nLab symmetric spectrum

Idea

Symmetric spectra are one version of highly structured spectra that support a symmetric monoidal smash product of spectra. A symmetric spectrum is a sequence of topological spaces/simplicial sets which are compatibly equipped with an action of the symmetric group.

The category of symmetric spectra is a presentation of the symmetric monoidal (∞,1)-category of spectra, with the special property that it implements the smash product of spectra such as to yield itself a symmetric monoidal model category of spectra: the model structure on symmetric spectra. This implies in particular that with respect to this symmetric smash product of spectra an E-∞ ring is presented simply as a plain commutative monoid in symmetric spectra. This is of course such that truncating down to the homotopy category yields equivalently the stable homotopy category with its usual smash product of spectra.

The main technical issue with symmetric spectra is that the naive definition of homotopy groups for them is not general homotopy correct, one needs to replace by a “semistable symmetric spectrum” first, see below. This problem goes away for orthogonal spectra (these however need to be based on topological spaces instead of simplicial sets).

Definition

In components

A symmetric spectrum is a sequential spectrum equipped with an action of the symmetric group on each component space, such that the structure maps intertwine these actions combined with the canonical permutation action on the n-spheres:

Definition

A symmetric spectrum $X$ in sSet is

1. a sequence $\{X_n| n \in \mathbb{N}\}$ of pointed simplicial sets;

2. a basepoint preserving left action of the symmetric group $\Sigma_n$ on $X_n$;

3. a sequence of morphisms of pointed simplicial sets $\sigma_n \colon X_n \wedge S^1 \longrightarrow X_{n+1}$

such that

• for all $n, k \in \mathbb{N}$ the composite

  $$X_n \wedge S^{k} \stackrel{\sigma_n \wedge id}{\longrightarrow} X_{n+1} \wedge S^{k-1} \stackrel{\sigma_{n+1}\wedge id}{\longrightarrow} \cdots \stackrel{\sigma_{n+k-1}}{\longrightarrow} X_{n+k}$$

  intertwines the $\Sigma_{n} \times \Sigma_k$-action.

A morphism of symmetric spectra $f\colon X \longrightarrow Y$ is

• a sequence of morphisms of pointed simplicial sets $f_n \colon X_n \longrightarrow Y_n$

such that

1. each $f_n$ intetwines the $\Sigma_n$-action;

2. the following diagrams commute

   $$\array{ X_n \wedge S^1 &\stackrel{f_n \wedge id}{\longrightarrow}& Y_n \wedge S^1 \\ \downarrow^{\mathrlap{\sigma^X_n}} && \downarrow^{\mathrlap{\sigma^Y_n}} \\ X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1} } \,.$$

(Hovey-Shipley-Smith 00, def. 1.2.1, Schwede 12, def. 1.1)

As $\mathbb{S}_{Sym}$-modules

A more concise equivalent formulation is the following:

Definition

Write $Sym$ for the standard skeleton of the core of FinSet, with objects labeled by natural numbers, all morphisms automorphisms and the automorphisms of $n$ forming the symmetric group $\Sigma_n$. The disjoint union monoidal structure on FinSet gives $Sym$ the structure of a monoidal category with tensor product given by addition of natural numbers on objects.

Write $[Sym,sSet^{\ast/}]$ for the functor category equipped with the induced Day convolution product. Write

$$\mathbb{S}_{Sym} \in [Sym,sSet^{\ast/}]$$

for the functor which sends $n$ to the minimal simplicial n-sphere $S^n = (S^1)^{\wedge^n}$ and which sends elements of the symmetric group to the corresponding endomorphisms of $S^n$ exchanging smash product factors.

Regard $\mathbb{S}_{Sym}$ as a monoid object with respect to Day convolution, whose monoidal structure is given via the discussion there by the system of canonical natural morphisms

$$S^{n_1}\wedge S^{n_2} \longrightarrow S^{n_1 + n_2} \,.$$

Proposition

Symmetric spectra in the sense of def. are equivalently right module objects in $([Sym,sSet^{\ast/}], \otimes_{Day})$ over the monoid object $\mathbb{S}_{Sym}$, according to def. :

$$SymSpec(sSet) \simeq \mathbb{S}_{Sym}Mod_r \,.$$

This point of view was highlighted and almost made explicit in (Mandell-May-Schwede-Shipley 01).

Proof

By the discussion there, right $\mathbb{S}_{Sym}$-modules with respect to Day convolution are equivalently right modules over monoidal functors over the monoidal functor corresponding to $\mathbb{S}_{Sym}$ as in def. . This means that they are functors $X \colon Sym \longrightarrow sSet^{\ast/}$ equipped with natural transformations

$$X_p \wedge S^{q} \longrightarrow X_{p+q}$$

satisfying the evident categorified action property. In the present case this action property says that these morphisms are determined by

$$X_p \wedge S^1 \longrightarrow X_{p+1}$$

under the isomorphisms $S^p \simeq S^1 \wedge S^{p-1}$. Naturality of all these morphisms as functors on $Sym$ is the equivariance under the symmetric group actions in def. .

For more on this see at Model categories of diagram spectra – part I.

Properties

Smash product

smash product of spectra modeled on symmetric spectra

e.g. (Schwede 12, I.3)

Homotopy groups

The main technical issue with symmetric spectra is that the evident definition of stable homotopy groups of symmetric spectra $X$ as

$$\pi_n(X) = \underset{\longrightarrow}{\lim}_k \pi_{k+n}(X_k)$$

does not in general come out in the homotopy correct way; instead one needs to replace by a “semistable symmetric spectrum” first, which however is hard to control (Hovey-Shipley-Smith 00, section 3.1, Schwede 12, chapter I, sections 2 and 6 and 8), survey includes (Malkiewich 14, section 2.3).

Among the various (now) common models for spectra, this issue is unique to symmetric spectra, see also the comment in (Mandell-May-Schwede-Shipley 01, p. 3).

In particular, this problem goes away for the concept of orthogonal spectra, which otherwise is very similar to that of symmetric spectra and shares most of its advantages (except that it does not admit a version based on simplicial sets).

Technically, the issue comes down to the fact that the quotients of symmetric groups $\Sigma_q/\Sigma_{q-n}$ do not become highly connected for large $q$ the way the quotients $O(q)/O(n-q)$ of orthogonal groups do (Mandell-May-Schwede-Shipley 01, proof of lemma 8.6, top of p. 26). See this proposition at Model categories of diagram spectra.

Related concepts

• symmetric sequence

• free symmetric spectrum

• symmetric ring spectrum

model structure on spectra, symmetric monoidal smash product of spectra

• symmetric spectrum, model structure on symmetric spectra

• orthogonal spectrum, model structure on orthogonal spectra

• S-module, model structure on S-modules

References

The orginal article is

• Mark Hovey, Brooke Shipley, Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149-208 [arXiv:math/9801077, doi:10.1090/S0894-0347-99-00320-3]

A comprehensive textbook account is in

• Stefan Schwede, Symmetric spectra, 2012 (pdf)

Further discussion of the model structure on symmetric spectra beyond (Hovey-Shipley-Smith 00) and Schwede 12, part III includes

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (pdf)

More or less brief reviews include

• Cary Malkiewich, section 2.3 of The stable homotopy category, 2014 (pdf)

• Alexander Kupers, Symmetric spectra (pdf)

• Mitya Boyarchenko, Introduction to symmetric spectra (prepared for a Geometric Langlands Seminar) Part I (pdf) and Part II (pdf)