homotopy theory - 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.