nLab S-module




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 “SS” stands for the sphere spectrum (other traditional notation being “QS 0Q S^0” for the suspension spectrum QQ 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.


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 𝕃\mathbb{L} on the category 𝒮\mathcal{S} of such spectra, and the category of 𝕃\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 𝕃\mathbb{L}-algebras for which the unit transformations are isomorphisms is again a well-behaved model for Ho𝕊Ho \mathbb{S}, which is moreover symmetric monoidal.

Since the unit transformation is of the form SEES\wedge E \to E, where SS is the sphere spectrum, and this map looks like the action of a ring on a module, the objects of this subcategory are called SS-modules and the category is called Mod SMod_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 Mod_\mathbb{Z} of \mathbb{Z}-modules, i.e. a \mathbb{Z}-algebra, an A A_\infty or E E_\infty ring spectrum is a (possibly commutative) monoid in the category of SS-modules, and thus referred to as an SS-algebra. More generally, for any A A_\infty-ring spectrum RR, there is a notion of RR-module spectra forming a category Mod RMod_R, which in turn carries an associative and commutative smash product R\wedge_R and a model category structure on Mod RMod_R such that R\wedge_R becomes unital in the homotopy category. All this is such that an A A_\infty-algebra over RR is a monoid object in (Mod R, R)(Mod_R, \wedge_R). Similarly E E_\infty-algebras are commutative monoid objects in (Mod R, R)(Mod_R, \wedge_R).

model structure on spectra, symmetric monoidal smash product of spectra


The construction originates in


Last revised on March 14, 2023 at 05:59:09. See the history of this page for a list of all contributions to it.