Commutative ring spectrum

In algebraic topology, a commutative ring spectrum, roughly equivalent to a $E_{\infty }$ -ring spectrum, is a commutative monoid in a good^[1] category of spectra. For example, the sphere spectrum is a commutative ring spectrum.

Almost all reasonable categories of commutative ring spectra can be shown to be Quillen equivalent to each other. Thus, from the point view of the stable homotopy theory, the term "commutative ring spectrum" may be used as a synonymous to a $E_{\infty }$ -ring spectrum.

Notes

^ symmetric monoidal with respect to smash product and perhaps some other conditions

References

J.P. May, What precisely are $E_{\infty }$ ring spaces and $E_{\infty }$ ring spectra? arXiv:0903.2813

[1] symmetric monoidal with respect to smash product and perhaps some other conditions

[1]