nLab
smooth infinity-stack

Idea

The notion of smooth \infty-stack is essentially that of

Following the logic described at

a smooth \infty-stack is the ∞-categorification of smooth space and differentiable stack. It is an ∞-stack on the (essentially small) site Diff of smooth manifolds, or correspondingly on BallDiffBall \subset Diff or CartSp Diff\subset Diff (see smooth space for more on that).

So smooth \infty-stacks are the objects in the (∞,1)-topos that computes smooth generalized cohomology. (See differential nonabelian cohomology and the disucssion under “Models” below for more on that).

Definition

Let CartSp={( n m)Diffn,m}DiffCartSp = \{ (\mathbb{R}^n \to \mathbb{R}^m) \in Diff| n,m \in \mathbb{N}\} \subset Diff be the full subcategory of Diff on the manifolds of the simple form n\mathbb{R}^n, equipped with the standard structure of a site with the coverage given by open covers of manifolds.

Then

H Diff:=(,1)Sh(CartSp) \mathbf{H}_{Diff} := (\infty,1)Sh(CartSp)

is the (∞,1)-topos given by the (∞,1)-category of (∞,1)-sheaves on CartSpCartSp.

Models

There is a large number of model structures presenting H Diff\mathbf{H}_{Diff}: all the model structures on simplicial (pre)sheaves on CartSpCartSp.

In terms of \infty-groupoids internal to smooth spaces

Notice for instance that there is the model structure on simplicial sheaves given by the category SSh(CartSp)SSh(CartSp) equipped with the injective local model structure on simplicial presheaves.

But sheaves on cartesian spaces

Sh(CartSp)=:SmoothSp Sh(CartSp) =: SmoothSp

is the category of smooth spaces, and SSh(CartSp)SSh(CartSp) is just the category of simplicial objects of that

SSh(CartSp)SmoothSp Δ op. SSh(CartSp) \simeq SmoothSp^{\Delta^{op}} \,.

So one model for smooth \infty-stacks is given by simplicial smooth spaces.

Notice that the fibrant object in SmoothSp Δ opSmoothSp^{\Delta^{op}} are the globally Kan complex-valued sheaves under the equivalence of categories

SmoothSp Δ opSh(CartSp,SSet), SmoothSp^{\Delta^{op}} \simeq Sh(CartSp, SSet) \,,

that satisfy descent (see descent for simplicial presheaves).

Being Kan complex-valued just means that the fibrant objects are sheaves on CartSpCartSp with values in ∞-groupoids.

Moreover, the descent-condition on CartSpCartSp is comparatively trivial, and in many cases (…details eventually here, but see examples below…) entirely empty, as every cartesian space is (smoothly, even) contractible.

This means that the fibrant objects in SSh(CartSp)SSh(CartSp) are pretty much nothing but ∞-groupoids internal to smooth spaces. (But notice that the requirement that she corresponding sheaf is Kan complex-valued is a bit weaker that other notions of ”\infty-groupoid internal to smooth spaces” that one may come up with).

In particular ∞-groupoids internal to diffeological spaces are therefore a model for smooth \infty-stacks.

Moreover, a morphism between smooth \infty-stacks modeled by such internal \infty-groupoids is modeled as an \infty-anafunctor (see simplicial localization, homotopy category and category of fibrant objects for details).

The model of smooth \infty-stacks given by \infty-groupoids internal to diffeological spaces with anafunctors as morphism between them is the model used in the Baez-ian school description of higher principal bundles and differential nonabelian cohomology.

Examples

Let GG be a Lie group. Using the embedding

DiffSmoothSp Diff \hookrightarrow SmoothSp

of manifolds into smooth spaces we may regard GG naturally as a sheaf on CartSp.

Write BG\mathbf{B} G for the delooping of GG, a one-object groupoid internal to SmoothSp. Postcomposing with the nerve functor N:N : Grpd \to SSet this yields a Kan complex-valued simplicial sheaf NBGN \mathbf{B} G which we shall by convenient and useful abuse of notation just call BG\mathbf{B} G itself.

Notice that BG\mathbf{B} G does not satisfy descent when regarded as a simplicial sheaf on all of Diff: there its ∞-stackification is instead GBund()G Bund(-), the stack of GG-principal bundles

GBund():UgroupoidofGbundlesonU G Bund(-) : U \mapsto groupoid of G-bundles on U

(or rather, in our context of simplicial sheaves, a rectification of that).

But restricted to the site CartSpCartSp the simplicial sheaf BG\mathbf{B} G does satisfy descent: there is up to isomorphism only a single GG-bundle on n\mathbb{R}^n, so that one finds an equivalence of categories

GBund( n)(BG)( n):=B(Diff( n,G)) G Bund(\mathbb{R}^n) \simeq (\mathbf{B} G)(\mathbb{R}^n) := \mathbf{B}(Diff(\mathbb{R}^n, G))

for each n\mathbb{R}^n. This means that BG\mathbf{B}G is a fibrant object in the injective model structure on simplicial sheaves. So in particular all the constructions and examples discussed at category of fibrant objects apply to BG\mathbf{B}G: we get the universal G-bundle EGBG\mathbf{E} G \to \mathbf{B}G regarded as a smooth \infty-stack as the pullback

EG * BG d 0 BG d 1 BG \array{ \mathbf{E}G &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{d_0}{\to}& \mathbf{B}G \\ \downarrow^{d_1} \\ \mathbf{B}G }

in SmoothSp Δ opSmoothSp^{\Delta^{op}}, which, do to the nerve being right adjoint is the same as the image under the nerve of the corresponding pullback in sheaves of groupoids (so that still our notational suppressing of NN is justified).

etc.

Revised on August 15, 2009 18:11:07 by Urs Schreiber (bogus address)