nLab orthogonal ring spectrum

Idea

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

There is a model structure for orthogonal ring spectra (MMSS 00) under which orthogonal ring spectra represent genuine A-infinity rings, and commutative orthogonal 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 orthogonal spectra.

Despite all this, the component expression of the structure in an orthgonal 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 orthogonal groups and of suspensions, and such that it is is associative and unital in the evident way.

Definition

The definition is directly analogous to that of symmetric ring spectrum, just with the symmetric groups replaced by orthogonal groups throughout.

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)