Definition 2.5.1.5 (Quasi-Isomorphisms). Let $\operatorname{\mathcal{A}}$ be an abelian category, let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in $\operatorname{\mathcal{A}}$, and let $f: C_{\ast } \rightarrow D_{\ast }$ be a chain map. We say that $f$ is a quasi-isomorphism if, for every integer $n$, the induced map of homology objects $\mathrm{H}_{n}(C) \rightarrow \mathrm{H}_{n}(D)$ is an isomorphism.
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