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)
The smooth Serre-Swan theorem (Nestruev 03, 11.33) states that over a smooth manifold $X$,
that sends smooth vector bundles over $X$ of finite rank to their spaces of smooth sections, regarded as modules over the algebra of smooth functions on $X$, is a fully faithful functor
its essential image consists precisely of the finitely generated projective modules
This is the variant for differential geometry of what the Serre-Swan theorem asserts in algebraic geometry and in topology.
(base smooth manifold not required to be compact)
Contrary to the original theorem of Swan 62 for topological vector bundles, here in the smooth case the base smooth manifold $X$ of the smooth vector bundle is not required to be compact. While for topological spaces compactness is needed to deduce that every topological vector bundle is a direct summand of a trivial vector bundle (this prop.) for smooth vector bundles this conclusion follows without assuming compactness, by embedding of smooth manifolds into Cartesian spaces (this prop.).
(other algebraic apects of differential geometry)
Together with the embedding of smooth manifolds into formal duals of R-algebras, the smooth Serre-Swan theorem states that that differential geometry is “more algebraic” than it may superficially seem. A third fact in this vein is that derivations of smooth functions are vector fields.