nLab brave new algebra

Idea

The term brave new algebra has been coined to refer to the higher algebra of E-infinity rings (“brave new rings”), amplifying the step from more traditional algebra.

Specifically, the term is typically used for the particular presentation of E-infinity rings by way of highly structured ring spectra such as (commutative) monoids with respect to a symmetric monoidal smash product of spectra (such as available on orthogonal spectra (orthogonal ring spectra), symmetric spectra (symmetric ring spectra) and S-modules).

According to Greenlees 06, p. 1:

The phrase ‘brave new rings’ was coined by F.Waldhausen, presumably to capture both an optimism about the possibilities of generalizing rings to ring spectra, and a proper awareness of the risk that the new step in abstraction would take the subject dangerously far from its justification in examples.

In consequence people also speak for instance of brave new Hopf algebroids when referring to the Hopf algebroids of (dual) $E$-Steenrod algebras for some ring spectrum $E$ as appearing in the Adams spectral sequence. See at Steenrod algebra – Hopf algebroid structure.

Related entries

• higher algebra, higher linear algebra

• E-infinity geometry

References

Introductions and expositions include

• Peter May, Brave new worlds in stable homotopy theory (pdf)

• John Greenlees, First steps in brave new commutative algebra (arXiv:math/0609453)

Monographs on the technical core of “brave new algebra” in terms of symmetric monoidal categories of spectra include

• Anthony Elmendorf, Igor Kriz, Peter May, Michael Mandell Modern foundations for stable homotopy theory (pdf)

• Anthony Elmendorf, Igor Kriz, Michael Mandell, P. May, Rings, modules and algebras in stable homotopy theory, AMS Mathematical Surveys and Monographs Volume 47 (1997) (pdf)

For more references see at ring spectrum and at higher algebra.