An object \(c\) of an \(\infty\)-category \(\mathcal C\) with geometric realizations (i.e. colimits indexed by \(\Delta^{\mathrm{op}}\)) is called projective [Lur09, Def. 5.5.8.18] if \(\mathrm{Hom}_{\mathcal C}(c,-) \colon \mathcal C\rightarrow\mathcal S\) preserves geometric realizations.
An object \(c\) of an \(\infty\)-category \(\mathcal C\) with sifted colimits [Lur09, Def. 5.5.8.1] is called compact-projective if \(\mathrm{Hom}_{\mathcal C}(c,-) \colon \mathcal C\rightarrow\mathcal S\) preserves sifted colimits [Lur09, Rem. 5.5.8.20].
3.2.1 Compact and projectively generated \(\infty\)-categories[003F]
Many presentable \(\infty\)-categories are generated by \(\omega\)-compact objects, where \(\omega\) is the cardinality of the natural numbers. We will henceforth refer to \(\omega\)-filtered diagrams simply as filtered diagrams and to \(\omega\)-compact objects as compact objects. Hence, an object \(c\) of an \(\infty\)-category \(\mathcal C\) with filtered colimits is compact if \(\mathrm{Hom}_{\mathcal C}(c,-)\colon \mathcal C\rightarrow\mathcal S\) preserves filtered colimits.
Similarly, we recall the following definitions:
Since sifted colimits are generated by filtered colimits and geometric realizations [Lur09, Cor. 5.5.8.17], an object is compact-projective if and only if it is compact and projective.
We refer to definition 3.6.1 and example 3.6.3 for a comparison with classical notions of compactness and projectivity in ordinary (abelian) categories.
We will use the following terminology.
A compactly generated \(\infty\)-category is an \(\infty\)-category with small colimits, for which there exists a small set of compact objects which generates \(\mathcal C\) under small colimits.
A projectively generated \(\infty\)-category is an \(\infty\)-category with small colimits, for which there exists a small set of compact-projective objects which generates \(\mathcal C\) under small colimits.
Let \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) (resp. \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\)) denote the subcategory of \(\mathrm{Pr}^\mathrm{L}\) on the compactly (resp. projectively) generated presentable \(\infty\)-categories and the cocontinuous functors which preserve compact (resp. compact-projective) objects.
Consider an adjunction between \(\infty\)-categories 
If \(\mathcal C\) is compactly generated and \(\mathcal D\) has filtered colimits, then the left adjoint \(L\) preserves compact objects if and only if the right adjoint \(R\) preserves filtered colimits.
If \(\mathcal C\) is projectively generated and \(\mathcal D\) has sifted colimits, then the left adjoint \(L\) preserves compact-projective objects if and only if the right adjoint \(R\) preserves sifted colimits.
Proof.
We prove the first statement; the proof of the second statement is analogous. Suppose \(R\) preserves filtered colimits and \(c\in \mathcal C\) is compact. Then, for every filtered diagram \(d\colon I \rightarrow\mathcal D\) we have \[\begin{gathered} \mathrm{Hom}_{\mathcal C}(Lc, \mathrm{colim}_i d_i) \simeq \mathrm{Hom}_{\mathcal D}(c, R\mathrm{colim}_i d_i) \simeq \mathrm{Hom}_{\mathcal D}(c,\mathrm{colim}_i Rd_i) \\ \hspace{3cm}\simeq \mathrm{colim}_i \mathrm{Hom}_{\mathcal D}(c, Rd_i) \simeq \mathrm{colim}_i \mathrm{Hom}_{\mathcal D}(Lc, d_i) \end{gathered}\] and hence \(Lc\) is compact. Conversely, suppose that \(L\) preserves compact objects. It follows that for a compact object \(c\in \mathcal C\) and a filtered diagram \(d\colon I \rightarrow\mathcal D\), we have \[\begin{gathered} \mathrm{Hom}_{\mathcal C}(c, R(\mathrm{colim}_i d_i))\simeq \mathrm{Hom}_{\mathcal D}(Lc, \mathrm{colim}_i d_i) \simeq \mathrm{colim}_i \mathrm{Hom}_{\mathcal D}(Lc, d_i) \\ \hspace{3cm} \simeq \mathrm{colim}_i \mathrm{Hom}_{\mathcal C}(c, Rd_i)\simeq \mathrm{Hom}_{\mathcal C}(c, \mathrm{colim}_i Rd_i). \end{gathered}\] Since \(\mathcal C\) is compactly generated, every object in \(\mathcal C\) is a small colimit of compact objects, and thus \(\mathrm{colim}_i Rd_i \simeq R\mathrm{colim}_i d_i\). ◻
The \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) is a subcategory of \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\).
Proof.
A projectively generated presentable \(\infty\)-category is also compactly generated since compact-projective objects are in particular compact. Thus, we only need to show that a left adjoint functor \(L\) between projectively generated presentable \(\infty\)-categories, which preserves compact-projective objects, also preserves compact objects. Indeed, by lemma 3.2.5.([003M]), \(L\) has a right adjoint which preserves sifted colimits and hence preserves filtered colimits. Applying the reverse direction of lemma 3.2.5.([003L]) now shows that \(L\) preserves compact objects. ◻
Let \(\mathcal C\) be a cocomplete \(\infty\)-category.
The full subcategory \(\mathcal C^{\mathrm{c}}\) of compact objects is closed under retracts and finite colimits [Lur09, Cor. 5.3.4.15 and Rem. 5.3.4.16] and hence yields an object \(\mathcal C^\mathrm{c}\in \mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\). This defines a functor \((-)^{\mathrm{c}} \colon \mathrm{Pr}^{\mathrm{L},\mathrm{c}}\rightarrow\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\).
The full subcategory \(\mathcal C^{\mathrm{cp}}\) of compact-projective objects is closed under retracts and finite coproducts [Lur09, Rem. 5.5.8.19] and hence defines an object \(\mathcal C^{\mathrm{cp}} \in \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\). This defines a functor \((-)^{\mathrm{cp}} \colon \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\).
In proposition 3.2.8 we will show that these functors are in fact equivalences.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2