Localization of spectra ($L_{E}X$)

Given a topological spectrum $E$, its localization of spectra sends $X$ to an $E$-local topological spectrum $L_{E}X$ through a map $X\to L_{E}X$ inducing an isomorphism on $E$-homology. Here $E$-local means that every map from an $E$-acyclic topological spectrum to $L_{E}X$ is nullhomotopic; $A$ is $E$-acyclic when $E\wedge A$ is trivial. This is Bousfield localization.^[ref]

The map is universal for maps from $X$ to $E$-local objects. For rational homology, localization rationalizes homotopy groups: ${\pi }_j(L_{H\mathbb{Q}}X)={\pi }_j(X)\otimes \mathbb{Q}$. Thus the sphere spectrum loses its finite torsion information and retains $\mathbb{Q}$ in degree zero. Localization deliberately preserves the information seen by a chosen generalized homology theory while discarding information invisible to it.