Higher algebra

Stable Homotopy theory

Higher linear algebra



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”.


Postnikov tower

The Postnikov tower of an E E_\infty-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 Γ-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

Revised on February 16, 2016 07:22:10 by Urs Schreiber (