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• 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 $H_n(-)$ 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

$H_n(Z_\bullet) \stackrel{H_n( cone(f)_\bullet \to Z_\bullet )^{-1}}{\to} H_{n}(cone(f)_\bullet) \to H_n(X[1]_\bullet)$

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 $X$ glued to $Y$. 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

1. fixed typos

Anonymous