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\begin{document}

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\section*{differential forms on simplices}

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

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

[[!include homological algebra - contents]]

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

[[!include homotopy - 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{smooth_differential_forms}{Smooth differential forms}\dotfill \pageref*{smooth_differential_forms} \linebreak
\noindent\hyperlink{Polynomial}{Polynomial differential forms}\dotfill \pageref*{Polynomial} \linebreak
\noindent\hyperlink{piecewise_polynomial_differential_forms}{Piecewise polynomial differential forms}\dotfill \pageref*{piecewise_polynomial_differential_forms} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

There are various [[simplicial object|simplicial]] [[dg-algebra]]s that assign to the standard $n$-[[simplex]] a kind of [[de Rham algebra]] on $\Delta^n$.

By the discussion at [[differential forms on presheaves]], each such extends to a notion of differential forms on simplicial sets.

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

\hypertarget{smooth_differential_forms}{}\subsubsection*{{Smooth differential forms}}\label{smooth_differential_forms}

\begin{defn}
\label{SmoothnSimplex}\hypertarget{SmoothnSimplex}{}
\textbf{(smooth $n$-simplex)}

For $n \in \mathbb{N}$ the \emph{smooth [[n-simplex]]} $\Delta^n_{smth}$ is the [[smooth manifold]] with [[manifold with boundary|boundary]] and [[manifold with corners|corners]] defined, up to [[isomorphism]], as the following locus inside the [[Cartesian space]] $\mathbb{R}^{n+1}$:

\begin{displaymath}
\Delta^n_{smth}
  \;\coloneqq\;
  \left\{
    (x_0, x_1, \cdots, x_n)
    \in 
    \mathbb{R}^{n+1}
     \;\vert\;
     0 \leq x_i \leq 1
     \;\text{and}\;
     \underoverset{i = 0}{n}{\sum}
     x_i 
     \; = 0
  \right\}
  \hookrightarrow
  \mathbb{R}^{n+1}
  \,.
\end{displaymath}
For $0 \leq i \leq n$ the [[function]]

\begin{displaymath}
x_i \;\colon\; \Delta^n_{smth} \to \mathbb{R}
\end{displaymath}
which picks the $i$th component in the above definition is called the $i$th \emph{barycentric coordinate function}.

For

\begin{displaymath}
f \;\colon\; [n_1] \longrightarrow [n_2]
\end{displaymath}
a morphism of finite non-empty [[linear orders]] $[n] \coloneqq \{0 \lt 1 \lt \cdots \lt n\}$, let

\begin{displaymath}
\Delta_{smth}(f) 
     \;\colon\; 
  \Delta^{n_1}_{smth} 
    \longrightarrow 
  \Delta^{n_2}_{smth}
\end{displaymath}
be the [[smooth function]] defined by $x_i \mapsto x_{f(i)}$.

\end{defn}
\begin{defn}
\label{SmoothDifferentialFormsOnSmoothnSimplex}\hypertarget{SmoothDifferentialFormsOnSmoothnSimplex}{}
\textbf{(smooth differential forms on the smooth $n$-simplex)}

For $k \in \mathbb{N}$ then a [[smooth differential k-form]] on the smooth $n$-simplex (def. \ref{SmoothnSimplex}) is a smooth differential form in the sense of [[smooth manifolds]] with [[manifold with boundary|boundary]] and [[manifold with corners|corners]]. Explicitly this means the following.

Let

\begin{displaymath}
F^n
  \;\coloneqq\;
  \left\{
    (x_0, x_1, \cdots, x_n)
    \in 
    \mathbb{R}^{n+1}
     \;\vert\;
     \underoverset{i = 0}{n}{\sum}
     x_i 
     \; = 0
  \right\}
  \hookrightarrow
  \mathbb{R}^{n+1}
\end{displaymath}
be the affine plane in $\mathbb{R}^{n+1}$ that contains $\Delta^n_{smth}$ in its defining inclusion from def. \ref{SmoothnSimplex}. This is a [[smooth manifold]] [[diffeomorphism|diffeomorphic]] to the [[Cartesian space]] $\mathbb{R}^{n}$.

A smooth differential form on $\Delta^n_{smth}$ of degree $k$ is a collection of [[linear functions]]

\begin{displaymath}
\wedge^k T_x F^n \longrightarrow \mathbb{R}
\end{displaymath}
out of the $k$-fold skew-symmetric [[tensor power]] of the [[tangent space]] of $F^n$ at some point $x$ to the [[real numbers]], for all $x \in \Delta^n_{smth}$ such that this extends to a smooth differential $k$-form on $F^n$.

