The following four categories are models for spectra with Eilenberg-MacLane spectra of the desired form.
- Kan's category of semisimplicial spectra [1]
- The category $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$ of sequential spectra of pointed simplicial sets together with the Kan suspension $\Sigma$
- The category $\mathbf{Sp}^\mathbb{N}(S^1\wedge -)$ of Bousfield-Friedlander spectra
- 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.
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.
$\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$: 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.
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 $A$.
$\mathbf{Sp}^\mathbb{N}(\mathcal{T})$: 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
