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](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf). In symmetric spectra, this is described in Example 1.14 of Stefan Schwede's [book project (version 3.0)](http://www.math.uni-bonn.de/people/schwede/SymSpec-v3.pdf).



> 
> **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](http://www.ams.org/mathscinet-getitem?mr=1695653)). 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.



Fibrant-cofibrant models of Eilenberg-MacLane spectra
The following four categories are models for spectra with Eilenberg-MacLane spectra of the desired form.


1. Kan's category of semisimplicial spectra [1]
2. The category $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$ of sequential spectra of pointed simplicial sets together with the Kan suspension $\Sigma$
3. The category $\mathbf{Sp}^\mathbb{N}(S^1\wedge -)$ of Bousfield-Friedlander spectra
4. The category $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$ of sequential spectra of pointed weak Hausdorff $k$-spaces $\mathcal{T}$


These four models are connected by Quillen equivalences
$$\mathbf{Sp}^\mathbb{N}(S^1\wedge -) \rightleftarrows \mathbf{Sp}^\mathbb{N}(\mathcal{T}) \leftrightarrows \mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma)} \rightleftarrows\{\text{semisimplicial spectra}\}$$
as described by Bousfield and Friedlander in [2]. (Strictly speaking, Bousfield and Friedlander work with the category of sequential spectra of pointed topological spaces instead of $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$, but the stable model structure and the corresponding Quillen equivalences also exist for $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$.)


We elaborate on each model.


1. Semisimplicial spectra: Ken Brown equipped the category of semisimplicial spectra with a model structure in which every object is cofibrant and group objects are fibrant in [3]. Kan's stable Dold-Kan correspondence asserts that the category of abelian group objects of semisimplicial spectra is equivalent to the category of unbounded chain complexes of abelian groups. Considering an abelian group $A$ as an unbounded chain complex concentrated in degree zero yields under this correspondence an abelian group object $HA$ in semisimplicial spectra that models the Eilenberg-MacLane spectrum and that is both cofibrant and fibrant.
2. $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$: There is a right Quillen equivalence $\mathrm{Ps}$ from the category of semisimplicial spectra to the category of sequential spectra $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$ with the stable model structure. For any semisimplicial spectrum $X$, the structure maps of $\mathrm{Ps}(X)$ are monomorphism. Thus $\mathrm{Ps}(X)$ is cofibrant. Hence if $X$ is a group object (and therefore in particular fibrant), then $\mathrm{Ps}(X)$ is cofibrant, fibrant and a group object as well, since $\mathrm{Ps}$ is a right Quillen functor. For the semisimplicial spectrum $HA$ above, the sequential spectrum $\mathrm{Ps}(HA)$ is an Eilenberg-MacLane spectrum of the desired form and is explicitly given by the sequence of pointed simplicial sets 
$$A, \overline{W}A, \overline{W}\overline{W}A, \ldots, \overline{W}^n A,\ldots$$
where $A$ is considered as a constant simplicial abelian group and $\overline{W}$ is "dual" to the right adjoint of Kan's loop group functor.
3. Bousfield-Friedlander spectra: A model for an Eilenberg-MacLane spectrum of the desired form is given by the sequence
$$A, BA, BBA, \ldots, B^nA,\ldots$$
where $B$ is the classifying space functor given by the diagonal of the bar construction. The structure map $S^1\wedge B^n A\to B^{n+1}A$ in level $k$ is just the inclusion of the $k$-fold wedge of $(B^n A)\_k$ into the $k$-fold product of $(B^n A)\_k$. In particular, this model is cofibrant. One way to show that this model is fibrant is to note that it is precisely the Bousfield-Friedlander spectrum construction of the $\Gamma$-space associated to $A$.
4. $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$: The left Quillen functor from any of the two categories of sequential spectra of pointed simplicial sets to $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$ is induced by geometric realization. In particular, it preserves finite products and thus group objects. As a left Quillen functor, it preserves cofibrant objects. It preserves fibrant objects as well. Thus the left Quillen functor applied to any model of an Eilenberg-MacLane spectrum of the desired form in $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$ or $\mathbf{Sp}^\mathbb{N}(S^1\wedge -)$ yields a model of the desired form in $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$.


References:


[1] Kan, Semisimplicial spectra


[2] Bousfield and Friedlander, Homotopy theory of $\Gamma $-spaces, spectra, and bisimplicial sets


[3] Brown, Abstract homotopy theory and generalized sheaf cohomology


