Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential 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& 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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Higher geometry




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;


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




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


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}.


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;


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).


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).


A development of differential geometry as as geometry modeled on CartSpCartSp is discussed, with an eye towards applications in physics, in geometry of physics.

The sheaf topos over CartSp smoothCartSp_{smooth} is that of smooth spaces.

The (∞,1)-sheaf (∞,1)-topos over CartSp topCartSp_{top} is discussed at ETop∞Grpd, that over CartSp smoothCartSp_{smooth} at Smooth∞Grpd, and that over CartSp synthdiffCartSp_{synthdiff} at SynthDiff∞Grpd.

The generalization of CartSpCartSp to formal smooth manifolds is FormalCartSp.


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

Revised on April 23, 2015 09:18:49 by Urs Schreiber (