nLab E-infinity-ring

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 spectra – highly 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.

Related concepts

• commutative ring, symmetric ring groupoid

• ring spectrum, functor with smash products

• symmetric ring spectrum

• model structure for ring spectra

• Hopf E-infinity ring

• ∞-group of units

• logarithmic cohomology operation

• étale morphism of E-∞ rings

• higher algebra

• E-∞ arithmetic geometry

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space $\simeq$ ∞-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

References

The theory is laid out in

• Peter May with contributions by Frank Quinn, Nigel Ray and Jørgen Tornehave, $E_\infty$-Ring spaces and $E_\infty$ ring spectra, Lecture Notes in Mathematics 577, Springer 1977 (pdf, cds:1690879)

• Jacob Lurie, Higher Algebra

For survey see also

• Moritz Groth, A short course on infinity-categories, pdf.

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

• Michael Ching, John Harper, Higher homotopy excision and Blakers-Massey theorems for structured ring spectra (arXiv:1402.4775)

In K(n)-local stable homotopy theory:

• Michael Hopkins, $K(1)$-local $E_\infty$-Ring spectra (pdf)

See also

• Benoit Fresse, The Bar Complex of an E-infinity algebra (pdf)

• Birgit Richter, Brooke Shipley, An algebraic model for commutative HZ-algebras, arXiv:1411.7238.