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)
A smooth locus is the formal dual of a finitely generated smooth algebra (or -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 , a smooth locus is the analog in well-adapted models for synthetic differential geometry for what in algebraic geometry is an affine variety over .
A finitely generated smooth algebra is one of the form , for an ideal of the ordinary underlying algebra.
Write for the category of finitely generated smooth algebras.
Then the opposite category is the category of smooth loci.
For one write for the corresponding object in .
Often one also write
for the real line regarded as an object of .
The category 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 : -valued functions on closed subobjects in have an extension.
There are various Grothendieck topologies on 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.
Last revised on July 7, 2023 at 19:54:31. See the history of this page for a list of all contributions to it.