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Are injective Omega-spectra the level-equivalenceS-local objects of symmetric spectra for some class S?

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) 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 setclass 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 level-equivalence-local objects of symmetric spectra?

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) 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 set 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?

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) 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?

replaced S with a better candidate
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skupers
skupers
  • 8.2k
  • 2
  • 44
  • 80

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) 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 set of levelstable equivalences. The corollary 3.1.8 that every level equivalence 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?

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) 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 set of level equivalences. The corollary 3.1.8 that every level equivalence 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?

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) 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 set 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?

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skupers
skupers
  • 8.2k
  • 2
  • 44
  • 80

Are injective Omega-spectra the level-equivalence-local objects of symmetric spectra?

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) 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 set of level equivalences. The corollary 3.1.8 that every level equivalence 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?