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9.5 Presentable $\infty $-Categories

Recall that a complete lattice is a partially ordered set $(Q, \leq )$ with the property that every subset $T \subseteq Q$ has a greatest lower bound $\mathrm{inf}(T)$. This condition is self-dual:

Proposition 9.5.0.1. Let $(Q, \leq )$ be a complete lattice. Then every subset $S \subseteq Q$ has a least upper bound $\mathrm{sup}(S)$.

Proof. Let $T \subseteq Q$ be the collection of all upper bounds for $S$: that is, the collection of all elements $t \in Q$ satisfying $s \leq t$ for each $s \in S$. We wish to show that $T$ has a smallest element. Our assumption that $(Q, \leq )$ is a complete lattice guarantees that $T$ has a greatest lower bound $t = \mathrm{inf}(T)$. Note that every element $s \in S$ is a lower bound for $T$, and therefore satisfies $s \leq t$. It follows that $t \in T$ is a least upper bound for $S$. $\square$

Proposition 9.5.0.1 has an $\infty $-categorical counterpart. Assume for simplicity that the partially ordered set $(Q, \leq )$ is small. Then $Q$ is a complete lattice if and only if the $\infty $-category $\operatorname{N}_{\bullet }(Q)$ is complete. Proposition 9.5.0.1 asserts that, if this condition is satisfied, then the $\infty $-category $\operatorname{N}_{\bullet }(Q)$ is also cocomplete. In §9.5.1 we show that, modulo set-theoretic issues, this phenomenon is quite general: an accessible $\infty $-category $\operatorname{\mathcal{C}}$ is complete if and only if it is cocomplete (Theorem 9.5.1.1). If these conditions are satisfied, we say that the $\infty $-category $\operatorname{\mathcal{C}}$ is presentable (Definition 9.5.1.2).

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, we have the following converse:

  • 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 §9.5.2 (Theorem 9.5.2.1). The proofs will use characterizations of representable and corepresentable functors on presentable $\infty $-categories, which we discuss in §9.5.1.

The collection of presentable $\infty $-categories can be organized into a (large) $\infty $-category in (at least) two different ways. We let $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ denote the $\infty $-category whose objects are presentable $\infty $-categories and whose morphisms are left adjoint functors (that is, functors which preserve small colimits), and we let $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ denote the $\infty $-category whose objects are presentable $\infty $-categories and whose morphisms are right adjoint functors (that is, accessible functors which preserve small limits); see Construction 9.5.3.1 and Variant 9.5.3.2. In §9.5.3, we show that the $\infty $-categories $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ and $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ are canonically anti-equivalent: more precisely, there is an equivalence of $(\operatorname{\mathcal{QC}}^{\operatorname{LPr}})^{\operatorname{op}}$ with $(\operatorname{\mathcal{QC}}^{\operatorname{RPr}})$ which is the identity on objects, and carries each colimit-preserving functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ to its right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ (see Corollary 9.5.3.9 and Remark 9.5.3.10).

In §9.5.4, we show that the collection of presentable $\infty $-categories is closed under various categorical constructions. In particular, we show that both $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ and $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ admit small limits which are computed at the level of the underlying $\infty $-categories (Proposition 9.5.4.12), and small colimits which which are more subtle to describe (Corollary 9.5.4.13 and Remark 9.5.4.15).

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). In §9.5.5, we show that properties of $\operatorname{\mathcal{C}}$ are reflected in the behavior 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 9.5.5.3).

  • 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 9.5.5.14).

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a Bousfield localization functor if it admits a fully faithful right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ (Definition 9.5.6.1). Bousfield localizations are useful for understanding the architecture of presentable $\infty $-categories:

  • Let $\operatorname{\mathcal{D}}$ be a fixed presentable $\infty $-category. In §9.5.6, we observe that $\operatorname{\mathcal{D}}$ can be realized as the Bousfield localization of a presentable $\infty $-category $\operatorname{\mathcal{C}}$ having a particularly simple form: we can arrange that $\operatorname{\mathcal{C}}= \operatorname{Fun}( \operatorname{\mathcal{K}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is the cocompletion of a small $\infty $-category $\operatorname{\mathcal{K}}$ (Remark 9.5.6.6).

  • Let $\operatorname{\mathcal{C}}$ be a fixed presentable $\infty $-category. In §9.5.7, we show that every small collection $W$ of morphisms of $\operatorname{\mathcal{C}}$ determines a Bousfield localization of $\operatorname{\mathcal{C}}$, given by the full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ spanned by the $W$-local objects (Theorem 9.5.7.1). Moreover, every Bousfield localization of $\operatorname{\mathcal{C}}$ can be obtained in this way (Proposition 9.5.7.5) and satisfies a universal mapping property (Proposition 9.5.7.2).

Remark 9.5.0.2. The theory of presentable $\infty $-categories has a classical counterpart, which 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.

Structure

  • Subsection 9.5.1: Presentable $\infty $-Categories
  • Subsection 9.5.2: The Adjoint Functor Theorem
  • Subsection 9.5.3: The $\infty $-Category of Presentable $\infty $-Categories
  • Subsection 9.5.4: Closure Properties of Presentable $\infty $-Categories
  • Subsection 9.5.5: Cocontinuous Functors
  • Subsection 9.5.6: Bousfield Localization
  • Subsection 9.5.7: Existence of Bousfield Localizations