CartSp (changes) in nLab

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Contents

Definition

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 CC, for CC (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↪CCartSp \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

The corresponding cohesive topos of sheaves is

• Sh (1,1)(CartSp smooth)Sh_{(1,1)}(CartSp_{smooth}), discussed at diffeological space.

• Sh (1,1)(CartSp synthdiff)Sh_{(1,1)}(CartSp_{synthdiff}), discussed at Cahiers topos.

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

• Sh (∞,1)(CartSp top)=Sh_{(\infty,1)}(CartSp_{top}) = ETop∞Grpd;

• Sh (∞,1)(CartSp smooth)=Sh_{(\infty,1)}(CartSp_{smooth}) = Smooth∞Grpd;

• Sh (∞,1)(CartSp synthdiff)=Sh_{(\infty,1)}(CartSp_{synthdiff}) = SynthDiff∞Grpd;

As a category with open maps

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

As an algebraic theory

The category CartSpCartSp 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, CartSpCartSp 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).

(…)

\,

geometries of physics

A\phantom{A}(higher) geometryA\phantom{A}	A\phantom{A}siteA\phantom{A}	A\phantom{A}sheaf toposA\phantom{A}	A\phantom{A}∞-sheaf ∞-toposA\phantom{A}
A\phantom{A}discrete geometryA\phantom{A}	A\phantom{A}PointA\phantom{A}	A\phantom{A}SetA\phantom{A}	A\phantom{A}Discrete∞GrpdA\phantom{A}
A\phantom{A}differential geometryA\phantom{A}	A\phantom{A}CartSpA\phantom{A}	A\phantom{A}SmoothSetA\phantom{A}	A\phantom{A}Smooth∞GrpdA\phantom{A}
A\phantom{A}formal geometryA\phantom{A}	A\phantom{A}FormalCartSpA\phantom{A}	A\phantom{A}FormalSmoothSetA\phantom{A}	A\phantom{A}FormalSmooth∞GrpdA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}	A\phantom{A}SuperFormalCartSpA\phantom{A}	A\phantom{A}SuperFormalSmoothSetA\phantom{A}	A\phantom{A}SuperFormalSmooth∞GrpdA\phantom{A}

\,

References

CartSp smthCartSp_{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](http://www.tac.mta.ca/tac/volumes/22/23/22-23abs.html), pdf]

CartSp smthCartSp_{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](http://arxiv.org/abs/0910.4001), doi:10.1007/s00220-012-1510-3]

• Urs Schreiber, Zoran Škoda, §6.2 of: Categorified symmetries [[arXiv:1004.2472](https://arxiv.org/abs/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](https://arxiv.org/abs/1011.4735), doi:10.1007/BF02104916]

• Urs Schreiber, §3.2.1 & §4.4 in: differential cohomology in a cohesive topos [[arXiv:1310.7930](https://arxiv.org/abs/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](http://www.numdam.org/item?id=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](http://www.numdam.org/item?id=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](https://cisinski.app.uni-regensburg.de/mtest2.pdf), pdf]

With an eye towards Frölicher spaces, CartSp synthdiffCartSp_{synthdiff} also briefly appears in:

• Hirokazu Nishimura, Section 5 of: Microlinearity in Frölicher – Beyond the Regnant Philosophy of Manifolds [[arXiv:0912.0827](http://arxiv.org/abs/0912.0827)]