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stable Dold-Kan correspondence

Context

Stable Homotopy theory

Higher algebra

Contents

Idea

The Dold-Kan correspondence stabilizes to identify unbounded chain complexes with certain spectra.

Statement

In terms of abelian combinatorial spectra

The following theorem was established in (Kan 63, prop. 5.8)

Similarly to the unstable case, the above categories, when interpreted as ∞-categories, are also equivalent to the ∞-category of module spectra over the Eilenberg-MacLane ring spectrum of the integers. See the section stable Dold-Kan correspondence at module spectrum.

In terms of EM-module spectra

Theorem

For RR any ring (or ringoid, even) there is a Quillen equivalence

HRModSpectraCh (RMod) H R ModSpectra \simeq Ch_\bullet(R Mod)

between a model structure on HRH R-module spectra over the Eilenberg-MacLane spectrum HRH R and the model structure on unbounded chain complexes of ordinary RR-modules.

This presents a corresponding equivalence of (∞,1)-categories. If RR is a commutative ring, then this is an equivalence of symmetric monoidal (∞,1)-categories.

This equivalence on the level of homotopy categories is due to (Robinson 87). The refinement to a Quillen equivalence is (Schwede-Shipley 03, theorem 5.1.6). See also the discussion at stable model categories. A direct description as an equivalence of (∞,1)-categories appears as (Lurie HA, theorem 7.1.2.13).

For R=R = \mathbb{Q} the rational numbers, then theorem may be thought of as a stable analog of classical rational homotopy theory, see at rational stable homotopy theory for more on this.

More generally:

Theorem

For RR a commutative ring, then there is a zig-zag of Quillen equivalences between a model structure for ring spectra over HRH R and model structure on dg-algebras over RR.

In particular the induced total derived functors constitute an equivalence of homotopy categories:

HRAlgSpecΘHdgAlg R H R AlgSpec \underoverset {\underset{\Theta}{\longrightarrow}} {\overset{H}{\longleftarrow}} {\simeq} dgAlg_R

(Shipley 02, theorem 1.1)

Theorem

For AA any dg-algebra, then there is a Quillen equivalence

(HA)ModSpec QuillenAMod (H A) ModSpec \simeq_{Quillen} A Mod

between HAH A-module spectra and dg-modules over AA.

Dually, for EE an HH \mathbb{Z}-algebra spectrum, then there is a Quillen equivalence

EModSpec Quillen(ΘE)Mod E ModSpec \simeq_{Quillen} (\Theta E) Mod

where H()H(-) and Θ()\Theta(-) are from theorem .

(Schwede-Shipley 03, theorem 5.1.6, Shipley 02, corollary 2.15)

Remark

The forgetful (∞,1)-functor HRModH R Mod \longrightarrow Spectra preserves (∞,1)-limits, so that (after simplicial localization LL) we have an (∞,1)-functor

DK:L qiCh (R)HRModSpectra DK \;\colon\; L_{qi} Ch_\bullet(R) \stackrel{\simeq}{\longrightarrow} H R Mod \stackrel{}{\longrightarrow} Spectra

from the (∞,1)-category of chain complexes to the (∞,1)-category of spectra which preseres (∞,1)-limits. In particular therefore a presheaf of chain complexes (as it appears in abelian sheaf cohomology/hypercohomology) which satisfies descent (for some given (∞,1)-site structure, hence which is an (∞,1)-sheaf/∞-stack of chain complexes) maps under the stable Dold-Kan correspondence DKDK to an (∞,1)-sheaf of spectra.

Example

For XX a topological space and RR a ring, let C (X,R)C_\bullet(X, R) be the standard chain complex for singular homology H (X,R)H_\bullet(X, R) of XX with coefficients in RR.

Under the stable Dold-Kan correspondence, prop. , this ought to be identified with the smash product (Σ + X)HR(\Sigma^\infty_+ X) \wedge H R of the suspension spectrum of XX with the Eilenberg-MacLane spectrum. Notice that by the general theory of generalized homology the homotopy groups of the latter are again singular homology

π ((Σ + X)HR)H (X,R). \pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R) \,.

While the correspondence (Σ + X)HRC (X,R)(\Sigma^\infty_+ X) \wedge H R \sim C_\bullet(X,R) under the above equivalence is suggestive, maybe nobody has really checked it in detail. It is sort of stated as true for instance on p. 15 of (BCT).

Monoidal version in terms of EM-module specta

More in detail we have the following statement.

Let RHR \coloneqq 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 02). 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 HA, prop. 7.1.4.6).

Remark

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

References

Last revised on May 18, 2018 at 04:46:15. See the history of this page for a list of all contributions to it.