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\begin{document}

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\section*{rational parameterized stable homotopy theory}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\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]]

\begin{quote}%
under construction


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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{chain_complexes}{Chain complexes}\dotfill \pageref*{chain_complexes} \linebreak
\noindent\hyperlink{dgalgebras}{dg-Algebras}\dotfill \pageref*{dgalgebras} \linebreak
\noindent\hyperlink{simplicial_lie_algebras}{Simplicial Lie algebras}\dotfill \pageref*{simplicial_lie_algebras} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \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}

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 tat 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}
Classical [[homotopy theory]] and [[stable homotopy theory]] are unified and jointly generalized in [[parameterized stable homotopy theory]], whose [[objects]] are [[parameterized spectra]], parameterized over a classical [[homotopy type]]. The \emph{rational parameterized stable homotopy theory} to be discussed here is supposed to be the rationalization of this joint generalization, unifiying and jointly generalizing the algebraic model of [[rational topological spaces]] by [[Sullivan algebras]] and of $H \mathbb{Q}$-[[module spectra]] by [[chain complexes]].

\begin{displaymath}
\itexarray{
    & \text{stable homotopy theory}
    && 
    \text{parameterized stable homotopy theory}
    &&
    \text{classical homotopy theory}
    \\
    &
    \text{spectra}
     &\overset{\phantom{\text{include}}}{\longrightarrow}&
    \text{parameterized spectra}
     &\overset{\phantom{\text{project}}}{\longrightarrow}&
    \text{spaces}
    \\
    &
    &\text{include}&
    &\text{project}&
    \\
    \text{rationally}
    &
    \text{chain complexes}
    &\overset{\phantom{\text{include}}}{\longrightarrow}&
    \text{dg-modules}
    &\overset{\phantom{\text{project}}}{\longrightarrow}&
    \text{dg(co)algebras}
  }
\end{displaymath}
Here we (intend to) show that, accordingly, rational parameterized homotopy theory is presented by the the [[opposite category|opposite]] of the [[homotopical category]] of [[dg-modules]] over cochain [[differential graded-commutative algebras]] in non-negative degrees.

\begin{quote}%
under construction


\end{quote}
\hypertarget{preliminaries}{}\subsection*{{Preliminaries}}\label{preliminaries}

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

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

\begin{itemize}%
\item $Ch_{\bullet,\mathbb{Q}}$ for the [[category of chain complexes]] of [[modules]]/[[vector spaces]] over $\mathbb{Q}$ (i.e. [[differential]] of degree -1)


\item $Ch^{\bullet}_{\mathbb{Q}}$ for the category of [[cochain complexes]] (i.e. differential of degree +1).



\end{itemize}
For $n \in \mathbb{N}$ write

\begin{itemize}%
\item $Ch_{\geq n, \mathbb{Q}} \hookrightarrow Ch_{\bullet,\mathbb{Q}}$ for the [[full subcategory]] of the [[chain complexes]] concentrated in degree $\geq n$;


\item $Ch^{\geq n}_{\mathbb{Q}} \hookrightarrow Ch^\bullet_{\mathbb{Q}}$ for the [[full subcategory]] of the [[cochain complexes]] concentrated in degree $\geq n$.



\end{itemize}
For $V \in \mathbb{Q} Mod$ a rational [[vector space]], and for $n \in \mathbb{N}$, we write $V[n]$ both for the [[chain complex]] as well as for the [[cochain complex]] concentrated on $V$ in degree $n$.

\end{defn}
\hypertarget{dgalgebras}{}\subsubsection*{{dg-Algebras}}\label{dgalgebras}

\begin{defn}
\label{dgcAlgebras}\hypertarget{dgcAlgebras}{}
Write $dgcAlg^{\geq 0}_{\mathbb{q}}$ for the [[category]] of cochain [[dgc-algebras]] over the [[rational numbers]] concentrated in non-negative degrees.

Say that a [[morphism]] in this category is

\begin{enumerate}%
\item a \emph{weak equivalence} if it is a [[quasi-isomorphisms]] on the [[forgetful functor|underlying]] [[chain complexes]];


\item a \emph{fibration} if it is degreewise [[surjection]];


\item a \emph{cofibration} it it is a [[relative Sullivan algebra]] inclusion,



\end{enumerate}
We write

\begin{displaymath}
(dgcAlg^{\geq 0}_{\mathbb{Q}})_{proj}
\end{displaymath}
for the category $dgcAlg^{\geq 0}_{\mathbb{Q}}$ equipped with these three [[classes]] of morphisms.

