3.3.2 Symmetric monoidal structure[004V]
The universal example of an stable presentable \(\infty\)-category is the \(\infty\)-category \(\mathrm{Sp}\) of spectra. Likewise, the universal example of an additive presentable \(\infty\)-category is the \(\infty\)-category \(\mathrm{Sp}_{\geq 0}\) of connective spectra, equivalent to the \(\infty\)-category \(\mathrm{Grp}_{\mathbb{E}_{\infty}}(\mathcal S)\) of grouplike \(\mathbb{E}_{\infty}\)-spaces, see [GGN15]. Both \(\mathrm{Sp}\) and \(\mathrm{Sp}_{\geq 0}\) are idempotent algebras in \(\mathrm{Pr}^\mathrm{L}\), i.e. commutative algebras \(A\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) so that the multiplication \(A\otimes A \rightarrow A\) is an isomorphism. It is shown in [Lur17, Prop. 4.8.2.18] and [GGN15, Cor. 4.8] that the full subcategories \(\mathrm{Mod}_{\mathrm{Sp}}(\mathrm{Pr}^\mathrm{L})\) and \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^\mathrm{L})\) of \(\mathrm{Pr}^\mathrm{L}\) are equivalent to \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) and \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\), respectively. As categories of modules of a commutative algebra, this induces symmetric monoidal structures on \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) and \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) respectively by proposition 3.1.8.([0032]).
By [Lur17, Prop. 1.4.3.7], the \(\infty\)-category \(\mathrm{Sp}\) is compactly generated (by the single object \(\mathbb{S}\), the sphere spectrum). It follows from lemma 3.2.9 that the compact objects in \(\mathrm{Sp}\) are finite spectra, i.e. finite colimits of the sphere spectrum (note that a retract of a finite spectrum is again finite). However, \(\mathrm{Sp}\) is not projectively generated (its only projective object is the zero spectrum, cf. [Lur17, Rem. 7.2.2.5]). On the other hand, \(\mathrm{Sp}_{\geq 0}\) is projectively generated by the sphere spectrum [Lur17, Cor. 7.1.4.13]. It therefore follows from lemma 3.2.9 that the compact-projective objects in \(\mathrm{Sp}_{\geq 0}\) are finite sums of the sphere spectrum (note that a retract of a finite sum of sphere spectra is again a finite sum of sphere spectra).
[004W]
Lemma 3.3.5.
The following hold.
The equivalence \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^\mathrm{L}) \xrightarrow{\simeq} {\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) restricts to an equivalence between the subcategories \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) \xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\).
The equivalence \(\mathrm{Mod}_{\mathrm{Sp}}(\mathrm{Pr}^\mathrm{L})\xrightarrow{\simeq} {\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) restricts to an equivalence between the subcategories \(\mathrm{Mod}_{\mathrm{Sp}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\).
[004X]
Proof.
We prove the first statement, the second is analogous. The \(\infty\)-category \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) may be understood as the subcategory of \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^\mathrm{L})\) on those presentable \(\mathrm{Sp}_{\geq 0}\)-module \(\infty\)-categories \(\mathcal C\) whose underlying \(\infty\)-category is projectively generated and for which the action functor \(\mathrm{Sp}_{\geq 0}\otimes \mathcal C\rightarrow\mathcal C\) preserves compact-projectives, and those cocontinuous \(\mathrm{Sp}_{\geq 0}\)-module functors \(\mathcal C\rightarrow\mathcal D\) for which the underlying functor preserves compact projectives. In particular, the equivalence \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^\mathrm{L}) \rightarrow
{\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) restricts to a fully faithful functor \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})
\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\). It therefore suffices to verify that for an additive presentable \(\infty\)-category \(\mathcal C\), the action \(\mathrm{Sp}_{\geq 0} \times \mathcal C\rightarrow\mathcal C\) sends a pair of compact-projective objects \((a, b) \in \mathrm{Sp}_{\geq 0}^{\mathrm{cp}} \times
\mathcal C^{\mathrm{cp}}\) to a compact-projective of \(\mathcal C\). This follows since any compact projective in \(\mathrm{Sp}_{\geq 0}\) is generated under finite coproducts and retracts by the unit object \(\mathbb{S}\); see lemma 3.2.9. ◻
Using the theory of commutative algebras in presentable categories, we immediately obtain the following stable and additive analogues of the first half of proposition 3.2.10 concerning symmetric monoidal structures on subcategories of \(\mathrm{Pr}^\mathrm{L}\). The passage from \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\simeq \mathrm{add}\) to \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}
\simeq \mathrm{st}\) will be treated in the next section.
[004Y]
Corollary 3.3.6.
The following hold.
The symmetric monoidal structure of \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) restricts to a presentably symmetric monoidal structure on the subcategory \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\), which induces a presentably symmetric monoidal structure on the \(\infty\)-category \(\mathrm{add}\) via the equivalence \(\mathcal P^{\Sigma}\colon \mathrm{add}\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\).
The symmetric monoidal structure of \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) restricts to a presentably symmetric monoidal structure on the subcategory \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\rightarrow{\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\), which induces a presentably symmetric monoidal structure on the \(\infty\)-category \(\mathrm{st}\) via the equivalence \(\operatorname{Ind}\colon \mathrm{st}\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\).
[004Z]
Proof.
Since \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\) and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) are module categories by lemma 3.3.5, they inherit via proposition 3.1.8.([0032]) presentably symmetric monoidal structures from the presentably symmetric monoidal categories \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) and \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) (see proposition 3.2.10), respectively. Symmetric monoidality of the functors \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\rightarrow{\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\rightarrow{\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) follows from symmetric monoidality of \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\rightarrow\mathrm{Pr}^\mathrm{L}\) and \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^\mathrm{L}\). ◻
Tracing through the proof, the symmetric monoidal structures on \(\mathrm{add}\) respectively \(\mathrm{st}\) may be characterized as follows (c.f. [BFN10, Prop. 4.4]): For \(\mathcal C, \mathcal D\in \mathrm{add}\) the tensor product \(\mathcal C\otimes \mathcal D\) is equipped with a functor \(\mathcal C\times \mathcal D\rightarrow\mathcal C\otimes \mathcal D\), additive in both variables, and satisfies the universal property that for any \(\mathcal E\in \mathrm{add}\) the induced functor \[\mathrm{Fun}^{\mathrm{add}}(\mathcal C\otimes \mathcal D, \mathcal E) \rightarrow\mathrm{Fun}^{\mathrm{add}\times\mathrm{add}}(\mathcal C\times \mathcal D, \mathcal E)\] is an equivalence, where \(\mathrm{Fun}^{\mathrm{add}\times\mathrm{add}}(\mathcal C\times \mathcal D, \mathcal E)\) denotes the full subcategory of \(\mathrm{Fun}(\mathcal C\times \mathcal D, \mathcal E)\) on the functors which are additive in both variables (i.e. which preserve finite coproducts separately in either variable).
For \(\mathcal C, \mathcal D\in \mathrm{st}\), the tensor product \(\mathcal C\otimes \mathcal D\) is characterized analogously in terms of functors \(\mathcal C\times \mathcal D\rightarrow\mathcal E\) which are exact in both variables (i.e. which preserve finite colimits separately in both variables).
[0050]
Warning 3.3.7.
As in warning 3.3.2, the \(\infty\)-categories \(\mathrm{add}\) and \(\mathrm{st}\) are the \(\infty\)-categories of additive, resp. stable, idempotent complete \(\infty\)-categories. In particular, the tensor product of additive/stable idempotent complete \(\infty\)-categories we consider here is automatically idempotent complete.