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\begin{document}

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\section*{Sullivan model}

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

\hypertarget{rational_homotopy_theory}{}\paragraph*{{Rational homotopy theory}}\label{rational_homotopy_theory}

[[!include differential graded objects - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_cofibrations}{As cofibrations}\dotfill \pageref*{as_cofibrations} \linebreak
\noindent\hyperlink{rationalization}{Rationalization}\dotfill \pageref*{rationalization} \linebreak
\noindent\hyperlink{relation_to_whitehead_products}{Relation to Whitehead products}\dotfill \pageref*{relation_to_whitehead_products} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

A \textbf{Sullivan model} of a [[rational space]] $X$ is a particularly well-behaved commutative [[dg-algebra]] [[quasi-isomorphism|quasi-isomorphic]] to the dg-algebra of Sullivan forms on $X$. These \emph{Sullivan algebras} are precisely the cofibrant objects in the standard [[model structure on dg-algebras]].

Sullivan models are a central tool in [[rational homotopy theory]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Sullivan models are particularly well-behaved [[differential graded-commutative algebras]] that are equivalent to the dg-algebras of [[piecewise polynomial differential forms on topological spaces]]. Conversely, every rational space can be obtained from a dg-algebra and the \emph{minimal} Sullivan algebras provide convenient representatives that correspond bijectively to rational homotopy types under this correspondence.

Abstractly, (relative) Sullivan models are the (relative) [[cell complexes]] in the standard [[model structure on dg-algebras]].

We now describe this in detail. First some notation and preliminaries:

\begin{defn}
\label{FiniteType}\hypertarget{FiniteType}{}
\textbf{(finite type)}

\begin{itemize}%
\item A [[graded vector space]] $V$ is \emph{of finite type} if in each degree it is finite dimensional. In this case we write $V^*$ for its degreewise dual.


\item A [[Grassmann algebra]] is of finite type if it is the Grassmann algebra $\wedge^\bullet V^*$ on a graded vector space of finite type

(the dualization here is just convention, that will help make some of the following constructions come out nicely).


\item A [[CW-complex]] is of finite type if it is built out of finitely many cells in each degree.



\end{itemize}
\end{defn}
For $V$ a $\mathbb{N}$-[[graded vector space]] write $\wedge^\bullet V$ for the [[Grassmann algebra]] over it. Equipped with the trivial differential $d = 0$ this is a [[semifree dgc-algebra]] $(\wedge^\bullet V, d=0)$.

With $k$ our ground [[field]] we write $(k,0)$ for the corresponding [[dg-algebra]], the tensor unit for the standard [[monoidal category|monoidal structure]] on $dgAlg$. This is the [[Grassmann algebra]] on the 0-vector space $(k,0) = (\wedge^\bullet 0, 0)$.

\begin{defn}
\label{SullivanAlgebra}\hypertarget{SullivanAlgebra}{}
\textbf{(Sullivan algebras)}

A \textbf{relatived Sullivan algebra} is a [[homomorphism]] of [[differential graded-commutative algebras]] that is an inclusion of the form

\begin{displaymath}
(A,d) \hookrightarrow (A \otimes_k \wedge^\bullet V, d')
\end{displaymath}
for $(A,d)$ any [[dgc-algebra]] and for $V$ some [[graded vector space]], such that

\begin{enumerate}%
\item there is a [[well ordered set]] $J$ indexing a [[linear basis]] $\{v_\alpha \in V| \alpha \in J\}$ of $V$;


\item writing $V_{\lt \beta} \coloneqq span(v_\alpha | \alpha \lt \beta)$ then for all basis elements $v_\beta$ we have that



\end{enumerate}
\begin{displaymath}
d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta}
    \,.
\end{displaymath}
This is called a \textbf{minimal} relative Sullivan algebra if in addition the condition

\begin{displaymath}
(\alpha \lt \beta) \Rightarrow (deg v_\alpha  \leq deg v_\beta)
\end{displaymath}
holds. For a Sullivan algebra $(k,0) \to (\wedge^\bullet V, d)$ relative to the tensor unit we call the [[semifree dgc-algebra]] $(\wedge^\bullet V,d)$ simply a \textbf{Sullivan algebra}, and we call it a \textbf{minimal Sullivan algebra} if $(k,0) \to (\wedge^\bullet V, d)$ is a minimal relative Sullivan algebra.

