As a small category of objects with a basis
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.
As 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 corresponding cohesive topos of sheaves is
, discussed at diffeological space.
, discussed at Cahiers topos.
The corresponding cohesive (∞,1)-topos of (∞,1)-sheaves is
As a category with open maps
There is a canonical structure of a category with open maps on (…)
As an algebraic theory
The category is (the syntactic category of ) a Lawvere theory: the theory for smooth algebras.
As a pre-geometry
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).
A development of differential geometry as as geometry modeled on is discussed, with an eye towards applications in physics, in geometry of physics.
The sheaf topos over is that of smooth spaces.
The (∞,1)-sheaf (∞,1)-topos over is discussed at ETop∞Grpd, that over at Smooth∞Grpd, and that over at SynthDiff∞Grpd.
The generalization of to formal smooth manifolds is FormalCartSp.
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
- Hirokazu Nishimura, Beyond the Regnant Philosophy of Manifolds (arXiv:0912.0827)