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nLab > Latest Changes: mapping cone

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2009

    Explained at mapping cone how the mapping cone is model for a homotopy cofiber. In fact I used that to define and motivate the mapping cone.

    Then I moved the example in Top to the top of the list, as that is the archetypical example.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2011

    have added to mapping cone a new Examples-subsection In cochain complexes with a bunch of explicit details.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2012
    • (edited Aug 30th 2012)

    I have been working on mapping cone:

    1. polished and expanded the Definition-section.

    2. added a section Examples - In chain complexes

    In principle of course that should run dually to the formerly existing section In cochain complexes but right now both sections are organized a bit differently. Maybe I find the time to re-structure the section on cochain complexes later. But maybe not.

    • CommentRowNumber4.
    • CommentAuthorjim_stasheff
    • CommentTimeAug 31st 2012
    @1 Urs: Explained at mapping cone how the mapping cone is model for a homotopy cofiber. In fact I used that to define and motivate the mapping cone.

    Then I moved the example in Top to the top of the list, as that is the archetypical example.

    well done - now that's an approach I can appreciate
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2012

    Thanks, Jim.

    Positive feedback is indeed also appreciated. :-)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 13th 2012
    • (edited Sep 13th 2012)

    I have further worked on the section Examples - In chain complexes.

    Now it includes also a detailed display of the differentials in cylinder/cone complexes and mapping cylcinder/mapping cone complexes and a detailed derivation and explanation of where the signs come from, systematically.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 13th 2012
    • (edited Sep 13th 2012)

    I have spelled out still more details of mapping cones in chain complex.

    Then I wrote a section Relation to homotopy fiber sequences which presents in full detail the proof that applying H n(−)H_n(-) to the long homotopy cofiber sequence of a monomorphism gives the long exact sequence in homology groups of its corresponding short exact sequence.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 18th 2012

    At mapping cone in Homology exact sequences and homotopy fiber sequences I tried to spell out (currently Lemma 1 there) more explicitly how

    H n(Z •)→H n(cone(f) •→Z •) −1H n(cone(f) •)→H n(X[1] •) H_n(Z_\bullet) \stackrel{H_n( cone(f)_\bullet \to Z_\bullet )^{-1}}{\to} H_{n}(cone(f)_\bullet) \to H_n(X[1]_\bullet)

    is the connecting homomorphism.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 18th 2012

    I have added a bit more glue-text to the section Distinguished triangles (which kept floating around in its form from the early days of this entry)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 18th 2012
    • (edited Sep 18th 2012)

    added at mapping cone below the main definition (which is Prop. 1 currently) another remark, currently remark 1, invoking the standard picture of a cone over XX glued to YY. Eventually maybe somebody feels inspired to add the canonical illustration as an SVG graphics.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2012

    It was pointed out to me by sombody attentive that my alleded proof of _this lemma (which asserts that a canonical map out of the mapping cone is a quasi iso) didn’t actually show injectivity on homology groups, but just on cycles. I have fixed that now.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2014

    added the statement that also the total complex of the double complex induced by a chain map is a model for the mapping cone, here

  13. fixed typos

    Anonymous

    diff, v42, current

1 to 13 of 13

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