Spectrum (topology)
From Wikipedia, the free encyclopedia
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory
,
there exist spaces such that evaluating the cohomology theory in degree
on a space
is equivalent to computing the homotopy classes of maps to the space
, that is
.
Note there are several different categories of spectra leading to many technical difficulties,[1] but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory.