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added two models, corrected the description of Wbar, added references

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edit approved Sep 18, 2015 at 18:50

Kan's category of semisimplicial spectra and the category of sequential spectra of pointed simplicial sets together with the Kan suspensionThe 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 $\mathcal{T}$

Ken Brown equipped the category of semisimplicial spectra with a model structure in which every object is cofibrant and group objectsThese four models are fibrantconnected 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 showed that there is a zig-zag of Quillen equivalences betweenwork with the model categoriescategory of semisimplicialsequential spectra and of Bousfield-Friedlander spectra withpointed topological spaces instead of $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$, but the stable model structure. In particular, the homotopy category of semisimplicial spectra is and the stable homotopy categorycorresponding Quillen equivalences also exist for $\mathbf{Sp}^\mathbb{N}(\mathcal{T})$.)

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 groupsWe elaborate on each model. Considering an abelian group $A$ as an unbounded chain complex concentrated in degree $0$ yields under this correspondence an abelian group object $HA$ in semisimplicial

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 that models the Eilenberg-MacLane spectrum and that is both cofibrant and fibrant.

The zig-zag of Quillen equivalences by[2] Bousfield and Friedlander passes through the category of sequential spectra of pointed simplicial sets with the Kan suspension. More precisely, there is a right Quillen equivalence $\mathrm{Ps}$ from the category of semisimplicial spectra to the category of sequential spectra with the stable model structure. For any semisimplicial spectrum $X$, the structure mapsHomotopy theory 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$\Gamma $-spaces, since $\mathrm{Ps}$ is a right Quillen functor. For the semisimplicial spectrum $HA$ abovespectra, the sequential spectrum $\mathrm{Ps}(HA)$ is an Eilenberg-MacLane spectrum of the desired form and is explicitly given by the sequence of pointed simplicialbisimplicial 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

[3] Brown, Abstract homotopy theory and $\overline{W}$ is the right adjoint of Kan's loop group functor.generalized sheaf cohomology

Kan's category of semisimplicial spectra and the category of sequential spectra of pointed simplicial sets together with the Kan suspension are models for spectra with Eilenberg-MacLane spectra of the desired form.

Ken Brown equipped the category of semisimplicial spectra with a model structure in which every object is cofibrant and group objects are fibrant. Bousfield and Friedlander showed that there is a zig-zag of Quillen equivalences between the model categories of semisimplicial spectra and of Bousfield-Friedlander spectra with the stable model structure. In particular, the homotopy category of semisimplicial spectra is the stable homotopy category.

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 $0$ 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.

The zig-zag of Quillen equivalences by Bousfield and Friedlander passes through the category of sequential spectra of pointed simplicial sets with the Kan suspension. More precisely, there is a right Quillen equivalence $\mathrm{Ps}$ from the category of semisimplicial spectra to the category of sequential spectra 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 the right adjoint of Kan's loop group functor.

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

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created Sep 11, 2015 at 18:46

Marc Stephan

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 $0$ 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.