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\section*{Models for Smooth Infinitesimal Analysis}

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

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

[[!include synthetic differential geometry - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

This entry is about the book

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Gonzalo Reyes|Gonzalo E. Reyes]], \emph{Models for Smooth Infinitesimal Analysis} Springer (1991)

\end{itemize}
about models of [[smooth topos]]es for [[synthetic differential geometry]] that have a [[full and faithful functor|full and faithful]] embedding of the [[category]] [[Diff]] of smooth [[manifold]]s.

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

\noindent\hyperlink{summary}{Summary}\dotfill \pageref*{summary} \linebreak
\noindent\hyperlink{Models}{Models}\dotfill \pageref*{Models} \linebreak
\noindent\hyperlink{the_topos_}{the topos $\mathcal{G}$}\dotfill \pageref*{the_topos_} \linebreak
\noindent\hyperlink{the_topos__2}{the topos $\mathcal{F}$}\dotfill \pageref*{the_topos__2} \linebreak
\noindent\hyperlink{the_topos__3}{the topos $\mathcal{Z}$}\dotfill \pageref*{the_topos__3} \linebreak
\noindent\hyperlink{the_topos__4}{the topos $\mathcal{B}$}\dotfill \pageref*{the_topos__4} \linebreak


\hypertarget{summary}{}\subsection*{{Summary}}\label{summary}

The book discusses the construction and the properties of [[smooth topos]]es $(\mathcal{T},R)$ that model the axioms of [[synthetic differential geometry]] and are \emph{well-adapted} to [[differential geometry]] in that there is a [[full and faithful functor]] $Diff \to \mathcal{T}$ embedding the [[category]] [[Diff]] of smooth [[manifold]]s into the more general category $\mathcal{T}$.

All models are obtained as [[category of sheaves|categories of sheaves]] on [[site]]s whose underlying category is a [[subcategory]] of that of [[smooth loci]].

\begin{itemize}%
\item [[de Rham theorem|synthetic de Rham theorem]]

\end{itemize}
\hypertarget{Models}{}\subsection*{{Models}}\label{Models}

The following tabulates various models for [[smooth topos]]es and lists their properties.

\hypertarget{the_topos_}{}\subsubsection*{{the topos $\mathcal{G}$}}\label{the_topos_}

\hypertarget{the_topos__2}{}\subsubsection*{{the topos $\mathcal{F}$}}\label{the_topos__2}

\hypertarget{the_topos__3}{}\subsubsection*{{the topos $\mathcal{Z}$}}\label{the_topos__3}

The [[smooth topos]] $\mathcal{Z}$ is that of [[sheaf|sheaves]] on the category $\mathbb{L}$ of [[smooth loci]] with respect to the [[Grothendieck topology]] given by \emph{finite} open covers of smooth loci.

\begin{udefn}
$\mathcal{Z} := Sh_{fin-open}(\mathbb{L})$ is the [[category of sheaves]] on the entire [[site]] $\mathbb{L}$ of [[smooth locus|smooth loci]] where the [[cover]]ing [[sieve]]s of any [[smooth locus]] $\ell A$ are those generated by covering families

\begin{displaymath}
\{f_i : \ell A_i  \to \ell A\}
\end{displaymath}
given by a \emph{finite} collection of elements $(a_i \in A)_{i=1}^n$ such that the ideal generated by these elements contains the unit, $1 \in (a_1, \cdots, a_n)$, and for each $i$ a commutative diagram

\begin{displaymath}
\itexarray{
    \ell A_i &&\stackrel{\simeq}{\to}&&  \ell (A\{a_i^{-1}\})
    \\
    & {}_{\mathllap{f_i}}\searrow && \swarrow
    \\
    &&
    \ell A
  }
  \,,
\end{displaymath}
where the right diagonal morphism is the canonical inclusion of a smooth locus corresponding to a [[generalized smooth algebra|smooth ring]] with one element inverted.

\end{udefn}
This is in chapter VI, 1. Inversion of elements is described around proposition 1.6 in chapter I.

For instance for $\ell A = R = \ell C^\infty(\mathbb{R})$ the real line, a covering family is given by maps from two further copies of the real line $f_{1,2} : R \to R$ determined by any two smooth functions $a_1, a_1 \in C^\infty(\mathbb{R})$ with support $(-\infty,1)$ and $(-1,\infty)$. By proposition 1.6 in chapter I we have $C^\infty(\mathbb{R})\{a_1^{-1}\} = C^\infty((-\infty,1))$ and $C^\infty(\mathbb{R})\{a_2^{-1}\} = C^\infty((-1,\infty))$. As both these open intervals are diffeomorphic to the real line, and as [[Diff]] embeds fully, we have isomorphisms $R \stackrel{\simeq}{\to} \ell(C^\infty(\mathbb{R}/{a_i^{-1}})$. Hence the cover defined by $(a_i, a_2)$ is the ordinary open cover of the real line by the two open subsets $(-\infty,1)$ and $(-1,\infty)$.

Similarly using I.1.6, one finds the general result:

\begin{ulemma}
Let $\ell C^\infty(U)/I$ be a [[smooth locus]] with $U \subset \mathbb{R}^n$ open and $I$ an ideal of $C^\infty(U)$. Then, up to isomorphisms, its covering families are precisely those families

\begin{displaymath}
\{\ell C^\infty(U_i)/(I|U_i) \to \ell C^\infty(U_i)/I\}_{i=1}^n
\end{displaymath}
such that the $U_i \subset U$ are an ordinary open cover of $U$.

