nLab A-infinity-ring

Idea

An $A_\infty$-ring is a monoid in an (∞,1)-category in an additive (that is, stable) (∞,1)-category. Alternatively one can take a model which is a (non-homotopic) additive monoidal category, but the monoid is replaced by an algebra over a resolution of the associative operad.

For example there is a variant for the stable (∞,1)-category of spectra. Sometimes this is called an associative ring spectrum.

This may be modeled equivalently as an ordinary monoid with respect to the symmetric monoidal smash product of spectra.

Notice the difference to an ordinary ring spectrum which which is not necessarily coherently homotopy-associative.

$A_\infty$-rings play the role of rings in higher algebra.

The higher analog of a commutative ring is an E-∞ ring.

Related concepts

• ring, ring groupoid

• ring spectrum, functor with smash products

• symmetric ring spectrum

• cohomology theory

• periodic ring spectrum

References

See the references at ring spectrum.

Another version of the $A_\infty$-ring is simply what is usually called the $A_\infty$-algebra in the case when the ground ring is the ring of integers. See

• Gerald Dunn, Lax operad actions and coherence for monoidal $n$-Categories, $A_{\infty}$ rings and modules, Theory Appl. Cat. 1997, n.4 (TAC)