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  "body": "\n## Definition\n\nA *spectrum* $E$ is a sequence $(E_n)_{n \\ge 0}$ of [[pointed spaces|T9DG8DDE75]] together with pointed maps\n$$\\sigma_n \\colon \\Sigma E_n \\to E_{n+1}, \\qquad n \\ge 0,$$\ncalled structure maps, where $\\Sigma$ is the reduced [[suspension|2VD7GR64JJ]].\nA *map of spectra* $f \\colon E \\to F$ is a sequence of pointed maps $f_n \\colon E_n \\to F_n$ with $f_{n+1} \\circ \\sigma_n = \\sigma_n \\circ \\Sigma f_n$ for all $n$.\nFor $k \\in \\Z$, the $k$-th *homotopy group* of $E$ is\n$$\\pi_k(E) = \\operatorname{colim}_n \\, \\pi_{k+n}(E_n),$$\nwhere the colimit runs over $n \\ge \\max(0, 2 - k)$ along the maps $\\pi_{k+n}(E_n) \\to \\pi_{k+n+1}(\\Sigma E_n) \\to \\pi_{k+n+1}(E_{n+1})$.\n\n## Remarks\n\nA spectrum is an *$\\Omega$-spectrum* if each adjoint map $E_n \\to \\Omega E_{n+1}$ into the [[loop space|2JFF5F3EGF]] is a [[weak homotopy equivalence|4Q3WY1RX91]].\nThen $\\pi_k(E) = \\pi_{k+n}(E_n)$ for all $n \\ge 0$ with $k + n \\ge 0$.\nA map of spectra is a *stable weak equivalence* if it induces isomorphisms on all $\\pi_k$.\n\nExample: for a pointed space $X$, the *suspension spectrum* $\\Sigma^\\infty X$ has $(\\Sigma^\\infty X)_n = \\Sigma^n X$ and identity structure maps.\nIts homotopy groups are the stable homotopy groups $\\pi_k^s(X)$ for $k \\ge 0$, and they vanish for $k \u003c 0$.\n\nExample: for an abelian group $G$, choose [[Eilenberg–MacLane spaces|AJHEYJE5XY]] $K(G, n)$ 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 $k \\neq 0$.\n\nNon-example: a sequence of pointed spaces alone is not a spectrum, since the structure maps are part of the data.\nFor instance, $E_n = S^n$ with every $\\sigma_n$ constant is a spectrum, but it is not the sphere spectrum.\nAll its homotopy groups are zero, because the maps in the colimit are zero.\n\nAn $\\Omega$-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.\nOther models, such as symmetric spectra and orthogonal spectra, add structure to get a well-behaved smash product.\n",
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