4.2 Graded linear \(\infty\)-categories[007M]
To incorporate \(\mathbb{K}\)-linearity for \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\) into the setup, we could define small \(\mathbb{K}\)-linear \(\infty\)-categories as small \(\infty\)-categories enriched in the presentably symmetric monoidal \(\infty\)-category \(\mathrm{Mod}_{\mathbb{K}}\) from notation 3.5.2. Due to proposition 4.1.7 and remark 4.1.9, it is technically easier to work with presentably enriched \(\infty\)-categories instead, as these can be expressed purely in the language of module categories. Our ‘presentable \(\mathbb{K}\)-linear’ terminology is justified by remark 4.2.3.
4.2.1 \(\mathbb{K}\)-linear \(\infty\)-categories[007N]
We start with some definitions which are crucial throughout the paper.
[007P]
Definition 4.2.1.
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define
the \(\infty\)-category \({\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{add}_{\mathbb{K}}}}}\) of additive presentable \(\mathbb{K}\)-linear \(\infty\)-categories as \[{\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{add}_{\mathbb{K}}}}}:= \mathrm{Mod}_{\mathrm{Mod}^{\geq 0}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L});\]
the \(\infty\)-category \(\mathrm{add}_{\mathbb{K}}\) of small additive, idempotent-complete \(\mathbb{K}\)-linear \(\infty\)-categories as \[\mathrm{add}_{\mathbb{K}}:=\mathrm{Mod}_{\mathrm{CProj}_{\mathbb{K}}}(\mathrm{add}).\]
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define
the \(\infty\)-category \({\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{st}_{\mathbb{K}}}}}\) of stable presentable \(\mathbb{K}\)-linear \(\infty\)-categories as \[{\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{st}_{\mathbb{K}}}}}:= \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L});\]
the \(\infty\)-category \(\mathrm{st}_{\mathbb{K}}\) of small stable, idempotent-complete \(\mathbb{K}\)-linear \(\infty\)-categories to be \[\mathrm{st}_{\mathbb{K}}:= \mathrm{Mod}_{\mathrm{Perf}_{\mathbb{K}}}(\mathrm{st}).\]
The following justifies the terminology ‘stable/additive presentable \(\mathbb{K}\)-linear’ in definition 4.2.1.
[007S]
Observation 4.2.4.
Since \(\mathrm{Mod}_{\mathbb{K}}\) is stable and \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0}\) is additive, we obtain the following equivalences from proposition 3.1.8.([0034]): \[\begin{aligned}
{\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{add}_{\mathbb{K}}}}}&:= \mathrm{Mod}_{\mathrm{Mod}^{\geq 0}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L})
\simeq \mathrm{Mod}_{\mathrm{Mod}^{\geq 0}_{\mathbb{K}}}({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}})\\
{\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{st}_{\mathbb{K}}}}}& := \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L})\simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}})
\end{aligned}\] In particular, any presentably \(\mathrm{Mod}_{\mathbb{K}}\)-enriched \(\infty\)-category is automatically stable, and any presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0}\)-enriched \(\infty\)-category is automatically additive. Combining proposition 3.1.8.([0034]) with the equivalences from §3.1 and subsection 3.3, we obtain the analogous characterizations of their small variants: \[\begin{aligned}
\mathrm{add}_{\mathbb{K}}& :=\mathrm{Mod}_{\mathrm{CProj}_{\mathbb{K}}}(\mathrm{add}) \simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0}}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}) \simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) \simeq \mathrm{Mod}_{\mathrm{CProj}_{\mathbb{K}}}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}})
\\
\mathrm{st}_{\mathbb{K}}&:=\mathrm{Mod}_{\mathrm{Perf}_{\mathbb{K}}}(\mathrm{st})\simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}) \simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \simeq \mathrm{Mod}_{\mathrm{Perf}_{\mathbb{K}}}(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}})
\end{aligned}\]
The next remark covers our main case of interest and connects to the framework from section 2.
[007U]
Observation 4.2.6.
