# nLab CartSp

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

###### Definition

Write $\mathrm{CartSp}$ for the category whose

• objects are Cartesian spaces ${ℝ}^{n}$ for $n\in ℕ$;

• morphisms are suitable structure-preserving functions between these spaces.

For definiteness we write

${\mathrm{CartSp}}_{\mathrm{lin}}$ for the category whose objects are Cartesian spaces regarded as real vector spaces and whose morphisms are linear functions between these;

## Properties

### 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 $C$

In all these cases, the inclusion $\mathrm{CartSp}↪C$ 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.

### As a site

###### Definition

Write

• CartSp${}_{\mathrm{top}}$ for the category whose objects are Cartesian spaces and whose morphisms are all continuous maps between these.

• CartSp${}_{\mathrm{smooth}}$ for the category whose objects are Cartesian spaces and whose morphisms are all smooth functions between these.

• CartSp${}_{\mathrm{synthdiff}}$ for the full subcategory of the category of smooth loci on those of the form ${ℝ}^{n}×D$ for $D$ an infinitesimal space (the formal dual of a Weil algebra).

###### Proposition

In all three cases there is the good open cover coverage that makes CartSp a site.

###### Proof

For CartSp${}_{\mathrm{top}}$ this is obvious. For CartSp${}_{\mathrm{smooth}}$ 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 ${\mathrm{CartSp}}_{\mathrm{synthdiff}}$.

###### Proposition

Equipped with this structure of a site, CartSp is an ∞-cohesive site.

The corresponding cohesive topos of sheaves is

• ${\mathrm{Sh}}_{\left(1,1\right)}\left({\mathrm{CartSp}}_{\mathrm{smooth}}\right)$, discussed at diffeological space.

• ${\mathrm{Sh}}_{\left(1,1\right)}\left({\mathrm{CartSp}}_{\mathrm{synthdiff}}\right)$, discussed at Cahiers topos.

The corresponding cohesive (∞,1)-topos of (∞,1)-sheaves is

• ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left({\mathrm{CartSp}}_{\mathrm{top}}\right)=$ ETop∞Grpd;

• ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left({\mathrm{CartSp}}_{\mathrm{smooth}}\right)=$ Smooth∞Grpd;

• ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left({\mathrm{CartSp}}_{\mathrm{synthdiff}}\right)=$ SynthDiff∞Grpd;

###### Corollary

We have equivalences of categories

• $\mathrm{Sh}\left({\mathrm{CartSp}}_{\mathrm{top}}\right)\simeq \mathrm{Sh}\left(\mathrm{TopMfd}\right)$

• $\mathrm{Sh}\left({\mathrm{CartSp}}_{\mathrm{smooth}}\right)\simeq \mathrm{Sh}\left(\mathrm{Diff}\right)$

• ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left({\mathrm{CartSp}}_{\mathrm{top}}\right)\simeq {\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{TopMfd}\right)$;

• ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left({\mathrm{CartSp}}_{\mathrm{smooth}}\right)\simeq {\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{Diff}\right)$.

###### Proof

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 ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{CartSp}\right)$ 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 $\mathrm{Sh}\left(\mathrm{CartSp}{\right)}_{\mathrm{loc}}^{{\Delta }^{\mathrm{op}}}$. The claim then follows with the first two statements.

### As a category with open maps

There is a canonical structure of a category with open maps on $\mathrm{CartSp}$ (…)

### As an algebraic theory

The category $\mathrm{CartSp}$ 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, $\mathrm{CartSp}$ 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 $\mathrm{CartSp}$ is discussed, with an eye towards applications in physics, in geometry of physics.

The sheaf topos over ${\mathrm{CartSp}}_{\mathrm{smooth}}$ is that of smooth space.

The (∞,1)-sheaf (∞,1)-topos over ${\mathrm{CartSp}}_{\mathrm{top}}$ is discussed at ETop∞Grpd, that over ${\mathrm{CartSp}}_{\mathrm{smooth}}$ at Smooth∞Grpd, and that over ${\mathrm{CartSp}}_{\mathrm{synthdiff}}$ at SynthDiff∞Grpd.

## References

In secton 2 of

$\mathrm{CartSp}$ is discussed as an example of a “cartesian differential category”.

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

in detal 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)

category: category

Revised on October 24, 2012 20:28:12 by Tim Porter (95.147.237.169)