# nLab stable Dold-Kan correspondence

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal 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)

###### Theorem

The category of unbounded chain complexes is equivalent to the category of combinatorial spectra internal to abelian groups.

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 $R$ any ring (or ringoid, even) there is a Quillen equivalence

$H R ModSpectra \simeq Ch_\bullet(R Mod)$

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

This presents a corresponding equivalence of (∞,1)-categories. If $R$ 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 $R = \mathbb{Q}$ 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 $R$ a commutative ring, then there is a zig-zag of Quillen equivalences between a model structure for ring spectra over $H R$ and model structure on dg-algebras over $R$.

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

$H R AlgSpec \underoverset {\underset{\Theta}{\longrightarrow}} {\overset{H}{\longleftarrow}} {\simeq} dgAlg_R$
###### Theorem

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

$(H A) ModSpec \simeq_{Quillen} A Mod$

between $H A$-module spectra and dg-modules over $A$.

Dually, for $E$ an $H \mathbb{Z}$-algebra spectrum, then there is a Quillen equivalence

$E ModSpec \simeq_{Quillen} (\Theta E) Mod$

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

###### Remark

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

$DK \;\colon\; L_{qi} Ch_\bullet(R) \stackrel{\simeq}{\longrightarrow} H R Mod \stackrel{}{\longrightarrow} Spectra$

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 $DK$ to an (∞,1)-sheaf of spectra.

###### Example

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

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

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

While the correspondence $(\Sigma^\infty_+ X) \wedge H R \sim C_\bullet(X,R)$ 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 $R \coloneqq H \mathbb{Z}$ 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:

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 02). 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 HA, prop. 7.1.4.6). The equivalence on the level of homotopy categories was proved in (Stanley 97, theorem 1.1.4).

###### Remark

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

## References

Last revised on August 8, 2022 at 18:23:21. See the history of this page for a list of all contributions to it.