Are dg-modules over a cofibrant dg-category cofibrant?

Question

Fix a commutative ring $k;$ all dg-categories will be dg-categories over $k.$ Throughout the question, I will be following the notation and conventions of Toën's "The homotopy theory of dg-categories and derived Morita theory." For a dg-category $C,$ let $[C]$ be the category whose objects are the same as the objects of $C,$ and whose morphisms are defined by $\operatorname{Hom}_{[C]}(X,Y) := H_0(C(X,Y)).$

Let $F : C\to D$ be a dg-functor between dg-categories, and recall that:

• $F$ is quasi-fully faithful if for all $X,Y\in C,$ $F_{X,Y} : C(X,Y)\to D(FX,FY)$ is a quasi isomorphism,
• $F$ is quasi-essentially surjective if $[F] : [C]\to [D]$ is essentially surjective,
• $F$ is a quasi-equivalence if it is quasi-fully faithful and quasi-essentially surjective.
• $F$ is a fibration if it satisfies the following two conditions:

1. For all $X,Y\in C,$ the morphism $F_{X,Y} : C(X,Y)\to D(FX,FY)$ is a fibration in the category $\mathsf{Ch}(k)$ of chain complexes over $k$ (i.e., a surjection), and
2. For all $X\in C,$ given any isomorphism $v : [F](X)\to Y'\in [D],$ there exists $Y\in C$ and an isomorphism $u : X\to Y$ in $[C]$ such that $[F](u) = v.$

Recall that there is a model structure on the category $\mathsf{dgCat}_k$ of dg-categories over $k$ and dg-functors between them, with fibrations as defined above, and with weak equivalences given by the quasi-equivalences.

For a dg-category $C,$ define also the dg-category $\widehat{C}$ to be the full sub-dg-category of $\mathsf{dgMod}_{C^{\textrm{op}}}$ consisting of the fibrant and cofibrant objects, where we define the fibrations and equivalences on $\mathsf{dgMod}_{C^{\textrm{op}}}$ to be the functors which are level-wise fibrations and equivalences in $\mathsf{Ch}(k).$

My question is: suppose that $C$ is a cofibrant dg-category. Then are either of $\widehat{C}$ or $\mathsf{dgMod}_{C^{\textrm{op}}}$ cofibrant dg-categories?

First, it is easy to show that $C$ is cofibrant if and only if $C^{\textrm{op}}$ is. Using this observation, the only way I've thought of to get a map $F : \mathsf{dgMod}_{C}\to A$ (or $\widehat{C}$) lifting a functor $\mathsf{dgMod}_C\to B$ along a trivial fibration $A\to B$ is to use the Yoneda embedding $$ \begin{align*} h^{-}:C^{\textrm{op}}&\to \widehat{C}\\ X&\mapsto\left(\begin{array}{lll} h^X:&C&\to\mathsf{Ch}(k) \\ &Y&\mapsto C(X,Y) \end{array}\right) \end{align*} $$ and write any dg-module $M$ as a colimit of representable functors $M\cong\varinjlim_i h^{X_i}$ to define $$F(M) := \varinjlim_i G(X_i),$$ where $G : C^{\textrm{op}}\to A$ is a lift of the composite $$C^{\textrm{op}}\to \mathsf{dgMod}_C\to B$$ along $A\to B.$

However, there are a few problems with the strategy: first, $A$ might not have colimits! Even if $A$ did have appropriate colimits, this would only define $F$ at the level of objects, and it seems that $A\to B$ would have to commute with colimits in order for this to be reasonable. Is there a way to salvage this strategy, and if not, is there another way to approach this?

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Edit: To add my main goal in asking this, I am asking this as a follow-up to my previous question about showing that the derived infinity category commutes with taking pushouts. I received a nice answer there addressing the situation in the $\infty$-categorical situation, but I was hoping to find a proof of this in the case of dg-categories which didn't pass through the $\infty$-categorical language. The proof sketch I came up with required the category of dg-modules over a cofibrant dg-category/algebra to be cofibrant in order to compute the derived tensor products that arise.

Answer 1

My question is: suppose that C is a cofibrant dg-category. Then are either of Ĉ or dgMod_C^op cofibrant dg-categories?

A cofibrant object in a cofibrantly generated model category (such as dgCat) is a retract of a transfinite composition of cobase changes of generating cofibrations. Generating cofibrations of dgCat are functors between small categories (see, for example, (4.7) and (4.13) in arXiv:1201.1575), and cobase change, transfinite composition, and retracts preserve this property. Since dgMod_C^op and Ĉ both have a proper class of objects (and even a proper class of weak equivalence classes of objects), there is no way dgMod_C^op or Ĉ could be made into cofibrant dg-categories.