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\section*{infinitesimal singular simplicial complex}

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

\hypertarget{synthetic_differential_geometry}{}\paragraph*{{Synthetic differential geometry}}\label{synthetic_differential_geometry}

[[!include synthetic differential geometry - 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{in_}{in $R^n$}\dotfill \pageref*{in_} \linebreak
\noindent\hyperlink{in_smooth_loci}{In smooth loci}\dotfill \pageref*{in_smooth_loci} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_formal_manifolds}{In formal manifolds}\dotfill \pageref*{in_formal_manifolds} \linebreak
\noindent\hyperlink{InclusionIntoFinPaths}{Inclusion into the finite singular simplicial complex}\dotfill \pageref*{InclusionIntoFinPaths} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{related_concepts_2}{Related concepts}\dotfill \pageref*{related_concepts_2} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{infinitesimal singular simplicial complex} of a [[space]] $X$ in a [[smooth topos]] $(\mathcal{T},R)$ is the infinitesimal analogue of the singular simplicial complex $X^{\Delta_R^\bullet}$ (see [[interval object]]) that in degree $k$ is the space of $k$-simplices $\Delta^k_R \to X$ in $X$: the infinitesimal singular simplicial complex has in degree $k$ the [[infinitesimal space|infinitesimal]] $k$-simplices in $X$.

There are several ways to make the notion of ``infinitesimal $k$-simplex in $X$'' precise. Here we describe a notion promoted by [[Anders Kock]], where an ``infinitesimal $k$-simplex'' in $X$ for $X$ a suitably \emph{locally linear space} , is a $(k+1)$-tuple $(x_0,\cdots, x_k) \in X^{\times^{k+1}}$ of points in $X$ that are pairwise \emph{infinitesimal neighbours} in $X$.

One central application of the singular simplicial complex is in the definition of [[differential forms in synthetic differential geometry]].

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

The basic definition applies to spaces of the form $R^n$ and is generalized from there to spaces that ``locally look like'' $R^n$ in one way or other.

Let here and in the following $(\mathcal{T},R)$ be a [[smooth topos]].

\hypertarget{in_}{}\subsection*{{in $R^n$}}\label{in_}

Write, as usual

\begin{displaymath}
D(n) := \{(\epsilon_i) \in n | \epsilon_i \epsilon_j = 0\} \hookrightarrow R^n
\end{displaymath}
for the [[infinitesimal space]] of first order infinitesimal neighbours of the origin of $R^n$, with its canonical inclusion into $R^n$.

Two elements $x , y \in R^n$ are called \textbf{first order infinitesimal neighbours}, denoted $x \sim_1 y$, if their difference is in the image of this inclusion.

\begin{displaymath}
(x \sim_1 y) \Leftrightarrow
  (\exists \epsilon \in D(n) :
  (x-y) = \epsilon) \,.
\end{displaymath}
Write

\begin{displaymath}
(R^n)^{\Delta^k_{inf}}
  :=
  \{
    (x_0 \in R^n, \cdots, x_k \in R^n) |
    x_i \sim_1 x_j
  \}
  \,.
\end{displaymath}
This naturally forms a [[simplicial object]] $X^{\Delta_{inf}^{bullet}} : \Delta^{op} \to \mathcal{T}$. This is the infinitesimal simplicial singular complex of $R^n$.

A more detailed discussion of this is in the entry [[infinitesimal object]] in the section \href{http://ncatlab.org/nlab/show/infinitesimal+object#SpacOfInfSimpl}{Spaces of infinitesimal simplices}.

\hypertarget{in_smooth_loci}{}\subsubsection*{{In smooth loci}}\label{in_smooth_loci}

\begin{quote}%
\textbf{warning} this section is as such not drawn from the literature, it seems


\end{quote}
Recall that a [[smooth locus]] in $\mathcal{T}$ is an object $\ell A$ that is the joint [[limit]] over some

\begin{displaymath}
R^n \stackrel{\stackrel{f \in J}{\to}}{\stackrel{0}{\to}}
  R
\end{displaymath}
here $J \subset Hom_{\mathcal{T}}(R^n,R)$ is an ideal.

