higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Write $CartSp$ for the category whose
objects are Cartesian spaces $\mathbb{R}^n$ for $n \in \mathbb{N}$;
morphisms are suitable structure-preserving functions between these spaces.
For definiteness we write
$CartSp_{lin}$ for the category whose objects are Cartesian spaces regarded as real vector spaces and whose morphisms are linear functions between these;
$CartSp_{top}$ for the category whose objects are Cartesian spaces regarded as topological spaces equipped with their Euclidean topology and morphisms are continuous maps between them.
$CartSp_{smooth}$ 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 $C$, for $C$ (any one of) the category of
In all these cases, the inclusion $CartSp \hookrightarrow C$ 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${}_{top}$ for the category whose objects are Cartesian spaces and whose morphisms are all continuous maps between these.
CartSp${}_{smooth}$ for the category whose objects are Cartesian spaces and whose morphisms are all smooth functions between these.
CartSp${}_{synthdiff}$ for the full subcategory of the category of smooth loci on those of the form $\mathbb{R}^n \times D$ for $D$ 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${}_{top}$ this is obvious. For CartSp${}_{smooth}$ 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 $CartSp_{synthdiff}$.
The site $CartSp_{top}$ is a dense subsite of the site of paracompact topological manifolds with the open cover coverage.
The site $CartSp_{smooth}$ 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
$Sh_{(1,1)}(CartSp_{smooth})$, discussed at diffeological space.
$Sh_{(1,1)}(CartSp_{synthdiff})$, discussed at Cahiers topos.
The corresponding cohesive (∞,1)-topos of (∞,1)-sheaves is
$Sh_{(\infty,1)}(CartSp_{top}) =$ ETop∞Grpd;
$Sh_{(\infty,1)}(CartSp_{smooth}) =$ Smooth∞Grpd;
$Sh_{(\infty,1)}(CartSp_{synthdiff}) =$ SynthDiff∞Grpd;
We have equivalences of categories
$Sh(CartSp_{top}) \simeq Sh(TopMfd)$
$Sh(CartSp_{smooth}) \simeq Sh(Diff)$
and equivalences of (∞,1)-categories
$Sh_{(\infty,1)}(CartSp_{top}) \simeq Sh_{(\infty,1)}(TopMfd)$;
$Sh_{(\infty,1)}(CartSp_{smooth}) \simeq Sh_{(\infty,1)}(Diff)$.
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 $Sh_{(\infty,1)}(CartSp)$ 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 $Sh(CartSp)^{\Delta^{op}}_{loc}$. The claim then follows with the first two statements.
There is a canonical structure of a category with open maps on $CartSp$ (…)
The category $CartSp$ 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, $CartSp$ 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 $CartSp$ is discussed, with an eye towards applications in physics, in geometry of physics.
The sheaf topos over $CartSp_{smooth}$ is that of smooth spaces.
The (∞,1)-sheaf (∞,1)-topos over $CartSp_{top}$ is discussed at ETop∞Grpd, that over $CartSp_{smooth}$ at Smooth∞Grpd, and that over $CartSp_{synthdiff}$ at SynthDiff∞Grpd.
The generalization of $CartSp$ to formal smooth manifolds is FormalCartSp.
In secton 2 of
$CartSp$ is discussed as an example of a “cartesian differential category”.
There are various slight variations of the category $CartSp$ (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 $CartSp_{synthdiff}$ 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