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\begin{document}

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\section*{stable Dold-Kan correspondence}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{in_terms_of_abelian_combinatorial_spectra}{In terms of abelian combinatorial spectra}\dotfill \pageref*{in_terms_of_abelian_combinatorial_spectra} \linebreak
\noindent\hyperlink{InTermsOfEMModuleSpectra}{In terms of EM-module spectra}\dotfill \pageref*{InTermsOfEMModuleSpectra} \linebreak
\noindent\hyperlink{monoidal_version_in_terms_of_emmodule_specta}{Monoidal version in terms of EM-module specta}\dotfill \pageref*{monoidal_version_in_terms_of_emmodule_specta} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The [[Dold-Kan correspondence]] [[stabilization|stabilizes]] to identify \emph{unbounded} [[chain complexes]] with certain [[spectra]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\hypertarget{in_terms_of_abelian_combinatorial_spectra}{}\subsubsection*{{In terms of abelian combinatorial spectra}}\label{in_terms_of_abelian_combinatorial_spectra}

The following theorem was established in (\hyperlink{Kan63}{Kan 63, prop. 5.8})

\begin{theorem}
\label{}\hypertarget{}{}
The [[category of unbounded chain complexes]] is [[equivalence of categories|equivalent]] to the category of [[combinatorial spectra]] [[internalization|internal]] to [[abelian groups]].

\end{theorem}
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 \emph{\href{module+spectrum#StableDoldKanCorrespondence}{stable Dold-Kan correspondence}} at \emph{[[module spectrum]]}.

\hypertarget{InTermsOfEMModuleSpectra}{}\subsubsection*{{In terms of EM-module spectra}}\label{InTermsOfEMModuleSpectra}

\begin{theorem}
\label{StableDoldKan}\hypertarget{StableDoldKan}{}
For $R$ any [[ring]] (or [[ringoid]], even) there is a [[Quillen equivalence]]

\begin{displaymath}
H R ModSpectra \simeq Ch_\bullet(R Mod)
\end{displaymath}
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]].

\end{theorem}
This equivalence on the level of [[homotopy categories]] is due to (\hyperlink{Robinson87}{Robinson 87}). The refinement to a [[Quillen equivalence]] is (\hyperlink{SchwedeShipley03}{Schwede-Shipley 03, theorem 5.1.6}). See also the discussion at \emph{[[stable model categories]]}. A direct description as an [[equivalence of (∞,1)-categories]] appears as ([[Higher Algebra|Lurie HA, theorem 7.1.2.13]]).

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

More generally:

\begin{theorem}
\label{HRAlgerasAreRdgAlgebras}\hypertarget{HRAlgerasAreRdgAlgebras}{}
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 categories|equivalence]] of [[homotopy categories]]:

\begin{displaymath}
H R AlgSpec
   \underoverset
     {\underset{\Theta}{\longrightarrow}}
     {\overset{H}{\longleftarrow}}
     {\simeq}
  dgAlg_R
\end{displaymath}
\end{theorem}
(\hyperlink{Shipley02}{Shipley 02, theorem 1.1})

\begin{theorem}
\label{HAModueSpectraAreAdgModules}\hypertarget{HAModueSpectraAreAdgModules}{}
For $A$ any [[dg-algebra]], then there is a [[Quillen equivalence]]

\begin{displaymath}
(H A) ModSpec \simeq_{Quillen} A Mod
\end{displaymath}
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]]

\begin{displaymath}
E ModSpec \simeq_{Quillen} (\Theta E) Mod
\end{displaymath}
where $H(-)$ and $\Theta(-)$ are from theorem \ref{HRAlgerasAreRdgAlgebras}.