Declare that two [[generalized element]]s $x,y \in \ell A$ are infinitesimal neighbours if their image under the injection

\begin{displaymath}
\ell A \hookrightarrow R^n
\end{displaymath}
is a pair of infinitesimal neighbour in $R^n$. Then let

\begin{displaymath}
(\ell A)^{\Delta^\bullet_{inf}}
  \hookrightarrow
  (R^n)^{\Delta_R^\bullet}
\end{displaymath}
be the sub-simplicial object of infinitesimal neighbours in $R^n$ that are points in $\ell A$.

\begin{ulemma}
\textbf{(linearity of space of infinitesimal neighbours)}

If $p, q \in \ell A$ are infinitesimal neighbours in the [[smooth locus]] $\ell A \subset R^n$, then for all $t \in R$ also the element $p + t(q-p)$ formed by linear combination in $R^n$ is in $\ell A$ and hence is an infinitesimal neighbour of $p$ there.

\end{ulemma}
\begin{proof}
Because by the [[Kock-Lawvere axiom]] valid in the [[smooth topos]] $(\mathcal{T},R)$ we have for all $f : R^n \to R$

\begin{displaymath}
0 = f(q) = f(p + (q-p)) = 
  f(p) + \sum_i (q-p)_i (\partial_i f)(q)
  = \sum_i (q-p)_i (\partial_i f)(q)
  \,.
\end{displaymath}
Therefore also

\begin{displaymath}
f(p + t(q-q)) = t \sum_i (q-p)_i (\partial_i f)(q) = 0
  \,.
\end{displaymath}
\end{proof}
\hypertarget{examples}{}\paragraph*{{Examples}}\label{examples}

Consider the circle $S^1$ regarded as the [[smooth locus]] $S^1 = \{(x, y) \in R^2 | x^2 + y^2 = 1\}$.

For $a \in S^1 \subset R^2$ an infinitesimal neighbour $(a + \epsilon)$ in $R^2$ is again a point on the circle, and hence an infinitesimal neighbour of $a$ in $S^1$, if

\begin{displaymath}
a_x^2 + 2 a_x \epsilon_x  + a_y^2 + 2 a_y \epsilon_y = 1
\end{displaymath}
which, due to $a_x^2 + a_y^2 = 1$ is equivalent to

\begin{displaymath}
2 a_x \epsilon_x + 2 a_y \epsilon_y = 0
  \,.
\end{displaymath}
This is solved by $\epsilon$ of the form

\begin{displaymath}
\epsilon = \delta \left(
    \itexarray{
       a_y
       \\
       - a_x
    }
  \right)
\end{displaymath}
for some fixed $\delta \in D$.

\hypertarget{in_formal_manifolds}{}\subsubsection*{{In formal manifolds}}\label{in_formal_manifolds}

\begin{quote}%
use that each manifold is locally isomorphic to an $R^n$ and that the neighbourhood relation only needs an infinitesimal neighbourhood. Proceed locally as above and then patch. See references below.


\end{quote}
\hypertarget{InclusionIntoFinPaths}{}\subsection*{{Inclusion into the finite singular simplicial complex}}\label{InclusionIntoFinPaths}

The [[lined topos]] $(\mathcal{T}, R)$ also comes canonically for every object $X \in \mathcal{T}$ with the finite singular simplicial complex $\Pi(X) : [n] \mapsto X^{R^n}$ induced from regarding

\begin{displaymath}
(0_*, 1_*) : {*} \coprod {*} \to R
\end{displaymath}
as an [[interval object]] (see there for details).

\begin{udef}
\textbf{(inclusion of infinitesimal into finite simplices)}

For $\ell A =: X \hookrightarrow R^n$ a [[smooth locus]] define for all $n \in \mathbb{N}$ a morphism

\begin{displaymath}
\iota_n
  :
  X^{\Delta^k_{inf}}
  \to 
  X^{\Delta_R^k} = X^{R^k}
\end{displaymath}
by defining it on [[generalized element]]s as

\begin{displaymath}
(x^0, \cdots, x^{k})
  \mapsto
  (\vec t \mapsto x^0 + \sum_{i=1}^{k} t_i (x^i - x^{i-1}) )
  \,.
\end{displaymath}
\end{udef}
\begin{uprop}
The morphisms $\iota_n$ constitute a morphism of [[simplicial object]]s

\begin{displaymath}
\iota : X^{\Delta^\bullet_{inf}}
  \hookrightarrow
  X^{\Delta_R^k}
\end{displaymath}
in that they respects the face and degenracy maps on each side.

