nLab
ring spectrum

Idea

A ring spectrum is a model for an A-∞ ring.

The notion of spectrum is the generalization of the notion of abelian group to homotopy theory/(∞,1)-category theory. The notion of ring spectrum is the corresponding generalization of the notion of ring.

Definition

A ring spectrum is a monoid in the stable homotopy category $\mathrm{Ho}(𝒮)$ equipped with the smash product of spectra. See there for more details.

This means that a ring spectrum is a monoid in the category $𝒮$ of spectra up to not-necessarily coherent homotopy.

A monoid-up-to-homotopy in the category of spectra for which the homotopies are coherent is called an $A_{\mathrm{\infty }}$-ring spectrum? or just an $A_{\mathrm{\infty }}$-ring. These may be modeled as monoids with respect to the symmetric monoidal smash product of spectra.

Abstractly these are E1-algebras in the symmetric monoidal (∞,1)-category Spec of spectra.

Properties

Not every ring spectrum may be refined to an $A_{\mathrm{\infty }}$-ring spectrum.

Related concepts

• module spectrum, algebra spectrum

Reference

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

• Jacob Lurie, Higher Algebra

An account in terms of model categories is in

• A. Elmendorf, I. Kriz, P. May, Modern foundations for stable homotopy theory (pdf)

See also the references at stable homotopy theory.