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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(𝒜)=Ho qi(Ch (𝒜))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

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