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Let $\mathscr{A}$ be an additive category and $f:X\rightarrow Y$ be a morphism of complexes in $\mathscr{A}$. The question is are there chain morphisms $h,g$ such that $f=gh$ where $h^{n}$ is a monomorphism with retraction(i.e. exists $i^n$ such that $i\circ h=id$) for each $n$ and $g$ is a homotopy equivalence?

I tried to show that $0\rightarrow X^{n}\rightarrow X^n\oplus Y^n\rightarrow Y^n\rightarrow 0$ with the first arrow $[id, f]^{T}$ to construct a monomorphism with retraction but it seems hard to find a homotopy equivalence g. May be my idea is improper or something.

Could anyone give hints on this? Any help will be appreciated.

Jason785
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  • The middle object should be something a bit more complicated. Look up mapping cylinders. – Zhen Lin Apr 07 '15 at 05:40
  • @ZhenLin Thank you for your help. I finally solve the problem with your hints and some references. My teacher hasn't introduced the concept of mapping cylinders yet so I have no idea about it. – Jason785 Apr 07 '15 at 16:15

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