nLab algebra spectrum

Idea

An algebra spectrum or A-∞ algebra-spectrum over a ring spectrum is the analog in the higher algebra of stable homotopy theory of an associative algebra over a ring on ordinary algebra.

Definition

Abstractly, an $A_\infty$-algebra spectrum over an E-∞ ring spectrum $R$ is algebra in an (∞,1)-category in the stable (∞,1)-category of $R$-module spectra.

Concretely this (∞,1)-category is presented by the model structure on monoids in the monoidal $R$-modules in the model structure on symmetric spectra.

(…)

Properties

Stable monoidal Dold-Kan correspondence

Let $R := H \mathbb{Z}$ be the Eilenberg-MacLane spectrum for the integers.

Proposition

There is a zig-zag of lax monoidal Quillen equivalences

$$H \mathbb{Z} Mod \stackrel{\overset{Z}{\longrightarrow}}{\underset{U}{\leftarrow}} Sp^\Sigma(sAb) \stackrel{\overset{L}{\leftarrow}}{\underset{\phi^* N}{\longrightarrow}} Sp^\Sigma(Ch_+) \stackrel{\overset{D}{\longrightarrow}}{\underset{R}{\leftarrow}} Ch_\bullet \,,$$

between monoidal model categories satisfying the monoid axiom in a monoidal model category:

• the model structure for $H \mathbb{Z}$-module spectra;

• the model structure on symmetric spectrum objects in simplicial abelian groups and in chain complexes;

• and the model structure on chain complexes (unbounded).

This induces a Quillen equivalence between the corresponding model structures on monoids in these monoidal categories, which on the left is the model structure on $H \mathbb{Z}$-algebra spectra and on the right the model structure on dg-algebras:

$$H \mathbb{Z} Alg \simeq dgAlg_\mathbb{Z} \,.$$

This is due to (Shipley). The corresponding equivalence of (∞,1)-categories for $R$ a commutative rings with the intrinsically defined (∞,1)-category of E1-algebra objects on the left appears as (Lurie, prop. 7.1.4.6).

Remark

This is a stable version of the monoidal Dold-Kan correspondence. See there for more details.

Related concepts

• ring spectrum, module spectrum

• symmetric algebra spectrum

• A-∞ algebra, algebra in an (∞,1)-category

References

An account in terms of (∞,1)-category theory is in section 7.1.4 of

• Jacob Lurie, Higher Algebra

The equivalence of $H \mathbb{Z}$-algebra spectra with dg-algebras is due to

• Brooke Shipley, $H \mathbb{Z}$-algebra spectra are differential graded algebras , Amer. Jour. of Math. 129 (2007) 351-379. (arXiv:math/0209215)

Eilenberg-MacLane spectra $H R$ for $R$ itself a dg-algebra are discussed in

• Daniel Dugger, Brooke Shipley, Topological equivalences for differential graded algebras (arXiv:math/0604259)

See also the references at stable homotopy theory.