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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 3rd 2012
   • (edited Sep 24th 2012)
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   Author: Urs
   Format: MarkdownItexI have touched _[[quasi-isomorphism]]_, expanded the Idea-section and polished the Definition-section, added References

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 3rd 2012
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   Author: Urs
   Format: MarkdownItexadded a brief section ideosyncratically titled _[Relation to chain homology type](http://ncatlab.org/nlab/show/quasi-isomorphism#RelationToChainHomologyType)_, currently mainly inhabited by the basic counter-example for non-symmetry of quasi-isomorphicness.

   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.

3. • CommentRowNumber3.
   • CommentAuthorjim_stasheff
   • CommentTimeSep 4th 2012
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   Author: jim_stasheff
   Format: Textin the context of (stable) homotopy theory. why (stable) - what are you intending to mean?

   in the context of (stable) homotopy theory.
   
   why (stable) - what are you intending to mean?
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 4th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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?

   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

   $$\array{ && AbelianHomotopyTheory &\stackrel{DK-correspondence}{\hookrightarrow}& HomotopyThey \\ && \downarrow && \downarrow \\ HomologicalAlgebra &=& StableAbelianHomotopyTheory &\stackrel{stable\;DK-correspondence}{\hookrightarrow}& StableHomotopyTheory }$$
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 24th 2012
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   Author: Urs
   Format: MarkdownItexAdded a basic remark on _[Relation to mapping cones and homotopy (co)fibers](http://ncatlab.org/nlab/show/quasi-isomorphism#RelationToMappingCone)_.

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