Here, we briefly review the theory of additive and stable \(\infty\)-categories. For more details, we refer to [Lur17], [BFN10] and [GGN15].
An \(\infty\)-category is called zero-pointed if it has an initial and a terminal object and if the unique morphism from the initial to the terminal object is an isomorphism. In this case, we call the initial/terminal object a zero object. A zero-pointed \(\infty\)-category is called semi-additive if it furthermore has finite products and finite coproducts and if the canonical morphism \(x\sqcup y \rightarrow x \times y\) is an isomorphism. In this case, we write the product/coproduct as \(x\oplus y\) and refer to it as a direct sum. A semi-additive \(\infty\)-category is called additive if furthermore the shear map \(( \pi_1, \nabla)\colon x\oplus x \rightarrow x \oplus x\) is an isomorphism, where \(\pi_1\colon x \oplus x \rightarrow x\) denotes the projection to the first factor (using that \(x \oplus x\) is a product) and \(\nabla\colon x \oplus x \rightarrow x\) is the fold map (using that \(x \oplus x\) is a coproduct). A functor between additive \(\infty\)-categories is called additive if it preserves finite coproducts. We denote the \(\infty\)-category of additive functors between two additive \(\infty\)-categories \(\mathcal A, \mathcal B\) by \(\mathrm{Fun}^{\sqcup}(\mathcal A,\mathcal B)\). This \(\infty\)-category is itself an additive \(\infty\)-category [GGN15, Cor. 2.9]. The notion of an additive \(\infty\)-category is a direct generalization of the ordinary \(1\)-categorical notion, and indeed an ordinary \(1\)-category is additive in the usual sense if and only if it(s nerve) is additive in the \(\infty\)-categorical sense. Conversely, a semi-additive \(\infty\)-category \(\mathcal C\) is additive if and only if its homotopy category \(h_1\mathcal C\) is additive as an ordinary \(1\)-category [GGN15, Prop. 2.8].
A zero-pointed \(\infty\)-category is called stable if it admits finite colimits and if any square is a pullback square if and only if it is a pushout square. A functor between stable \(\infty\)-categories is called exact if it preserves finite colimits. Given two stable \(\infty\)-categories \(\mathcal C, \mathcal D\), we denote the \(\infty\)-category of exact functors between them by \(\mathrm{Fun}^{\mathrm{ex}}(\mathcal C,\mathcal D)\). This \(\infty\)-category \(\mathrm{Fun}^{\mathrm{ex}}(\mathcal C,\mathcal D)\) is itself stable since it is a full subcategory of \(\mathrm{Fun}(\mathcal C, \mathcal D)\) (which is stable by [Lur17, Prop. 1.1.3.1]) that contains the zero object and is stable under forming fibers and cofibers, as a straightforward computation shows. In what follows, we will only consider idempotent complete stable categories.
We use the following notation:
\(\mathrm{add}\) for the full \(\infty\)-subcategory of \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) consisting of additive, idempotent complete, small \(\infty\)-categories.
\(\mathrm{st}\) for the full \(\infty\)-subcategory of \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) consisting of stable, idempotent complete, small \(\infty\)-categories.
Since stable \(\infty\)-categories are additive and exact functors preserve finite coproducts, there is a forgetful functor \(\mathrm{st}\rightarrow\mathrm{add}\).
All additive and stable \(\infty\)-categories will be implicitly assumed to be idempotent complete. In particular, we have defined \(\mathrm{add}\) and \(\mathrm{st}\) as full subcategories of \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) and \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\).
As in subsection 3.1 (in particular proposition 3.2.8), it will be useful to characterize small additive or stable \(\infty\)-categories in terms of projectively resp. compactly generated presentable \(\infty\)-categories.
Following definition 3.2.3, we use the following notations:
\({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) for the full subcategory of \(\mathrm{Pr}^\mathrm{L}\) on the stable, presentable \(\infty\)-categories and \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) for the full subcategory on the additive, presentable \(\infty\)-categories.
\(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) for the full subcategory of \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) on the stable presentable \(\infty\)-categories which are compactly generated as \(\infty\)-categories, and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\) for the full subcategory of \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) on the additive presentable \(\infty\)-categories which are projectively generated as \(\infty\)-categories.
The following hold.
The equivalence \(\mathcal P^{\Sigma}\colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) restricts to an equivalence between full subcategories \[\mathcal P^{\Sigma}\colon\mathrm{add}\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}.\] Its inverse is \((-)^{\mathrm{cp}}\) which takes a projectively generated additive presentable \(\infty\)-category \(\mathcal C\) to its full subcategory \(\mathcal C^{\mathrm{cp}}\) on the compact-projective objects.
The equivalence \(\operatorname{Ind}\colon \mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) restricts to an equivalence between full subcategories \[\operatorname{Ind}\colon\mathrm{st}\xrightarrow{\simeq}\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}.\] Its inverse is \((-)^{\mathrm{c}}\) which takes a compactly generated stable presentable \(\infty\)-category \(\mathcal C\) to its full subcategory \(\mathcal C^{\mathrm{c}}\) on the compact objects.
Proof.
We will prove part ([004S]), the proof of part ([004T]) is entirely analogous and can for example be found in [BGT13, Lem. 2.20]. Recall from proposition 3.2.8 that \(\mathcal P^{\Sigma}\colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) is an equivalence, whose inverse is \((-)^{\mathrm{cp}}\). To prove statement (1), it therefore suffices to show that the essential image of the composite \(\mathrm{add}\hookrightarrow \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\simeq \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) is the full subcategory \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\).
If \(\mathcal C\) is a small additive \(\infty\)-category, then \(\mathcal P^{\Sigma}(\mathcal C) \simeq \mathrm{Fun}^{\sqcup}(\mathcal C^{\mathrm{op}}, \mathcal S)\) is additive by [GGN15, Cor. 2.9]. On the other hand, if \(\mathcal D\) is any projectively generated additive presentable category, then the full subcategory on its compact-projective objects is closed under finite coproducts and hence is again additive. Therefore, \(\mathcal D\) is in the image of \(\mathrm{add}\hookrightarrow \mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\). ◻
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).
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 \(\mathrm{Mod}_{\mathrm{Sp}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\).
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.
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 \(\operatorname{Ind}\colon \mathrm{st}\xrightarrow{\simeq} \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\).
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).
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.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2