\end{uprop}
\begin{proof}
Straightforward checking:

For instance

\begin{itemize}%
\item The inner face maps $d_i$ on $X^{\Delta_{inf}^{k}}$ omit the $i$th point in the $(k+1)$-tuple of points, while on $X^{\Delta^{k}}$ they act by pullback along $(t_1, \cdots, t_k) \mapsto (t_1, \cdots, t_{i-1}, t_{i}, t_i, t_{i+1}, cdots, t_k)$. That means that in the sum above $t_i$ appears twice to yield

\begin{displaymath}
\cdots + t_i(x^i - x^{i-1}) + t_i(x^{i+1} - x^i) + \cdots
  =
  \cdots + t_i(x^{i+1} - x^{i-1} ) + \cdots
\end{displaymath}
which indeed corresponds to omission of the $i$th point $x^i$.



\end{itemize}
\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

The collection of first order infinitesimal neighbours of a space $X$ arranges itself into the [[schreiber:infinitesimal path ∞-groupoid]] $\Pi^{inf}(X)$. Various concepts derive from this one:

of [[differential form]]s may be understood in terms of functions on $\Pi(x)^{inf}$. This is described at

\begin{itemize}%
\item [[differential forms in synthetic differential geometry]].

\end{itemize}
A [[deRham space]] is the colimit over a $\Pi^{inf}(X)$.

\hypertarget{related_concepts_2}{}\subsection*{{Related concepts}}\label{related_concepts_2}

\begin{itemize}%
\item [[formal neighbourhood of the diagonal]]

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

In the language of [[synthetic differential geometry]] the infinitesimal singular complex for ``formal manifolds'' (internally defined manifolds with an infinitesimal thickening to all orderes) is described (with the simplicial structure not made explicit) in

section I.18 of

\begin{itemize}%
\item [[Anders Kock]], \emph{Synthetic Differential Geometry} (\href{http://home.imf.au.dk/kock/sdg99.pdf}{pdf})

\end{itemize}
and in \href{http://home.imf.au.dk/kock/SGM-final.pdf#page=89}{section 2.8} of

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

\end{itemize}
Discussion of this that does make the simplicial structure explicit and relates it to the [[Dold-Kan correspondence]] is in

\begin{itemize}%
\item Herman Stel, \emph{[[schreiber:master thesis Stel|∞-Stacks and their Function Algebras ? with applications to ∞-Lie theory]]}.


\item Herman Stel, \emph{Cosimplicial C-infinity rings and the de Rham complex of Euclidean space} (\href{http://arxiv.org/abs/1310.7407}{arXiv:1310.7407})



\end{itemize}
The details of what $X^{\Delta^k_{inf}}$ is like concretely on representables in the [[smooth topos]] $PSh(k-Alg^{op})$ of [[algebraic geometry]], i.e. on [[affine schemes]] is worked out in detail in

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

\end{itemize}
The formulas given there should more or less directly carry over to [[smooth topos]]es with [[smooth locus|smooth loci]] by replacing ordinary rings with [[smooth algebra]]s.

\begin{quote}%
to be discussed


\end{quote}
As the title suggests, the infinitesimal singular simplicial complex is tightly related to [[differential forms in synthetic differential geometry]]: the deRham complex is the normalized [[Moore complex|Moore cochain complex]] of the [[cosimplicial algebra]] $C^\infty(X^{\Delta^\bullet_{inf}})$ of functions on the spaces of infinitesimal simplices.

There is also

\begin{itemize}%
\item [[Eduardo Dubuc]], [[Anders Kock|Kock]], \emph{On 1-form classifiers} , Communications in Algebra 12 (1984)


\item Dubuc, $C^\infty$-schemes, Amer. J. of Math. 103 (1981)


\item Kumpera, Spencer, \emph{Lie Equations} , Annals of Math. Studies 73 (1973)



\end{itemize}
There is also a version of the infinitesimal singular simplicial context in the context of [[nonstandard analysis]]. See

\begin{itemize}%
\item Zivaljevic, \emph{On a cohomology theory based on hyperfinite sums of microcomplexes} .

\end{itemize}
[[!redirects infinitesimal path ∞-groupoid in a lined topos]] [[!redirects infinitesimal path ∞-groupoid in a smooth topos]] [[!redirects infinitesimal path infinity-groupoid in a smooth topos]] [[!redirects infinitesimal path infinity-groupoid in a lined topos]] [[!redirects infinitesimal neighbour]]



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