Rational homotopy theory

Remarks

The condition on $f$ is equivalent to: $H_n(f; \Q)$ is an isomorphism in singular homology for all $n$.

Example: the rationalization $S^3 \to K(\Q, 3)$ is a rational homotopy equivalence, since both sides have rational homotopy $\Q$ in degree $3$ and $0$ in other degrees. So $S^3$ and $K(\Q, 3)$ have the same rational homotopy type.

Non-example: the map $S^2 \to K(\Q, 2)$ that represents a generator of $H^2(S^2; \Q)$ is an isomorphism on $\pi_2 \otimes \Q$. It is not a rational homotopy equivalence: $\pi_3(S^2) \otimes \Q \cong \Q$, generated by the Hopf map, while $\pi_3 K(\Q, 2) = 0$.

Sullivan showed that this homotopy category, with finiteness conditions, is equivalent to a homotopy category of commutative differential graded algebras over $\Q$. Quillen showed that it is also equivalent to a homotopy category of differential graded Lie algebras over $\Q$.