nLab ring spectrum

Context

Stable Homotopy theory

stable homotopy theory

Contents

Higher algebra

higher algebra

universal algebra

Contents

Idea

In older literature, the term (commutative) ring spectrum refers to a (commutative) monoid in the stable homotopy category, hence to a spectrum equipped with a product operation, which is associative (and commutative) up to unspecified homotopy.

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.

In terms of modern homotopy theory the monoid structure alone is in general not quite appropriate, since one really needs A-∞ ring struture or even E-∞ ring structure. On the other hand, with a suitably symmetric monoidal smash product of spectra on the given category of spectra, this does follow.

For more on this see at brave new algebra and higher algebra.

Definition

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

This means that a ring spectrum is a monoid in the category $\mathcal{S}$ 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 $A_\infty$-ring spectrum or just an $A_\infty$-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 $A_\infty$-ring spectrum.

Reference

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

An account in terms of model categories is in

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