TITLE: Is dgCat a category or a 2-category?
QUESTION [8 upvotes]: Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e.  dgCat has small dg categories as objects and the dg functors as morphisms. They then give a model structure on dgCat.
However, as pointed in What do DG-categories form?, it is more likely that dgCat is a 2-category: it does not make too much sense to say whether two dg-functors $F$ and $G$ are equal.  Instead we can talk about morphisms between dg-functors. 
How to integrate these two viewpoint? In particular, could we still have a model category structure on dgCat if we treat it as a 2-category?

REPLY [11 votes]: Adeel answer is perfect, I will be more basic. 
There is many notions of "2-category" structure here around. 
1) the category of small dg-categories $\mathbf{dgCat}$ is symmetric monoidal closed category. Closed means that there is an internal $HOM$ in the sense that for any two small dg-categories $A$ and $B$ there is $HOM(A,B)\in \mathbf{dgCat} $ such that there is a natural isomorphism of sets $\mathbf{dgCat}(X,HOM(A,B))\cong \mathbf{dgCat}(X\otimes A,B)$. That means $\mathbf{dgCat}$ is enriched over it self. 
There is a functor from $H^{0}:\mathbf{dgCat}\rightarrow \mathbf{Cat}$ which gives you an enrichment of the category $\mathbf{dgCat}$ over $\mathbf{Cat}$, hence you can see $\mathbf{dgCat}$ as a 2-category. 
On an other hand, the internal $HOM(-,-)$ described before has the wrong homotopy type, you can not derive it since it does not take Dwyer-Kan equivalences (between fibrant-cofibrant objects) to  Dwyer-Kan equivalences (the $\mathbf{dgCat}$ is not symmetric monoidal model category in the sense of Hovey). Bertand Toen constructed, for the model category $\mathbf{dgCat}$, the right notion of the derived internal hom denoted by $RHOM(A,B)\in \mathbf{dgCat}$ (using bimodules, I will not write the details). Moreover this new derived $RHOM(A,B)$ induces the derived Mapping space $Map
_{\mathbf{dgCat}}(A,B)$ via the nerve functor of some well choosen subcategory of $RHOM(A,B)$). This new derived internal allows you to see the category $\mathbf{dgCat}$ as $(2,\infty)$-category and in the same time as symmetric monoidal $(1,\infty)$-category.  
An important consequence is the following isomorphism in $Ho(\mathbf{sSet})$:
$$Map_{\mathbf{dgCat}}(A\otimes^{L}B,C)\cong  Map_{\mathbf{dgCat}}(A,RHOM(B,C))$$