Let us consider dgCat, the "collection" of all small dg-categories. In [On differential graded categories](https://atlas.mat.ub.edu/grgta/articles/Keller.pdf) and [Lectures on dg categories](http://www.mi.ras.ru/~akuznet/dgcat/Toen%20Lectures%20on%20DG-categories.pdf) 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?](http://sites.math.northwestern.edu/~tamarkin/Papers1/dg.pdf), 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?



Is dgCat a category or a 2-category?
The model structure on the category of dg-categories presents an $(\infty,1)$-category DGCat. This structure is essentially provided by the existence of mapping spaces (or mapping $\infty$-groupoids) between dg-categories.


To see why DGCat admits the further structure of an $(\infty,2)$-category, it is sufficient to see why these mapping $\infty$-groupoids can be refined to mapping $(\infty,1)$-categories, which give back our mapping $\infty$-groupoids when we pass to the sub-$\infty$-groupoids of invertible morphisms.


Such structure is provided by functor dg-categories, which are the internal hom objects in DGCat (in the "derived" sense -- these are denoted $R\underline{Hom}$ by Toen). These functor dg-categories give $(\infty,1)$-categories (by using for example the Dold-Kan correspondence), which provide the mapping $(\infty,1)$-categories in the $(\infty,2)$-category DGCat.


(Toen showed that these mapping $(\infty,1)$-categories can also be described as the $(\infty,1)$-categories of right quasi-representable bimodules.)


Edit: I just noticed that the preprint [Giovanni Faonte, *$A\_\infty$-functors and homotopy theory of dg-categories*, [arXiv:1412.1255](http://arxiv.org/abs/1412.1255)] appears to contain a precise construction of the $(\infty,2)$-category of dg-categories.


