Stable homotopy category

Remarks

The stable weak equivalences are the weak equivalences of a model structure on spectra. So the localization exists with small morphism sets, and it is the homotopy category of that model structure.

Basic properties:

1. Every morphism set $[E, F]$ is an abelian group, and composition is bilinear.
2. The suspension $\Sigma$ is an equivalence of categories, with inverse the loop functor $\Omega$.
3. With $\Sigma$ as shift and cofibre sequences as distinguished triangles, $\mathrm{Ho}(\mathbf{Sp})$ is a triangulated category.
4. The smash product makes it a symmetric monoidal category, with the sphere spectrum as unit.

Example: for finite pointed CW complexes $X$ and $Y$, $[\Sigma^\infty X, \Sigma^\infty Y] \cong \operatorname{colim}_k [\Sigma^k X, \Sigma^k Y]_*$, the group of stable maps.

Non-example: in the homotopy category of pointed CW complexes, suspension is not an equivalence. The suspension map $\pi_3(S^2) = [S^3, S^2]_* \to [S^4, S^3]_* = \pi_4(S^3)$ goes from $\Z$ onto $\Z/2$, so it is not injective, and $\Sigma$ is not fully faithful there.

The first construction of a stable homotopy category with a well-behaved smash product is due to Boardman. Modern models, such as symmetric spectra and orthogonal spectra, have homotopy categories equivalent to it.