Editing Spectrum (topology) (section)

{{Short description|Mathematical object}}
In [[algebraic topology]], a branch of [[mathematics]], a '''spectrum''' is an object [[representable functor|representing]] a [[Cohomology#Generalized cohomology theories|generalized cohomology theory]]. Every such cohomology theory is representable, as follows from [[Brown's representability theorem]]. This means that, given a cohomology theory<blockquote><math>\mathcal{E}^*:\text{CW}^{op} \to \text{Ab}</math>,</blockquote>there exist spaces <math>E^k</math> such that evaluating the cohomology theory in degree <math>k</math> on a space <math>X</math> is equivalent to computing the homotopy classes of maps to the space <math>E^k</math>, that is<blockquote><math>\mathcal{E}^k(X) \cong \left[X, E^k\right]</math>.</blockquote>Note there are several different [[category (mathematics)|categories]] of spectra leading to many technical difficulties,<ref name=":0">{{Cite journal|last=Lewis|first=L. Gaunce|date=1991-08-30|title=Is there a convenient category of spectra?|journal=Journal of Pure and Applied Algebra|language=en|volume=73|issue=3|pages=233–246|doi=10.1016/0022-4049(91)90030-6|issn=0022-4049|doi-access=free}}</ref> 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.