symmetric monoidal (∞,1)-category of spectra
A symmetric smash product of spectra is a realization of the smash product of spectra such as to make a symmetric monoidal model category presentation of the symmetric monoidal (infinity,1)-category of spectra.
In higher algebra and stable homotopy theory one is interested in monoid objects in the stable (∞,1)-category of spectra – called $A_\infty$-rings – and commutative monoid objects – called $E_\infty$-rings. These monoid objects satisfy associativity, uniticity and, in the $E_\infty$-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 ordinary commutative monoids in a suitable ordinary category – one speaks of highly structured ring spectra.
Historically, this had been desired but out of reach for a long time, due to the initial focus on the model by plain sequential spectra. By this remark at smash product of spectra, plain sequential spectra do not reflect the graded-commutativity implicit in the braiding of the smash product of n-spheres and thus do not admit a symmetric smash product of spectra.
When the relevant highly structured ring spectra were finally found that do admit symmetric smash products, the relief was substantial and led to terminology such as “brave new algebra”. More recently maybe the term higher algebra is becoming more popular.
Then, model structures were found which also admit symmetric monoidal smash products, but which are not of the form “highly structured spectra”: model structure for excisive functors.
As a first step one wants a model category of spectra $\mathcal{S}$ 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 $Ho \mathcal{S}$.
It was also known that the stable homotopy category $Ho \mathcal{S}$ is a symmetric monoidal category, via a “derived smash product of spectra.” Ordinary (commutative) monoids in $Ho \mathcal{S}$ 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 (see at H-space).
One could, however, build in coherent associativity (resp. commutativity) homotopies by the use of an operad, by using an $A_\infty$-operad (resp. an $E_\infty$-operad). This resulted in the notions of $A_\infty$-ring spectrum and $E_\infty$-ring spectrum, which have a much better-behaved theory.
Now, for some of the model categories $\mathcal{S}$ of spectra, the smash product on $Ho \mathcal{S}$ can be lifted to a functor
but for the most part these functors were neither associative nor unital nor commutative at the level of the 1-category $\mathcal{S}$. In fact (Lewis 91) proved a theorem that there could be no symmetric monoidal category $\mathcal{S}$ 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-May 97, 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 $\mathcal{S}$ absorbs all the higher coherent homotopies that used to be supplied by the action of an $A_\infty$ or $E_\infty$ operad. Thus, honest (commutative) monoids in $\mathcal{S}$ model the same “(commutative) ring objects up to all coherent higher homotopies” that are modeled by the classical $A_\infty$ and $E_\infty$ ring spectra, and for this reason they are often still referred to as $A_\infty$ or $E_\infty$ ring spectra, respectively.
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 $\mathbb{L}$ on the category $\mathcal{S}$ of such spectra, and the category of $\mathbb{L}$-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 $\mathbb{L}$-algebras for which the unit transformations are isomorphisms is again a well-behaved model for $Ho \mathbb{S}$, 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 $Mod_S$. 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 $Mod_\mathbb{Z}$ of $\mathbb{Z}$-modules, i.e. a $\mathbb{Z}$-algebra, an $A_\infty$ or $E_\infty$ 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 $A_\infty$-ring spectrum $R$, there is a notion of $R$-module spectra forming a category $Mod_R$, which in turn carries an associative and commutative smash product $\wedge_R$ and a model category structure on $Mod_R$ such that $\wedge_R$ becomes unital in the homotopy category. All this is such that an $A_\infty$-algebra over $R$ is a monoid object in $(Mod_R, \wedge_R)$. Similarly $E_\infty$-algebras are commutative monoid objects in $(Mod_R, \wedge_R)$.
Also the
carries a symmetric monoida smash product.
The original no-go theorem for a well-behave smash product of spectra is
In the mid-1990s, several categories of spectra with nice smash products were discovered, and simultaneously, model categories experienced a major renaissance.
The definition of S-modules and their theory originates in
and around 1993 Jeff Smith gave the first talks about symmetric spectra; the details of the model structure were later worked out and written up in
Discussion that makes the Day convolution structure on the symmetric smash product of spectra manifest is in
Surveys of the history are in
Anthony Elmendorf, Igor Kriz, Peter May, Modern foundations for stable homotopy theory,1995 (pdf)
Stefan Schwede, p.214-216 of Symmetric spectra (2012)
A textbook account of the theory of symmetric spectra is
Seminar notes on symmetric spectra are in
See also