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, (4.13) 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.

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.

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, 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.

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, (4.13) 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.