nLab
smooth infinity-stack

Idea

The notion of smooth $\mathrm{\infty }$-stack is essentially that of

• Lie ∞-groupoid.

• ∞-orbifold.

Following the logic described at

• space and quantity

  • generalized smooth space

• motivation for sheaves, cohomology and higher stacks

• (∞,1)-topos

a smooth $\mathrm{\infty }$-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 $\mathrm{Ball}\subset \mathrm{Diff}$ or CartSp $\subset \mathrm{Diff}$ (see smooth space for more on that).

So smooth $\mathrm{\infty }$-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 $\mathrm{CartSp}=\{({ℝ}^n\to {ℝ}^m)\in \mathrm{Diff}\mid n,m\in \mathbb{N}\}\subset \mathrm{Diff}$ be the full subcategory of Diff on the manifolds of the simple form ${ℝ}^n$, equipped with the standard structure of a site with the coverage given by open covers of manifolds.

Then

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

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

Models

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

In terms of $\mathrm{\infty }$-groupoids internal to smooth spaces

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

But sheaves on cartesian spaces

$$\mathrm{Sh}(\mathrm{CartSp})=:\mathrm{SmoothSp}$$Sh(CartSp) =: SmoothSp

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

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

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

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

$${\mathrm{SmoothSp}}^{{\Delta }^{\mathrm{op}}}\simeq \mathrm{Sh}(\mathrm{CartSp},\mathrm{SSet}),$$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 $\mathrm{CartSp}$ with values in ∞-groupoids.

Moreover, the descent-condition on $\mathrm{CartSp}$ 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 $\mathrm{SSh}(\mathrm{CartSp})$ 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 ”$\mathrm{\infty }$-groupoid internal to smooth spaces” that one may come up with).

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

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

The model of smooth $\mathrm{\infty }$-stacks given by $\mathrm{\infty }$-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

$$\mathrm{Diff}\hookrightarrow \mathrm{SmoothSp}$$Diff \hookrightarrow SmoothSp

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

Write $BG$ 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 $NBG$ which we shall by convenient and useful abuse of notation just call $BG$ itself.

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

$$G\mathrm{Bund}(-):U\mapsto \mathrm{groupoid}\mathrm{of}G-\mathrm{bundles}\mathrm{on}U$$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 $\mathrm{CartSp}$ the simplicial sheaf $BG$ does satisfy descent: there is up to isomorphism only a single $G$-bundle on ${ℝ}^n$, so that one finds an equivalence of categories

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

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

$$\begin{array}{ccc}EG& \to & *\\ \downarrow & & \downarrow \\ BG& \stackrel{d_0}{\to }& BG\\ {\downarrow }^{d_1}\\ BG\end{array}$$\array{ \mathbf{E}G &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{d_0}{\to}& \mathbf{B}G \\ \downarrow^{d_1} \\ \mathbf{B}G }

in ${\mathrm{SmoothSp}}^{{\Delta }^{\mathrm{op}}}$, 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.