nLab smooth infinity-stack

For more see at smooth ∞-groupoid.

Idea

The concept of smooth $\infty$-stack is essentially that of

• smooth ∞-groupoid.

• ∞-orbifold.

Following the logic described at

• motivation for sheaves, cohomology and higher stacks

• (∞,1)-topos

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 $Ball \subset Diff$ or CartSp $\subset Diff$ (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 $CartSp = \{ (\mathbb{R}^n \to \mathbb{R}^m) \in Diff| n,m \in \mathbb{N}\} \subset Diff$ be the full subcategory [of on the of the simple form , equipped with the standard structure of a with the given by of manifolds.]

Then

$$Smooth\infty Grpd := (\infty,1)Sh(CartSp)$$

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

This is the cohesive (∞,1)-topos Smooth∞Grpd.

Models

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

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 $SSh(CartSp)$ equipped with the injective local model structure on simplicial presheaves.

But sheaves on cartesian spaces

$$Sh(CartSp) =: SmoothSp$$

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

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

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

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

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

that satisfy descent (see descent for simplicial presheaves).

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

Moreover, the descent-condition on $CartSp$ 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 $SSh(CartSp)$ 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 $G$ be a Lie group. Using the embedding

$$Diff \hookrightarrow SmoothSp$$

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

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

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

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

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

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

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

for each $\mathbb{R}^n$. This means that $\mathbf{B}G$ 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 $\mathbf{B}G$: we get the universal G-bundle $\mathbf{E} G \to \mathbf{B}G$ regarded as a smooth $\infty$-stack as the pullback

$$\array{ \mathbf{E}G &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{d_0}{\to}& \mathbf{B}G \\ \downarrow^{d_1} \\ \mathbf{B}G }$$

in $SmoothSp^{\Delta^{op}}$, 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 $N$ is justified).

etc.