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.

• $CartSp_{smth}$ 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

$CartSp_{smth}$ as an example of a “cartesian differential category”:

• R. Blute, J.R.B. Cockett, Robert Seely, section 2 of Cartesian differential categories, Theory and Applications of Categories 22 23 (2009) 622-672 [tac:22-23, pdf]

$CartSp_{smth}$ as a convenient site for diffeological spaces, smooth sets, smooth groupoids, … smooth $\infty$-groupoids (and the term “CartSp”, or similar, for it) was first considered in:

• Hisham Sati, Urs Schreiber, Jim Stasheff, p. 22 in: Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics 315 1 (2012) 169-213 [arXiv:0910.4001, doi:10.1007/s00220-012-1510-3]

• Urs Schreiber, Zoran Škoda, §6.2 of: Categorified symmetries [arXiv:1004.2472]

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Appendix of: Čech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Physics, 16 1 (2012) 149-250 [arXiv:1011.4735, doi:10.1007/BF02104916]

• Urs Schreiber, §3.2.1 & §4.4 in: differential cohomology in a cohesive topos [arXiv:1310.7930]

following the site $CartSp_{synthdiff}$ of infinitesimally thickened Cartesian spaces, previously claimed (then without proof, it seems) to be a site for the Cahiers topos in:

• Anders Kock, Section 5 of: Convenient vector spaces embed into the Cahiers topos, Cahiers de Topologie et Géométrie Différentielle Catégoriques 27 1 (1986) 3-17 [numdam:CTGDC_1986__27_1_3_0]

• Anders Kock, Gonzalo Reyes, Corrigendum and addenda to: Convenient vector spaces embed into the Cahiers topos, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 28 2 (1987) 99-110 [numdam:CTGDC_1987__28_2_99_0]

The idea is also implicit in

• Denis-Charles Cisinski, Exp. 6.1.2 in: Faisceaux localement asphériques (2003) [pdf, pdf]

With an eye towards Frölicher spaces, $CartSp_{synthdiff}$ also briefly appears in:

• Hirokazu Nishimura, Section 5 of: Microlinearity in Frölicher – Beyond the Regnant Philosophy of Manifolds [arXiv:0912.0827]

category: category