Spectrum (topology)

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

${\displaystyle {\mathcal {E}}^{*}:{\text{CW}}^{op}\to {\text{Ab}}}$,

there exist spaces ${\displaystyle E^{k}}$ such that evaluating the cohomology theory in degree ${\displaystyle k}$ on a space ${\displaystyle X}$ is equivalent to computing the homotopy classes of maps to the space ${\displaystyle E^{k}}$, that is

${\displaystyle {\mathcal {E}}^{k}(X)\cong \left[X,E^{k}\right]}$.

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.