nLab symmetric ring spectrum

Idea

A symmetric ring spectrum is a ring spectrum modeled as a symmetric spectrum. Equivalently this is a monoid object with respect to the symmetric monoidal smash product of spectra, for symmetric spectra. Morever, if we regard symmetric spectra as $\mathbb{S}_{sym}$-modules, as discussed at Model categories of diagram spectra, then this, in turn, is equivalent to a $\mathbb{S}_{sym}$-algebra, where $\mathbb{S}_{sym}$ is the standard model of the sphere spectrum as a symmetric spectrum.

There is a model structure for symmetric ring spectra (MMSS 00) under which symmetric ring spectra represent genuine A-infinity rings, and commutative symmetric ring spectra represent genuine E-infinity rings. This fact is one of the key motivation for passing from sequential spectra to the richer model of symmetric spectra (and possibly further to other highly structured spectra such as orthogonal spectra and excisive functors).

Despite all this, the component expression of the structure in a symmetric ring spectrum, in the fashion of functors with smash product, is rather straightforward, and directly analogous to the structure in a dg-algebra: essentially there is for all pairs of indices $n_1, n_2$ a pairing between the component spaces of the spectrum in these degrees

$$E_{n_1}\wedge E_{n_2}\longrightarrow E_{n_1 + n_2}$$

such that this respects the given action of the symmetric groups and of suspensions, and such that it is is associative and unital in the evident way.

Definition

Write $Top_{cg}^{\ast}$ for the category of pointed compactly generated topological spaces.

One minimal way to state the definition is as follows. We follow conventions as used at model structure on orthogonal spectra.

Definition

A commutative symmetric ring spectrum $E$ is

1. a sequence of component spaces $E_n \in Top^{\ast/}_{cg}$ for $n \in \mathbb{N}$;

2. a basepoint preserving continuous left action of the symmetric group $\Sigma(n)$ on $E_n$;

3. for all $n_1,n_2\in \mathbb{N}$ a multiplication map

   $$\mu_{n_1,n_2} \;\colon\; E_{n_1} \wedge E_{n_2} \longrightarrow E_{n_1 + n_2}$$

   (a morphism in $Top^{\ast/}_{cg}$)

4. two unit maps

   $$\iota_0 \;\colon\; S^0 \longrightarrow E_0$$
   $$\iota_1 \;\colon\; S^1 \longrightarrow E_1$$

such that

1. $\mu_{n_1,n_2}$ intertwines the $\Sigma(n_1) \times \Sigma(n_2)$-action;

2. (associativity) for all $n_1, n_2, n_3 \in \mathbb{N}$ the following diagram commutes (where we are notationally suppressing the associators of $(Top^{\ast/}_{cg}, \wedge, S^0)$)

   $$\array{ E_{n_1} \wedge E_{n_2} \wedge E_{n_3} &\overset{id \wedge \mu_{n_2,n_3}}{\longrightarrow}& E_{n_1} \wedge E_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1,n_2}\wedge id }}\downarrow && \downarrow^{\mathrlap{\mu_{n_1, n_2 + n_3}}} \\ E_{n_1 + n_2} \wedge E_{n_3} &\underset{\mu_{n_1 + n_2, n_3}}{\longrightarrow}& E_{n_1 + n_2 + n_3} } \,;$$

3. (unitality) for all $n \in \mathbb{N}$ the following diagram commutes

   $$\array{ S^0 \wedge E_n &\overset{\iota_0 \wedge id}{\longrightarrow}& E_0 \wedge E_n \\ &{}_{\mathllap{\ell^{Top^{\ast/}_{cg}}_{E_n}}}\searrow& \downarrow^{\mathrlap{\mu_{0,n}}} \\ && E_n }$$

   and

   $$\array{ E_n \wedge S^0 &\overset{id \wedge \iota_0 }{\longrightarrow}& E_n \wedge E_0 \\ &{}_{\mathllap{r^{Top^{\ast/}_{cg}}_{E_n}}}\searrow& \downarrow^{\mathrlap{\mu_{n,0}}} \\ && E_n }$$

4. (commutativity) for all $n_1, n_2 \in \mathbb{N}$ the following diagram commutes

   $$\array{ E_{n_1} \wedge E_{n_2} &\overset{\tau^{Top^{\ast/}_{cg}}_{E_{n_1}, E_{n_2}}}{\longrightarrow}& E_{n_2} \wedge E_{n_1} \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_2,n_1}}} \\ E_{n_1 + n_2} &\underset{\chi_{n_1,n_2}}{\longrightarrow}& E_{n_2 + n_1} } \,,$$

where $\chi_{n_1,n_2} \in \Sigma(n_1 + n_2)$ denotes the permutation which shuffles the first $n_1$ elements past the last $n_2$ elements.

A homomorphism of symmetric commutative ring spectra $f \colon E \longrightarrow E'$ is a sequence $f_n \;\colon\; E_n \longrightarrow E'_n$ of $\Sigma(n)$-equivariant pointed continuous functions such that the following diagrams commute for all $n_1, n_2 \in \mathbb{N}$

$$\array{ E_{n_1} \wedge E_{n_2} &\overset{f_{n_1} \wedge f_{n_2}}{\longrightarrow}& E'_{n_1} \wedge E'_{n_2} \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mu_{n_2,n_1}} \\ E_{n_1 + n_2} &\underset{\chi_{n_1, n_2}}{\longrightarrow}& E_{n_2 + n_1} }$$

and $f_0 \circ \iota_0 = \iota_0$ and $f_1\circ \iota_1 = \iota_1$.

Write

$$CRing(SymSpec(Top_{cg}))$$

for the resulting category of symmetric commutative ring spectra.

There is an analogous definition of orthogonal ring spectrum and we write

$$CRing(OrthSpec(Top_{cg}))$$

for the category that these form.

(Schwede 12, def. 1.3)

Related concepts

• orthogonal ring spectrum

References

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

• Stefan Schwede, Symmetric spectra, 2012 (pdf)

The structure of a symmetric ring spectrum on KO is discussed in

• Michael Joachim, A symmetric ring spectrum representing

  $KO$-theory_, Topology 40 (2001) 299-308