I am trying to read the Hovey-Shipley-Smith article as defining the stable model structure on symmetric spectra as a left Bousfield localization (as explained on [nLab](http://ncatlab.org/nlab/show/Bousfield+localization+of+model+categories)) of the projective level model structure on symmetric spectra, which has level equivalences as weak equivalences.


Almost all ingredients are there in the article. All I have left to show is that the injective Omega-spectra are indeed the S-local objects, where S is the class of stable equivalences. By defintion any map in S induces a weak equivalence of simplicial hom-sets $Map\_{Sp^\Sigma}(f,E)$ for E a injective Omega-spectrum. Conversely, lemma 3.1.5 and example 3.1.10 conspire to tell you that if a symmetric spectrum is S-local and injective, it is an Omega-spectrum. So, what remains: is any S-local symmetric spectrum injective?



Are injective Omega-spectra the S-local objects of symmetric spectra for some class S?
You have to realize it has been a long time since we wrote that paper. But I'll give it my best shot. 


I think we intentionally chose the injective Omega-spectra because they are "extra fibrant", so to speak. That is, I think S-local spectra don't have to be injective, just Omega-spectra. 


The injective Omega-spectra should be the fibrant objects in a different model structure. There should be an injective level structure, which I guess we did not construct, where the cofibrations are monomorphisms and the weak equivalences are level equivalences. The fibrant objects would then be the injective spectra. The injective Omega-spectra are then the fibrant objects in the left Bousfield localization of this category with respect to the stable equivalences. 


