If $fg$ and $gf$ are quasi-isomorphisms, are $f,g$ a chain homotopy equivalence? $ \newcommand\A{\mathcal{A}} \newcommand\comp{^{\bullet}} $ Suppose $\A$ is an Abelian category and $A\comp$, $B\comp$ two cochain complexes with cochain maps $f\colon A\comp\to B\comp$ and $g\colon B\comp\to A\comp$ such that $fg$ and $gf$ are both quasi-isomorphisms. Are $f,g$ a chain homotopy equivalence?

No. For instance, let $A^*=0$ and let $B^*$ be a short exact sequence $0\to B^0\stackrel{d_0}\to B^1\stackrel{d_1}\to B^2\to 0$ that does not split, with $f$ and $g$ the zero maps. Then $fg$ and $gf$ are both quasi-isomorphisms, since $A^*$ and $B^*$ are both exact.

If $f$ and $g$ were a chain homotopy equivalence, then $fg=0$ would be chain-homotopic to the identity $1_{B^*}$. Such a chain homotopy would consist of maps $h_1:B^1\to B^0$ and $h_2:B^2\to B^1$ such that $h_1d_0=1_{B^0}$, $d_0h_1+h_2d_1=1_{B^1}$, and $d_1h_2=1_{B^2}$. But this is exactly the data of a splitting of the short exact sequence $B^*$, and no such splitting exists.