4.3 From gradings to actions[0081]
Given a ring \(k\), and a \(k\)-linear category \(\mathcal C\) with an action by a discrete monoid \(Z\), then the category \(\mathcal C\) is canonically enriched in the ordinary category \(\mathrm{mod}_k^Z\) of \(Z\)-graded \(k\)-modules. Indeed, given objects \(c, d\in \mathcal C\) we define the \(k\)-module of degree-\(z\) morphisms to be \[\mathrm{Hom}_{\mathcal C}(c,d)_z \coloneqq \mathrm{Hom}_{\mathcal C}(c, d[z])\] where \((-)[z] \colon \mathcal C\rightarrow\mathcal C\) denotes the action of \(z\in Z\) on \(\mathcal C\). Conversely, if \(\mathcal C\) is a category enriched in \(Z\)-graded \(k\)-modules and if moreover for every \(z\in Z\), the inner-hom functor \(\underline{\mathrm{Hom}}_{\mathcal C}(c,-) \colon \mathcal C\rightarrow\mathrm{mod}_k^Z\) is corepresentable (e.g. if \(\mathcal C\) is presentably enriched), then the enrichment arises from a \(Z\)-action on the underlying category. These constructions provide an equivalence between the category of presentable \(k\)-linear categories with an \(Z\)-action and the category of categories presentably enriched in \(Z\)-graded \(k\)-modules. In this section, we generalize these constructions to our \(\infty\)-categorical setting.
[0082]
Definition 4.3.1.
Let \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\).
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define the \(\infty\)-category \(\mathrm{add}_{\mathbb{K}}^J\) of \(J\)-graded additive, idempotent-complete \(\mathbb{K}\)-linear categories as \[\mathrm{add}_{\mathbb{K}}^J\coloneqq \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}}).\]
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define the \(\infty\)-category \(\mathrm{st}_{\mathbb{K}}^J\) of \(J\)-graded stable, idempotent-complete \(\mathbb{K}\)-linear categories as \[\mathrm{st}_{\mathbb{K}}^J \coloneqq \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{st}_{\mathbb{K}}).\]
We need the following compatibilities of structures with the functor \({\mathbf K}^b\) from proposition 4.2.7.
[0083]
Proposition 4.3.2.
Let \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\).
Day convolution induces presentably symmetric monoidal structures on \(\mathrm{add}_{\mathbb{K}}^J\) and on \(\mathrm{st}_\mathbb{K}^J\) for \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\) and \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), respectively.
Composing with the symmetric monoidal left adjoint \({\mathbf K}^b\colon \mathrm{add}_{\mathbb{K}} \rightarrow\mathrm{st}_{\mathbb{K}}\) from proposition 4.2.7 induces a symmetric monoidal functor \[{\mathbf K}^b: \mathrm{add}_{\mathbb{K}}^J = \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}}) \rightarrow\mathrm{Fun}(J^{\mathrm{op}}, \mathrm{st}_{\mathbb{K}}) =\mathrm{st}_{\mathbb{K}}^J\] left adjoint to the forgetful functor. Moreover, for \(\mathcal C\in \mathrm{add}_{\mathbb{K}}^J = \mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}})\), the unit of the adjunction \(\mathcal C\rightarrow{\mathbf K}^b(\mathcal C)\) is pointwise (i.e. for every \(j\in J\)) fully faithful.
[0085]
Proof.
