nLab
algebra spectrum

Context

Higher algebra

Stable Homotopy theory

Contents

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

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

(…)

Properties

Stable monoidal Dold-Kan correspondence

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

Proposition

There is a zig-zag of lax monoidal Quillen equivalences

HModUZSp Σ(sAb)ϕ *NLSp Σ(Ch +)RDCh , 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:

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 HH \mathbb{Z}-algebra spectra and on the right the model structure on dg-algebras:

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

This is due to (Shipley). The corresponding equivalence of (∞,1)-categories for RR 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.

References

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

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

Eilenberg-MacLane spectra HRH R for RR itself a dg-algebra are discussed in

See also the references at stable homotopy theory.

Revised on November 15, 2013 04:50:22 by Urs Schreiber (82.169.114.243)