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, 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.