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In their book, Elmendorf, Kriz, May and Mandell describe a useful category of spectra, called S-modules, where S is the sphere spectrum. Ring objects in this category can be identified with spectra with an action of an $A_\infty$-operad (if I understand correctly) and commutative rings can be identified with spectra with an action of the $E_\infty$-operad. Has anyone written about $E_n$-rings in this category? In particular, are there nice model category structures on categories of $A$-modules and $A$-algebras if $A$ is only an $E_n$-algebra? Can this perhaps be shown by somehow trapping this model structure between the nice model category structures on commutative rings and associative rings?

Thanks!

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  • $\begingroup$ Modules over an algebra over an operad have a model structure. You can then try to put a tensor product compatible with the model structure on $A$-modules. It seems sensible, but I don't know of any reference. $\endgroup$
    – Fernando Muro
    Sep 11, 2014 at 12:12
  • $\begingroup$ Have you seen Mike Mandell's paper on this? He looks at what an $E_1$ through $E_4$ structure on $A$ buy you in terms of the structures on the derived category of $A$-modules. $\endgroup$
    – Sean Tilson
    Dec 11, 2014 at 16:03

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You can pick a model for the operad $E_n$ which receives a map from the associative operad. For instance, the Boardman-Vogt tensor product of the associative operad with $E_{n-1}$ has this property. Then, if you have an algebra $A$ over that operad, it is in particular an associative algebra and you can put a model structure on its category of modules using EKMM's method. The only problem with this method is that it is not clear that the resulting model category has a monoidal structure. According to Lurie, it should have and $E_{n-1}$-monoidal structure but this is quite hard to prove using model categories.

To put a model structure on $E_n$-algebras over $A$ is easy. It suffices to observe that an $E_n$-algebra over $A$ is an $E_n$-algebra in spectra together with a map of $E_n$-algebra from $A$. You can put a model structure on that category by taking the model structure on $E_n$-algebras and then taking the under-category model structure.

Constructing the model structure on $E_n$-algebras is a bit technical and I don't know of a reference for EKMM spectra. However, it has been worked out in details by John Harper in symmetric spectra. In fact John Harper proves it for any operad (see http://www.msp.warwick.ac.uk/agt/2009/09-03/p057.xhtml).

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  • $\begingroup$ So it's not clear that one can actually tensor together two $A$-modules and get another $A$-module back, is that what you're saying? $\endgroup$
    – Jonathan Beardsley
    Sep 11, 2014 at 19:45
  • $\begingroup$ There are (at least) two reasonable definitions of a module over an E_n-algebra. I assumed that you meant module over the underlying associative algebra but maybe you meant module in the operadic sense. In both cases, constructing the monoidal structure is a bit tricky. $\endgroup$
    – Geoffroy Horel
    Sep 11, 2014 at 19:56
  • $\begingroup$ Nah I'm just talking about an $A$-module in the traditional sense. $\endgroup$
    – Jonathan Beardsley
    Sep 11, 2014 at 19:58
  • $\begingroup$ Then the first problem is to construct the tensor product. You want to send $(M,N)$ to $M\otimes_AN$. This does not even make sense since $M$ is not a right $A$-module. If A is at least $E_2$, then $A$ and $A^{op}$ are equivalent and it is not too hard to construct the monoidal structure in the homotopy category. But there is a lot of technicality needed to construct a good point set-level construction of this tensor product. $\endgroup$
    – Geoffroy Horel
    Sep 11, 2014 at 20:03
  • $\begingroup$ I see. Ah, that's frustrating. I mean, yeah, I'm interested entirely in $E_2$ and above, but I guess I was hoping this had been worked out. Thanks! $\endgroup$
    – Jonathan Beardsley
    Sep 11, 2014 at 20:05

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