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




DiffDiff (also called ManMan or MfdMfd) is the category whose


As a site


The category DiffDiff becomes a large site by equipping it with the coverage consisting of open covers.

This is an essentially small site: a dense sub-site for DiffDiff is given by CartSp smooth{}_{smooth}.


The first statement follows trivially as for Top: the preimage of an open subset under a continuous function is again open (by definition of continuouss function).

For the second statement one needs that every paracompact manifold admits a differentially good open cover : an open cover by open balls that are diffeomorphic to a Cartesian spaces. The proof for this is spelled out at good open cover.


The sheaf topos over DiffDiff is a cohesive topos.

The hypercompletion of the (∞,1)-sheaf (∞,1)-topos over DiffDiff is a cohesive (∞,1)-topos.


For the first statement, use that by the comparison lemma discussed at dense sub-site we have an equivalence of categories

Sh(Diff)Sh(CartSp smooth). Sh(Diff) \simeq Sh(CartSp_{smooth}) \,.

By the discussion at CartSp we have that CartSp smoothCartSp_{smooth} is a cohesive site. By the discussion there the claim follows.

For the second statement observe that the Joyal-Jardine model structure on simplicial sheaves Sh(Diff) loc Δ opSh(Diff)^{\Delta^{op}}_{loc} is a presentation for the hypercompletion of the (∞,1)-category of (∞,1)-sheaves Sh^ (,1)(Diff)\hat Sh_{(\infty,1)}(Diff) (see presentations of (∞,1)-sheaf (∞,1)-toposes). By the above result it follows that there is an equivalence of (∞,1)-categories between the hypercompletions

Sh^ (,1)(Diff)Sh^ (,1)(CartSp smooth). \hat Sh_{(\infty,1)}(Diff) \simeq \hat Sh_{(\infty,1)}(CartSp_{smooth}) \,.

Now CartSp smooth{}_{smooth} is even an ∞-cohesive site. By the discussion there it follows that Sh (,1)(CartSp smooth)Sh_{(\infty,1)}(CartSp_{smooth}) (before hypercompletion) is a cohesive (∞,1)-topos. This means that it is in particular a local (∞,1)-topos. But this implies (as discussed there), that the (∞,1)-category of (∞,1)-sheaves already is the hypercomplete (∞,1)-topos. Therefore finally

Sh (,1)(CartSp smooth). \cdots \simeq Sh_{(\infty,1)}(CartSp_{smooth}) \,.

The cohesive topos Sh(Diff)Sh(CartSp smooth)Sh(Diff) \simeq Sh(CartSp_{smooth}) is in particular the home of diffeological spaces. See there for more details.

The cohesive (∞,1)-topos

SmoothGrp:=Sh (,1)(Diff)Sh (,1)(CartSp smooth) Smooth \infty Grp := Sh_{(\infty,1)}(Diff) \simeq Sh_{(\infty,1)}(CartSp_{smooth})

is that of smooth ∞-groupoids. Discussed at Smooth∞Grpd.

The theory of differentiable stacks is that of geometric stacks in the (2,1)-sheaf (2,1)-topos

Sh (2,1)(Diff)Sh (2,1)(CartSp smooth)τ 1Sh (,1)(CartSp smooth) Sh_{(2,1)}(Diff) \simeq Sh_{(2,1)}(CartSp_{smooth}) \simeq \tau_{\leq 1} Sh_{(\infty,1)}(CartSp_{smooth})

category: category

Revised on May 8, 2017 03:57:37 by Urs Schreiber (