There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct Eilenberg-MacLane spectra $HA$, where $A$ is an abelian group. In $S$-modules, this is described in Section IV.2 of this version of [EKMM][1]. In symmetric spectra, this is described in Example 1.14 of Stefan Schwede's [book project (version 3.0)][2].

> **Question:** Are there models of Eilenberg-MacLane spectra that are
> fibrant, cofibrant, and (strict) abelian group objects with respect to
> the addition map $+ \colon HA \times HA \to HA$?

My first candidate was symmetric spectra because there, the construction of $HA$ follows directly from a standard construction of Eilenberg-MacLane spaces $K(A,n)$ as topological abelian groups (or simplicial abelian groups, if working in simplicial sets). In particular, $HA$ is an abelian group object. Moreover, $HA$ is an $\Omega$-spectrum, and those are the fibrant objects in, for instance, the absolute projective stable model structure (Theorem III.4.11 in Schwede's book, or Theorem 3.4.4 in [Hovey—Shipley—Smith][3]). However, I'm still missing cofibrancy, and I suspect that a cofibrant replacement would mess up the abelian group object structure.

Another idea would be to use the various stable model structures on symmetric spectra. One could also try in $S$-modules, where every object is fibrant.

For the record, an associative smash product is not crucial to my purposes, though of course it would be nice.


  [1]: http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf
  [2]: http://www.math.uni-bonn.de/people/schwede/SymSpec-v3.pdf
  [3]: http://www.ams.org/mathscinet-getitem?mr=1695653