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  "body": "\n## Definition\n\nLet $S^1 = \\Delta[1] / \\partial \\Delta[1]$ be the simplicial circle and $S^p = S^1 \\wedge \\cdots \\wedge S^1$ ($p$ factors), with the [[symmetric group|0RJMNEN3BJ]] $\\Sigma_p$ acting by permuting the factors.\nA *symmetric spectrum* $X$ consists of:\n\n1. for each $n \\ge 0$, a pointed simplicial set $X_n$ with a base-point-preserving left action of $\\Sigma_n$;\n2. for each $n \\ge 0$, a pointed map $\\sigma_n \\colon S^1 \\wedge X_n \\to X_{n+1}$;\n3. such that for all $n, p \\ge 0$ the iterated structure map $S^p \\wedge X_n \\to X_{n+p}$ is $\\Sigma_p \\times \\Sigma_n$-equivariant, where $\\Sigma_p \\times \\Sigma_n$ acts on $X_{n+p}$ through the inclusion $\\Sigma_p \\times \\Sigma_n \\subseteq \\Sigma_{p+n}$.\n\nA map $f \\colon X \\to Y$ of symmetric spectra is a sequence of $\\Sigma_n$-equivariant pointed maps $f_n \\colon X_n \\to Y_n$ that commute with the structure maps.\n\n## Remarks\n\nForgetting the actions gives a sequential [[spectrum|1GHF4EKBXS]], after geometric realization.\nThe same definition with [[pointed|T9DG8DDE75]] topological spaces in place of simplicial sets gives symmetric spectra of spaces.\n\nExample: the *symmetric sphere spectrum* $\\mathbb{S}$ has $\\mathbb{S}_n = S^n$ with the permutation action of $\\Sigma_n$.\nIts structure maps are the identifications $S^1 \\wedge S^n = S^{n+1}$.\nThe equivariance in clause 3 holds because both actions permute the same circle factors.\n\nNon-example: on the sequence $X_n = S^n$ with the same structure maps, let every $\\Sigma_n$ act as the identity.\nClauses 1 and 2 hold, but clause 3 fails for $p = 2$ and $n = 0$.\nThe map $S^2 \\wedge S^0 \\to X_2$ is the identity of $S^2$.\nIt does not carry the swap of the two circle factors, which is not the identity of $S^2$, to the identity action on $X_2$.\n\nSymmetric spectra form a closed symmetric monoidal category under a smash product with unit $\\mathbb{S}$.\nThey carry a stable model structure whose homotopy category is equivalent to the [[stable homotopy category|26GTNMAF7B]]; this is the theorem of Hovey, Shipley and Smith.\nMonoids and commutative monoids in symmetric spectra model $A_\\infty$ and $E_\\infty$ ring spectra.\n",
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