nLab
CartSp

Definition

Definition

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

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

• 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;

• ${\mathrm{CartSp}}_{\mathrm{top}}$ for the category whose objects are Cartesian spaces regarded as topological spaces equipped with their Euclidean topology and morphisms are continuous maps between them.

• ${\mathrm{CartSp}}_{\mathrm{smooth}}$ for the category whose objects are Cartesian spaces regarded as smooth manifolds with their standard smooth structure and morphisms are smooth functions.

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 $C$

• vector spaces,
• affine spaces,
• normed vector spaces,
• inner product spaces,
• Euclidean spaces.

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

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 ${ℝ}^n\times D$ 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

• The site ${\mathrm{CartSp}}_{\mathrm{top}}$ is a dense subsite of the site of paracompact topological manifolds with the open cover coverage.

• The site ${\mathrm{CartSp}}_{\mathrm{smooth}}$ is a dense subsite of the site Diff of paracompact smooth manifolds equipped with the open cover coverage.

Proposition

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

The corresponding cohesive topos of sheaves is

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

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

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

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

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

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

Corollary

We have equivalences of categories

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

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

and equivalences of (∞,1)-categories?

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

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

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 hat that ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{CartSp})$ 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 $\mathrm{Sh}(\mathrm{CartSp}{)}_{\mathrm{loc}}^{{\Delta }^{\mathrm{op}}}$. 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).

(…)

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)

category: category