Localization of a space

Remarks

Every simply connected CW complex has a $P$-localization, unique up to homotopy equivalence under $X$. The condition on homotopy groups is equivalent to: $f$ induces isomorphisms $\tilde H_n(X; \Z) \otimes \Z_{(P)} \to \tilde H_n(X_P; \Z)$ in reduced singular homology for all $n$. The localization is universal: every map from $X$ to a $P$-local space factors through $f$, uniquely up to homotopy. The theory extends to nilpotent spaces.

Example: for $n$ odd, the map $S^n \to K(\Q, n)$ that represents a generator of $H^n(S^n; \Q)$ is a rationalization. Indeed, $\pi_k(S^n) \otimes \Q$ is $\Q$ for $k = n$ and $0$ otherwise, by Serre's finiteness theorem, which matches the homotopy of $K(\Q, n)$.

Non-example: the identity of $S^2$ is not a rationalization, because $S^2$ is not $P$-local for $P = \emptyset$: $\pi_2(S^2) \cong \Z$ is not a $\Q$-vector space.