Since \(\mathrm{st}_{\mathbb{K}}\) and \(\mathrm{add}_{\mathbb{K}}\) are in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) by proposition 4.2.7 and \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\), corollary 3.5.10 induces a presentably symmetric monoidal structure on \(\mathrm{Fun}(J^{\mathrm{op}}, \mathrm{st}_{\mathbb{K}})\) and \(\mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}})\). Under the equivalence \(\mathrm{Fun}(J^{\mathrm{op}}, \mathrm{add}_{\mathbb{K}}) \simeq \mathrm{add}_{\mathbb{K}} \otimes \mathcal P(J)\) of lemma 3.5.9, the postcomposition functor becomes the functor \({\mathbf K}^b\otimes \mathrm{id}_{\mathcal P(J)}\) and hence is a morphism in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Given \(\mathcal C\in \mathrm{add}_{\mathbb{K}}^J\), i.e. \(\mathcal C_{-}\colon J^{\mathrm{op}} \rightarrow\mathrm{add}_{\mathbb{K}}\), the unit of the adjunction \(\mathcal C\rightarrow{\mathbf K}^b(\mathcal C)\) is given by the natural transformation which at an object \(j\in J\) is the unit \(\mathcal C_j \rightarrow{\mathbf K}^b(\mathcal C_j)\) of the adjunction \({\mathbf K}^b\colon \mathrm{add}_{\mathbb{K}} \rightarrow\mathrm{st}_{\mathbb{K}}\). This is fully faithful by proposition 4.2.7. ◻
We are interested in \(\infty\)-categories with an action by a commutative monoid. For \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\), let \(B \mathcal Z\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) denote its delooped symmetric monoidal \(\infty\)-category. Since \(\mathcal Z\) is commutative, there is a symmetric monoidal equivalence \(B\mathcal Z\simeq B\mathcal Z^{\mathrm{op}}\). Then an object of \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z} = \mathrm{Fun}(B\mathcal Z, \mathrm{add}_{\mathbb{K}})\) is precisely a small additive, idempotent complete \(\mathbb{K}\)-linear \(\infty\)-category with an action by \(\mathcal Z\) via \(\mathbb{K}\)-linear additive functors (and similarly for \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\)).
The equivalence between gradings and actions derives from the following proposition:
[0086]
Proposition 4.3.3.
Let \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\) with delooping \(B\mathcal Z\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\). Then, there is a symmetric monoidal equivalence between \(\mathrm{Fun}(B\mathcal Z, \mathcal S)\) with its Day convolution symmetric monoidal structure and \(\mathrm{Mod}_{\mathcal Z}(\mathcal S)\) with symmetric monoidal structure given by relative tensor product over \(\mathcal Z\).
[0087]
Proof.
For \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) for which \(\mathrm{Hom}_{\mathcal C}(I,-)\colon \mathcal C\rightarrow\mathcal S\) preserves all small colimits and is conservative, it follows from [Lur17, Prop. 4.8.5.21] that \(\mathcal C\) is symmetric monoidally equivalent to \(\mathrm{Mod}_{\mathrm{End}_{\mathcal C}(I)}(\mathcal S)\) where \(\mathrm{End}_{\mathcal C}(I) \in \mathrm{CAlg}(\mathcal S)\) is equipped with the commutative monoid structure induced from symmetric monoidality of \(\mathcal C\).
If \(J \in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\), then the Yoneda embedding \(J \rightarrow\mathrm{Fun}(J^{\mathrm{op}}, \mathcal S)\) is symmetric monoidal for the Day convolution symmetric monoidal structure. In particular, the monoidal unit of \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal S)\) is the image under the Yoneda embedding of \(I \in J\), and its endomorphism algebra agrees with \(\mathrm{End}_J(I)\). In particular, as a representable presheaf, \(\mathrm{Hom}_{\mathrm{Fun}(J^{\mathrm{op}}, \mathcal S)}(I,-) \colon \mathrm{Fun}(J^{\mathrm{op}}, \mathcal S) \rightarrow\mathcal S\) preserves all small colimits.
Let now \(J=B\mathcal Z\) for a \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\). The functor \(\mathrm{Hom}_{\mathrm{Fun}(B \mathcal Z^{\mathrm{op}}, \mathcal S)}(I,-) \colon \mathrm{Fun}(B \mathcal Z^{\mathrm{op}}, \mathcal S) \rightarrow\mathcal S\) then forgets the \(\mathcal Z\) action and is hence conservative. Since the unit of \(B\mathcal Z\) is the basepoint \({\sf pt}\) with \(\mathrm{End}_{B\mathcal Z}({\sf pt}) \simeq \mathcal Z\) as commutative algebras in spaces, and using the symmetric monoidal equivalence \(B\mathcal Z^{\mathrm{op}} \simeq B\mathcal Z\) induced by commutativity of \(\mathcal Z\), it therefore follows [Lur17, Prop. 4.8.5.21] that we have symmetric monoidal equivalences \(\mathrm{Fun}(B\mathcal Z, \mathcal S) \simeq \mathrm{Fun}(B\mathcal Z^{\mathrm{op}}, \mathcal S) \simeq \mathrm{Mod}_{\mathrm{End}_{B\mathcal Z}({\sf pt})}(\mathcal S) = \mathrm{Mod}_{\mathcal Z}(\mathcal S)\). ◻
We now prove the main proposition of this subsection: For a homotopy coherent abelian monoid \(\mathcal Z\), the \(\infty\)-category of compactly generated presentably \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\)-enriched \(\infty\)-categories is equivalent to the category \(\mathrm{Fun}(B\mathcal Z, \mathrm{st}_{\mathbb{K}})\), i.e. to the category of \(\mathbb{K}\)-linear stable \(\infty\)-categories with an action by \(\mathcal Z\). This equivalence is symmetric monoidal for the Day convolution structure on \(\mathrm{Fun}(B\mathcal Z, \mathrm{st}_{\mathbb{K}})\).