Write $\Omega^\bullet(\Delta^n_{smth})$ for the graded [[real vector space]] defined this way. By definition there is then a canonical linear map

\begin{displaymath}
\Omega^\bullet(F^n) \longrightarrow \Omega^\bullet(\Delta^n_{smth})
\end{displaymath}
from the [[de Rham complex]] of $F^n$ and there is a unique structure of a [[differential graded-commutative algebra]] on $\Omega^\bullet(\Delta^n_{smth})$ that makes is a [[homomorphism]] of [[dg-algebras]] form the [[de Rham algebra]] of $F^n$. This is the de Rham algebra of smooth differential forms on the smooth $n$-simplex.

For $f \colon [n_1] \to [n_2]$ a homomorphism of finite non-empty [[linear orders]] with $\Delta_{smth}(f) \colon \Delta^{n_1}_{smth} \to \Delta^{n_2}_{smth}$ the corresponding smooth function according to def. \ref{SmoothnSimplex}, there is the induced [[homomorphism]] of [[differential graded-commutative algebras]]

\begin{displaymath}
(\Delta_{smth}(f))^\ast
    \;\colon\;
  \Omega^\bullet(\Delta^{n_2}_{smth})
    \longrightarrow
  \Omega^\bullet(\Delta^{n_1}_{smth})
\end{displaymath}
induced from the usual [[pullback of differential forms]] on $F^n$. This makes smooth differential forms on smooth simplices be a [[simplicial object]] in [[differential graded-commutative algebras]]:

\begin{displaymath}
\Omega^\bullet(\Delta^{(-)}_{smth})
    \;\colon\;
   \Delta^{op}
     \longrightarrow
   dgcAlg_{\mathbb{R}}
  \,.
\end{displaymath}
\end{defn}
The standard proof of the [[Poincaré lemma]] applies to show that

\begin{displaymath}
H^\bullet(\Omega^\bullet(\Delta^n_{smth})) \simeq \mathbb{R}
  \,.
\end{displaymath}
Each element of $\Omega^p_{poly}(\Delta^n)$ may be uniquely written

\begin{displaymath}
\Phi =\sum_{1\leq i_1\lt\ldots\lt i_p\leq n}\Phi_{i_1\ldots i_p}d b_{i_1}\wedge\ldots d b_{i_p},
\end{displaymath}
where $b_j$ is as above the $j^{th}$ barycentric coordinate function and each $\Phi_{i_1\ldots i_p}$ is a $C^\infty$-function on $\mathbf{\Delta}^n$.

With this representation the multiplication and differential are given by the usual formulae. The multiplication is defined by $\Phi \wedge \Psi$ and extends linearly the product

\begin{displaymath}
(d b_{i_1}\wedge\ldots d b_{i_p})\wedge (d b_{j_1}\wedge\ldots d b_{j_q}) = (d b_{i_1}\wedge\ldots d b_{i_p}\wedge d b_{j_1}\wedge\ldots d b_{j_q})
\end{displaymath}
on the generating forms. Now if $f$ is a differentiable function

\begin{displaymath}
d f = \sum_{i=1}^{n} \frac{\partial f}{\partial x^i}d x_i,
\end{displaymath}
so if

\begin{displaymath}
\Phi =\sum_{1\leq i_1\lt \ldots\lt i_p\leq n}\Phi_{i_1\ldots i_p}d b_{i_1}\wedge\ldots d b_{i_p},
\end{displaymath}
then

\begin{displaymath}
d\Phi =\sum_{1\leq i_1\lt\ldots\lt i_p\leq n} d\Phi_{i_1\ldots i_p} \wedge d b_{i_1} \wedge \ldots d b_{i_p},
\end{displaymath}
\hypertarget{Polynomial}{}\subsubsection*{{Polynomial differential forms}}\label{Polynomial}

\begin{defn}
\label{PolynomialDifferentialForms}\hypertarget{PolynomialDifferentialForms}{}
For $n \in \mathbb{N}$ write

\begin{displaymath}
\Omega_{poly}^{\bullet}(\Delta^n)
   \;\coloneqq\;
  Sym^\bullet_{\mathbb{Q}} 
   \langle t_0, \cdots, t_n, d t_0, \cdots, d t_n\rangle/\left(\sum t_i -1, \sum d t_i \right)
\end{displaymath}
for the [[quotient]] of the $\mathbb{Z}$-graded [[symmetric algebra]] over the [[rational numbers]] on $n+1$ generators $t_i$ in degree 0 and $n+1$ generators $d t_i$ of degree 1.

