Added a basic remark on *Relation to mapping cones and homotopy (co)fibers*.

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

$\array{ && AbelianHomotopyTheory &\stackrel{DK-correspondence}{\hookrightarrow}& HomotopyThey \\ && \downarrow && \downarrow \\ HomologicalAlgebra &=& StableAbelianHomotopyTheory &\stackrel{stable\;DK-correspondence}{\hookrightarrow}& StableHomotopyTheory }$ ]]>why (stable) - what are you intending to mean? ]]>

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.

I have touched *quasi-isomorphism*, expanded the Idea-section and polished the Definition-section, added References