= Localization of spectra
{title2=$L_EX$}
Given a <topological spectrum> $E$, its <localization of spectra> sends $X$ to an $E$-local <topological spectrum> $L_EX$ through a map $X\to L_EX$ inducing an <isomorphism> on $E$-<homology>. Here $E$-local means that every map from an $E$-acyclic <topological spectrum> to $L_EX$ is nullhomotopic; $A$ is $E$-acyclic when $E\wedge A$ is trivial. This is Bousfield localization.<Bousfield Localization by Jacob Lurie>[Lecture 20, Definition 2 and Example 4]{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.
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