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\begin{document}

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\section*{rational stable homotopy theory}

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

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

[[!include stable homotopy theory - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{rational_spectra}{Rational spectra}\dotfill \pageref*{rational_spectra} \linebreak
\noindent\hyperlink{RationalStabilization}{Rational stabilization}\dotfill \pageref*{RationalStabilization} \linebreak
\noindent\hyperlink{appendix}{Appendix}\dotfill \pageref*{appendix} \linebreak
\noindent\hyperlink{chain_complexes}{Chain complexes}\dotfill \pageref*{chain_complexes} \linebreak
\noindent\hyperlink{dgcalgebras}{dgc-Algebras}\dotfill \pageref*{dgcalgebras} \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}

Rational stable homotopy theory is the simple special case of [[stable homotopy theory]] under [[rationalization]].

It is a classical fact that the [[rationalization]] of classical [[homotopy theory]] (of [[topological spaces]] or [[simplicial sets]]) -- called \emph{[[rational homotopy theory]]} -- is considerably more tractable than general [[homotopy theory]], as exhibited by the existence of small concrete [[dg-algebra|dg-algebraic]] models for [[rational homotopy types]]: [[minimal Sullivan algebras]] or equivalently their [[dual]] [[dg-coalgebras]]. A similar statement holds for the rationalization of [[stable homotopy theory]] i.e. the [[homotopy theory]] of [[spectra]] (of [[topological spaces]] or [[simplicial sets]]): rational spectra are equivalent to rational [[chain complexes]], i.e. to [[dg-modules]] over $\mathbb{Q}$. This is a dg-model for \emph{rational stable homotopy theory} compatible with that of classical rational homotopy theory in that the [[stabilization]] adjunction that connects classical [[homotopy theory]] to [[stable homotopy theory]] is, under these identifications, modeled by the [[forgetful functor]] from dg-(co-)algebras to [[chain complexes]]

\begin{displaymath}
\itexarray{
    & \text{classical homotopy theory}
     &&
    \text{stable homotopy theory}
    \\
    & spaces & \underoverset{\underset{\Sigma^\infty}{\longrightarrow}}{\overset{\Omega^\infty}{\longleftarrow}}{} & spectra
    \\
    & & \text{stabilization}
    \\
    \text{rationally}
    &
    \text{dg(co)algebras}
    &
     \underoverset{\underset{underlying}{\longrightarrow}}{\overset{free}{\longleftarrow}}{}
    &
    \text{chain complexes}
  }
\end{displaymath}
\hypertarget{rational_spectra}{}\subsection*{{Rational spectra}}\label{rational_spectra}

\begin{theorem}
\label{RationalStableDoldKanCorrespondence}\hypertarget{RationalStableDoldKanCorrespondence}{}
By the [[stable Dold-Kan correspondence]], the ([[stable homotopy theory|stable]]) [[homotopy theory]] of \emph{[[rationalization|rationalized]] [[spectra]]}, namely of $H \mathbb{Q}$-[[module spectra]] is [[equivalence of (infinity,1)-categories|equivalent]] to that of [[chain complexes]] of [[modules]]/[[vector space]] over the [[rational numbers]]

\begin{displaymath}
H \mathbb{Q} ModSpectra
   \;\simeq\;
  Ch_\bullet(\mathbb{Q})
  \,.
\end{displaymath}
\end{theorem}
(\hyperlink{SchwedeShipley03}{Schwede-Shipley 03, theorem 5.1.6})

\begin{remark}
\label{}\hypertarget{}{}
Observe that $H \mathbb{Q}$-module spectra are just the rational spectra. Since [[rationalization]] of spectra is the [[smashing localization]] $(-)\wedge H \mathbb{Q}$ (\href{rationalization#RationalizationOfSpectra}{here}) every rational spectrum $X \simeq H \mathbb{Q} \wedge X$ carries an $H \mathbb{Q}$-module structure. Conversely, just as a $\mathbb{Q}$-[[module]] structure on an [[abelian group]] is unique if it exists, so an $H \mathbb{Q}$-[[module spectrum]] structure on a [[spectrum]] is essentially unique if it exists, due to the fact that the multiplication map $H \mathbb{Q} \wedge H \mathbb{Q} \to H \mathbb{Q}$ is an [[equivalence in an (infinity,1)-category|equivalence]].

