universal algebra Algebraic theories
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Geometry on formal duals of algebras
Stable homotopy theory
-ring is a 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.
-rings play the role of A ∞ rings in higher algebra.
The higher analog of a commutative ring is an
- E ∞ ring. References
Another version of the
-ring is simply what is usually called the A ∞ - A ∞ algebra in the case when the ground ring is the ring of integers. See
Lax operad actions and coherence for monoidal , Theory Appl. Cat. 1997, n.4 ( -Categories, n rings and modules A ∞ TAC)