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I am wondering how the singular chain complex functor from the category of topological spaces to the category of chain complexes of abelian groups takes a mapping cone to a mapping cone in the sense of chain complexes as it is claimed in https://ncatlab.org/nlab/show/mapping+cone. Also, in Rotman's book "An introduction to algebraic topology" p. 350, it is pointed out "One can show that this geometric construction corresponds to the algebraic mapping cone".

I am interested in a clear explanation.

Many thanks

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    $\begingroup$ It is completely formal that the free simplicial abelian group functor preserves homotopy cofibres and the explicit mapping cone construction, and the category of simplicial abelian groups is equivalent to the category of nonnegatively graded chain complexes by the Dold-Kan correspondence, so the only tricky part is to show that the singular complex of the topological mapping cone is a homotopy cofibre in simplicial sets. This is not immediately obvious, but I'd guess that this follows from the Quillen equivalence of Kan-Quillen and Quillen-Serre model structures between sSet & Top rspctivly $\endgroup$
    – Harry Gindi
    Commented Nov 11, 2018 at 13:47
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    $\begingroup$ @Harry That's unnecessarily fancy. It is easy to construct a chain level map of short exact sequences from the algebraic construction to the singular chains on the geometric construction which is clearly a quasi-isomorphism on the first two factors. Thus you conclude. (note that this equivalence is natural on the arrow category of TOP.) $\endgroup$
    – mme
    Commented Nov 11, 2018 at 17:27
  • $\begingroup$ @MikeMiller True, but in my sketch, naturality comes immediately by universal property and no maps need to be constructed. It's the easier proof if you're more familiar with simplicial sets than spaces like I am. $\endgroup$
    – Harry Gindi
    Commented Nov 11, 2018 at 17:45
  • $\begingroup$ Thank you for your ideas. @MikeMiller Could you please clarify your chain level map? $\endgroup$
    – ARA
    Commented Nov 12, 2018 at 17:13
  • $\begingroup$ For the mapping cylinder: simplices in $C_*(X)$ are sent to the simplices on the "far end" of the cylinder, simplices in $C_*(Y)$ are sent to simplices on the "near end" of the cylinder, and elements of $C_*(X)[1]$ are sent to (a triangulation of) $\Delta \times I$ in the cylinder $X \times I$. I would post a detailed answer on MSE. I don't really think this is appropriate for MO. $\endgroup$
    – mme
    Commented Nov 12, 2018 at 17:17

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