Quasi-isomorphism

Remarks

Every chain homotopy equivalence is a quasi-isomorphism, because chain homotopic maps induce the same maps on homology. Quasi-isomorphisms are closed under composition, but in general a quasi-isomorphism has no chain map inverse up to homotopy.

Example: let $C_\bullet$ have $C_1 = C_0 = \Z$ with $d_1$ multiplication by $2$, and let $D_\bullet$ be $\Z/2\Z$ in degree $0$. Let $f_0 \colon \Z \to \Z/2\Z$ be the reduction and $f_1 = 0$; this is a chain map since $f_0(2x) = 0$. Both complexes have homology $\Z/2\Z$ in degree $0$ and $0$ elsewhere, and $H_0(f)$ sends the class of $1$ to $1$, so $f$ is a quasi-isomorphism. It is not a chain homotopy equivalence: every chain map $D_\bullet \to C_\bullet$ is $0$, because $\Z/2\Z$ has no nonzero map to $\Z$.

Non-example: the zero chain map from $\Z$ in degree $0$ to itself is not a quasi-isomorphism, since $H_0$ of it is the zero map $\Z \to \Z$.