This entry is about the book
about models of smooth toposes for synthetic differential geometry that have a full and faithful embedding of the category Diff of smooth manifolds.
The book discusses the construction and the properties of smooth toposes $(\mathcal{T},R)$ that model the axioms of synthetic differential geometry and are well-adapted to differential geometry in that there is a full and faithful functor $Diff \to \mathcal{T}$ embedding the category Diff of smooth manifolds into the more general category $\mathcal{T}$.
All models are obtained as categories of sheaves on sites whose underlying category is a subcategory of that of smooth loci.
The following tabulates various models for smooth toposes and lists their properties.
The smooth topos $\mathcal{Z}$ is that of sheaves on the category $\mathbb{L}$ of smooth loci with respect to the Grothendieck topology given by finite open covers of smooth loci.
$\mathcal{Z} := Sh_{fin-open}(\mathbb{L})$ is the category of sheaves on the entire site $\mathbb{L}$ of smooth loci where the covering sieves of any smooth locus $\ell A$ are those generated by covering families
given by a 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
where the right diagonal morphism is the canonical inclusion of a smooth locus corresponding to a smooth ring with one element inverted.
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:
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
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 $1 \in I|_V$.
This is lemma 1.2 in chapter VI.
We now list central properties of this topos.
(properties)
For the topos $\mathcal{Z}$ the following is true.
the Grothendieck topology is subcanonical
(chapter VI, lemma 1.3)
the category Diff of smooth manifolds embeds full and faithfully, $Diff \hookrightarrow \mathcal{Z}$
(chapter VI, corollary 1.4)
the general Kock-Lawvere axiom holds
(chapter VI, 1.9)
the integration axiom holds
(chapter VI, 1.10)
it models nonstandard analysis in that
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 germs of functions at 0 to $\mathbb{R}-{0}$ is an object: the object of invertible infinitesimals.
However, even though this object exists, in the intuitionistic internal logic of the topos one cannot prove that there are any infinitesimal 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).
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 finite covers it follows that maps into the sheafification of the presheaf constant on $\mathbb{N}$ are 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 smooth natural numbers . It may be thought of as containing “infinite natural numbers”.
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.
(chapter VI, 5.1)
$\mathcal{B} := Sh_{fin-open/proj}(\mathbb{L})$ is the category of sheaves on the site $\mathbb{L}$ of smooth loci with covering sieves given by
finite open covers as above for $\mathbb{Z}$
and in addition letting projections $\ell A \times \ell B \to \ell A$ out of products in $\mathbb{L}$ (for $B \neq 0$ if $A \neq 0$) be singleton covers.
(This is not a Grothendieck topology, as it is not closed under composition, but still a coverage.)
(properties)
The topos inherits most of the properties of $\mathcal{Z}$, notably:
the Grothendieck topology is subcanonical
(chapter VI, 5.2)
the category Diff of smooth manifolds embeds full and faithfully, $Diff \hookrightarrow \mathcal{B}$
(chapter VI, 5.2)
the general Kock-Lawvere axiom holds
(chapter VI, below 5.4)
the integration axiom holds
(chapter VI, below 5.4)
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).
A main difference is that in $\mathcal{B}$ every smooth locus, i.e. every representable, is an inhabited object. In particular therefore there exist, in the internal logic, elements of the object of invertible infinitesimals:
(chpater VI, prop 5.4).