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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(CatSp smooth). Sh(Diff) \simeq Sh(CatSp_{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 September 13, 2011 21:16:35 by Urs Schreiber (82.93.78.115)