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I am reading section 1.5 of Weibel's book on homological algebra. I understand the intuition of the mapping cone of a chain complex if we are talking about simplicial homology but I would like to know how broadly the argument generalizes.

Let $f: X\to Y$ ne a continuous map of topological spaces, and $C_{\bullet}(X),C_{\bullet}(Y)$ are the associated singular chain complexes, with $f_\sharp :C_{\bullet}(X)\to C_{\bullet}(Y)$ the induced morphism of chain complexes.

Let $\operatorname{cone}{f}$ be the mapping cone (as a topological space) formed by gluing the cone $C(X)$ to $Y$ along the relation $(x,1)\sim 0$ if $f(x)=y$.

Let $\operatorname{cone}{f_\sharp}$ be the chain complex mapping cone from homological algebra, as constructed in i.e. Weibel.

Are $C_\bullet(\operatorname{cone} f)$ and $\operatorname{cone}(f_{\sharp})$ chain homotopy equivalent? How can I prove this?

It is clear that the map $X\to Y\to \operatorname{cone}f$ is nullhomotopic. I have figured out how to construct a map from $C_n(X)$ to $C_{n+1}(\operatorname{cone}f)$ which implements this chain nullhomotopy. I am not sure how this helps me.

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