[0088]
Proposition 4.3.4.
Fix \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\).
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), the equivalence \((-)^{\mathrm{cp}}\colon \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\simeq \mathrm{add}_{\mathbb{K}} \colon \mathcal P^{\Sigma}\) induces a symmetric monoidal equivalence: 
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), the equivalence \((-)^{\mathrm{c}} \colon \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\simeq \mathrm{st}_{\mathbb{K}} \colon \operatorname{Ind}\) induces a symmetric monoidal equivalence: 
[008B]
Proof.
We prove statement ([008A]), the proof of statement ([0089]) is entirely analogous. Consider the symmetric monoidal equivalences \[\mathrm{st}_{\mathbb{K}}^{B\mathcal Z} \coloneqq \mathrm{st}_{\mathbb{K}} \otimes \mathcal P(B\mathcal Z) \simeq \mathrm{st}_{\mathbb{K}} \otimes \mathrm{Mod}_{\mathcal Z}(\mathcal S) \simeq \mathrm{Mod}_{\mathrm{Mod}_\mathbb{K}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \otimes \mathrm{Mod}_{\mathcal Z}(\mathcal S),\] where the first equivalence is given by proposition 4.3.3 and the second equivalence follows from observation 4.2.4. It then follows from corollary 3.1.9 and lemma 3.2.11 that \[\mathrm{Mod}_{\mathrm{Mod}_\mathbb{K}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \otimes \mathrm{Mod}_{\mathcal Z}(\mathcal S) \simeq \mathrm{Mod}_{\mathcal P(\mathcal Z) \otimes \mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) = \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}).\qedhere\] ◻
[008C]
Observation 4.3.5.
As presentably symmetric monoidal \(\infty\)-categories, \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\) are self-enriched. Transporting these self-enrichments along the equivalences from proposition 4.3.4 provides enrichments in \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\) and \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\), respectively, i.e. \[\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}])\hspace{0.25cm}\text{ and }\hspace{0.25cm} \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}]).\]
It follows from lemma 4.1.6 applied to \(\mathrm{Mod}_{\left(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\right)^{\mathrm{cp}}}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}})\) that given \(\mathcal C, \mathcal D\in \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\), their \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\)-enriched hom is the small idempotent-complete additive \(\infty\)-category \[\mathrm{Fun}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, B\mathcal Z}}\left( \mathcal C, \mathcal D\right)\in \mathrm{add}_{\mathbb{K}}^{B\mathcal Z}\] with \(\mathrm{CProj}_{\mathbb{K}}\) and \(\mathcal Z\)-action induced by the \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\)-action on \(\mathcal D\).
Similarly, it follows from lemma 4.1.6 applied to \(\mathrm{Mod}_{\left(\mathrm{Mod}_{\mathbb{K}}^{ \mathcal Z}\right)^{\mathrm{cp}}}(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}})\) that given \(\mathcal C, \mathcal D\in \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\), their \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\)-enriched hom is the small idempotent-complete stable \(\infty\)-category \[\mathrm{Fun}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{B\mathcal Z}}\left( \mathcal C, \mathcal D\right) \in \mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\] with \(\mathrm{Perf}_{\mathbb{K}}\) and \(\mathcal Z\)-action induced by the \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\)-action on \(\mathcal D\).