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]$^2$ 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] 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], 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]).

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)?


  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] 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.

[1] Toën, B. The homotopy theory of dg-categories and derived Morita theory.

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

[3] Cohn, L. Differential Graded Categories are k-linear Stable Infinity Categories.

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



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.