Juvitop on Bousfield Localization

Homotopy theory studies topological spaces up to homotopy, so it must study the functor π∗ : Top → Π−algebras. Passage to the homotopy category is declaring π∗-isomorphisms to be isomorphisms. Because this functor is difficult to compute, one way to do homotopy theory is to study simpler functors which also map to graded abelian categories, e.g. πQ ∗ , H∗(−;Z), K∗, or more generally E∗ for E some generalized homology theory. The study of these functors reveals information about spaces as seen through the eyes of these various homology theories. It also reveals information about the spectra representing these homology theories, namely how much of Top they can “see.” To study this we must work in a category where the isomorphisms are exactly the E∗-isomorphisms, i.e. the maps f such that E∗ is an isomorphism. These are the maps seen as isomorphisms by E. Is there always such a category? What about if we’re working in Spectra instead of Top?