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  "body": "\n## Definition\n\nLet $\\mathbf{Sp}$ be the category of [[spectra|1GHF4EKBXS]] and maps of spectra.\nA map $f \\colon E \\to F$ of spectra is a *stable weak equivalence* if $\\pi_k(f) \\colon \\pi_k(E) \\to \\pi_k(F)$ is an isomorphism of [[homotopy groups|P976WYEFMB]] for every $k \\in \\Z$.\nThe *stable homotopy category* $\\mathrm{Ho}(\\mathbf{Sp})$ is the localization of $\\mathbf{Sp}$ at the stable weak equivalences.\nIt has the same objects as $\\mathbf{Sp}$.\nIt is the universal category under $\\mathbf{Sp}$ in which every stable weak equivalence becomes an isomorphism.\n\n## Remarks\n\nThe stable weak equivalences are the weak equivalences of a model structure on spectra.\nSo the localization exists with small morphism sets, and it is the homotopy category of that model structure.\n\nBasic properties:\n\n1. Every morphism set $[E, F]$ is an abelian group, and composition is bilinear.\n2. The suspension $\\Sigma$ is an equivalence of categories, with inverse the loop functor $\\Omega$.\n3. With $\\Sigma$ as shift and cofibre sequences as distinguished triangles, $\\mathrm{Ho}(\\mathbf{Sp})$ is a triangulated category.\n4. The smash product makes it a symmetric monoidal category, with the sphere spectrum as unit.\n\nExample: 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.\n\nNon-example: in the homotopy category of pointed CW complexes, suspension is not an equivalence.\nThe 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.\n\nThe first construction of a stable homotopy category with a well-behaved smash product is due to Boardman.\nModern models, such as symmetric spectra and orthogonal spectra, have homotopy categories equivalent to it.\n",
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