Spectrum

A spectrum $\mathcal{E}$ is (roughly) a sequence of based spaces $E_{n}, n \in \mathbb{N}$, provided with maps $f_{n}:\Sigma E_{n} \rightarrow E_{n+1}$ (where $\Sigma$ denotes suspension). There are many different definitions of the category of spectra, but they all yield the same homotopy category, known as the stable homotopy category. The homotopy category of spectra forms a triangulated category (with shifts given by suspension and looping); if we associate to a space $X$ the suspesion spectrum $\Sigma^\infty X$ with $n^{th}$-space $(\Sigma^\infty X)_{n} = \Sigma^{n}X$, the homotopy classes of maps between the suspension spectra of $X$ and $Y$ are the stable homotopy classes of maps between $X$ and $Y$. There is a correspondence between generalized (co)homology theories and spectra as follows. Given a generalized cohomology theory $h^{n}$, the Brown representability theorem gives a (universal) space $E_{n}$ such that $h^{n}(X) = [X,E_{n}]$; the suspension axiom provides the required structure maps for $E_{n}$ to form a spectrum. Conversely, for any spectrum $\mathcal{E}$, the functor $h^n(X) = [X,\Omega^n \mathcal{E}]$ is a generalized cohomology theory, and $h_n(X) = \pi_n(X \wedge \mathcal{E})$ is a generalized homology theory.