In particular in degree 0 this are called the \emph{polynomial functions}

\begin{displaymath}
\Omega^0_{poly}(\Delta^n)
  \;=\;
  \mathbb{Q}[t_0, t_1, \cdots t_n]/\left( \underset{i}{\sum} t_i = 0 \right)
\end{displaymath}
due to the canonical inclusion

\begin{displaymath}
\Omega^0_{poly}(\Delta^n)
    \hookrightarrow
  C^\infty(\Delta^n_{smth})
\end{displaymath}
into the [[smooth functions]] on the $n$-simplex according to def. \ref{SmoothDifferentialFormsOnSmoothnSimplex}, obtained by regarding the generator $t_i$ as the $i$th barycentric coordinate function.

Observe that the [[tensor product]] of the polynomial differential forms over these polynomial functions with the [[smooth functions]] on the $n$-simplex, is canonically [[isomorphism|isomorphic]] to the space $\Omega^\bullet(\Delta^n_{smth})$ of smooth [[differential forms]], according to def. \ref{SmoothDifferentialFormsOnSmoothnSimplex}:

\begin{displaymath}
\Omega^\bullet(\Delta^n_{smth})
    \simeq
  C^\infty(\Delta^n_{smth}) \otimes_{\Omega^0_{poly}(\Delta^n)}
  \Omega^\bullet_{poly}(\Delta^n)
\end{displaymath}
where moreover the generators $d t_i$ are identified with the de Rham differential of the $i$th barycentric coordinate functions.

This defines a canonical inclusion

\begin{displaymath}
\Omega^\bullet_{poly}(\Delta^n)
    \hookrightarrow
  \Omega^\bullet(\Delta^n_{smth})
\end{displaymath}
and there is uniquely the structure of a [[differential graded-commutative algebra]] on $\Omega^\bullet_{poly}(\Delta^n)$ that makes this a [[homomorphism]] of [[dg-algebras]]. This is the \emph{dg-algebra of polynomial differential forms}.

For $f \colon [n_1] \to [n_1]$ a [[morphism]] of finite non-empty [[linear orders]], let

\begin{displaymath}
\Omega^\bullet_{poly}(f) 
   \;\colon\; 
  \Omega^\bullet_{poly}(\Delta^{n_2}) \to \Omega^\bullet_{poly}(\Delta^{n_1})
\end{displaymath}
be the morphism of dg-algebras given on generators by

\begin{displaymath}
\Omega^\bullet_{poly}(f) : t_i \mapsto \sum_{f(j) = i} t_j
  \,.
\end{displaymath}
This yields a [[simplicial object|simplicial]] [[differential graded-commutative algebra]]

\begin{displaymath}
\Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k
\end{displaymath}
which is a sub-simplicial object of that of smooth differential form

\begin{displaymath}
\Omega^\bullet_{poly}(\Delta^{(-)})
    \hookrightarrow
  \Omega^\bullet(\Delta_{smth}^{(-)})
  \,.
\end{displaymath}
\end{defn}
\hypertarget{piecewise_polynomial_differential_forms}{}\subsubsection*{{Piecewise polynomial differential forms}}\label{piecewise_polynomial_differential_forms}

By left [[Kan extension]] the functor of polynomial differential forms from def. \ref{PolynomialDifferentialForms} yields a functor on all [[simplicial sets]]

\begin{displaymath}
\Omega^\bullet_{poly} \colon sSet \longrightarrow cdgAlg_k^{op}
  \,.
\end{displaymath}
This is the [[left adjoint]] in a [[nerve and realization]] [[adjunction]]

\begin{displaymath}
(\Omega^\bullet_{poly} \dashv \mathcal{K}_{poly})
    \;\colon\;
  (dgcAlg_{\mathbb{Q}, \geq 0})^{op}
    \underoverset
      {\underset{K_{poly}}{\longrightarrow}}
      {\overset{\Omega^\bullet_{poly}}{\longleftarrow}}
      {\bot}
  sSet 
  \,.
\end{displaymath}
Composing with the [[singular simplicial complex]] functor

\begin{displaymath}
Sing \;\colon\; Top \longrightarrow sSet
\end{displaymath}
on [[topological spaces]], this yields a functor on topological spaces

\begin{displaymath}
\Omega^\bullet_{pwpoly}
   \;\colon\;
  Top
   \overset{Sing}{\longrightarrow}
  sSet
   \overset{\Omega^\bullet_{poly}}{\longrightarrow}
  (dgcAlg_{\mathbb{Q}, \geq 0})^{op}
\end{displaymath}
which we may think of as assigning ``piecewise polynomial'' differential forms.

