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)
higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
is the category whose
morphisms are smooth functions between these.
Similarly, for
is the category whose
morphisms are differentiable functions between these
for -fold differentiabiliy.
Each of these categories is also commonly denoted or or Diff etc.
The category becomes a large site by equipping it with the coverage consisting of open covers.
This is an essentially small site: a dense sub-site for is given by CartSp.
The first statement follows trivially as for Top: the preimage of an open subset under a continuous function is again open (by definition of continuouss function).
For the second statement one needs that every paracompact manifold admits a differentially good open cover : an open cover by open balls that are diffeomorphic to a Cartesian spaces. The proof for this is spelled out at good open cover.
The sheaf topos over is a cohesive topos.
The hypercompletion of the (∞,1)-sheaf (∞,1)-topos over is a cohesive (∞,1)-topos.
For the first statement, use that by the comparison lemma discussed at dense sub-site we have an equivalence of categories
By the discussion at CartSp we have that is a cohesive site. By the discussion there the claim follows.
For the second statement observe that the Joyal-Jardine model structure on simplicial sheaves is a presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves (see presentations of (∞,1)-sheaf (∞,1)-toposes). By the above result it follows that there is an equivalence of (∞,1)-categories between the hypercompletions
Now CartSp is even an ∞-cohesive site. By the discussion there it follows that (before hypercompletion) is a cohesive (∞,1)-topos. This means that it is in particular a local (∞,1)-topos. But this implies (as discussed there), that the (∞,1)-category of (∞,1)-sheaves already is the hypercomplete (∞,1)-topos. Therefore finally
The cohesive topos is in particular the home of diffeological spaces. See there for more details.
is that of smooth ∞-groupoids. Discussed at Smooth∞Grpd.
The theory of differentiable stacks is that of geometric stacks in the (2,1)-sheaf (2,1)-topos
Last revised on June 26, 2022 at 11:54:34. See the history of this page for a list of all contributions to it.