nLab S-module

Idea

S-modules is the name (due to EKMM 97) of one model of highly structured spectra that supports a symmetric monoidal smash product of spectra.

Here “$S$” stands for the sphere spectrum (other traditional notation being “$Q S^0$” for the suspension spectrum $Q$ of the 0-sphere) regarded as a ring spectrum. Namely, in homotopification of how general abelian groups may equivalently be understood as $\mathbb{Z}$-modules, so general spectra may (once their brave new algebraic stable homotopy theory is set up) equivalently be understood as module spectra over the sphere spectrum.

Concretely, the 1-category of S-modules as set up in EKMM 97 is a presentation (see at model structure on S-modules) of the symmetric monoidal (∞,1)-category of spectra, with the special property that it implements the smash product of spectra such as to yield itself a symmetric monoidal model category of spectra: the model structure on symmetric spectra. This implies in particular that with respect to this symmetric smash product of spectra an E-∞ ring is presented simply as a plain commutative monoid in S-modules.

Other presentations sharing this property are symmetric spectra and orthogonal spectra.

Construction

The construction of S-modules by (EKMM 97) 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 $(Mod_R, \wedge_R)$.

Related concepts

model structure on spectra, symmetric monoidal smash product of spectra

• symmetric spectrum, model structure on symmetric spectra

• orthogonal spectrum, model structure on orthogonal spectra

• S-module, model structure on S-modules

References

The construction originates in

• Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, Rings, modules and algebras in stable homotopy theory, AMS 1997 (2014) [ISBN:978-0-8218-4303-1, pdf]

Review:

• Anthony Elmendorf, Igor Kriz, Peter May, sections 2 and 3 of Modern foundations for stable homotopy theory (pdf)