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\begin{document}

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\section*{smooth infinity-stack}

\begin{quote}%
For more see at \emph{[[smooth ∞-groupoid]]}.


\end{quote}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{in_terms_of_groupoids_internal_to_smooth_spaces}{In terms of $\infty$-groupoids internal to smooth spaces}\dotfill \pageref*{in_terms_of_groupoids_internal_to_smooth_spaces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

\begin{itemize}%
\item [[smooth ∞-groupoid]].


\item [[orbifold|∞-orbifold]].



\end{itemize}
Following the logic described at

\begin{itemize}%
\item [[motivation for sheaves, cohomology and higher stacks]]


\item [[(∞,1)-topos]]



\end{itemize}
a \emph{smooth $\infty$-stack} is the [[∞-categorification]] of [[smooth space]] and [[differentiable stack]]. It is an [[∞-stack]] on the ([[essentially small category|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 \emph{smooth} generalized [[cohomology]]. (See [[schreiber:Differential Nonabelian Cohomology|differential nonabelian cohomology]] and the disucssion under ``Models'' below for more on that).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

Then

\begin{displaymath}
Smooth\infty Grpd := (\infty,1)Sh(CartSp)
\end{displaymath}
is the [[(∞,1)-topos]] given by the [[(∞,1)-category of (∞,1)-sheaves]] on $CartSp$.

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

\hypertarget{models}{}\subsection*{{Models}}\label{models}

There is a large number of [[model category|model structures]] [[presentable (infinity,1)-category|presenting]] $\mathbf{H}_{Diff}$: all the [[model structure on simplicial presheaves|model structures on simplicial (pre)sheaves]] on $CartSp$.

\hypertarget{in_terms_of_groupoids_internal_to_smooth_spaces}{}\subsubsection*{{In terms of $\infty$-groupoids internal to smooth spaces}}\label{in_terms_of_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

\begin{displaymath}
Sh(CartSp)
  =: SmoothSp
\end{displaymath}
is the category of [[smooth spaces]], and $SSh(CartSp)$ is just the category of [[simplicial objects]] of that

\begin{displaymath}
SSh(CartSp) 
  \simeq
  SmoothSp^{\Delta^{op}}
  \,.
\end{displaymath}
So one model for smooth $\infty$-stacks is given by [[simplicial object|simplicial]] [[smooth spaces]].

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

\begin{displaymath}
SmoothSp^{\Delta^{op}}
  \simeq
  Sh(CartSp, SSet)
  \,,
\end{displaymath}
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 (\ldots{}details eventually here, but see examples below\ldots{}) 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]] [[internalization|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 [[John Baez|Baez]]-ian school description of [[principal infinity-bundle|higher principal bundles]] and [[schreiber:Differential Nonabelian Cohomology|differential nonabelian cohomology]].

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

Let $G$ be a [[Lie group]]. Using the embedding

\begin{displaymath}
Diff \hookrightarrow SmoothSp
\end{displaymath}
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]] [[internalization|internal to]] [[smooth space|SmoothSp]]. Postcomposing with the [[nerve]] functor $N :$ [[Grpd]] $\to$ [[SSet]] this yields a [[Kan complex]]-valued [[simplicial presheaf|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]]

\begin{displaymath}
G Bund(-) : U \mapsto groupoid of G-bundles on U
\end{displaymath}
(or rather, in our context of simplicial sheaves, a [[rectified infinity-stack|rectification]] of that).

But restricted to the [[site]] $CartSp$ the [[simplicial presheaf|simplicial sheaf]] $\mathbf{B} G$ \emph{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]]

\begin{displaymath}
G Bund(\mathbb{R}^n) \simeq (\mathbf{B} G)(\mathbb{R}^n)
  := \mathbf{B}(Diff(\mathbb{R}^n,  G))
\end{displaymath}
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 [[generalized universal bundle|universal G-bundle]] $\mathbf{E} G \to \mathbf{B}G$ regarded as a smooth $\infty$-stack as the [[pullback]]

\begin{displaymath}
\itexarray{
    \mathbf{E}G &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{d_0}{\to}& \mathbf{B}G
    \\
    \downarrow^{d_1}
    \\
    \mathbf{B}G
  }
\end{displaymath}
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.

[[!redirects smooth infinity-stacks]] [[!redirects smooth ∞-stack]] [[!redirects smooth ∞-stacks]]



\end{document}