\end{defn}
\begin{prop}
\label{dgcAlgProjectiveModelStructure}\hypertarget{dgcAlgProjectiveModelStructure}{}
The [[homotopical category]] $(dgcAlg^{\geq 0}_{\mathbb{Q}})_{proj}$ from def. \ref{dgcAlgebras} is a [[model category]], to be called the \emph{[[projective model structure on dgc-algebras]] in non-negative degrees}.

\end{prop}
(\hyperlink{BousfieldGugenheim76}{Bousfield-Gugenheim 76, theorem 4.3})

\begin{example}
\label{DifferentialFormsPolynomial}\hypertarget{DifferentialFormsPolynomial}{}
For $S \in sSet$ a [[simplicial set]], write

\begin{displaymath}
\Omega^\bullet_{poly}(S)
   \in
  dgcAlg^{\geq 0}_{\mathbb{Q}}
\end{displaymath}
for the [[polynomial differential forms]] with [[rational number|rational]] [[coefficients]] on $S$.

\end{example}
(\hyperlink{BousfieldGugenheim76}{Bousfield-Gugenheim 76, def. 2.1})

\begin{defn}
\label{SliceModelStructureOnAugmenteddgcalg}\hypertarget{SliceModelStructureOnAugmenteddgcalg}{}
Write $\mathbb{Q}[0] \coloneqq (\mathbb{Q}[0], d = 0)$ for the [[dgc-algebra]] concentrated on the [[ground field]] in degree 0, necessarily with vanishing [[differential]]. This is the [[initial object]] in $dgcAlg^{\geq 0}_{\mathbb{Q}}$.

Write

\begin{displaymath}
(dgcAlg^{\geq 0}_{\mathbb{Q}})_{/\mathbb{Q}[0]}
\end{displaymath}
for the [[slice category]] of that of all [[dgc-algebras]] (def. \ref{dgcAlgebras}) over $\mathbb{Q}[0]$. Hence an [[object]] in this category is a pair consisting of a [[dgc-algebra]] $A$ and a dg-algebra [[homomorphism]] of the form

\begin{displaymath}
\epsilon_A \;\colon\; A \longrightarrow \mathbb{Q}[0]
  \,.
\end{displaymath}
This is equivalently called a \emph{$\mathbb{Q}[0]$-[[augmented algebra|augmented]] dgc-algebra}. The [[kernel]] of the augmentation map $\epsilon$

\begin{displaymath}
ker_{\epsilon_{A}} \in Ch^{\geq 0}_{\mathbb{Q}}
\end{displaymath}
is the \emph{[[augmentation ideal]]} of $(A,\epsilon)$.

Since $\mathbb{Q}[0] \in dgcAlg^{\geq 0}_{\mathbb{Q}}$ carries a unique augmentation $\epsilon = id$, we still write $\mathbb{Q}[0]$ for the [[ground field]] regarded as an augmented dgc-algebra. As such this is now a [[zero object]].

Furthermore write

\begin{displaymath}
\left(
   (dgcAlg^{\geq 0}_{\mathbb{Q}})_{/\mathbb{Q}[0]}
  \right)_{proj}
\end{displaymath}
for the [[slice model structure]] induced on this by the [[projective model structure on dgc-algebras]] according to prop. \ref{dgcAlgProjectiveModelStructure}.

\end{defn}
See also \hyperlink{BousfieldGugenheim76}{Bousfield-Gugenheim 76, 4.11}

\hypertarget{simplicial_lie_algebras}{}\subsubsection*{{Simplicial Lie algebras}}\label{simplicial_lie_algebras}

\begin{displaymath}
(LieAlg_k)^{\Delta^{op}}_{proj}
    \underoverset
      {\underset{N}{\longrightarrow}}
      {\overset{N^\ast}{\longleftarrow}}
      {\bot}
  (dgLieAlg_k)_{proj}
    \underoverset
      {\underset{CE}{\longrightarrow}}
      {\overset{\mathcal{L}}{\longleftarrow}}
      {\bot}
  dgcoAlg_k
\end{displaymath}
(\ldots{})