As \(\infty\)-categories of modules of commutative algebras in presentably symmetric monoidal \(\infty\)-categories, both \(\mathrm{add}_{\mathbb{K}}\) and \(\mathrm{st}_{\mathbb{K}}\) are presentably symmetric monoidal. The symmetric monoidal structure on \(\mathrm{add}_{\mathbb{K}}\) can be characterized as follows: for \(\mathcal C, \mathcal D\in \mathrm{add}_{\mathbb{K}}\), there is a functor \(\mathcal C\times \mathcal D\rightarrow\mathcal C\otimes \mathcal D\) which is additive and \(\mathbb{K}\)-linear in either variable, and which for all \(\mathcal E\in \mathrm{add}_{\mathbb{K}}\) induces an equivalence between the \(\infty\)-category of additive \(\mathbb{K}\)-linear functors \(\mathcal C\otimes \mathcal D\rightarrow\mathcal E\) and the \(\infty\)-category of functors \(\mathcal C\times \mathcal D\rightarrow\mathcal E\) that are additive and \(\mathbb{K}\)-linear in either variable. An analogous characterization with additive replaced by exact holds for \(\mathrm{st}_{\mathbb{K}}\).
[007V]
Proposition 4.2.7.
Let \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\). Recall the functor \({\mathbf K}^b\colon \mathrm{st}\rightarrow\mathrm{add}\) from notation 3.4.11.
This functor induces a symmetric monoidal functor \({\mathbf K}^b\colon st_{\mathbb{K}} \rightarrow\mathrm{add}_{\mathbb{K}}\) which is left adjoint to the forgetful functor \(\mathrm{add}_{\mathbb{K}} \rightarrow\mathrm{st}_{\mathbb{K}}\).
For \(\mathcal C\in \mathrm{add}_{\mathbb{K}}\), the unit of the adjunction \(\mathcal C\rightarrow{\mathbf K}^b(\mathcal C)\) is fully faithful.
[007W]
Proof.
By proposition 3.1.8.([0035]), the symmetric monoidal left adjoint \({\mathbf K}^b: \mathrm{add}\rightarrow\mathrm{st}\) induces a symmetric monoidal left adjoint functor \(\mathrm{add}_{\mathbb{K}} = \mathrm{Mod}_{\mathrm{CProj}_{\mathbb{K}}}(\mathrm{add}) \rightarrow\mathrm{Mod}_{{\mathbf K}^b(\mathrm{CProj}_{\mathbb{K}})} (\mathrm{st})\). Composing with the equivalence \({\mathbf K}^b(\mathrm{CProj}_{\mathbb{K}}) \simeq \mathrm{Perf}_{\mathbb{K}}\) from proposition 3.5.8 results in the desired functor proing the first part. Fully faithfulness of the unit of the adjunction follows from proposition 3.4.5. ◻
4.2.2 \(\infty\)-categories enriched in graded modules[007X]
remark 4.1.9 motivates the following terminology:
[007Y]
Definition 4.2.8.
Let \(\mathcal Z\) be a homotopy coherent abelian monoid, i.e. \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\).
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define
the \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) of presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\)-enriched \(\infty\)-categories, \[\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}(\mathrm{Pr}^\mathrm{L}) ~;\]
the \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) of projectively generated presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\)-enriched \(\infty\)-categories, \[\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) ~.\]
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define
the \(\infty\)-category \({\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}}\) of presentably \(\mathrm{Mod}_{\mathbb{K}}^{ \mathcal Z}\)-enriched \(\infty\)-categories, \[{\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}(\mathrm{Pr}^\mathrm{L}) ~;\]
the \(\infty\)-category \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\) of compactly generated presentably \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\)-enriched \(\infty\)-categories, \[\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\coloneqq \mathrm{Mod}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})~.\]
As \(\infty\)-categories of modules of commutative algebras in presentably symmetric monoidal categories, both, \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\), are presentably symmetric monoidal \(\infty\)-categories.
[0080]
Example 4.2.10.
As discussed in example 3.5.15, if \(\mathcal Z\) is a discrete (i.e. ordinary) commutative monoid \(Z\), then \((\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z})^{\mathrm{cp}}\) and \((\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z})^c\) are the \(\infty\)-categories of finitely supported functors \(\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{CProj}_{\mathbb{K}})\) and \(\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{Perf}_{\mathbb{K}})\), i.e. of functors that vanish on all but finitely many elements of \(Z\). In particular, in the case of grading by a discrete monoid \(Z\), we obtain the following equivalences:\[\begin{aligned}
\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, Z}} &\simeq \mathrm{Mod}_{\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{CProj}_{\mathbb{K}})}(\mathrm{add})
\\
\mathrm{Pr}^{\mathrm{L},\mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{ Z}} &\simeq \mathrm{Mod}_{\mathrm{Fun}^{\text{fin.supp.}}(Z, \mathrm{Perf}_{\mathbb{K}})}(\mathrm{st})
\end{aligned}\]