fixed typos

Anonymous

]]>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

]]>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.

]]>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.

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)

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

is the connecting homomorphism.

]]>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.

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.

]]>Thanks, Jim.

Positive feedback is indeed also appreciated. :-)

]]>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 ]]>

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.

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

]]>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.

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