Math 2.0

Algebraic topology Definition

Symmetric spectrum

Builds on SpectrumSymmetric groupPointed spaceStable homotopy category

Definition

Let S1=Δ[1]/∂Δ[1]S^1 = \Delta[1] / \partial \Delta[1] be the simplicial circle and Sp=S1∧⋯∧S1S^p = S^1 \wedge \cdots \wedge S^1 (pp factors), with the symmetric group Σp\Sigma_p acting by permuting the factors. A symmetric spectrum XX consists of:

  1. for each n≥0n \ge 0, a pointed simplicial set XnX_n with a base-point-preserving left action of Σn\Sigma_n;
  2. for each n≥0n \ge 0, a pointed map σn ⁣:S1∧Xn→Xn+1\sigma_n \colon S^1 \wedge X_n \to X_{n+1};
  3. such that for all n,p≥0n, p \ge 0 the iterated structure map Sp∧Xn→Xn+pS^p \wedge X_n \to X_{n+p} is Σp×Σn\Sigma_p \times \Sigma_n-equivariant, where Σp×Σn\Sigma_p \times \Sigma_n acts on Xn+pX_{n+p} through the inclusion Σp×Σn⊆Σp+n\Sigma_p \times \Sigma_n \subseteq \Sigma_{p+n}.

A map f ⁣:X→Yf \colon X \to Y of symmetric spectra is a sequence of Σn\Sigma_n-equivariant pointed maps fn ⁣:Xn→Ynf_n \colon X_n \to Y_n that commute with the structure maps.

Remarks

Forgetting the actions gives a sequential spectrum, after geometric realization. The same definition with pointed topological spaces in place of simplicial sets gives symmetric spectra of spaces.

Example: the symmetric sphere spectrum 𝕊\mathbb{S} has 𝕊n=Sn\mathbb{S}_n = S^n with the permutation action of Σn\Sigma_n. Its structure maps are the identifications S1∧Sn=Sn+1S^1 \wedge S^n = S^{n+1}. The equivariance in clause 3 holds because both actions permute the same circle factors.

Non-example: on the sequence Xn=SnX_n = S^n with the same structure maps, let every Σn\Sigma_n act as the identity. Clauses 1 and 2 hold, but clause 3 fails for p=2p = 2 and n=0n = 0. The map S2∧S0→X2S^2 \wedge S^0 \to X_2 is the identity of S2S^2. It does not carry the swap of the two circle factors, which is not the identity of S2S^2, to the identity action on X2X_2.

Symmetric spectra form a closed symmetric monoidal category under a smash product with unit 𝕊\mathbb{S}. They carry a stable model structure whose homotopy category is equivalent to the stable homotopy category; this is the theorem of Hovey, Shipley and Smith. Monoids and commutative monoids in symmetric spectra model A∞A_\infty and E∞E_\infty ring spectra.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptPointed space, concept, no evidence levelPointed spaceconceptSpectrum, concept, no evidence levelSpectrumconceptStable homotopy category, concept, no evidence levelStable homotopy categ…conceptSymmetric group, definition, no evidence levelSymmetric groupdefinitionSymmetric spectrum, definition, no evidence levelSymmetric spectrumdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Group concept
  2. Pointed space concept
  3. Spectrum concept
  4. Stable homotopy category concept
  5. Symmetric group definition
  6. Symmetric spectrum definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Symmetric_spectrum
About this page
TagBH7C1Y1F0E
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Kinddefinition
WritingW0 stub
MSC55P42, 55P43, 55U35
DefinesSymmetric spectrum
UsesSpectrum, Symmetric group, Pointed space, Stable homotopy category
Changed2 November 2026

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