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. 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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