Equivalently, such families where the $(U_i)$ need not cover all of $U$, but where there is $V \subset U$ open such that the $(U_i)$ together with $V$ do cover and $1 \in I|_V$.

\end{ulemma}
\begin{proof}
This is lemma 1.2 in chapter VI.

\end{proof}
We now list central properties of this topos.

\hypertarget{proposition}{}\paragraph*{{Proposition}}\label{proposition}

\textbf{(properties)}

For the [[topos]] $\mathcal{Z}$ the following is true.

\begin{itemize}%
\item the [[Grothendieck topology]] is [[subcanonical coverage|subcanonical]]

(chapter VI, lemma 1.3)


\item the [[category]] [[Diff]] of smooth [[manifold]]s embeds [[full and faithful functor|full and faithfully]], $Diff \hookrightarrow \mathcal{Z}$

(chapter VI, corollary 1.4)


\item the general [[Kock-Lawvere axiom]] holds

(chapter VI, 1.9)


\item the [[integration axiom]] holds

(chapter VI, 1.10)


\item it models [[nonstandard analysis]] in that

\begin{itemize}%
\item since the topology is subcanonical, in particular the [[smooth locus]] $\mathbb{I} := \ell(C^\infty(\mathbb{R}-{0})/(f|germ_0(f) 
  = 0))$ of the ring of restrictions of [[germ]]s of functions at 0 to $\mathbb{R}-{0}$ is an object: the object of \textbf{invertible infinitesimals}.

However, even though this object exists, in the [[intuitionistic logic|intuitionistic]] [[internal logic]] of the topos one cannot prove that there are any infinitesimal \emph{elements} : all one can prove is that it is false that there are no elements: $\not \not \exists x : x \in \mathbb{I}$.

(chapter VI, section 1.8)

This changes when one refines to the topos $\mathcal{B}$, discussed below (section VI.5).


\item due the conditions that covers are finite, the [[smooth locus]] $N := \ell C^\infty(\mathbb{N})$ -- which is such that functions to it are arbitrary locally constant $\mathbb{N}$-valued functions -- does not coincide with the [[natural numbers object]] of the topos, which is the [[sheafification]] of the [[presheaf]] constant on $\mathbb{N} \in Set$:

since covering families are by \emph{finite} covers it follows that maps into the sheafification of the presheaf constant on $\mathbb{N}$ are \emph{bounded} smooth $\mathbb{N}$-valued functions, instead of all such functions.

(chapter VI, 1.6)

The object $N = \ell C^\infty(\mathbb{N})$ is called the object of \textbf{[[smooth natural numbers]]} . It may be thought of as containing ``infinite natural numbers''.



\end{itemize}


\end{itemize}
\hypertarget{the_topos__4}{}\subsubsection*{{the topos $\mathcal{B}$}}\label{the_topos__4}

The [[smooth topos]] $\mathcal{B}$ may be motivated as a slight refinement of the topos $\mathcal{Z}$ designed such that in the [[internal logic]] of $\mathcal{B}$ it does become true that for $\mathbb{I}$ the object of invertible infinitesimals, we have $\exists x : x \in \mathbb{I}$, internally.

\begin{udefn}
\textbf{(chapter VI, 5.1)}

$\mathcal{B} := Sh_{fin-open/proj}(\mathbb{L})$ is the [[category of sheaves]] on the [[site]] $\mathbb{L}$ of [[smooth locus|smooth loci]] with [[cover]]ing [[sieve]]s given by

\begin{itemize}%
\item finite open covers as above for $\mathbb{Z}$


\item and in addition letting projections $\ell A \times \ell B \to \ell A$ out of [[product]]s in $\mathbb{L}$ (for $B \neq 0$ if $A \neq 0$) be singleton covers.



\end{itemize}
\end{udefn}
(This is not a [[Grothendieck topology]], as it is not closed under composition, but still a [[coverage]].)

\hypertarget{proposition_2}{}\paragraph*{{Proposition}}\label{proposition_2}

\textbf{(properties)}

The topos inherits most of the properties of $\mathcal{Z}$, notably:

\begin{itemize}%
\item the [[Grothendieck topology]] is [[subcanonical coverage|subcanonical]]

(chapter VI, 5.2)


\item the [[category]] [[Diff]] of smooth [[manifold]]s embeds [[full and faithful functor|full and faithfully]], $Diff \hookrightarrow \mathcal{B}$

(chapter VI, 5.2)


\item the general [[Kock-Lawvere axiom]] holds

(chapter VI, below 5.4)


\item the [[integration axiom]] holds

(chapter VI, below 5.4)


\item invertible infinitesimally small and infinitesimally big numbers are realized (with some vague similarity to [[nonstandard analysis]], but without its basic features like transfer principle in full generality)

(chapter VI, below 5.4).



\end{itemize}
A main difference is that in $\mathcal{B}$ every [[smooth locus]], i.e. every [[representable functor|representable]], is an [[inhabited object]]. In particular therefore there exist, in the [[internal logic]], elements of the object of invertible infinitesimals:

\begin{displaymath}
\exists x \in \mathbb{I}
  \,.
\end{displaymath}
(chpater VI, prop 5.4).

category: reference



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