nLab E-infinity-ring

Contents

Context

Higher algebra

higher algebra

universal algebra

Contents

Idea

An $E_\infty$-ring is a commutative monoid in the stable (∞,1)-category of spectra, an E-∞ algebra in spectra. This is (up to equivalence) also called a highly structured ring spectrum.

This means that an $E_\infty$-ring is an A-∞ ring that is commutative up to coherent higher homotopies. $E_\infty$-rings are the analogue in higher algebra of the commutative rings in ordinary algebra.

In terms of model categories, and $E_\infty$-rings may be modeled as ordinary commutative monoids with respect to the symmetric monoidal smash product of spectrahighly structured ring spectra, a fact sometimes referred to as “brave new algebra”. For details see Introduction to Stable homotopy theory, Part 1-2 – Structured spectra.

Properties

Postnikov tower

The Postnikov tower of a connective E-infinity-ring? is a sequence of square-zero extensions (Lurie, section 8.4).

Examples

• The sphere spectrum $\mathbb{S}$ becomes an $E_\infty$-ring via the canonical maps $S^{n_1} \wedge S^{n_2} \stackrel{\simeq}{\longrightarrow} S^{n_1 + n_2}$. As such the sphere spectrum is the initial object in $E_\infty$-rings.

• Given any ∞-group, there is its ∞-group ∞-ring.

A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ ∞-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
stabilizationspectrumspectrum object

References

The theory is laid out in

Discussion of a Blakers-Massey theorem for $E_\infty$-rings is in
• Michael Hopkins, $K(1)$-local $E_\infty$-Ring spectra (pdf)