\end{defn}
(e.g. \hyperlink{Hess06}{Hess 06, def. 1.10, remark 1.11})

See also the section \href{model+structure+on+dg-algebras#SullivanAlgebras}{Sullivan algebras} at [[model structure on dg-algebras]].

\begin{remark}
\label{}\hypertarget{}{}
The special condition on the ordering in the relative Sullivan algebra says that these morphisms are composites of [[pushouts]] of the [[cofibrantly generated model category|generating cofibrations]] for the [[model structure on dg-algebras]], which are the inclusions

\begin{displaymath}
S(n) \hookrightarrow D(n)
  \,,
\end{displaymath}
where

\begin{displaymath}
S(n) = (\wedge^\bullet \langle c \rangle, d = 0)
\end{displaymath}
is the dg-algebra on a single generator in degree $n$ with vanishing differential, and where

\begin{displaymath}
D(n) = (\wedge^\bullet (\langle b \rangle \oplus \langle c \rangle), d b = c, d c = 0)
\end{displaymath}
with $b$ an additional generator in degree $n-1$.

Therefore for $A \in dgcAlg$, a pushout

\begin{displaymath}
\itexarray{
     S(n) &\stackrel{\phi}{\to}& A
     \\
     \downarrow && \downarrow
     \\
     D(n) &\to& (A \otimes \wedge^ \bullet \langle b \rangle, d b = \phi) 
  }
\end{displaymath}
is precisely a choice $\phi \in A$ of a $d_A$-closed element in degree $n$ and results in adjoining to $A$ the element $b$ whose differential is $d b = \phi$. This gives the condition in the above definition: the differential of any new element has to be a sum of wedge products of the old elements.

Notice that it follows in particular that the cofibrations in $dgAlg_{proj}$ are precisely all the [[retracts]] of relative Sullivan algebra inclusions.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{($L_\infty$-algebras)}

Because they are [[semifree dga]]s, Sullivan dg-algebras $(\wedge^\bullet V,d)$ are (at least for degreewise finite dimensional $V$) [[Chevalley-Eilenberg algebra]]s of [[L-∞-algebra]]s.

The co-commutative differential co-algebra encoding the corresponding [[L-∞-algebra]] is the free cocommutative algebra $\vee^\bullet V^*$ on the degreewise dual of $V$ with differential $D = d^*$, i.e. the one given by the formula

\begin{displaymath}
\omega(D(v_1 \vee v_2 \vee \cdots v_n)) =
  - (d \omega) (v_1, v_2, \cdots,  v_n)
\end{displaymath}
for all $\omega \in V$ and all $v_i \in V^*$.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{(Sullivan models)}

For $X$ a [[simply connected]] [[topological space]] $X$, a \textbf{Sullivan (minimal) model} for $X$ is a Sullivan (minimal) algebra $(\wedge^\bullet V^\ast, d_V)$ equipped with a [[quasi-isomorphism]]

\begin{displaymath}
(\wedge^\bullet V^*, d_V) 
      \stackrel{\simeq}{\longrightarrow}
    \Omega^\bullet_{pwpoly}(X)
\end{displaymath}
to the dg-algebra of [[piecewise polynomial differential forms]].

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{as_cofibrations}{}\subsubsection*{{As cofibrations}}\label{as_cofibrations}

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(cofibrations are relative Sullivan algebras)}

The cofibrations in the [[projective model structure on differential graded-commutative algebras]] $(dgcAlg_{\mathbb{N}})_{proj}$ are precisely the [[retracts]] of relative Sullivan algebra inclusions (def. \ref{SullivanAlgebra}).

Accordingly, the cofibrant objects in $(dgcAlg_{\mathbb{N}})_proj{}$ are precisely the [[retracts]] of Sullivan algebras.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
Minimal Sullivan models are unique up to [[isomorphism]].

\end{prop}
e.g \hyperlink{Hess06}{Hess 06, prop 1.18}.

