nLab
symmetric monoidal smash product of spectra

Idea

In higher algebra and stable homotopy theory one is interested in monoid objects in the stable (∞,1)-category of spectra – called $A_{\mathrm{\infty }}$-rings – and commutative monoid objects – called $E_{\mathrm{\infty }}$-rings. These monoid objects satisfy associativity, uniticity and, in the $E_{\mathrm{\infty }}$-case, commutativity up to coherent higher homotopies.

For concretely working with these objects, it is often useful to have concrete 1-categorical algebraic models for these intricate higher categorical/homotopical entities. The symmetric monoidal smash product of spectra is a structure that allows to model A-infinity rings as ordinary monoids and E-infinity rings as ordinray commutative monoids in a suitable category.

Prehistory

As a first step one wants a model category $𝒮$ of spectra that presents the full (infinity,1)-category of spectra. This allows to model the notion of equivalence of spectra and of homotopies between maps of spectra. Several such model categories have been known for a long time; all are Quillen equivalent and their common homotopy category is called “the” stable homotopy category $\mathrm{Ho}𝒮$.

It was also known that the stable homotopy category $\mathrm{Ho}𝒮$ is a symmetric monoidal category, via a “derived smash product of spectra.” Ordinary (commutative) monoids in $\mathrm{Ho}𝒮$ were called (commutative) ring spectra. While their product has associativity and uniticity up to homotopy, these homotopies are not specified and not required to satisfy higher coherence laws up to higher homotopies themselves.

One could, however, try to build in coherent associativity (resp. commutativity) homotopies by the use of an operad, by using an $A_{\mathrm{\infty }}$-operad (resp. an $E_{\mathrm{\infty }}$-operad). This resulted in the notions of $A_{\mathrm{\infty }}$-ring spectrum and $E_{\mathrm{\infty }}$-ring spectrum, which have a much better-behaved theory.

Symmetric monoidal categories of spectra

Now, for some of the model categories $𝒮$ of spectra, the smash product on $\mathrm{Ho}𝒮$ can be lifted to a functor

$$\wedge :𝒮\times 𝒮\to 𝒮,$$\wedge\colon \mathcal{S} \times \mathcal{S} \to \mathcal{S} \,,

but for the most part these functors were neither associative nor unital nor commutative at the level of the 1-category $𝒮$. In fact, Gaunce Lewis proved a theorem that there could be no symmetric monoidal category $𝒮$ modeling the stable homotopy category and satisfying a couple of other natural requirements.

However, in the 1990s it was realized that by dropping one or another of Lewis’ other requirements, symmetric monoidal categories of spectra could be produced. The first such category was the category of S-modules described by Elmendorf, Kriz, Mandell, and May, but others soon followed, including symmetric spectra and orthogonal spectra. All of these form symmetric monoidal model categories which are symmetric-monoidally Quillen equivalent.

Moreover, in all of these cases, the monoidal structure on the model category $𝒮$ absorbs all the higher coherent homotopies that used to be supplied by the action of an $A_{\mathrm{\infty }}$ or $E_{\mathrm{\infty }}$ operad. Thus, honest (commutative) monoids in $𝒮$ model the same “(commutative) ring objects up to all coherent higher homotopies” that are modeled by the classical $A_{\mathrm{\infty }}$ and $E_{\mathrm{\infty }}$ ring spectra, and for this reason they are often still referred to as $A_{\mathrm{\infty }}$ or $E_{\mathrm{\infty }}$ ring spectra, respectively.

$S$-modules

The construction of $S$-modules by EKMM begins with the notion of coordinate free Lewis-May spectra. Using the linear isometries operad?, one can construct a monad $𝕃$ on the category $𝒮$ of such spectra, and the category of $𝕃$-algebras is a well-behaved model for the stable homotopy category, and moreover admits a smash product which is associative up to isomorphism, but unital only up to weak equivalence. However, the subcategory of the $𝕃$-algebras for which the unit transformations are isomorphisms is again a well-behaved model for $\mathrm{Ho}𝕊$, which is moreover symmetric monoidal.

Since the unit transformation is of the form $S\wedge E\to E$, where $S$ is the sphere spectrum?, and this map looks like the action of a ring on a module, the objects of this subcategory are called $S$-modules and the category is called ${\mathrm{Mod}}_S$. The intuition is that just as an abelian group is a module over the archetypical ring $\mathbb{Z}$ of integers?, a spectrum should be regarded as a module over the archetypal ring spectrum, namely the sphere spectrum.

Similarly, just as an ordinary ring is a monoid in the category ${\mathrm{Mod}}_{\mathbb{Z}}$ of $\mathbb{Z}$-modules, i.e. a $\mathbb{Z}$-algebra, an $A_{\mathrm{\infty }}$ or $E-\mathrm{\infty }$ ring spectrum is a (possibly commutative) monoid in the category of $S$-modules, and thus referred to as an $S$-algebra. More generally, for any $A_{\mathrm{\infty }}$-ring spectrum $R$, there is a notion of $R$-module spectra forming a category ${\mathrm{Mod}}_R$, which in turn carries an associative and commutative smash product ${\wedge }_R$ and a model category structure on ${\mathrm{Mod}}_R$ such that ${\wedge }_R$ becomes unital in the homotopy category. All this is such that an $A_{\mathrm{\infty }}$-algebra over $R$ is a monoid object in $({\mathrm{Mod}}_R,{\wedge }_R)$. Similarly $E_{\mathrm{\infty }}$-algebras? are commutative monoid objects in $({\mathrm{Mod}}_R,{\wedge }_R)$.

Symmetric spectra

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Orthogonal spectra

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References

A survey of the general theory, also of its history, is

• A. Elmendorf, I. Kriz, P. May, Modern foundations for stable homotopy theory (pdf)

The definition of $S$-modules and their theory ca nbe found in

• A. Elmendorf, I. Kriz, M. Mandell, P. May, Rings, modules and algebras in stable homotopy theory (aka “EKMM”)

The definition of the symmetric monoidal category of symmetric spectra is compiled in the seminar notes

• Sander Kupers?, Symmetric spectra