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 -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 -ring is an A-∞ ring that is commutative up to coherent higher homotopies. -rings are the analogue in higher algebra of the commutative rings in ordinary algebra.
In terms of model categories, and -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 becomes an -ring via the canonical maps . As such the sphere spectrum is the initial object in -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 Jørgen Tornehave, -Ring spaces and ring spectra, Lecture Notes in Mathematics 577, Springer 1977 (pdf, cds:1690879)
For survey see also
Discussion of a Blakers-Massey theorem for -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.
Last revised on May 16, 2022 at 07:26:54. See the history of this page for a list of all contributions to it.