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

We want to claim the following:

For every $\mathfrak{g} \in (LieAlg_{\mathbb{Q}})^{\Delta^{op}}$ there is a [[Quillen equivalence]]

\begin{displaymath}
SeqSpec\left(
    \mathfrak{g}Mod
  \right)_{stable}
    \underoverset
      {\underset{SeqSpec((Sym)^{op})}{\longrightarrow}}
      {
        \overset{SeqSpec( ker_{\epsilon(-)}^{op} )}{\longleftarrow}
      }
      {\simeq_{Qu}}
  SeqSpec\left(
    \mathfrak{g} / (LieAlg_{\mathbb{Q}})^{\Delta^{op}} / \mathfrak{g}
  \right)_{stable}
\end{displaymath}
Idea of proof: the analogous statement for simplicial Lie algebras replaced by rational [[simplicial algebras]] $cAlg_{\mathbb{A}}^{\Delta^{op}}$ is \hyperlink{Schwede97}{Schwede 97, theorem 3.2.3}. Apart from the connectivity of the $Sym$-construction, all that this proof uses is that simplicial commutative algebras form a [[proper model category|right proper]] [[simplicial model category]]. But also the [[model structure on simplicial Lie algebras]] is right proper and simplicial.

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

\begin{itemize}%
\item [[rational model for a suspension]]


\item [[rational homotopy theory]]


\item [[rational stable homotopy theory]]


\item [[rational equivariant homotopy theory]]


\item [[rational equivariant stable homotopy theory]]



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

A classical reference on plain [[rational homotopy theory]] is

\begin{itemize}%
\item [[Aldridge Bousfield]], V. K. A. M. Gugenheim, \emph{On PL deRham theory and rational homotopy type} , Memoirs of the AMS, vol. 179 (1976)

\end{itemize}
The equivalence between $H R$-module spectra (unparametrized) and $R$-chain complexes is due to

\begin{itemize}%
\item [[Stefan Schwede]], section 3 of \emph{Spectra in model categories and applications to the algebraic cotangent complex}, Journal of Pure and Applied Algebra 120 (1997) 104 (\href{http://www.math.uni-bonn.de/people/schwede/modelspec.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})



\end{itemize}
Discussion of rational fiberwise [[suspension spectra]] is in

\begin{itemize}%
\item Michael Charles Crabb, [[Ioan James]], around Prop. 15.8 of \emph{Fibrewise Homotopy Theory}, Springer Monographs in Mathematics, 1998


\item [[Yves Félix]], Aniceto Murillo [[Daniel Tanré]], \emph{Fibrewise stable rational homotopy}, Journal of Topology, Volume 3, Issue 4, 2010, Pages 743–758 (\href{https://doi.org/10.1112/jtopol/jtq023}{doi:10.1112/jtopol/jtq023})



\end{itemize}
A discussion of full blown rational parametrized stable homotopy theory is due to

\begin{itemize}%
\item [[Vincent Braunack-Mayer]], \emph{[[schreiber:thesis Braunack-Mayer|Rational parameterized stable homotopy theory]]}, Zurich, 2018


\item [[Vincent Braunack-Mayer]], \emph{Strict algebraic models for rational parametrised spectra I} (\href{https://arxiv.org/abs/1910.14608}{arXiv:1910.14608})



\end{itemize}
Application to mathematical analysis of [[duality between M-theory and type IIA string theory]]:

\begin{itemize}%
\item [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes rational parameterized stable homotopy theory]]}, Communications in Mathematical Physics 371: 197 (2019) (\href{https://arxiv.org/abs/1806.01115}{arXiv:1806.01115}, \href{https://doi.org/10.1007/s00220-019-03441-4}{doi:10.1007/s00220-019-03441-4})


\item [[Vincent Braunack-Mayer]], \emph{[[schreiber:Parametrised homotopy theory and gauge enhancement]]}, talk at \emph{\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory}} Durham Symposium 2018 , Fortschritte der Physik (2019) (\href{https://onlinelibrary.wiley.com/doi/abs/10.1002/prop.201910003}{doi:10.1002/prop.201910003}m \href{https://arxiv.org/abs/1903.02862}{arXiv:1903.02862})



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



\end{document}