nLab
Diff

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

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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Higher geometry

Contents

Definition

Definition

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

Properties

As a site

Proposition

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}.

Proof

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.

Corollary

The sheaf topos over DiffDiff is a cohesive topos.

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

Proof

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}) \,.
Remark

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 (92.218.150.85)