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)
Synthetic differential topology (SDT) is a synthetic axiomatization of differential topology in analogy to how synthetic differential geometry (SDG) is synthetic axiomatization of differential geometry.
Where in SDG the concept of infinitesimal neighbourhoods is encoded by the axioms (the Kock-Lawvere axiom) in SDT it is germs of spaces that are being encoded by the axioms.
Notice that the germ of a manifold around a point is in general “larger” than the formal neighbourhood of that point, reflecting, dually, the fact that there are smooth functions which are non-vanishing in every open neighbourhood of that point but all whose partial derivatives vanish at that point (see also at bump function).
SDT uses the representability of germs by an object defined intrinsically as the points not well-separated from $0 \in R^n$, $\neg \neg \{0\}$.
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
Marta Bunge and Eduardo Dubuc, Local Concepts in Synthetic Differential Geometry and Germ Representability, article
Marta Bunge, Felipe Gago, Synthetic aspects of $C^\infty$-mapping II: Mather’s theorem for infinitesimally represented germs, Journal of Pure and Applied Algebra 55 (1988) 213-250 North-Holland (doi:10.1016/0022-4049(88)90117-X)
Marta Bunge, Felipe Gago, Ana Maria San Luis, Synthetic Differential Topology, Cambridge, to appear April 2018