\end{remark}
Theorem \ref{RationalStableDoldKanCorrespondence} parallels that of classical [[rational homotopy theory]]:

\begin{theorem}
\label{ClassicalRationalHomotopyTheory}\hypertarget{ClassicalRationalHomotopyTheory}{}
There is an [[equivalence of (infinity,1)-categories|equivalence of homotopy theories]] between the [[homotopy theory]] of [[nilpotent topological space|nilpotent]] [[rational topological spaces]] of [[finite type]] with that of cochain [[dgc-algebras]] over $\mathbb{Q}$ in non-negative degree

\begin{displaymath}
\infty Grpd_{\mathbb{Q}, nil,fin}
    \;\simeq\;
  dgcAlg^{\geq 0}_{\mathbb{Q},nil,fin}
\end{displaymath}
\end{theorem}
\hypertarget{RationalStabilization}{}\subsection*{{Rational stabilization}}\label{RationalStabilization}

\begin{prop}
\label{RationalStabilizationModel}\hypertarget{RationalStabilizationModel}{}
The following composite total [[derived functors]]

\begin{displaymath}
\itexarray{
    \mathrm{Ho}( \mathrm{Spectra}_{\mathbb{Q}, \mathrm{fin}} )
    \\
    \downarrow \simeq \uparrow
    \\
    \mathrm{Ho}( \mathrm{Ch}_{\mathbb{Q},\bullet,\mathrm{fin}} )
     &
      \underoverset
        \underset{\mathbb{R}\mathrm{cn}_2}{\longrightarrow}
        \overset{\mathbb{L} i_2}{\longleftarrow}
        {\bot}
     &
    \mathrm{Ho}( \mathrm{Ch}_{\mathbb{Q}, \gt 1, \mathrm{fin}} )
    \\
    &&
    \downarrow \simeq \uparrow^{(-)^\ast}
    \\
    &&
    \mathrm{Ho}( \mathrm{Ch}^{\gt 1}_{\mathbb{Q}, \mathrm{fin}}   )^{\mathrm{op}}
     &
      \underoverset
       \underset{ (\mathbb{L}\mathrm{Sym}_{/\mathbb{Q}[0]})^{\mathrm{op}} }{\longrightarrow}
       \overset{ (\mathbb{R}( U \circ \mathrm{ker}(\epsilon_{(-)}) ))^{\mathrm{op}} }{\longleftarrow}    
       {\bot}
     &
   \mathrm{Ho}( \mathrm{dgcAlg}^{\gt 0}_{\mathbb{Q}, \mathrm{fin}})_{/\mathbb{Q}[0]} )^{\mathrm{op}}
    \\
   && && \updownarrow \simeq
    \\
    && &&
    \mathrm{Ho}(\mathrm{Top}_{\mathbb{Q}, \gt 1 , \mathrm{fin}})
  }
\end{displaymath}
(where the key part in the middle right, involving $Sym$, is from prop. \ref{dgcAlgChAdjunction}, the left middle part involving connectivity is from prop. \ref{ChainComplexesConnectiveCover}, and the equivalences on the far left and far right are those from theorem \ref{RationalStableDoldKanCorrespondence} and theorem \ref{ClassicalRationalHomotopyTheory}, respectively)

agree with the restriction of the [[stabilization]] [[adjoint (infinity,1)-functor|infinity-adjunction]]

\begin{displaymath}
Spectra
    \underoverset
      {\underset{\Omega^\infty}{\longrightarrow}}
      {\overset{\Sigma^\infty}{\longleftarrow}}
      {\bot}
  \infty Grpd^{\ast/}
\end{displaymath}
to simply connected rational homotopy types of finite type.

