For \(J\in \mathrm{Cat}_{\infty}\) and \(\mathcal C\in \mathrm{Pr}^\mathrm{L}\), the functor \(\mathcal C\times J \rightarrow\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\), \[ (c,j)\mapsto c \otimes \mathrm{Hom}_{J}(-, j) \in \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] (where \(\otimes\) denotes the action of \(\mathcal S\) on \(\mathcal C\) inherited from the presentability of \(\mathcal C\)) induces an equivalence \[ \mathcal C\otimes \mathcal P(J) \simeq \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] in \(\mathrm{Pr}^\mathrm{L}\) (where \(\otimes\) denotes the tensor product of \(\mathrm{Pr}^\mathrm{L}\)).
3.5.3 \(\mathcal Z\)-graded \(\mathbb{K}\)-modules[0061]
Given an ordinary monoid \(Z\) and a commutative ring \(k\), the category \(\mathrm{Fun}(Z, \mathrm{mod}_k)\) of \(Z\)-graded \(k\)-modules admits a convolution monoidal structure, for which the tensor product of \(Z\)-graded modules \((M_z)_{z \in Z}\) and \((N_z)_{z\in Z}\) is given by the \(Z\)-graded module which in degree \(z\in Z\) is \(\oplus_{z_1z_2 = z} M_{z_1} \otimes N_{z_2}\). This construction is a special case of the Day convolution monoidal structure on a functor category [Lur17, § 2.2.6]. Here, we focus on the symmetric monoidal case.
We briefly recall this construction of a symmetric monoidal structure on \(\mathrm{Fun}(J, \mathcal C)\) in the case where \(J\) is a small symmetric monoidal \(\infty\)-category and \(\mathcal C\) is a presentably symmetric monoidal \(\infty\)-category.
Proof.
Consider the chain of equivalences \[\mathcal C\otimes \mathcal P(J) \simeq \mathrm{Fun^L}(\mathcal P(J), \mathcal C^{\mathrm{op}})^{\mathrm{op}} \simeq \mathrm{Fun}(J, \mathcal C^{\mathrm{op}})^{\mathrm{op}} \simeq \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] Here, the first equivalence follows from ([002Z]), the second equivalence is the universal property of the Yoneda embedding [Lur09, Thm. 5.1.5.6], and the last records the interplay between functor categories and opposites. Precomposing this equivalence with the inclusion functor \(\mathcal C\times J \rightarrow\mathcal C\otimes \mathcal P(J)\) (which is cocontinuous in its second argument) unpacks to the functor ([0063]). ◻
Assume \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) and \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Since \(\mathcal P\colon \mathrm{Cat}_{\infty}\rightarrow\mathrm{Pr}^\mathrm{L}\) is symmetric monoidal by proposition 3.1.6, it follows that for \(J \in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\), the \(\infty\)-category \(\mathcal P(J)\) inherits a presentably symmetric monoidal structure, i.e. \(\mathcal P(J) \in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Then ([0064]) provides the following Day convolution monoidal structure on \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\).
Let \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) and \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Then \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\) inherits a presentably symmetric monoidal structure from the tensor product \(\mathcal C\otimes \mathcal P(J)\) of commutative algebras in \(\mathrm{Pr}^\mathrm{L}\).
By [BS24, Prop. 3.10], this construction agrees with the Day convolution structure on functor categories, as e.g. defined in [Lur17, Rem. 2.2.6.8], also see [BS24, Thm. 3.1]. Explicitly, the tensor product of functors \(F\colon J^{\mathrm{op}} \rightarrow\mathcal C\) and \(G \colon J^{\mathrm{op}} \rightarrow\mathcal C\) is given by the left Kan extension of the functor \(J^{\mathrm{op}} \times J^{\mathrm{op}} \xrightarrow{F\otimes G}\mathcal C\) along the tensor product \(J^{\mathrm{op}} \times J^{\mathrm{op}} \rightarrow J^{\mathrm{op}}\).
If \(J \in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) and \(\mathcal C\) is in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) or \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), then \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\) with its Day convolution monoidal structure is also in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) or \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), respectively.
Proof.
