synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
The Cahier topos is a cohesive topos that constitutes a well-adapted model for synthetic differential geometry (a “smooth topos”).
It is the sheaf topos on the site FormalCartSp of infinitesimally thickened Cartesian spaces.
Let FormalCartSp be the full subcategory of the category of smooth loci on those of the form
consisting of a product of a Cartesian space with an infinitesimally thickened point, i.e. a formal dual of a Weil algebra .
Dually, the opposite category is the full subcategory $FormalCartSp^{op} \hookrightarrow SmoothAlg$ of smooth algebras on those of the form
This appears for instance in Kock Reyes (1).
Define a structure of a site on FormalCartSp by declaring a covering family to be a family of the form
where $\{U_i \stackrel{p_i}{\to} U\}$ is an open cover of the Cartesian space $U$ by Cartesian spaces $U_i$.
This appears as Kock (5.1).
The Cahiers topos $\mathcal{CT}$ is the category of sheaves on this site:
This site of definition appears in Kock, Reyes. The original definition is due to Dubuc 79
The Cahiers topos is a well-adapted model for synthetic differential geometry.
This is due to Dubuc 79.
The Cahiers topos is a cohesive topos. See synthetic differential infinity-groupoid for details.
The category of convenient vector spaces with smooth functions between them embeds as a full subcategory into the Cahiers topos.
The embedding is given by sending a convenient vector space $V$ to the sheaf given by
This result was announced in Kock. See the corrected proof in (KockReyes).
Together with prop. 1 this means that the differential geometry on convenient vector spaces may be treated synthetically in the Cahiers topos.
We discuss here induced synthetic tangent spaces of smooth spaces in the sense of diffeological spaces and more general sheaves on the site of smooth manifolds after their canonical embedding into the Cahiers topos.
Write $SmoothLoc$ for the category of smooth loci. Write
for the full subcategory on the Cartesian spaces $\mathbb{R}^n$ ($n \in \mathbb{N}$). Write
for the full subcategory on the infinitesimally thickened points, and write
for the full subcategory on those smooth loci which are the cartesian product of a Cartesian space $\mathbb{R}^n$ ($n \in \mathbb{N}$) and an infinitesimally thickened point.
We regard CartSp as a site by equipping it with the good open cover coverage. We regard $InfThPoint$ as equipped with the trivial coverage and $CartSp_{synthdiff}$ as equipped with the induced product coverage.
The sheaf topos $Sh(CartSp)$ is that of smooth spaces. The sheaf topos $Sh(CartSp_{synthdiff})$ is the Cahier topos.
We write
for the infinitesimal interval, the smooth locus dual to the smooth algebra “of dual numbers”.
For $X \in Sh(CartSp_{synthdiff})$ any object in the Cahier topos, its synthetic tangent bundle in the sense of synthetic differential geometry is the internal hom space $X^D$, equipped with the projection map
The canonical inclusion functor $i \colon CartSp \to CartSp_{synthdiff}$ induces an adjoint pair
where $i^\ast$ is given by precomposing a presheaf on $CartSp_{synthdiff}$ with $i$. The left adjoint $i_\ast$ has the interpretation of the inclusion of smooth spaces as reduced objects in the Cahiers topos.
This is discussed in more detail at synthetic differential infinity-groupoid.
For $X \in Sh(CartSp)$ a smooth space, and for $\ell(W) \in InfThPoint$ an infinitesimally thickened point, the morphisms
in $Sh(CartSp_{synthdiff})$ are in natural bijection to equivalence classes of pairs of morphisms
consisting of a morphism in $CartSp_{synth}$ on the left and a morphism in $Sh(CartSp)$ on the right (which live in different categories and hence are not composable, but usefully written in juxtaposition anyway). The equivalence relation relates two such pairs if there is a smooth function $\phi \colon \mathbb{R}^n \to \mathbb{R}^{n'}$ such that in the diagram
the left triangle commutes in $CartSp_{synthdiff}$ and the right one in $Sh(CartSp)$.
By general properties of left adjoints of functors of presheaves, $i_\ast X$ is the left Kan extension of the presheaf $X$ along $i$. By the Yoneda lemma and the coend formula for these (as discussed there), we have that the set of maps $\ell(W) \to i_\ast X$ is naturally identified with
Unwinding the definition of this coend as a coequalizer yields the above description of equivalence classes.
When restricting the site of infinitesimally thickened Cartesian spaces to that of plain Cartesian spaces one obtaines the topos discussed at smooth space. This is still a cohesive topos, but no longer a model for synthetic differential geometry.
The (∞,1)-sheaf (∞,1)-topos over $CartSp_{th}$ is disucssed at synthetic differential ∞-groupoid. It contains that Cahiers topos as the sub-(1,1)-topos of 0-truncated objects.
The Cahiers topos was introduced in
and got its name from this journal publication. The definition appears in theorem 4.10 there, which asserts that it is a well-adapted model for synthetic differential geometry.
A review discussion is in section 5 of
and with a corrected definition of the site of definition in
It appears briefly mentioned in example 2) on p. 191 of the standard textbook
With an eye towards Frölicher spaces the site is also considered in section 5 of
The (∞,1)-topos analog of the Cahiers topos (synthetic differential ∞-groupoids) is discussed in section 3.4 of