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
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Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
higher geometry / derived geometry
Ingredients
Concepts
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geometric big (∞,1)-toposes
Constructions
Examples
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Theorems
Write for the category whose
objects are Cartesian spaces for ;
morphisms are suitable structure-preserving functions between these spaces.
For definiteness we write
for the category whose objects are Cartesian spaces regarded as real vector spaces and whose morphisms are linear functions between these;
for the category whose objects are Cartesian spaces regarded as topological spaces equipped with their Euclidean topology and morphisms are continuous maps between them.
for the category whose objects are Cartesian spaces regarded as smooth manifolds with their standard smooth structure and morphisms are smooth functions.
A Cartesian space carries a lot of structure, for instance CartSp may be naturally regarded as a full subcategory of the category , for (any one of) the category of
In all these cases, the inclusion is an equivalence of categories: choosing an isomorphism from any of these objects to a Cartesian space amounts to choosing a basis of a vector space, a coordinate system.
Write
CartSp for the category whose objects are Cartesian spaces and whose morphisms are all continuous maps between these.
CartSp for the category whose objects are Cartesian spaces and whose morphisms are all smooth functions between these.
CartSp for the full subcategory of the category of smooth loci on those of the form for an infinitesimal space (the formal dual of a Weil algebra).
In all three cases there is the good open cover coverage that makes CartSp a site.
For CartSp this is obvious. For CartSp this is somewhat more subtle. It is a folk theorem (see the references at open ball). A detailed proof is at good open cover. This directly carries over to .
The site is a dense subsite of the site of paracompact topological manifolds with the open cover coverage.
The site is a dense subsite of the site Diff of paracompact smooth manifolds equipped with the open cover coverage.
Equipped with this structure of a site, CartSp is an ∞-cohesive site.
The corresponding cohesive topos of sheaves is
, discussed at diffeological space.
, discussed at Cahiers topos.
The corresponding cohesive (∞,1)-topos of (∞,1)-sheaves is
The first two statements follow by the above proposition with the comparison lemma discussed at dense sub-site.
For the second condition notice that since an ∞-cohesive site is in particular an ∞-local site we have that is a local (∞,1)-topos. As discussed there, this implies that it is a hypercomplete (∞,1)-topos. By the discussion at model structure on simplicial presheaves this means that it is presented by the Joyal-Jardine-model structure on simplicial sheaves . The claim then follows with the first two statements.
There is a canonical structure of a category with open maps on (…)
The category is (the syntactic category of ) a Lawvere theory: the theory for smooth algebras.
Equipped with the above coverage-structure, open map-structure and Lawvere theory-property, is essentially a pregeometry (for structured (∞,1)-toposes).
(Except that the pullback stability of the open maps holds only in the weaker sense of coverages).
(…)
In secton 2 of
is discussed as an example of a “cartesian differential category”.
There are various slight variations of the category (many of them equivalent) that one can consider without changing its basic properties as a category of test spaces for generalized smooth spaces. A different choice that enjoys some popularity in the literature is the category of open (contractible) subsets of Euclidean spaces. For more references on this see diffeological space.
The site of infinitesimally thickened Cartesian spaces is known as the site for the Cahiers topos. It is considered in detail in section 5 of
and briefly mentioned in example 2) on p. 191 of
following the original article
With an eye towards Frölicher spaces the site is also considered in section 5 of
Last revised on July 18, 2022 at 04:05:52. See the history of this page for a list of all contributions to it.