Symmetric spectra.

*(English)*Zbl 0931.55006The stable homotopy category of spectra has a nice symmetric monoidal smash product. A central problem since the sixties has been whether it is possible to construct a category of spectra having a symmetric monoidal smash product without passing to the homotopy category.

Recently this problem has been resolved successfully by various authors. They use different models for the category of spectra, but all models come equipped with a Quillen model category structure whose homotopy category is equivalent to the usual category of spectra.

A. D. Elmendorf, I. Kriz, M. A. Mandell and J. P. May’s theory of \(S\)-modules [Rings, modules, and algebras in stable homotopy theory, Math. Surv. Monogr. 47 (1997; Zbl 0894.55001)] is well developed and already widely used. In [Smash products and \(\Gamma\)-spaces, Math. Proc. Camb. Philos. Soc. 126, No. 2, 311-328 (1999)] M. Lydakis gives a smash product on Segal’s \(\Gamma\)-spaces, and more recently he also gave such a structure on simplicial functors. In [Theory Appl. Categ. 1, No. 5, 78-118 (1995; Zbl 0876.55009)] R. W. Thomason claims that there is a smash product on the category of symmetric monoidal categories. In the paper under review, the authors develop a model called symmetric spectra discovered by the third author. Questions about priority may be disputed, but this theory can make the claim to be at least one of the first on the market, if not the first to be published.

The theory has a number of advantages and disadvantages when compared with its competitors, and serious users of smash products of spectra do well not to rely solely on any one of the models. One of the main strengths of the theory behind symmetric spectra is that it is rather transparent, and one gets a hands-on feeling about the spectra. It has the drawback that not all symmetric spectra are fibrant, but the advantage that the model for the sphere spectrum is cofibrant. The weak equivalences are not as easily described as one could wish.

A symmetric spectrum is a sequence of simplicial sets \(X_n\) with \(\Sigma_n\) actions together with structure maps \(S^1\wedge X_n\to X_{n+1}\) such that all iterations \(S^p\wedge X_n\to X_{n+p}\) are \(\Sigma_p\times\Sigma_n\)-equivariant.

To define the smash product, one considers symmetric sequences (do without the structure maps). Here we have a symmetric monoidal structure readily available to us, which the authors call the tensor product. The sphere spectrum \(S\) is a symmetric monoid with respect to the tensor product, and symmetric spectra are nothing but \(S\)-modules. The tensor product over \(S\) is defined in the usual way, and serves as the smash product.

The authors then prove that this theory has the required homotopical properties. They define a closed model structure on symmetric spectra in which the stable equivalences are maps that induce an isomorphism in all cohomology theories. The fibrant symmetric spectra are the \(\Omega\)-spectra. They also prove the crucial result that smashing with any cofibrant symmetric spectrum preserves stable equivalences.

The paper keeps a leisurely pace, and is almost self-contained. It is well written and apart for building new machinery it can almost serve as an introduction to this part of stable homotopy theory for graduate students.

Recently this problem has been resolved successfully by various authors. They use different models for the category of spectra, but all models come equipped with a Quillen model category structure whose homotopy category is equivalent to the usual category of spectra.

A. D. Elmendorf, I. Kriz, M. A. Mandell and J. P. May’s theory of \(S\)-modules [Rings, modules, and algebras in stable homotopy theory, Math. Surv. Monogr. 47 (1997; Zbl 0894.55001)] is well developed and already widely used. In [Smash products and \(\Gamma\)-spaces, Math. Proc. Camb. Philos. Soc. 126, No. 2, 311-328 (1999)] M. Lydakis gives a smash product on Segal’s \(\Gamma\)-spaces, and more recently he also gave such a structure on simplicial functors. In [Theory Appl. Categ. 1, No. 5, 78-118 (1995; Zbl 0876.55009)] R. W. Thomason claims that there is a smash product on the category of symmetric monoidal categories. In the paper under review, the authors develop a model called symmetric spectra discovered by the third author. Questions about priority may be disputed, but this theory can make the claim to be at least one of the first on the market, if not the first to be published.

The theory has a number of advantages and disadvantages when compared with its competitors, and serious users of smash products of spectra do well not to rely solely on any one of the models. One of the main strengths of the theory behind symmetric spectra is that it is rather transparent, and one gets a hands-on feeling about the spectra. It has the drawback that not all symmetric spectra are fibrant, but the advantage that the model for the sphere spectrum is cofibrant. The weak equivalences are not as easily described as one could wish.

A symmetric spectrum is a sequence of simplicial sets \(X_n\) with \(\Sigma_n\) actions together with structure maps \(S^1\wedge X_n\to X_{n+1}\) such that all iterations \(S^p\wedge X_n\to X_{n+p}\) are \(\Sigma_p\times\Sigma_n\)-equivariant.

To define the smash product, one considers symmetric sequences (do without the structure maps). Here we have a symmetric monoidal structure readily available to us, which the authors call the tensor product. The sphere spectrum \(S\) is a symmetric monoid with respect to the tensor product, and symmetric spectra are nothing but \(S\)-modules. The tensor product over \(S\) is defined in the usual way, and serves as the smash product.

The authors then prove that this theory has the required homotopical properties. They define a closed model structure on symmetric spectra in which the stable equivalences are maps that induce an isomorphism in all cohomology theories. The fibrant symmetric spectra are the \(\Omega\)-spectra. They also prove the crucial result that smashing with any cofibrant symmetric spectrum preserves stable equivalences.

The paper keeps a leisurely pace, and is almost self-contained. It is well written and apart for building new machinery it can almost serve as an introduction to this part of stable homotopy theory for graduate students.

Reviewer: Bjørn Dundas (Trondheim)

##### MSC:

55P42 | Stable homotopy theory, spectra |

18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

55U10 | Simplicial sets and complexes in algebraic topology |

55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |

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\textit{M. Hovey} et al., J. Am. Math. Soc. 13, No. 1, 149--208 (2000; Zbl 0931.55006)

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