# 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

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

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

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

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.