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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.
have added to mapping cone a new Examples-subsection In cochain complexes with a bunch of explicit details.
I have been working on mapping cone:
polished and expanded the Definition-section.
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.
Thanks, Jim.
Positive feedback is indeed also appreciated. :-)
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.
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.
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.
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)
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.
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) didnâ€™t actually show injectivity on homology groups, but just on cycles. I have fixed that now.
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
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