Remark 6.6.9. The localization functor identifies \[ \Ch ^-(\Aa )[\{\text {quasi-isomorphisms}\}^{-1}] \iso \D ^-(\Aa ), \] where \(\D ^-(\Aa )\) is the full subcategory of bounded below objects from Definition 6.3.33. Indeed, Proposition 6.1.22 identifies the localization of \(\Ch (\Aa )_{\geq m}\) with the full subcategory \(\D (\Aa )_{\geq m}\) of \(\D (\Aa )\) for every \(m\). Taking the union over all \(m\) gives the displayed identification. We use it without further comment.
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