Ring spectrum (homotopy theory)
Encyclopedia
In stable homotopy theory
, a ring spectrum is a spectrum
E together with a multiplication map
and a unit map
where S is the sphere spectrum
. These maps have to satisfy associativity
and unitality conditions up to homotopy much in the same way as the multiplication of a ring is associative and unital. That is,
and
Examples of ring spectra include singular homology
with coefficients in a ring, complex cobordism
, K-theory
, and Morava K-theory
.
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
, a ring spectrum is a spectrum
Spectrum (homotopy theory)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory....
E together with a multiplication map
- μ:E ∧ E → E
and a unit map
- η:S → E,
where S is the sphere spectrum
Sphere spectrum
In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the smallest nontrivial spectrum. It is the suspension spectrum of S0, i.e., a set of two points. Explicitly, the nth space in the sphere spectrum is the n-dimensional sphere Sn, and the structure maps from the suspension...
. These maps have to satisfy associativity
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
and unitality conditions up to homotopy much in the same way as the multiplication of a ring is associative and unital. That is,
- μ (id ∧ μ) ∼ μ (μ ∧ id)
and
- μ (id ∧ η) ∼ id ∼ μ(η ∧ id).
Examples of ring spectra include singular homology
Singular homology
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
with coefficients in a ring, complex cobordism
Complex cobordism
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories...
, K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
, and Morava K-theory
Morava K-theory
In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s...
.