Definition 6.1.19. We define the derived \(\infty \)-category of \(\Aa \) as the (\(\infty \)-categorical) localization of the 1-category \(\Ch (\Aa )\) of chain complexes at the class of quasi-isomorphisms: \[ \D (\Aa ) := \Ch (\Aa )[\{\text {quasi-isomorphisms}\}^{-1}]. \] We will sometimes refer to objects of \(\D (\Aa )\) as complexes. This is slightly abusive terminology: objects of \(\D (\Aa )\) should best be thought of as ‘chain complexes up to quasi-isomorphism’ and in particular do not determine a well-defined chain complex in \(\Aa \).
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