\kappa ,\lambda )$-compactly generated $\infty $-categories for $\kappa \trianglelefteq \lambda $, which we establish in §9.5.1 and §9.5.2, respectively.
In §9.5.3, we show that the collection of presentable $\infty $-categories is closed under various categorical constructions:
If $\operatorname{\mathcal{C}}$ is a presentable $\infty $-category and $f: K \rightarrow \operatorname{\mathcal{C}}$ is a small diagram, then the slice and coslice $\infty $-categories $\operatorname{\mathcal{C}}_{/f}$ and $\operatorname{\mathcal{C}}_{f/}$ are presentable (Proposition 9.5.3.1).
If $\operatorname{\mathcal{C}}$ is a presentable $\infty $-category and $K$ is a small simplicial set, then the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is presentable (Proposition 9.5.3.2).
If $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ is a small collection of presentable $\infty $-categories, then the product $\operatorname{\mathcal{C}}= \prod _{i \in I} \operatorname{\mathcal{C}}_ i$ is presentable (Example 9.5.3.7). More generally, the limit of any small diagram of presentable $\infty $-categories is presentable, provided that the transition functors are cocontinuous (Proposition 9.5.3.6).
According to Definition 9.5.0.1, an $\infty $-category $\operatorname{\mathcal{C}}$ is presentable if it is both accessible and cocomplete. Both of these conditions pertain to the behavior (and existence) of colimits in the $\infty $-category $\operatorname{\mathcal{C}}$. Perhaps unexpectedly, they have nontrivial consequences concerning limits in $\operatorname{\mathcal{C}}$. In §9.5.4, we show that every presentable $\infty $-category $\operatorname{\mathcal{C}}$ is complete: that is, every small diagram in $\operatorname{\mathcal{C}}$ admits a limit (Theorem 9.5.4.2).
Recall that, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, then the functor $F$ preserves all colimit diagrams which exist in $\operatorname{\mathcal{C}}$ (and the functor $G$ preserves all limit diagrams which exist in $\operatorname{\mathcal{D}}$). If the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are presentable, these observations admit converses
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint if and only if it preserves small colimits.
A functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ admits a left adjoint if and only if it is accessible and preserves small limits.
We prove both of these results in §9.5.6 (Theorem 9.5.6.1). The proofs will use characterizations of representable and corepresentable functors on presentable $\infty $-categories, which we discuss in §9.5.5.
Structure
- Subsection 9.5.1: Presentable $\infty $-Categories
- Subsection 9.5.2: Cocontinuous Functors
- Subsection 9.5.3: Stability Properties of Presentable $\infty $-Categories
- Subsection 9.5.4: Limits in Presentable $\infty $-Categories
- Subsection 9.5.5: Representable and Corepresentable Functors
- Subsection 9.5.6: The Adjoint Functor Theorem