\end{prop}
\begin{proof}
By the nature of the classical model structure on topological spaces, every simply connected object of $\mathrm{Ho}(\mathrm{Top}^{\ast/})$ is represented by one that is a [[retract]] of a [[transfinite composition]] of [[pushouts]] of the pointed connected generating cofibrations $S^n \to D^n$, for $n \geq 1$. More abstractly, the [[(∞,1)-category]] of simply connected homotopy types is generated under [[(∞,1)-colimits]] from the [[n-spheres]] of dimension $n \geq 2$. (See the analogous argument used in the [[Brown representability theorem]] \href{Brown%20representability%20theorem#TheClassicalPointedConnectedHomotopyCategoryAsDomainForTheAbstractBrownRepresentabilityTheorem}{here}).

Since $\Sigma^\infty$ is a left adjoint functor it is sufficient to check that the two functors agree on $S^n$ for $n \geq 1$. That they agree on $D^n$ is immediate. We hence need to consider their value on the positive dimensional $n$-spheres:

Let $n$ be odd. Then the [[minimal Sullivan model]] for $S^n$ is $\mathrm{Sym}( \mathbb{R}[n] )$. Since every dgc-algebra is fibrant in the projective model structure, the value of $\mathbb{R}(U \circ \mathrm{ker}(\epsilon_{(-)}))$ on this model is represented by the value of $U \circ \mathrm{ker}(\epsilon_{(-)})$ on that model, which is the cochain complex $\mathbb{Q}[n]$, hence equivalently the corresponding chain complex. This is indeed the rationalization of $\Sigma^\infty S^n$.

Next let $n$ be even with $n \geq 1$. Then the minimal Sullivan model for $S^n$ is $\mathrm{Sym}( \mathbb{R}[n] \oplus \mathbb{R}[2n-1], d c_{2n-1} = c_{n} \wedge c_n  )$. The underlying chain complex of the augmentation ideal is spanned by the elements $c_{n}^{\wedge^k}$ in degree $n k$ and by the elements $c_{n}^{\wedge^k} \wedge c_{2n-1}$ in degree $nk+2n-1$. The former elements are all closed, but except for $k = 1$ are all in the image of the latter elements, none of which is closed. Hence this chain complex is quasi-isomorphic to $\mathbb{R}[n]$. Again, this is indeed the rationalization of $\Sigma^\infty S^{n}$

\end{proof}
\hypertarget{appendix}{}\subsection*{{Appendix}}\label{appendix}

Here collect some background definitions, notations and facts for ease of reference in the main text above.

\hypertarget{chain_complexes}{}\subsubsection*{{Chain complexes}}\label{chain_complexes}

Throughout, let $k$ be a [[field]] of [[characteristic zero]].

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{enumerate}%
\item $\mathrm{Ch}_{k,\bullet}$ for the [[category of chain complexes]] of $k$-[[vector spaces]], i.e. the category of $\mathbb{Z}$-graded $k$-vector spaces $V$ equipped with a linear map $\partial_V : V \to V$ of degree -1 such that $\partial_V \circ \partial_V = 0$ (the [[differential]]);


\item $\mathrm{Ch}_k^\bullet$ for the category of [[cochain complexes]] of $k$-vector spaces, i.e. as before, but with differential $d_V$ of degree +1.



\end{enumerate}
For $V$ a $k$-vector space, and $n \in \mathbb{N}$ we write $V[n]$ for the (co-)chain complex concentrated on $V$ in degree $k$.

For $n \in \mathbb{Z}$ write

\begin{displaymath}
\mathrm{Ch}_{k, \geq n}
      \overset{i_n}{\hookrightarrow}
    \mathrm{Ch}_{k,\bullet}
\end{displaymath}
\begin{displaymath}
\mathrm{Ch}_k^{\geq n}
      \overset{i^n}{\hookrightarrow}
    \mathrm{Ch}_k^\bullet
\end{displaymath}
for the [[full subcategories]] on the (co-)chain complexes that are concentrated in degrees $\geq n$.

