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Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but morphisms between two objects $x$ and $y$ are defined as $$ H^0(\mathcal{C})(x,y):=Z^0(\mathcal{C})(x,y)/B^0(\mathcal{C})(x,y). $$

We could also consider the category $Z^0(\mathcal{C})$ and the class $W$ of homotopic invertible morphisms in $Z^0(\mathcal{C})$. More precisely, $$ W=\{f\in Z^0(\mathcal{C})(x,y) \text{ for some }x \text{ and }y~|~f \text{ is invertible in } H^0(\mathcal{C})(x,y)\}. $$

We could localize the category $Z^0(\mathcal{C})$ by formally inverting $W$ and obtain a category $W^{-1}Z^0(\mathcal{C})$. There is a natural functor $W^{-1}Z^0(\mathcal{C})\to H^0(\mathcal{C})$.

My question is: Is $W^{-1}Z^0(\mathcal{C})$ equivalent to $H^0(\mathcal{C})$?

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    $\begingroup$ This certainly needs some hypotheses on the dg-category (consider a one object dg-category: not all quotients by ideals are localizations!). Maybe pretriangulated is enough? $\endgroup$
    – Denis Nardin
    Commented Dec 28, 2018 at 18:04
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    $\begingroup$ Proposition 5.17 in arxiv.org/abs/1602.01515 has one possible answer to your question. (One must pass to the homotopy category on both sides.) $\endgroup$
    – Dmitri Pavlov
    Commented Dec 29, 2018 at 1:14

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