nLab CartSp

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Higher geometry

Contents

Definition

Definition

Write CartSpCartSp for the category whose

For definiteness we write

CartSp linCartSp_{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 CC, for CC (any one of) the category of

In all these cases, the inclusion CartSpCCartSp \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

Proposition

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

Proof

For CartSp top{}_{top} this is obvious. For CartSp smooth{}_{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 synthdiffCartSp_{synthdiff}.

Proposition
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)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;

Corollary

We have equivalences of categories

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

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

and equivalences of (∞,1)-categories

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

  • Sh (,1)(CartSp smooth)Sh (,1)(Diff)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 (,1)(CartSp)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) loc Δ opSh(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 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”:

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:

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

The idea is also implicit in

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]
category: category

Last revised on November 10, 2023 at 12:20:02. See the history of this page for a list of all contributions to it.