How explicit are the model structures for various categories of spectra?


Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit generating (acyclic) cofibrations, so we have explicit generating cofibrations for them.


I'm thinking it's too much to ask for explicit generating acyclic cofibrations, but it would be nice to at least have a pseudo-generating set -- i.e. an explicit set of generating acyclic cofibrations, lifting against which characterizes the fibrant objects and fibrations between fibrant objects.


**Questions:** Let $\mathcal C$ be a model category modeling spectra, (e.g. naive, [symmetric](https://ncatlab.org/nlab/show/Model+categories+of+diagram+spectra), [orthogonal](https://ncatlab.org/nlab/show/model+structure+on+orthogonal+spectra), [EKMM](https://www.math.uchicago.edu/~may/PAPERS/mmLMSDec30.pdf), [combinatorial](https://ncatlab.org/nlab/show/combinatorial+spectrum#ModelCategoryStructure)...)


1. Are explicit generating cofibrations available for $\mathcal C$?
2. How about explicit generating acyclic cofibrations?
3. If not (2), how about an explicit pseudo-generating set of acyclic cofibrations?
4. If not (3), is there at least an explicit description of the fibrant objects via lifting properties?



For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?
You have explicit generating (acyclic) cofibrations for pretty much any model of spectra you can think of. As you point out, most begin as levelwise model structures, and so you have explicit generating (acyclic) cofibrations before left Bousfield localization, hence still have the same generating cofibrations after. You are right that the general machinery of Hirschhorn's book does not tend to give you explicit generating acyclic cofibrations, but for all the models of spectra I can think of, there is a trick that gives you the generating acyclic cofibrations. I learned this trick from Theorem 3.11 in the paper [Stable left and right Bousfield localisations](https://arxiv.org/pdf/1204.5384.pdf) by Barnes and Roitzheim. The proof has a lovely trick that uses the fact that the levelwise model structure is proper. The set of generating acyclic cofibrations is $J \cup \Lambda S$, where $J$ is the old set of generating acyclic cofibrations, and $\Lambda S$ is the set of horns on $S$, where $S$ is the set of maps you're inverting. This result is the best Hirschhorn could have hoped for, but as examples in his book show, you don't get this result without the assumption of a stable, proper model category as input. Before this result of Barnes and Roitzheim, the usual idea was to build the left Bousfield localization "by hand", without reference to Hirschhorn's book. That is done in several of Mark Hovey's papers, and it works because you know you want the new fibrant objects to be the $\Omega$-spectra. When you do it that way, you often have reasonably good control over the generating acyclic cofibrations, but the approach of Barnes and Roitzheim is even better.