\end{defn}
\begin{prop}
\label{ChainComplexesConnectiveCover}\hypertarget{ChainComplexesConnectiveCover}{}
The inclusions from def. \ref{Complexes} have right resp. left adjoints, which we denote by

\begin{displaymath}
\mathrm{Ch}_{k,\bullet}
      \underoverset
        {\underset{\mathrm{cn}_n}{\longrightarrow}}
        {\overset{i_n}{\longleftarrow}}
        {\bot}
    \mathrm{Ch}_{k, \geq n}
\end{displaymath}
and

\begin{displaymath}
\mathrm{Ch}^{\geq 0}_k
      \underoverset
       {\underset {i_n}{\longrightarrow}}
       {\overset{\mathrm{cn}^n}{\longleftarrow}}
       {\bot}
    \mathrm{Ch}^\bullet_k
  \,.
\end{displaymath}
These are [[Quillen adjunctions]] with respect to the [[projective model structure on chain complexes]].

\end{prop}
\hypertarget{dgcalgebras}{}\subsubsection*{{dgc-Algebras}}\label{dgcalgebras}

\begin{defn}
\label{dgcalgebras}\hypertarget{dgcalgebras}{}
We write $\mathrm{dgcAlg}^\bullet_k$ for the category of [[differential graded-commutative algebras]] over $k$, i.e. commutative monoids in $\mathrm{Ch}^\bullet_k$. We write $(\mathrm{dgcAlg}^\bullet_k)_{/k[0]}$ for its [[slice category]] over $k = (k[0], d = 0) \in \mathrm{dgcAlg}^{\geq 0}$, hence for the category of [[augmented algebra|augmented]] dgc-algebras, hence for dgc-algebras $A$ equipped with a homomorphism $\epsilon_A : A \to (k[0], d = 0)$.

Finally we write

\begin{displaymath}
\mathrm{dgcAlg}^{\geq 0}_k
    \hookrightarrow
  \mathrm{dgcAlg}^\bullet_k
\end{displaymath}
for the full subcategory of the dgc-algebras in non-negative degree, hence for the commutative monoids in $\mathrm{Ch}^{\geq 0}$, and similarly for the augmented case

\begin{displaymath}
(\mathrm{dgcAlg}^{\geq 0}_k)_{/k[0]}
    \hookrightarrow
  (\mathrm{dgcAlg}^{\bullet}_k)_{/k[0]}
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{dgcAlgChAdjunction}\hypertarget{dgcAlgChAdjunction}{}
There are [[adjunctions]] between the categories from def. \ref{dgcalgebras} of the form

\begin{displaymath}
\mathrm{dgcAlg}^{\geq 0}_k
     \underoverset
      {\underset{U}{\longrightarrow}}
      {\overset{\mathrm{Sym}}{\longleftarrow}}
      {\bot}
    \mathrm{Ch}^{\geq 0}_k
\end{displaymath}
and

\begin{displaymath}
(\mathrm{dgcAlg}^{\geq 0}_k)_{/k[0]}
     \underoverset
      {\underset{U \circ \mathrm{ker}(\epsilon_{(-)})}{\longrightarrow} }
      {\overset{\mathrm{Sym}_{/k[0]}}{\longleftarrow}}
      {\bot}
    \mathrm{Ch}^{\geq 0}_k
\end{displaymath}
where

\begin{enumerate}%
\item $U$ forms the underlying chain complex;


\item $\mathrm{ker}(\epsilon_{(-)})$ forms the [[augmentation ideal]];


\item $\mathrm{Sym}$ forms the free graded-commutative dg-algebra;


\item $\mathrm{Sym}_{/k[0]}$ forms the free graded-commutative dg-algebra and regards it with its canonical augmentation over $k$.



\end{enumerate}
Moreover, these are [[Quillen adjunctions]]: the first with respect to the [[projective model structure on chain complexes]] and the [[projective model structure on dgc-algebras]] and the second with respect to the corresponding [[slice model structure]].

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

\begin{itemize}%
\item [[rational homotopy theory]]


\item [[rational parameterized stable homotopy theory]]


\item [[rational equivariant homotopy theory]]


\item [[rational equivariant stable homotopy theory]]



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

\begin{itemize}%
\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})

\end{itemize}
[[!redirects stable rational homotopy theory]]



\end{document}