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I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on the underlying spaces - simplicial sets in Hovey/Shipley/Smith and motivic spaces in Jardine - with monomorphisms as cofibrations. Then both establish two level model structures on symmetric (resp. motivic symmetric) spectra with weak equivalences given levelwise, an injective one with levelwise cofibrations and a projective one with levelwise fibrations.

However, both then proceed using only the projective model structure, and Bousfield localizing it with respect to an appropriate class of maps. Presumably they do so because it is much more straightforward to verify that the projective model structure satisfies the prerequisites for a Bousfield localization, in particular one has easy candidates for generating cofibrations which then turn out to do the job.

My question is: Is the injective level model structure on symmetric spectra (resp. motivic symmetric spectra) known or expected to be cellular or combinatorial? How would you try to prove this?

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You should probably have a look at – Lennart Meier Oct 22 '10 at 19:31
Indeed, that contained what I wanted to know about symmetric spectra - thank you! – Peter Arndt Oct 22 '10 at 23:37
up vote 4 down vote accepted

The answer is yes. As Lennart Meier points out, part of this answer is contained in Schwede´s book project. I´m writing to flesh out those references and discuss the topological injective case, which doesn´t appear in Schwede. Plus, old questions shouldn´t linger around without answers. Let´s start with the case where we build spectra from simplicial sets. Then Theorem 1.9 on page 129 of the Schwede pdf discusses how to do the levelwise injective model structure. Observe that cofibrations are monomorphisms, so all objects are cofibrant, so it´s immediately left proper. Furthermore, because it´s built via sequences of simplicial sets it will remain combinatorial. The same considerations apply for motivic symmetric spectra, i.e. you´ll get something left proper and combinatorial (both cases are actually right proper as well, but this doesn´t matter right now). Thus, Bousfield localization applies and you get the stable injective structures. This is mentioned in Theorem 2.2.

Now let´s think about symmetric spectra built from Top rather than sSet. For motivic symmetric spectra this is the wrong question, since you seem to really need there that everything is built from sSet. For non-motivic one would be tempted to look into the paper Model Categories of Diagram Spectra by Mandell-May-Schwede-Shipley. This paper does a nice job of treating topological symmetric spectra, but doesn´t make mention of the injective structure at all. Similarly Hovey´s Spectra and Symmetric Spectra in General Model Categories also doesn´t mention injective structures. So we turn back to Schwede, but in Theorem 1.13 on page 132, he claims not to know if topological symmetric spectra admit a levelwise injective structure. Of course you can define the classes of maps, and once you have them you can see that if this forms a model structure then it´ll be left proper and cellular (because Top is cellular, see Hovey), so it would admit left Bousfield localization. I see no reason why following Schwede´s proof for the simplicial case would break down here. True, you don´t have combinatoriality, but you should be able to use cellularity to get the necessary smallness to make the small object argument work. This notion doens´t appear in Schwede´s book, which is probably why he didn´t do it. If it´s very important to you, email me and we can try to write down all the details. Otherwise I´ll leave it as an exercise in cellular model categories

Incidentally, if you´re interested in whether or not you can ever get away with fewer hypotheses and still have Bousfield localization, check out this MO question. It´s related to a current project of mine.

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Thanks! If I ever come back to the project which drove me to this question, it will be nice to have this recorded. – Peter Arndt May 15 '13 at 22:20

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