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](https://arxiv.org/abs/math/0408337)." 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?




---


**Edit:** To add my main goal in asking this, I am asking this as a follow-up to my [previous question](https://mathoverflow.net/questions/370227/does-formation-of-the-derived-infty-category-preserve-pushouts) 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.



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

> 
> 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](https://arxiv.org/abs/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.


