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.

The category of commutative ring spectra over the field $\mathbb {Q}$ of rational numbers is Quillen equivalent to the category of differential graded algebras over $\mathbb {Q}$ .

Example: The Witten genus may be realized as a morphism of commutative ring spectra MString → tmf.

Terminology

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

Notes

^ symmetric monoidal with respect to smash product and perhaps some other conditions; one choice is the category of symmetric spectra

References

Goerss, P. (2010). "1005 Topological Modular Forms [after Hopkins, Miller, and Lurie]" (PDF). Séminaire Bourbaki : volume 2008/2009, exposés 997–1011. Société mathématique de France.
May, J.P. (2009). "What precisely are $E_{\infty }$ ring spaces and $E_{\infty }$ ring spectra?". arXiv: 0903.2813.

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[1] symmetric monoidal with respect to smash product and perhaps some other conditions; one choice is the category of symmetric spectra

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