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 “” stands for the sphere spectrum (other traditional notation being “” for the suspension spectrum of the 0-sphere) regarded as a ring spectrum. Namely, in homotopification of how general abelian groups may equivalently be understood as -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 on the category of such spectra, and the category of -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 -algebras for which the unit transformations are isomorphisms is again a well-behaved model for , which is moreover symmetric monoidal.
Since the unit transformation is of the form , where is the sphere spectrum, and this map looks like the action of a ring on a module, the objects of this subcategory are called -modules and the category is called . The intuition is that just as an abelian group is a module over the archetypical ring 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 of -modules, i.e. a -algebra, an or ring spectrum is a (possibly commutative) monoid in the category of -modules, and thus referred to as an -algebra. More generally, for any -ring spectrum , there is a notion of -module spectra forming a category , which in turn carries an associative and commutative smash product and a model category structure on such that becomes unital in the homotopy category. All this is such that an -algebra over is a monoid object in . Similarly -algebras are commutative monoid objects in .
model structure on spectra, symmetric monoidal smash product of spectra
S-module, model structure on S-modules
The construction originates in
Review:
Last revised on March 14, 2023 at 05:59:09. See the history of this page for a list of all contributions to it.