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I have touched quasi-isomorphism, expanded the Idea-section and polished the Definition-section, added References
added a brief section ideosyncratically titled Relation to chain homology type, currently mainly inhabited by the basic counter-example for non-symmetry of quasi-isomorphicness.
why (stable) - what are you intending to mean?
By the Dold-Kan correspondence, a chain complex in non-negative degree is an abelian approximation or model of a topological space in homotopy theory.
By the stable Dold-Kan correspondence and unbounded chain complex is an abelian approximation or model of a spectrum in stable homotopy theory.
The fact that the derived category $D(\mathcal{A}) = Ho_{qi}(Ch_\bullet(\mathcal{A}))$ is a triangulated category is a shadow of the fact that unbounded chain complexes form a stable (∞,1)-category: in unbounded chain complexes we may arbitrarily loop and deloop (shift degrees), which is not possible with the “unstable” chain complexes in non-negative degree.
As a diagram
$\array{ && AbelianHomotopyTheory &\stackrel{DK-correspondence}{\hookrightarrow}& HomotopyThey \\ && \downarrow && \downarrow \\ HomologicalAlgebra &=& StableAbelianHomotopyTheory &\stackrel{stable\;DK-correspondence}{\hookrightarrow}& StableHomotopyTheory }$Added a basic remark on Relation to mapping cones and homotopy (co)fibers.
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