# Does the Dwyer-Kan model structure make dgCat a model $2$-category?

Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.

Now we consider dgCat as a 2-category, which objects small dg-categories, $$1$$-morphisms dg-functors, and $$2$$-morphisms (degree $$0$$, closed) natural transformations between dg-functors.

My question is: does the Dwyer-Kan model structure make dgCat a model $$2$$-category in the same of this post?

No. If dgCat were a model 2-category, then the 2-functor from dgCat to Cat that sends a dg-category $$A$$ to its underlying category (which has the same objects as $$A$$, and whose morphisms are the $$0$$-cycles in the hom chain complexes of $$A$$) would send DK-equivalences to equivalences of categories, since it is represented by the cofibrant "unit" dg-category $$\mathbf{1}$$ (which has a single object, and whose single hom chain complex is the group of integers $$\mathbb{Z}$$ concentrated in degree $$0$$) and since every dg-category is fibrant. But this is false: for example, consider the "two-object suspension" (i.e. the functor that sends a chain complex $$C$$ to the dg-category with two objects $$\bot,\top$$, and whose hom-object from $$\bot$$ to $$\top$$ is $$C$$, all other hom-objects being trivial, either $$0$$ or $$\mathbf{1}$$) of any quasi-isomorphism of chain complexes that is not bijective on $$0$$-cycles, such as the canonical morphism $$\mathrm{Cyl}(\mathbf{1}) \to \mathbf{1}$$.