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
Models
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 , (“logical topology”).
Examples of sequences of local structures
Marta Bunge and Eduardo Dubuc, Local Concepts in Synthetic Differential Geometry and Germ Representability, article
Marta Bunge, Felipe Gago, Synthetic aspects of -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, 2018, (CUP) (excerpt)
The logical topology goes back to
Review in the context of cohesive toposes, modal type theory and cohesive homotopy type theory includes
Last revised on July 26, 2022 at 14:40:42. See the history of this page for a list of all contributions to it.