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)
A smooth locus is the formal dual of a finitely generated smooth algebra (or $C^\infty$-ring):
a space that behaves as if its algebra of functions is a finitely generated smooth algebra.
Given that a smooth algebra is a smooth refinement of an ordinary ring with a morphism from $\mathbb{R}$, a smooth locus is the analog in well-adapted models for synthetic differential geometry for what in algebraic geometry is an affine variety over $\mathbb{R}$.
A finitely generated smooth algebra is one of the form $C^\infty(\mathbb{R}^n)/J$, for $J$ an ideal of the ordinary underlying algebra.
Write $C^\infty Ring^{fin}$ for the category of finitely generated smooth algebras.
Then the opposite category $\mathbb{L} := (C^\infty Ring^{fin})^{op}$ is the category of smooth loci.
For $A \in C^\infty Ring^{fin}$ one write $\ell A$ for the corresponding object in $\mathbb{L}$.
Often one also write
for the real line regarded as an object of $\mathbb{L}$.
The category $\mathbb{L}$ has the following properties:
The canonical inclusion functor
from the category SmthMfd of smooth manifolds is a full subcategory embedding (i.e. a full and faithful functor. Moreover, it preserves pullbacks along transversal maps.
The Tietze extension theorem holds in $\mathbb{L}$: $R$-valued functions on closed subobjects in $\mathbb{L}$ have an extension.
There are various Grothendieck topologies on $\mathbb{L}$ and various of its subcategories, such that categories of sheaves on these are smooth toposes that are well-adapted models for synthetic differential geometry.
For more on this see
See the references at C-infinity-ring.