Spectrum

Remarks

A spectrum is an $\Omega$-spectrum if each adjoint map $E_n \to \Omega E_{n+1}$ into the loop space is a weak homotopy equivalence. Then $\pi_k(E) = \pi_{k+n}(E_n)$ for all $n \ge 0$ with $k + n \ge 0$. A map of spectra is a stable weak equivalence if it induces isomorphisms on all $\pi_k$.

Example: for a pointed space $X$, the suspension spectrum $\Sigma^\infty X$ has $(\Sigma^\infty X)_n = \Sigma^n X$ and identity structure maps. Its homotopy groups are the stable homotopy groups $\pi_k^s(X)$ for $k \ge 0$, and they vanish for $k < 0$.

Example: for an abelian group $G$, choose Eilenberg–MacLane spaces $K(G, n)$ and weak equivalences $K(G, n) \to \Omega K(G, n+1)$; their adjoints make $(K(G, n))_n$ an $\Omega$-spectrum $HG$ with $\pi_0(HG) = G$ and $\pi_k(HG) = 0$ for $k \neq 0$.

Non-example: a sequence of pointed spaces alone is not a spectrum, since the structure maps are part of the data. For instance, $E_n = S^n$ with every $\sigma_n$ constant is a spectrum, but it is not the sphere spectrum. All its homotopy groups are zero, because the maps in the colimit are zero.

An $\Omega$-spectrum $E$ defines a generalized cohomology theory by $E^n(X) = [X, E_n]_*$ for $n \ge 0$, extended to all $n \in \Z$ by suspension. Other models, such as symmetric spectra and orthogonal spectra, add structure to get a well-behaved smash product.