symmetric monoidal (∞,1)-category of spectra
The terminology (commutative) ring spectrum refers either to a (commutative) monoid in the stable homotopy category regarded as monoidal category via the smash product of spectra, or to the richer structure of a monoid in a model structure for spectra equipped with a symmetric smash product of spectra.
In the first case a ring spectrum is a spectrum equipped with a unit and product operation, which is associative, unital (and commutative) just up to unspecified homotopy, such as in an H-space structure. Accordingly, these might be called “H-ring spectra”, but it is traditional to just call them “ring specta.” (An H-infinity ring spectrum (Bruner-May-McLure-Steinberger 86) is such an H-ring spectrum equipped with some extra structure modeling extended power operations).
In the second case the structure is much richer; and in good cases, such as for highly structured spectra, is equivalent to A-∞ ring structure (E-∞ ring structure).
To distinguish the two situations further qualification is being used. Sometimes one says homotopy ring spectrum to explicitly refer to the first case (e.g. Schwede 12, chapter II 4.1) or one says “highly structured ring spectrum” to refer explicitly to the second case. For more on this see at brave new algebra and higher algebra.
Since the concept of spectrum is the refinement of the concept of abelian group to homotopy theory/(∞,1)-category theory. The concept of ring spectrum is the corresponding generalization of the notion of (commutative) ring.
algebra | homological algebra | higher algebra |
---|---|---|
abelian group | chain complex | spectrum |
ring | dg-ring | ring spectrum |
module | dg-module | module spectrum |
For details see Introduction to Stable homotopy theory, Part 1-2 -- Structured spectra.
Under the Brown representability theorem, the generalized cohomology theory represented by a ring spectrum inherits the structure of a multiplicative cohomology theory.
Conversely via the Brown representability theorem a spectrum representing a multiplicative cohomology theory inherits the structure of (at least) an H-ring spectrum. See there.
Frank Adams, part III, section 10 of Stable homotopy and generalised homology, 1974
Dai Tamaki, Akira Kono, appendix C of Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (pdf)
R. Bruner, Peter May, James McLure, M. Steinberger, $H_\infty$-Ring Spectra and their Applications, Lecture Notes in Mathematics 1176, Springer 1986 (pdf)
Quick survey includes
Discussion of connective ring spectra as monoids with respect to a smash product on Gamma-spaces is in
Stefan Schwede, Stable homotopical algebra and $\Gamma$-spaces, Math. Proc. Camb. Phil. Soc. (1999), 126, 329 (pdf)
Tyler Lawson, Commutative Γ-rings do not model all commutative ring spectra, Homology Homotopy Appl. Volume 11, Number 2 (2009), 189-194. (Euclid)
Discussion of ring spectra as rings with respect to the symmetric smash product of spectra on S-modules includes
Anthony Elmendorf, Igor Kriz, Peter May, sections 2 and 3 of Modern foundations for stable homotopy theory (pdf)
Anthony Elmendorf, Igor Kriz, Michael Mandell, P. May, Rings, modules and algebras in stable homotopy theory, AMS Mathematical Surveys and Monographs Volume 47 (1997) (pdf)
See also
A comprehensive account for for symmetric spectra is in
and for orthogonal spectra in
An account in terms of (∞,1)-category theory is in section 7.1 of
Discussion of simplicial ring spectra is in
See also the references at stable homotopy theory.