ScalingStacks

A.7 Presentable \(\infty\)-categories[00IN]

Many (\(\infty\)-)categories of lasting interest are not small, but are nevertheless “controlled by small data” – namely, they are presentable. By definition, an \(\infty\)-category \(\mathcal C\) is presentable if it admits all small colimits and moreover there exists some regular cardinal \(\kappa\) such that \(\mathcal C\) is the completion of its full subcategory \(\mathcal C^\kappa \subseteq \mathcal C\) of \(\kappa\)-compact objects under \(\kappa\)-filtered colimits. For this we recall that a \(\kappa\)-filtered colimit means a colimit indexed by a \(\kappa\)-filtered category, i.e. an \(\infty\)-category, in which every diagram of cardinality \(<\kappa\) has a cocone, and an object is called \(\kappa\)-compact if the associated representable functor preserves \(\kappa\)-filtered colimits. If we can take \(\kappa\) to be the cardinality \(\omega\) of the natural numbers, we say that \(\mathcal C\) is compactly generated (as \(\omega\)-compact objects are generally just called “compact objects”).

An extremely convenient feature of presentable \(\infty\)-categories is their adjoint functor theorem [Lur09, Cor. 5.5.2.9]: a functor between presentable \(\infty\)-categories is a left adjoint if and only if it preserves small colimits, and it is a right adjoint if and only if it is accessible (i.e. preserves \(\kappa\)-filtered colimits for some \(\kappa\)) and preserves small limits. Presentable \(\infty\)-categories naturally define two subcategories \[{\Pr}^L \subset \widehat{\mathrm{Cat}}_\infty \supset {\Pr}^R\] of the huge \(\infty\)-category of large \(\infty\)-categories, in which the morphisms are the left (resp. right) adjoint functors. Evidently, passing to adjoints defines an equivalence \(\Pr^L \simeq (\Pr^R)^\mathrm{op}\). These actually define \((\infty,2)\)-categories (by taking all natural transformations as 2-morphisms), and we write \(\mathrm{Fun}^L(-,-)\) and \(\mathrm{Fun}^R(-,-)\) for their respective hom-\((\infty,1)\)-categories.

An accessible localization is by definition a reflective localization among presentable \(\infty\)-categories. The left adjoint of an accessible localization is a localization not just in \(\widehat{\mathrm{Cat}}_\infty\) but also in \(\Pr^L\) [Lur09, Prop. 5.5.4.20]. Moreover, given any accessible localization the left adjoint \(L\) witnesses \(\mathcal D\) as the localization \(\mathcal C[S^{-1}]\) for some small set \(S\) of morphisms in \(\mathcal C\), and hence \(R\) is the fully faithful inclusion of the subcategory of \(S\)-local objects [Lur09, Prop. 5.5.4.1].

Presentable \(\infty\)-categories admit presentations by generators and relations, in the following sense. First of all, for any small \(\infty\)-category \(\mathcal C\in \mathrm{Cat}_\infty\), its \(\infty\)-category \(\mathcal P(\mathcal C)\) of presheaves is presentable. This is the free presentable \(\infty\)-category on \(\mathcal C\): for any \(\mathcal D\in \Pr^L\), restriction along the Yoneda embedding defines an equivalence \(\mathrm{Fun}(\mathcal C,\mathcal D) \xleftarrow{\sim} \mathrm{Fun}^L(\mathcal P(\mathcal C),\mathcal D)\) [Lur09, Thm. 5.1.5.6]. And then, any presentable \(\infty\)-category is an accessible localization of \(\mathcal P(\mathcal C)\) for some \(\mathcal C\in \mathrm{Cat}_\infty\) [Lur09, Thm. 5.5.1.1].

There exists a symmetric monoidal structure on \(\Pr^L\), which is characterized by the fact that morphisms \(\mathcal C\otimes \mathcal D\rightarrow\mathcal E\) in \(\Pr^L\) (i.e. left adjoint functors) are equivalent to functors \(\mathcal C\times \mathcal D\rightarrow\mathcal E\) that are bicocontinuous (i.e. cocontinuous (or equivalently, left adjoints) separately in each variable), whose unit object is \(\mathcal S\simeq \mathcal P({\sf pt}) \in \Pr^L\). A presentably (symmetric) monoidal \(\infty\)-category is a (resp. commutative) algebra object in \((\Pr^L,\otimes)\), i.e. a presentable \(\infty\)-category equipped with a (resp. symmetric) monoidal structure that is cocontinuous separately in each variable.⁵² Most (symmetric) monoidal presentable \(\infty\)-categories of lasting interest (e.g. \(\mathrm{Cat}_{(\infty, {n})}\) (and in particular \(\mathcal S\) and \(\mathrm{Cat}_\infty\)) and \(\mathrm{Sp}\)) are presentably (resp. symmetric) monoidal.