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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 3rd 2012
• (edited Sep 24th 2012)

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeSep 3rd 2012

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.

• CommentRowNumber3.
• CommentAuthorjim_stasheff
• CommentTimeSep 4th 2012
in the context of (stable) homotopy theory.

why (stable) - what are you intending to mean?
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 4th 2012

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

$\array{ && AbelianHomotopyTheory &\stackrel{DK-correspondence}{\hookrightarrow}& HomotopyThey \\ && \downarrow && \downarrow \\ HomologicalAlgebra &=& StableAbelianHomotopyTheory &\stackrel{stable\;DK-correspondence}{\hookrightarrow}& StableHomotopyTheory }$
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 24th 2012