The Day convolution monoidal structure was defined by identifying \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\) with \(\mathcal C\otimes \mathcal P(J)\). The presheaf category \(\mathcal P(J)\) is an object of \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) (in fact, it is generated by a small set of objects which commute with all small colimits). Hence, if \(\mathcal C\) is in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) or in the subcategory \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\), then so is \(\mathcal C\otimes \mathcal P(J)\). ◻
The monoidal unit \(I \in J\) of any symmetric monoidal \(\infty\)-category \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) induces a symmetric monoidal functor \(\mathcal S\rightarrow\mathcal P(J)\) left adjoint to the evaluation functor \(\mathrm{ev}_{I} \colon \mathcal P(J) \rightarrow\mathcal S\), and explicitly given by sending a space \(X\) to the functor \(\mathrm{Hom}_{J}(-, I) \times X\colon J^{\mathrm{op}} \rightarrow\mathcal S\). It follows that for any presentably symmetric monoidal category \(\mathcal C\), there is a symmetric monoidal left adjoint \[\mathcal C\simeq \mathcal C\otimes \mathcal S\rightarrow\mathcal C\otimes\mathcal P(J) \simeq \mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\] to the evaluation functor \(\mathrm{ev}_{I}\colon\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C) \rightarrow\mathcal C\), explicitly given by sending \(c\in \mathcal C\) to the functor \(\mathrm{Hom}_{J}(-, I) \otimes c \colon J^{\mathrm{op}} \rightarrow\mathcal C\).
We will particularly focus on gradings by a homotopy coherent abelian monoid, i.e. a \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\).
Let \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\) and recall Day convolution from corollary 3.5.10.
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define the \(\infty\)-category of \(\mathcal Z\)-graded connective \(\mathbb{K}\)-modules \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) as the functor category \(\mathrm{Fun}(\mathcal Z, \mathrm{Mod}^{\geq 0}_{\mathbb{K}})\) with the Day convolution structure.
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define the \(\infty\)-category of \(\mathcal Z\)-graded \(\mathbb{K}\)-modules \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z} \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) to be the functor category \(\mathrm{Fun}(\mathcal Z, \mathrm{Mod}_{\mathbb{K}})\) with the Day convolution structure.
Following example 3.5.3, if \(\mathcal Z\) is a discrete (i.e. ordinary) commutative monoid \(Z\) and \(\mathbb{K}= Hk\) the Eilenberg-MacLane spectrum of an ordinary commutative ring \(k\), the \(\infty\)-category \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) is the unbounded derived \(\infty\)-category \(\mathcal D(\mathrm{mod}_k^Z)\) of the ordinary abelian \(1\)-category \(\mathrm{mod}_k^{Z}\coloneqq \mathrm{Fun}(Z, \mathrm{mod}_k)\) of \(Z\)-graded \(k\)-modules. This will be discussed in more detail in subsection 3.6.
Unpacking Day convolution from corollary 3.5.10 in these terms, the tensor product of an ordinary \(k\)-module \(M\) concentrated in degree \(z\in Z\) and an ordinary \(k\)-module \(N\) concentrated in degree \(w\in Z\) is given by the derived tensor product \(M\otimes_k^L N\) concentrated in degree \(z+w\in Z\).
Still in the setup of example 3.5.15, the \(\infty\)-categories \(\left(\mathrm{Mod}_{Hk}^{\geq 0, Z}\right)^{\mathrm{cp}}\) and \(\left(\mathrm{Mod}_{Hk}^{Z}\right)^{\mathrm{c}}\) may be identified with the full subcategories \(\mathrm{Fun}^{\mathrm{fin.supp.}}(Z, \mathrm{CProj}_{k})\) and \(\mathrm{Fun}^{\mathrm{fin.supp.}}(Z, \mathrm{Perf}_{k})\) of the functor \(\infty\)-categories \(\mathrm{Fun}(Z, \mathrm{CProj}_{k})\) and \(\mathrm{Fun}(Z, \mathrm{Perf}_{k})\), respectively, on the finitely supported functors, i.e. functors that vanish on all but finitely many elements of \(Z\).
Assume \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\) and \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\). The symmetric monoidal functor \(- \otimes \mathrm{Sp}\colon \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) from construction 3.4.4 takes \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\) with its Day convolution monoidal structure to \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) with its Day convolution monoidal structure. Indeed, we have the following sequence\[\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \otimes \mathrm{Sp}\simeq \mathcal P(\mathcal Z) \otimes \mathrm{Mod}_{\mathbb{K}}^{\geq 0} \otimes \mathrm{Sp}\simeq \mathcal P(\mathcal Z) \otimes \mathrm{Mod}_{\mathbb{K}} \simeq \mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\] of symmetric monoidal equivalences. In particular, it follows that the fully faithful inclusion \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \hookrightarrow \mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) is symmetric monoidal and hence a morphism in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2