\hypertarget{rationalization}{}\subsubsection*{{Rationalization}}\label{rationalization}

\begin{theorem}
\label{SullivanRationalizationEquivalence}\hypertarget{SullivanRationalizationEquivalence}{}
Consider the [[adjunction]] of [[derived functors]]

\begin{displaymath}
Ho(Top)
   \simeq
  Ho(sSet)
   \underoverset
     {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}}
     {\overset{\mathbb{L} K_{poly} }{\longleftarrow}}
     {\bot}
  Ho( (dgcAlg_{\mathbb{Q}, \geq 0})^{op} )
\end{displaymath}
induced from the [[Quillen adjunction]]

\begin{displaymath}
(dgcAlg_{\mathbb{Q}, \geq 0}_{proj})^{op}
    \underoverset
      {\underset{K_{poly}}{\longrightarrow}}
      {\overset{\Omega^\bullet_{poly}}{\longleftarrow}}
      {\bot}
  sSet_{Quillen}
\end{displaymath}
(\href{rational+homotopy+theory#SullivanRationalizationAdjunction}{this theorem}).

Then: On the [[full subcategory]] $Ho(Top_{\mathbb{Q}, \geq 1}^{fin})$ of [[simply connected topological space|simply connected]] [[rational topological spaces]] of [[finite type]] this adjunction restricts to an [[equivalence of categories]]

\begin{displaymath}
Ho(Top_{\mathbb{Q}, \gt 1}^{fin})
   \underoverset
     {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}}
     {\overset{\mathbb{L} K_{poly} }{\longleftarrow}}
     {\simeq}
  Ho( (dgcAlg_{\mathbb{Q}, \gt 1}^{fin})^{op} )  
  \,.
\end{displaymath}
In particular the [[adjunction unit]]

\begin{displaymath}
X \longrightarrow K_{poly}(\Omega^\bullet_{pwpoly}(X))
\end{displaymath}
exhibits the [[rationalization]] of $X$.

\end{theorem}
This is a central theorem of [[rational homotopy theory]], see for instance \hyperlink{Hess06}{Hess 06, corollary 1.26}.

It follows that the [[cochain cohomology]] of the cochain complex of [[piecewise polynomial differential forms]] on any topological, hence equivalently that of any of its [[Sullivan models]], coincides with its [[ordinary cohomology]] with coefficients in the [[rational numbers]]:

\begin{theorem}
\label{}\hypertarget{}{}
Let $(\wedge^\bullet V^*, d_V)$ be a minimal Sullivan model of a simply connected rational topological space $X$. Then there is an [[isomorphism]]

\begin{displaymath}
\pi_\bullet(X)
    \simeq
  V
\end{displaymath}
between the [[homotopy groups]] of $X$ and the generators of the minimal Sullivan model.

\end{theorem}
e.g. \hyperlink{Hess06}{Hess 06, theorem 1.24}.

\hypertarget{relation_to_whitehead_products}{}\subsubsection*{{Relation to Whitehead products}}\label{relation_to_whitehead_products}

See at \emph{[[the co-binary Sullivan differential is the dual Whitehead product]]}.

$\backslash$linebreak

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

[[!include Sullivan models -- examples]]

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

\begin{itemize}%
\item [[formal dg-algebra]]


\item [[minimal dg-module]]


\item [[minimal fibration]]



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

\begin{itemize}%
\item [[Steve Halperin]], \emph{Lectures on minimal models}, Mem. Soc. Math. Franc. no 9/10 (1983) (\href{https://eudml.org/doc/94833}{web})


\item [[Kathryn Hess]], around def 1.10 of \emph{Rational homotopy theory: a brief introduction} (\href{http://arxiv.org/abs/math.AT/0604626}{arXiv:math.AT/0604626})



\end{itemize}
[[!redirects Sullivan models]] [[!redirects Sullivan algebra]] [[!redirects Sullivan algebras]] [[!redirects Sullivan minimal algebra]] [[!redirects Sullivan minimal algebras]]

[[!redirects relative Sullivan algebra]] [[!redirects relative Sullivan algebras]]

[[!redirects minimal Sullivan algebra]] [[!redirects minimal Sullivan algebras]]

[[!redirects minimal Sullivan model]] [[!redirects minimal Sullivan models]]

[[!redirects Sullivan minimal model]] [[!redirects Sullivan minimal models]]



\end{document}