# Symmetric monoidal structure(s) on the $\infty$-category of dg-categories

Let $$k$$ be a commutative ring with $$1$$, and let $$\mathsf{dgCat}_k$$ be the category of $$k$$-linear dg-categories, as defined in [1, Section 2]. We may equip $$\mathsf{dgCat}_k$$ with the Morita model structure [2, Théorème 2.27], which we will denote by $$\mathsf{dgCat}_k^\textrm{Mor}$$.

The category $$\mathsf{dgCat}_k$$ is symmetric monoidal: if $$\mathcal{C},\mathcal{D}\in\operatorname{Obj}(\mathsf{dgCat}_k)$$, then we define $$\mathcal{C}\otimes_{\mathsf{dgCat}_k}\mathcal{D}$$ to be the $$k$$-linear dg-category with objects $$\operatorname{Obj}(\mathcal{C}\otimes_{\mathsf{dgCat}_k}\mathcal{D}) := \operatorname{Obj}(\mathcal{C})\times\operatorname{Obj}(\mathcal{D})$$ and morphism complexes $$\left(\mathcal{C}\otimes_{\mathsf{dgCat}_k}\mathcal{D}\right)((X,Y),(X',Y'))_* := \mathcal{C}(X,X')_*\otimes_k \mathcal{D}(Y,Y')_*.$$

While the tensor product defined above does not respect the Morita model category structure$$^1$$, we may derive it to obtain a symmetric monoidal model category structure on $$\mathsf{dgCat}_k^{\textrm{Mor}}$$, which we will denote by $$\left(\mathsf{dgCat}_k^\textrm{Mor},\otimes_{\mathsf{dgCat}_k^\textrm{Mor}}^L\right)$$ (see [2, Remarque 2.40]). Explicitly, $$\mathcal{C}\otimes_{\mathsf{dgCat}_k^\textrm{Mor}}^L\mathcal{D} \simeq Q(\mathcal{C})\otimes_{\mathsf{dgCat}_k}\mathcal{D},$$ where $$Q$$ is a cofibrant replacement functor for the $$\mathsf{dgCat}_k^{\textrm{Mor}}$$.

Cohn has shown [3, Corollary 5.5] that the underlying $$\infty$$-category as defined in [4, Definition 1.3.4.15]$$^2$$ of $$\mathsf{dgCat}_k^\textrm{Mor}$$ is equivalent to the $$\infty$$-category of small idempotent-complete $$k$$-linear stable $$\infty$$-categories: $$N(\mathsf{dgCat}_k^\textrm{Mor})[W^{-1}]\simeq\operatorname{Mod}_{\operatorname{Perf}(Hk)}((\mathcal{Cat}_\infty^\textrm{perf})^\otimes).$$ In fact [3, Corollary 5.7], it is also equivalent to the $$\infty$$-category of compactly-generated presentable $$k$$-linear stable $$\infty$$-categories with functors that preserve colimits and compact objects: $$N(\mathsf{dgCat}_k^\textrm{Mor})[W^{-1}]\simeq\operatorname{Mod}_{Hk\textrm{-}\operatorname{Mod}}((\mathcal{Pr}_{\textrm{st},\omega}^L)^\otimes).$$

We also have symmetric monoidal structures on both these $$\infty$$-categories. The Lurie tensor product [4, Proposition 4.8.1.15] induces a symmetric monoidal structure on $$\operatorname{Mod}_{\operatorname{Perf}(Hk)}((\mathcal{Cat}_\infty^\textrm{perf})^\otimes)$$ and $$\operatorname{Mod}_{Hk\textrm{-}\operatorname{Mod}}((\mathcal{Pr}_{\textrm{st},\omega}^L)^\otimes)$$ (I believe this follows from [4, Theorem 3.3.3.9], at least in the case of $$\operatorname{Mod}_{Hk\textrm{-}\operatorname{Mod}}((\mathcal{Pr}_{\textrm{st},\omega}^L)^\otimes)$$), and the symmetric monoidal structure $$\otimes^L_{\mathsf{dgCat}_k^\textrm{Mor}}$$ induces a symmetric monoidal structure on $$N(\mathsf{dgCat}_k^\textrm{Mor})[W^{-1}]$$ (see [4, Example 4.1.7.6]).

My question: is the equivalence above an equivalence of symmetric monoidal $$\infty$$-categories? In other words, does the symmetric monoidal model category structure on $$\mathsf{dgCat}_k^\textrm{Mor}$$ induce the expected symmetric monoidal structure given by the Lurie tensor product on the underlying $$\infty$$-category of dg-categories? If the symmetric monoidal structures do not coincide, what is the relationship between them (if any)?

Footnotes.

1. The linked MO question deals with the Dwyer-Kan model structure on $$\mathsf{dgCat}_k$$ (see [1, Section 2] and [2, Théorème 1.8]), but the issue is present in the Morita model structure as well.
2. It appears Cohn does not restrict to cofibrant objects as in the cited definition, but [4, Remark 1.3.4.16] implies that this is not a problem, as the Morita model structure on $$\mathsf{dgCat}_k$$ admits functorial fibrant-cofibrant factorizations of morphisms.

This question is very much related to my question 2 here. In some sense, both are about the same underlying question/confusion, but I wanted to ask a more focused question about the symmetric monoidal structures in particular. I also hope that the (perhaps excessive) citations will be useful to anyone attempting to make their way through this literature and find precise definitions and references for the first time.

[2] Tabuada, G. Théorie homotopique des DG categories.

[4] Lurie, J. Higher Algebra. (September 18, 2017 version).