\end{theorem}
(\hyperlink{SchwedeShipley03}{Schwede-Shipley 03, theorem 5.1.6}, \hyperlink{Shipley02}{Shipley 02, corollary 2.15})

\begin{remark}
\label{}\hypertarget{}{}
The [[forgetful functor|forgetful]] [[(∞,1)-functor]] $H R Mod \longrightarrow$ [[Spectra]] preserves [[(∞,1)-limits]], so that (after [[simplicial localization]] $L$) we have an [[(∞,1)-functor]]

\begin{displaymath}
DK
  \;\colon\;
  L_{qi} Ch_\bullet(R) 
    \stackrel{\simeq}{\longrightarrow}
  H R Mod
   \stackrel{}{\longrightarrow}
  Spectra
\end{displaymath}
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]].

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
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. \ref{StableDoldKan}, 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

\begin{displaymath}
\pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R)
  \,.
\end{displaymath}
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  of (\hyperlink{BCT}{BCT}).

\end{example}
\hypertarget{monoidal_version_in_terms_of_emmodule_specta}{}\subsubsection*{{Monoidal version in terms of EM-module specta}}\label{monoidal_version_in_terms_of_emmodule_specta}

More in detail we have the following statement.

Let $R \coloneqq H \mathbb{Z}$ be the [[Eilenberg-MacLane spectrum]] for the [[integers]].

\begin{prop}
\label{}\hypertarget{}{}
There is a zig-zag of [[monoidal Quillen adjunction|lax monoidal]] [[Quillen equivalences]]

\begin{displaymath}
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
  \,,
\end{displaymath}
between [[monoidal model categories]] satisfying the [[monoid axiom in a monoidal model category]]:

\begin{itemize}%
\item the model structure for $H \mathbb{Z}$-[[module spectra]];


\item the [[model structure on symmetric spectrum objects]] in [[simplicial abelian group]]s and in [[chain complex]]es;


\item and the [[model structure on chain complexes]] (unbounded).



\end{itemize}
This induces a [[Quillen equivalence]] between the corresponding [[model structures on monoids]] in these [[monoidal category|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]]:

\begin{displaymath}
H \mathbb{Z} Alg \simeq dgAlg_\mathbb{Z}
  \,.
\end{displaymath}
\end{prop}
This is due to (\hyperlink{Shipley02}{Shipley 02}). The corresponding [[equivalence of (∞,1)-categories]] for $R$ a commutative rings with the intrinsically defined [[(∞,1)-category]] of [[En-algebra|E1-algebra]] objects on the left appears as ([[Higher Algebra|Lurie HA, prop. 7.1.4.6]]).

\begin{remark}
\label{}\hypertarget{}{}
This is a stable version of the [[monoidal Dold-Kan correspondence]]. See there for more details.

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[∞-Dold-Kan correspondence]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Daniel Kan]], \emph{Semisimplicial spectra}, Illinois J. Math. Volume 7, Issue 3 (1963), 463-478. (\href{http://projecteuclid.org/euclid.ijm/1255644953}{Euclid})


\item Alan Robinson, \emph{The extraordinary derived category}, Math. Z. 196 (2) (1987) 231--238.


\item [[Stefan Schwede]], [[Brooke Shipley]], \emph{Stable model categories are categories of modules}, Topology 42 (2003), 103-153 (\href{http://www.math.uic.edu/~bshipley/classTopFinal.pdf}{pdf})


\item [[Brooke Shipley]], \emph{$H \mathbb{Z}$-algebra spectra are differential graded algebras}, Amer. Jour. of Math. 129 (2007) 351-379. (\href{http://arxiv.org/abs/math/0209215}{arXiv:math/0209215})


\item [[John Frederick Jardine]], \emph{Stable Dold-Kan theory}, section 4.6 of \emph{Generalized \'E{}tale cohomology theories}, Modern Birkh\"a{}user classics (1991)


\item [[Andrew Blumberg]], [[Ralph Cohen]], [[Constantin Teleman]], \emph{Open-closed field theories, string topology and Hochschild homology} (\href{http://arxiv.org/abs/0906.5198}{arXiv:0906.5198})


\item [[Jacob Lurie]], section 7.1.4 of \emph{[[Higher Algebra]]}


\item [[Jacob Lurie]], \emph{[[Deformation Theory]]}



\end{itemize}


\end{document}