symmetric monoidal (∞,1)-category of spectra
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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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.
The Postnikov tower of a connective E-infinity-ring? is a sequence of square-zero extensions (Lurie, section 8.4).
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.
The theory is laid out in
Peter May with contributions by Frank Quinn, Nigel Ray and Jorgen Tornehave, $E_\infty$-Ring spaces and $E_\infty$ ring spectra (pdf)
For survey see also
Discussion of a Blakers-Massey theorem for $E_\infty$-rings is in
In K(n)-local stable homotopy theory:
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.