In
stable homotopy theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the F ...
, a ring spectrum is a
spectrum ''E'' together with a multiplication map
:''μ'': ''E'' ∧ ''E'' → ''E''
and a unit map
: ''η'': ''S'' → ''E'',
where ''S'' is the
sphere spectrum In stable homotopy theory, a branch of mathematics, the sphere spectrum ''S'' is the monoidal unit in the category of spectra. It is the suspension spectrum of ''S''0, i.e., a set of two points. Explicitly, the ''n''th space in 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,
: ''μ'' (id ∧ ''μ'') ∼ ''μ'' (''μ'' ∧ id)
and
: ''μ'' (id ∧ ''η'') ∼ id ∼ ''μ''(''η'' ∧ id).
Examples of ring spectra include
singular homology with coefficients in a
ring,
complex cobordism,
K-theory, and
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Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number ''p'' (which is su ...
.
See also
*
Highly structured ring spectrum
References
*
Algebraic topology
Homotopy theory
de:Ringspektrum
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