nLab A-infinity-ring



Higher algebra

Higher linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Stable homotopy theory



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

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


See the references at ring spectrum.

Another version of the A A_\infty-ring is simply what is usually called the A 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 nn-Categories, A A_{\infty} rings and modules, Theory Appl. Cat. 1997, n.4 (TAC)

Last revised on May 16, 2022 at 07:26:07. See the history of this page for a list of all contributions to it.