In the article *Grothendieck Ring of pretriangulated categories* by Bondal-Larsen-Lunts, two dg-categories $\mathcal A$ and $\mathcal B$ are called *quasi-equivalent* if there is a chain of dg-categories and quasi-equivalences 


$\mathcal A \leftarrow \mathcal C\_1 \rightarrow \cdots \leftarrow \mathcal C\_n \rightarrow \mathcal B$.


Another possibility, I think, would be to call $\mathcal A$ and $\mathcal B$ quasi-equivalent if they are isomorphic in the homotopy category $\mathrm{Ho}(\mathbf{dgcat})$. Clearly if $\mathcal A$ and $\mathcal B$ are quasi-equivalent as in the definition of Bondal-Larsen-Lunts, they are also isomorphic in $\mathrm{Ho}(\mathbf{dgcat})$. Is the converse true? What is the "better" definition of quasi-equivalent dg-categories?


Thanks in advance!



About the definition of quasi-equivalent dg-categories
The converse is true. 


You can see this for example as follows: first since DG-categories admit a model structure each morpism $A \to B$ in Ho(dgCat) can be realized as a 3-step span like this
$A \leftarrow A' \to B' \leftarrow B$ (here $A'$ is a cofibrant replacement and $B'$ is a fibrant replacement). Then such a morpism is an isomorphism in the homotopy category be the 2-out-of-3 property if and only if the middle morphism $A'\to B'$ is a weak equivalence. Thus each isomorphism in the homotopy category comes from a Zig-Zag of weak equivalences (and this argument is valid in each model category).


