# Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?

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 $$H^0(\mathcal{C})$$?

• 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? – Denis Nardin Dec 28 '18 at 18:04
• 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.) – Dmitri Pavlov Dec 29 '18 at 1:14