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?
A site of formal Cartesian spaces.
Let $FormalCartSp$ (or $FormalCartSp$) 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
The (∞,1)-topos of (∞,1)-sheaves over $FormalCartSp$ is that of formal smooth ∞-grouopoids?
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