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 function $f : X \to Y$ between two smooth manifolds is a local diffeomorphism if the following equivalent conditions hold
$f$ is both a submersion and an immersion;
for each point $x \in X$ the derivative $d f : T_x X \to T_{f(x)} Y$ is an isomorphism of tangent vector spaces;
the canonical diagram
(with the differential between the tangent bundles) on top is a pullback;
for each point $x \in X$ there exists an open subset $x \in U \subset X$ such that
the image $f(U)$ is an open subset in $Y$;
$f$ restricted to $U$ is a diffeomorphism onto its image
The equivalence of the conditions on tangent space with the conditions on open subsets follows by the inverse function theorem.
An analogous characterization of étale morphisms between affine algebraic varieties is given by tangent cones. See there.
The category SmoothMfd of smooth manifolds may naturally be thought of as sitting inside the more general context of the cohesive (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids. This is canonically equipped with a notion of differential cohesion exhibited by its inclusion into SynthDiff∞Grpd. This implies that there is an intrinsic notion of formally étale morphisms of smooth $\infty$-groupoids in general and of smooth manifolds in particular
A smooth function is a formally étale morphism in this sense precisely if it is a local diffeomorphism.
See this section for more details.
Discussion in the synthetic differential geometry of the Cahiers topos is in