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  "body": "\n## Summary\n\nA symmetric spectrum is a [[spectrum|1GHF4EKBXS]] in which the $n$-th space carries an action of the symmetric group on $n$ letters.\nThese actions are compatible with the structure maps.\nThis extra symmetry gives a smash product of spectra that is associative, commutative and unital on the nose, not only up to homotopy.\nSymmetric spectra form a model for the [[stable homotopy category|26GTNMAF7B]] in which ring spectra with fully coherent multiplications are simply monoids.\n\n## Explanations\n\n### L1 human\n\nThe smash product $S^n \\wedge S^m = S^{n+m}$ hides a choice: the coordinates of the first factor come before those of the second.\nSwapping the factors permutes coordinates, which is a nontrivial map.\nA symmetric spectrum keeps track of these permutations by letting the symmetric group act on each space.\nWith this bookkeeping the smash product of two symmetric spectra is defined directly, and it is strictly commutative.\nThe sphere spectrum, with the symmetric group acting on $S^n = S^1 \\wedge \\cdots \\wedge S^1$ by permuting the factors, is the basic example.\n",
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