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\begin{document}

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\section*{differential forms in synthetic differential geometry}

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

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{content}{}\section*{{Content}}\label{content}

\noindent\hyperlink{content}{Content}\dotfill \pageref*{content} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{differential_forms}{differential forms}\dotfill \pageref*{differential_forms} \linebreak
\noindent\hyperlink{coboundary_operator}{coboundary operator}\dotfill \pageref*{coboundary_operator} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In the context of [[synthetic differential geometry]] a [[differential form]] $\omega$ of degree $k$ on a [[manifold]] $X$ is literally a function on the space of \emph{infinitesimal cubes} or \emph{infinitesimal simplices} in $X$.

We give the definition as available in the literature and then interpret this in a more unified way in terms of the [[schreiber:Chevalley-Eilenberg algebra|Chevalley-Eilenberg algebra]] of the [[infinitesimal singular simplicial complex]].

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

\begin{quote}%
missing here are details on what axioms the space we are working on has to satisfy for the following to make sense. See the case distinction at [[infinitesimal singular simplicial complex]].


\end{quote}
\hypertarget{differential_forms}{}\subsection*{{differential forms}}\label{differential_forms}

An \textbf{infinitesimal $k$-simplex} in a synthetic differential space $X$ is a collection of $k+1$-points in $X$ that are pairwise infinitesimal neighbours.

The spaces $X^{\Delta^k_{diff}}$ of infinitesimal $k$-simplices arrange to form the [[infinitesimal singular simplicial complex]] $X^{\Delta^\bullet_{diff}}$.

The functions on the space of infinitesimal $k$-simplices form a [[generalized smooth algebra]] $C^\infty(X^{\Delta^k_{inf}})$.

A \textbf{differential $k$-form} (often called simplicial $k$-form or, less accurately, \textbf{combinatorial $k$-form} to distinguish it from similar but cubical definitions) on $X$ is an element in this function algebra that has the property that it vanishes on degenerate infinitesimal simplices.

See definition 3.1.1 in

\begin{itemize}%
\item Anders Kock, \emph{Synthetic geometry of manifolds} (\href{http://home.imf.au.dk/kock/SGM-kopi.pdf}{pdf})

\end{itemize}
for this simplicial definition. A detailed account of this is in the entry [[infinitesimal object]] in the section \href{http://ncatlab.org/nlab/show/infinitesimal+object#SpacOfInfSimpl}{Spaces of infinitesimal simplices}.

This is a very simple-looking statement. The reason is the [[topos]]-theoretic language at work in the background, which takes care that we may talk about [[infinitesimal object]]s as if they were just plain ordinary sets. For a very detailed account of how the above statement is implemented concretely in terms of concrete models for synthetic differential spaces see section 1 of

\begin{itemize}%
\item Breen, Messing, \emph{Combinatorial differential forms} (\href{http://arxiv.org/abs/math/0005087}{arXiv})

\end{itemize}
There are also cubical variants of the above definition

\begin{itemize}%
\item Anders Kock, \emph{Cubical version of combinatorial differential forms} (\href{http://www.springerlink.com/content/87tj80l51h138177/fulltext.pdf}{pdf for fee})

\end{itemize}
See also section 4.1 of

\begin{itemize}%
\item Moerdijk-Reyes, [[Models for Smooth Infinitesimal Analysis]]

\end{itemize}
for a realization of the cubical version in models based on sheaves on [[generalized smooth algebra]]s.

We may characterize the object $\Omega^k(X) \subset C^\infty(X^{\Delta^k_{inf}})$ as follows:

for $k \geq 1$ there are the obvious images

\begin{displaymath}
s_i^*
  :
  C^\infty(X^{\Delta^{k}_{inf}})
  \to 
  C^\infty(X^{\Delta^{k-1}_{inf}})
\end{displaymath}
of the [[simplicial set|degeneracy maps]]. As one can see, these act by restricting a function on infinitesimal $k$-simplices to the degenerate ones and regarding these then as a $(k-1)$-simplex.

Therefore we may characterize the subobject $\Omega^k(X) \hookrightarrow C^\infty(X^{\Delta^k_{inf}})$ as the joint kernel of the degeneracy maps

\begin{displaymath}
\Omega^k(X) = \cap_{i = 0}^{k-1} ker(s_i^*)
  \,.
\end{displaymath}
\hypertarget{coboundary_operator}{}\subsection*{{coboundary operator}}\label{coboundary_operator}

According to section 3.2 of Andres Kock's book, the coboundary operator $d : \Omega^k(X) \to \Omega^{k+1}(X)$ sends a differential $k$-form $\omega$ to the $(k+1)$-form $d \omega$ that on an infinitesimal $(k+1)$-simplex $(x_0, x_1, \cdots, x_{k+1})$ in $X$ evaluates to

\begin{displaymath}
d\omega(x_0, x_1, \cdots, x_{k+1})
  :=
  \sum_{i=0}^{k+1}
  \omega(x_0, \cdots , \hat{x_i}, \cdots, x_{k+1})
  \,,
\end{displaymath}
where the hat indicates that the corresponding variable is omitted, as usual.

We recognize this as the alternating sum of the [[simplicial identities|face maps]] $\partial_i^*$ of the [[simplicial object|cosimplicial object]] $C^\infty(X^{\Delta_{inf}^\bullet})$.

\begin{displaymath}
d := \sum_{i=0}^{k+1} \partial_i^* : 
  \Omega^k(X) \to \Omega^{k+1}(X)
  \,.
\end{displaymath}
These constructions remind one and should be compared with the [[Dold-Kan correspondence]]. In particular with its \emph{dual} (cosimplicial) version as recalled in section 4 of \href{http://arxiv.org/PS_cache/math/pdf/0306/0306289v3.pdf}{CastiglioniCortinas}

In total this should show the following

\begin{uprop}
Let $X$ be a synthetic differential space and $C^\infty(X^{\Delta_{inf}^\bullet})$ the [[simplicial object|cosimplicial object]] of [[generalized smooth algebra]]s of functions on the spaces of infinitesimal $k$-simplices in $X$.

Then the deRham complex $(\Omega^\bullet(X), d)$ of differential forms on $X$ is the normalized [[Moore complex]] of the cosimplicial object $C^\infty(X^{\Delta_{inf}^\bullet})$.

\end{uprop}
In other words, in as far as the Dold-Kan correspondence is an equivalence, we find that:

the object of differential forms on $X$ \emph{is} the cosimplicial [[generalized smooth algebra]] $C^\infty(X^{\Delta_{inf}^k})$.

\hypertarget{references}{}\section*{{References}}\label{references}

\begin{itemize}%
\item [[Anders Kock]], \emph{Synthetic geometry of manifolds} (\href{http://home.imf.au.dk/kock/SGM-kopi.pdf}{pdf})


\item Moerdijk-Reyes, [[Models for Smooth Infinitesimal Analysis]]


\item Castiglioni, Cortinas, \emph{Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence} (\href{http://arxiv.org/PS_cache/math/pdf/0306/0306289v3.pdf}{pdf})



\end{itemize}


\end{document}