symmetric monoidal (∞,1)-category of spectra
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.
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.
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Let $R := H \mathbb{Z}$ be the Eilenberg-MacLane spectrum for the integers.
There is a zig-zag of lax monoidal Quillen equivalences
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:
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).
This is a stable version of the monoidal Dold-Kan correspondence. See there for more details.
An account in terms of (∞,1)-category theory is in section 7.1.4 of
The equivalence of $H \mathbb{Z}$-algebra spectra with dg-algebras is due to
Eilenberg-MacLane spectra $H R$ for $R$ itself a dg-algebra are discussed in
See also the references at stable homotopy theory.
Last revised on March 16, 2017 at 10:21:36. See the history of this page for a list of all contributions to it.