nLab CartSp

Definition

Definition

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.

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 $C$, for $C$ (any one of) the category of

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

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.

As a site

Definition

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).

Proposition

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

Proof

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 $CartSp_{synthdiff}$.

Proposition

• 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.

Proposition

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;

Corollary

We have equivalences of categories

• $Sh(CartSp_{top}) \simeq Sh(TopMfd)$

• $Sh(CartSp_{smooth}) \simeq Sh(Diff)$

and equivalences of (∞,1)-categories

• $Sh_{(\infty,1)}(CartSp_{top}) \simeq Sh_{(\infty,1)}(TopMfd)$;

• $Sh_{(\infty,1)}(CartSp_{smooth}) \simeq Sh_{(\infty,1)}(Diff)$.

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 $Sh_{(\infty,1)}(CartSp)$ 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.

As a category with open maps

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

As an algebraic theory

The category $CartSp$ 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, $CartSp$ 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).

(…)

$\,$

Related concepts

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$	$\phantom{A}$site$\phantom{A}$	$\phantom{A}$sheaf topos$\phantom{A}$	$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$	$\phantom{A}$Point$\phantom{A}$	$\phantom{A}$Set$\phantom{A}$	$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$CartSp$\phantom{A}$	$\phantom{A}$SmoothSet$\phantom{A}$	$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$	$\phantom{A}$FormalCartSp$\phantom{A}$	$\phantom{A}$FormalSmoothSet$\phantom{A}$	$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$SuperFormalCartSp$\phantom{A}$	$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$	$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

$\,$

References

In secton 2 of

• R. Blute, J.R.B. Cockett, Robert Seely, Cartesian differential categories, Theory and Applications of Categories, Vol. 22, 2009, No. 23, pp 622-672. (journal, pdf)

$CartSp$ 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

• 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