nLab stable Dold-Kan correspondence

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

Proposition

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

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

between a model category of $H R$-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.

(Notice that the homotopy category of the model structure on $R$-chain complexes is the “derived category” of $R Mod$.)

This equivalence on the level of homotopy categories is due to Robinson 1987. The refinement to a Quillen equivalence is due to Shipley 2007, Thm. 1.1 and Schwede & Shipley 2003, 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, Thm. 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:

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

(Shipley 02, theorem 1.1)

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 .

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

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:

• 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 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.

Related concepts

• ∞-Dold-Kan correspondence

• rational stable homotopy theory

References

• Daniel Kan, Semisimplicial spectra, Illinois J. Math. Volume 7, Issue 3 (1963), 463-478. (euclid:ijm/1255644953)

• Alan Robinson, The extraordinary derived category, Math. Z. 196 2 (1987) 231-238 [doi:10.1007/BF01163657]

• Stefan Schwede, Brooke Shipley, Stable model categories are categories of modules, Topology 42 (2003), 103-153 (pdf, doi:10.1016/S0040-9383(02)00006-X)

• Brooke Shipley, $H \mathbb{Z}$-algebra spectra are differential graded algebras, Amer. Jour. of Math. 129 (2007) 351-379. [arXiv:math/0209215, jstor:40068065]

• John Frederick Jardine, Stable Dold-Kan theory, section 4.6 of Generalized Étale cohomology theories, Modern Birkhäuser classics (1991)

• Andrew Blumberg, Ralph Cohen, Constantin Teleman, Open-closed field theories, string topology and Hochschild homology (arXiv:0906.5198)

• Jacob Lurie, section 7.1.4 of Higher Algebra

• Jacob Lurie, Deformation Theory

• Donald Stanley, Closed model categories and monoidal categories, Ph.D. Thesis, University of Toronto (1997) (pdf)