nLab ring spectrum

Idea

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 spectra.” (An H-infinity ring spectrum (Bruner-May-McClure-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

Definition

For details see Introduction to Stable homotopy theory, Part 1-2 – Structured spectra.

Properties

Relation to multiplicative generalized cohomology

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.

Related concepts

• ultracommutative ring spectrum

• symmetric ring spectrum

• symmetric algebra spectrum

• module spectrum, algebra spectrum

• Gorenstein ring spectrum

• functor with smash products

• periodic ring spectrum

• Hopf ring spectrum

• model structure for ring spectra

• Boardman homomorphism, Adams spectral sequence

• multiplicative spectral sequence

References

The notion of (commutative) homotopy ring spectra, i.e. (commutative) monoids in the stable homotopy category with respect to the smash product of spectra:

• Frank Adams, part III, section 10 of Stable homotopy and generalised homology, 1974 (pdf)

  (attributed there to George Whitehead)

Review:

• John Michael Boardman, Sections 3,7 of: Stable Operations in Generalized Cohomology [pdf, pdf] in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

• Andrew Baker, Birgit Richter Structured ring spectra, London Mathematical Society Lecture Notes Series 315, Springer 2004 (ISBN:9780521603058)

• Dai Tamaki, Akira Kono, Appendix C.2 of: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (pdf, ISBN: 978-0-8218-3514-2)

• Cary Malkiewich, section 1.3 of The stable homotopy category, 2014 (pdf)

• Urs Schreiber, Introduction to Stable homotopy theory – Homotopy ring spectra, 2016

• Birgit Richter, Commutative ring spectra (arXiv:1710.02328, pdf), in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds.), Stable categories and structured ring spectra, MSRI Book Series, Cambridge University Press.

On H-infinity ring spectra:

• R. Bruner, Peter May, James McClure, M. Steinberger, $H_\infty$-Ring Spectra and their Applications, Lecture Notes in Mathematics 1176, Springer 1986 (doi:10.1007/BFb0075405, pdf)

Discussion of connective ring spectra as monoids with respect to a smash product on Gamma-spaces:

• 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)

A comprehensive account for symmetric spectra is in

• Stefan Schwede, Symmetric spectra, 2012 (pdf)

and for orthogonal spectra in

• Stefan Schwede, Global homotopy theory

An account in terms of (∞,1)-category theory is in section 7.1 of

• Jacob Lurie, Higher Algebra

Discussion of simplicial ring spectra is in

• Paul Goerss, Michael Hopkins, Simplicial Structured Ring Spectra (1999) (web)

See also the references at stable homotopy theory.

• Glossary for stable and chromatic homotopy theory (pdf)