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

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          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

          In secton 2 of

          CartSpCartSp is discussed as an example of a “cartesian differential category”.

          There are various slight variations of the category CartSpCartSp (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 synthdiffCartSp_{synthdiff} of infinitesimally thickened Cartesian spaces is known as the site for the Cahiers topos. It is considered in detail in section 5 of

          and briefly mentioned in example 2) on p. 191 of

          following the original article

          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

          Last revised on June 25, 2018 at 09:04:07. See the history of this page for a list of all contributions to it.