$\mathrm{Cart}\mathrm{Sp}$

Idea

The category $\mathrm{CartSp}$ is the category of cartesian spaces ${ℝ}^n$ and smooth functions between them. This is the basic category of test spaces on which differential geometry and its generalizations is modeled.

Definition

By $\mathrm{CartSp}$ we denote the category

• whose objects are the cartesian spaces ${ℝ}^n$, $n\in \mathbb{N}$ – the real line to its $n$th cartesian power – equipped with their standard smooth structure;

• whose morphisms are all smooth functions between these spaces.

So $\mathrm{CartSp}$ may be thought of as the full subcategory of Diff on the smooth manifolds of the form ${ℝ}^n$ for $n\in \mathbb{N}$.

Some variants of this are of interest:

• write $\mathrm{ThCartSp}$ for the category of infinitesimally thickened Cartesian spaces: the full subcategory on the category of smooth loci on those of the form ${ℝ}^n\times d$, where $d$ is an infinitesimal space.

Geometry induced from $\mathrm{CartSp}$.

Along the general lines of notions of space we have the following notions of spaces modeled on $\mathrm{CartSp}$.

Consider CartSp as a site with the standard notion of coverage (for instance good open covers, using that fact that a Cartesian space is diffeomorphic to an open ball).

Then

• the category of very general spaces modeled on $\mathrm{CartSp}$ is the sheaf topos $\mathrm{Sh}(\mathrm{CartSp})$;

  Some objects in here are indeed very general spaces. For instance there is the sheaf ${\Omega }^p:{ℝ}^n\mapsto {\Omega }_{\mathrm{closed}}^p({ℝ}^n)$ that assigns sets of closed differential forms. This is a very non-classical object. For instance in that this space has just a single point, a single curve, a single surface, and so on, up to a single $(p-1)$-dimensional plot, but then has infinitely many $p$-dimensional plots. Despite this non-classicality, this is a very natural and useful object. To some extent it plays the role of an Eilenberg-MacLane space $K(ℝ,p)$, even though it is far from having an underlying topological space.

• Accordingly, the first major subcategories inside $\mathrm{Sh}(\mathrm{Cart})$ of more tame objects are those sheaves that do have underlying topological spaces: these are the concrete sheaves

  $$\mathrm{DiffeologicalSpaces}\subset \mathrm{Sh}(\mathrm{CartSp})$$DiffeologicalSpaces \subset Sh(CartSp)

  called diffeological spaces. Among them objects like Frölicher spaces play roughly the role of locally CartSp-ringed spaces, vaguely in the sense of structured (∞,1)-toposes.

• Refining one step further to even more tame objects inside $\mathrm{Sh}(\mathrm{CartSp})$ we arrive at those concrete sheaves that are locally isomorphic to a representable functor. i.e. to a Cartesian space itsef. These are the smooth manifolds

  $$\mathrm{SmoothManifolds}\hookrightarrow \mathrm{DiffeologicalSpaces}\hookrightarrow \mathrm{Sh}(\mathrm{CartSp}).$$SmoothManifolds \hookrightarrow DiffeologicalSpaces \hookrightarrow Sh(CartSp) \,.

• Finally the Cartesian test spaces themselves sind inside this hierarchy via the Yoneda embedding $\mathrm{CartSp}\hookrightarrow \mathrm{Sh}(\mathrm{CartSp})$ , which we have hence factored as

  $$\mathrm{CartSp}\hookrightarrow \mathrm{SmoothManifolds}\hookrightarrow \mathrm{DiffeologicalSpaces}\hookrightarrow \mathrm{Sh}(\mathrm{CartSp}).$$CartSp \hookrightarrow SmoothManifolds \hookrightarrow DiffeologicalSpaces \hookrightarrow Sh(CartSp) \,.

As already indicated, this story in ordinary category theory may be lifted to higher topos theory. The (∞,1)-topos ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})$ is a model for generalized Lie ∞-groupoids. The fact that a Cartesian space is effectively just a smooth open ball makes this a locally contractible (∞,1)-topos.

Dually, using that $\mathrm{CartSp}$ is evidently the syntactic category of a Lawvere theory (even of a Fermat theory) there are the generalized quantities modeled on $\mathrm{CartSp}$, algebras over this theory:

• a product-preserving functor $A\in {\mathrm{Func}}_{\times }(\mathrm{CartSp},\mathrm{Set})$ is a smooth algebra.

• a product-preserving (∞,1)-functor $A\in (\mathrm{\infty }\mathrm{,1}){\mathrm{Func}}_{\times }(\mathrm{CartSp},\mathrm{\infty }\mathrm{Grpd})$ , i.e. an algebra over $\mathrm{CartSp}$ regarded as an (∞,1)-algebraic theory is a smooth (∞,1)-algebra.

Iterating this way of passing from simple test spaces to spaces and quantities modeled on them, we can next consider

• $C^{\mathrm{\infty }}{\mathrm{Alg}}^{\mathrm{op}}:={\mathrm{Func}}_{\times }(\mathrm{CartSp},\mathrm{Set}{)}^{\mathrm{op}}$

and

• $(\mathrm{\infty }\mathrm{,1})C^{\mathrm{\infty }}{\mathrm{Alg}}^{\mathrm{op}}:={\mathrm{Func}}_{\times }(\mathrm{CartSp},\mathrm{\infty }\mathrm{Grpd}{)}^{\mathrm{op}}$

as new categories of test spaces, and pass to generalized spaces modeled on these.

The very general spaces modeled on $C^{\mathrm{\infty }}{\mathrm{Alg}}^{\mathrm{op}}$ form the smooth toposes used in the well-adapted Models for Smooth Infinitesimal Analysis in the context of synthetic differential geometry.

On the other hand, considering locally ringed spaces with local rings of functions in $(\mathrm{\infty }\mathrm{,1})C^{\mathrm{\infty }}{\mathrm{Alg}}^{\mathrm{op}}$ leads to the notion of derived smooth manifolds.

References

There are various slight variations of the category $\mathrm{CartSp}$ 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 $\mathrm{ThCartSp}$ of infinitesimally thickened Cartesian spaces is known as the site for the Cahiers topos. It is considered

in detal in section 5 of

• Anders Kock, Convenient vector spaces embed into the Cahiers topos (numdam)

and briefly mentioned in example 2) on p. 191 of

• Anders Kock, Synthetic differential geometry (pdf)

following the original article

• Eduardo Dubuc, Sur les modeles de la geometrie differentielle synthetique (numdam).

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)