Shift preserves homotopy equivalences, contractibility, and quasi-isomorphisms

If $f:C\to D$ has homotopy inverse $g$, then [L1] shifts the homotopies $gf\simeq 1_C$ and $fg\simeq 1_D$ to homotopies $g[k]f[k]\simeq 1_{C[k]},f[k]g[k]\simeq 1_{D[k]}\mathrm{.}$ The inverse shift $[-k]$ from [L2] shows this construction stays inside the same homotopy-equivalence class of objects. Thus [L7] shows that shift preserves chain homotopy equivalences. By [L6], the special case of a homotopy equivalence $C\simeq 0$ shows that shift also preserves contractible complexes.

Let $f$ be a quasi-isomorphism. By [L3], the map $H_n(f[k])$ identifies with $H_{n-k}(f)$ for every $n$, so $H_n(f[k])$ is an isomorphism whenever $H_{n-k}(f)$ is. Then [L4] makes $f[k]$ a quasi-isomorphism. This is compatible with step 1.1 and [L5], since every shifted homotopy equivalence is again a quasi-isomorphism.