Definition: A spectrum \(E\) represents a cohomology theory \(h\) iff \begin{align*} h^{\color{blue} n}(X) = \lim_{{\color{red}k}\to\infty} [ \Sigma^{{\color{red} k} - {\color{blue} n }} X, E_{\color{red} k} ] \end{align*}
Theorem: Any cohomology theory defined on the category of compact topological spaces can be extended to a cohomology theory on \({\mathsf{Top}}\).
To represent singular cohomology with coefficients in \(G\), take the suspension spectrum of \(K(G, 1)\) denoted \(HG\) (note: probably not right!!!) then \(H^n(X; G) = [X, HG_n] = [X, K(G, n)]\).
Homotopy groups of spectra, e.g. \begin{align*} \pi_{k}(H G)=\left\{\begin{array}{ll} G & k=0 \\ 0 & k \neq 0 \end{array}\right. \end{align*} Motivating example: the cobordism spectrum.