nLab E-infinity-ring



Higher algebra

Stable Homotopy theory

Higher linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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

In terms of model categories, and E 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.


Postnikov tower

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


  • The sphere spectrum 𝕊\mathbb{S} becomes an E E_\infty-ring via the canonical maps S n 1S n 2S n 1+n 2S^{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 E_\infty-rings.

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

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
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


The theory is laid out in

For survey see also

Discussion of a Blakers-Massey theorem for E E_\infty-rings is in

In K(n)-local stable homotopy theory:

See also

Last revised on May 16, 2022 at 07:26:54. See the history of this page for a list of all contributions to it.