This is the starting point of the Sullivan approach to [[rational homotopy theory]]. See there for more

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

Let $k$ be a [[field]] of [[characteristic]] 0. Let $\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}$ be the left [[Kan extension]] of $\Omega^\bullet_{poly} : \Delta \to cdgAlg_k^{op}$ from \hyperlink{Polynomial}{above}.

\begin{defn}
\label{}\hypertarget{}{}
For $S \in sSet$, define a morphism of graded $k$-vector spaces

\begin{displaymath}
\int : \Omega^\bullet_{poly}(S) \to C^\bullet(S, k)
\end{displaymath}
from polynomial differential forms on simplices to [[cochains on simplicial sets]] by sending $\omega \in \Omega^n_{poly}(K)$ to the cochain that sends $\sigma \in K_n$ to

\begin{displaymath}
\int_\sigma f := \int_{\Delta^n} f_{max}(\sigma) d t_1 \wedge \cdots d t_n
  \,,
\end{displaymath}
where on the right we have the ordinary [[integral]] of the $1,\cdots,n$-component of the restriction of $f$ to $\sigma$.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The morphism $\int$ is a [[quasi-isomorphism]] of cochain complexes.

\end{prop}
This is (\hyperlink{BousfieldGugenheim}{Bousfield-Gugenheim, theorem 2.2, corollary 3.4}).

The following is the central fact of the Sullivan approach to [[rational homotopy theory]]:

\begin{prop}
\label{}\hypertarget{}{}
The functor $\Omega^\bullet_{poly}$ is the [[left adjoint]] of a [[Quillen adjunction]]

\begin{displaymath}
(\Omega^\bullet_{poly} \dashv R)
  \;\colon\;
  sSet 
    \underoverset
      {\underset{\Omega^\bullet_{poly}}{\to}}
      {\overset{R}{\leftarrow}}
      {\bot}
  cdgAlg_k^{op}
\end{displaymath}
for the standard [[model structure on simplicial sets]] and the projective [[model structure on dg-algebras|model structure on commutative dg-algebras]].

\end{prop}
This is shown in (\hyperlink{BousfieldGugenheim}{Bousfield-Gugenheim, section 8}).

So in particular $\Omega^\bullet_{poly}$ sends cofibrations of simplicial sets to fibrations of dg-algebras. Hence for $i : \partial \Delta[k] \hookrightarrow \Delta[k]$ a boundary inclusion the corresponding restriction

\begin{displaymath}
i^* : \Omega^\bullet_{poly}(\Delta^k) \to \Omega^\bullet_{poly}(\partial \Delta^k)
\end{displaymath}
is degreewise surjective.

\begin{prop}
\label{RespectForProduct}\hypertarget{RespectForProduct}{}
The functor $\Omega^\bullet_{poly}$ is a [[lax monoidal functor]] whose lax monoidal structure map

\begin{displaymath}
\nabla_{X,Y}
  :
  \Omega^\bullet_{poly}(X)
  \otimes
  \Omega^\bullet_{poly}(Y)
  \to 
  \Omega^\bullet_{poly}(X \times Y)
\end{displaymath}
is a [[quasi-isomorphism]].

\end{prop}
This is reviewed for instance in (\hyperlink{Hess}{Hess, page 12}).

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

Applications include

\begin{itemize}%
\item the [[Sullivan construction]] in [[rational homotopy theory]];


\item the [[sSet]]-[[enriched category|enrichment]] of the [[model structure on dg-algebras over an operad]].


\item higher [[Lie integration]]



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

An original reference is

\begin{itemize}%
\item [[Aldridge Bousfield]] and V. K. A. M. Gugenheim, \S{}1 and \S{}2 of: \emph{On PL De Rham Theory and Rational Homotopy Type} , Memoirs of the A. M. S., vol. 179, 1976.

\end{itemize}
A standard textbook is

\begin{itemize}%
\item [[Stephen Halperin]], \emph{Lecture Notes on Minimal Models}, Publications de l'U.E.R. Math\'e{}matiques Pures et Appliqu\'e{}es, Universit\'e{} des Sciences et techniques, Lille, Vol 3 (1981) Fasc.3.

\end{itemize}
This is based on

\begin{itemize}%
\item [[Dennis Sullivan]], \emph{Infinitesimal computations in topology}, Publications Math\'e{}matiques de l'IH\'E{}S, 47 (1977), p. 269-331 (\href{http://www.numdam.org/item/PMIHES_1977__47__269_0/}{numdam})

\end{itemize}
A useful survey is in

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

\end{itemize}
[[!redirects polynomial differential form]] [[!redirects polynomial differential forms]]

[[!redirects piecewise polynomial differential form]] [[!redirects piecewise polynomial differential forms]]

[[!redirects polynomial differential forms on the n-simplex]] [[!redirects polynomial differential forms on n-simplices]]

[[!redirects smooth differential forms on simplices]]

[[!redirects Sullivan differential form]] [[!redirects Sullivan differential forms]]



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