3.1.2 Presentable \(\infty\)-categories[002S]
In general, the presheaf category \(\mathcal P(\mathcal C)\) of a small \(\infty\)-category \(\mathcal C\) is a large category. The sense in which \(\mathcal P(\mathcal C)\) is nevertheless still controlled by a small amount of data is formalized by the notion of a presentable \(\infty\)-category. We recall the definition and some basic facts from [Lur09, Sec. 5.4 and 5.5].
[002T]
Definition 3.1.2.
Let \(\mathcal D\) be a (possibly large) \(\infty\)-category.
Let \(\mathcal K\) be a collection of \(\infty\)-categories and \(S\) a small set of objects of \(\mathcal D\). Then \(\mathcal D\) is generated by \(S\) under \(\mathcal K\)-indexed colimits if \(\mathcal D\) has all colimits indexed by categories in \(\mathcal K\) and is the smallest full subcategory of \(\mathcal D\) which contains the objects in \(S\) and is closed under \(\mathcal K\)-indexed colimits.
Let \(\kappa\) be an infinite regular cardinal and assume \(\mathcal D\) admits \(\kappa\)-filtered colimits. Then an object \(d\in \mathcal D\) is called \(\kappa\)-compact if the functor \(\mathrm{Hom}_{\mathcal D}(d,-)\colon \mathcal D\rightarrow\mathcal S\) preserves \(\kappa\)-filtered colimits.
The \(\infty\)-category \(\mathcal D\) is called accessible if it is locally small and there exists a regular cardinal \(\kappa\) and a small set \(S\) of \(\kappa\)-compact objects in \(\mathcal C\) that generates \(\mathcal C\) under \(\kappa\)-filtered colimits.
The \(\infty\)-category \(\mathcal D\) is called presentable if it has all small colimits and is accessible.
[002U]
Example 3.1.3.
By Simpson’s characterisation of presentable \(\infty\)-categories as localizations of presheaf categories, [Lur09, Thm. 5.5.1.1], we obtain presentability of \(\mathcal P(\mathcal C)\) for any small \(\infty\)-category \(\mathcal C\) [Lur09, Ex. 5.4.2.7, Ex. 5.5.1.8.], and more generally the presentability of \(\mathrm{Fun}(\mathcal C, \mathcal D)\) for a small \(\infty\)-category \(\mathcal C\) and a presentable \(\infty\)-category \(\mathcal D\).
A main application of the notion of presentable \(\infty\)-category is the adjoint functor theorem:
[002V]
Proposition 3.1.4. ([Lur09, Cor. 5.5.2.9 and Rem. 5.5.2.10]).
A functor from a presentable \(\infty\)-category to a locally small \(\infty\)-category preserves small colimits if and only if it is a left adjoint.
[002W]
Notation 3.1.5.
We denote by
\(\mathrm{Pr}^\mathrm{L}\) the \(\infty\)-category of presentable \(\infty\)-categories and small colimit preserving functors, i.e left adjoint functors by the adjoint functor theorem.
\(\mathrm{Fun^L}(\mathcal C,\mathcal D)\), for \(\mathcal C, \mathcal D\in \mathrm{Pr}^\mathrm{L}\), the full subcategory of \(\mathrm{Fun}(\mathcal C, \mathcal D)\) of left adjoint (equivalently cocontinuous) functors. Dually, full subcategories of right adjoint functors will be denoted \(\mathrm{Fun^R}(-,-)\).
When denoting an adjunction
between \(\infty\)-categories, we use the convention that the top arrow is the left adjoint and the bottom arrow the right adjoint.
For more details, we refer to [Lur09, § 5.5.3] and [Cis19, § 7].