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\begin{document}

\begin{abstract}
We show that there is a functor from stacks over smooth manifolds to the category of concrete sheaves of sets over manifolds that is the left adjoint of the Grothendieck construction taking concrete sheaves to stacks.  The category of concrete sheaves is known to be equivalent to the category of diffeological spaces.  With this perspective, the functor sends each stack to its ``underlying'' diffeological coarse moduli space.  Moreover, in the case of a differentiable stack, we show that the underlying diffeology matches the quotient diffeology on the orbit space of a representative Lie groupoid.  As a consequence, the quotient diffeology on the orbit space only depends on the isomorphism class of the stack.

As an application, we define differential forms for sheaves or stacks over manifolds and show that this definition agrees with basic differential forms in the case of a differentiable stack and de Rham differential forms in the case of a diffeology.\\
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\emph{\textbf{Keywords:} diffeology, stack, concrete sheaf, Lie groupoid, geometric stack, differentiable stack, coarse moduli space}\\
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\emph{\textbf{AMS Subject Classification Numbers:} 14D23, 58A99, 22A22}
\end{abstract}

\maketitle

\renewcommand*{\thefootnote}{\fnsymbol{footnote}}
\footnotetext{*This author acknowledges support from National Science Foundation grant DMS 08-38434 ``EMSW21-MCTP: Research Experience for Graduate Students''.}

\section{Introduction}\labell{s:intro}

The category of smooth manifolds lacks many structures that naturally arise in geometry.  Important examples include (critical) level sets of smooth real-valued functions and orbit spaces of Lie group actions, among many others.  Many categories extending the notion of smoothness have been introduced to account for this deficiency; for example, that of diffeological spaces, introduced by Souriau \cite{souriau} (see \hypref{d:diffeology}{Definition}).

On the other hand, the language of stacks arose out of the need to better understand orbit and moduli spaces (see, for example, \cite{grothendieck}).  Indeed, the quotient stack associated to a compact Lie group action contains more information than the quotient topology, the ring of smooth functions on the orbit space in the sense of Schwarz (see \cite{schwarz}, \cite{CS}), or even the associated diffeology.  For example, consider $\SO(n)$ acting on $\RR^n$ via rotations.  The orbit space is homeomorphic to $[0,\infty)$ for all $n$, and the rings of invariant functions for each $n$ are all isomorphic.  The diffeology detects which $n$ we started with (this is the ``diffeological dimension'' at the singularity).  However, the orbit space obtained from the action of $\SO(2n)$ on $\RR^{2n}$ via rotations is diffeomorphic as a diffeological space to that obtained from the action of $U(n)$ on the same space.  The corresponding stacks, of course, are not isomorphic in these examples.

The deficiencies of manifolds in modeling geometric structures has an effect on stacks over manifolds as well.  In the study of stacks over (compactly generated) topological spaces, it has been noted (see \cite{noohi}, for instance) that any stack over a topological space admits a natural coarse moduli space.  This serves as a left adjoint to the Yoneda embedding of topological spaces into stacks over topological spaces.  Unfortunately, in the case of stacks over manifolds, no such left adjoint exists.  As noted by Hepworth in \cite[Section 2]{hepworth:morse}, there are examples of stacks for which no manifold coarse moduli space may be chosen.

A paper of Baez and Hoffnung \cite{BH} shows that the category of diffeological spaces is equivalent to the category of so-called concrete sheaves over open subsets of Euclidean space with smooth maps between them.  A concrete sheaf is a sheaf of sets that not only encodes an underlying set, but a family of maps that induces a diffeology on the set.  Moreover, Baez and Hoffnung show that the inclusion of concrete sheaves into all sheaves of sets has a left adjoint functor called concretization $\concr:\sheaves \to \csheaves$, which sends every sheaf of sets to its ``underlying'' concrete sheaf (or, equivalently, diffeology).

The purpose of this paper is to show that one can go even further, extending this relationship to stacks over manifolds (both differentiable and otherwise).  In particular, there is a functor $\coarse:\stacks \to \csheaves$ from the category of stacks over manifolds to the category of concrete sheaves, sending each stack to its diffeological coarse moduli space.  This is a left adjoint to $\int:\csheaves\to\stacks$, the Grothendieck construction taking concrete sheaves to stacks.  $\coarse$ is a 2-functor with respect to the 2-structure of stacks and the degenerate 2-structure of concrete sheaves (\emph{i.e.} that for which every 2-arrow is trivial).  This 2-functor factors through a functor $\discr:\stacks\to\sheaves$ called discretization, which sends each stack to an ``underlying'' sheaf of sets, before being concretized in the sense described above.  $\coarse$ has the useful property of being ``co-continuous'' in the sense that it sends (small) 2-colimits to (1-)colimits.

Since manifolds are a full subcategory of diffeological spaces they naturally form a full subcategory of concrete sheaves.  The Grothendieck construction embedding concrete sheaves into stacks over manifolds restricted to this full subcategory corresponds exactly to the Yoneda embedding of manifolds into stacks.  In other words, for $\iota:\mfld\to\csheaves$ the aforementioned embedding of manifolds into concrete sheaves, $\int:\csheaves \to \stacks$ the Grothendieck construction, and $y:\mfld\to \stacks$ the Yoneda embedding, the diagram:
\[\xymatrix{& \csheaves \ar[dr]^{\int} & \\ \mfld \ar[ur]^{\iota} \ar[rr]_y& & \stacks}\]
commutes.  For this reason, we may properly think of $\int:\csheaves \to \stacks$ as an extension of the Yoneda embedding.  As $\coarse$ and $\int$ form an adjunction, we recover a result similar to that for topological stacks (hence, the name ``diffeological coarse moduli space'' for the object $\coarse(\mathcal{X})$ for any stack $\mathcal{X}$).

$\coarse$ has interesting properties when restricted to differentiable stacks.  If $G=(G_1\toto G_0)$ is a Lie groupoid representing a differentiable stack $\mathcal{X}$, we show that $\coarse(\mathcal{X})$ is (isomorphic to) the concrete sheaf associated to the quotient diffeology on $G_0/G_1$, the orbit space of the groupoid.  In the case where $\mathcal{X}=BG$,  the stack of right principal $G$-bundles, we show that discretization ($\discr$) identifies principle $G$-bundles over each manifold that are locally $G$-equivariantly bundle-diffeomorphic, and concretization ($\concr$) further identifies such bundles that are fiberwise $G$-equivariantly bundle-diffeomorphic.

This paper is organized as follows.  \hypref{s:diffeology}{Section} reviews diffeology and concrete sheaves, and extends their categorical equivalence to concrete sheaves over manifolds (\hypref{l:concrete comparison}{Lemma}).  \hypref{s:concrete}{Section} reviews that the inclusion of concrete sheaves into sheaves of sets over manifolds has a left adjoint functor (concretization), and that the inclusion of sheaves of sets into stacks over manifolds (also called the Grothendieck construction) has a left adjoint functor (discretization).  Together, these left adjoints compose to form $\coarse$ (see \hypref{d:coarse}{Definition} and \hypref{t:coarse adj}{Theorem}).  We prove that it is co-continuous in \hypref{t:coarse co-continuous}{Theorem}, in the sense that it takes 2-colimits (more prevalently used in stacks) to colimits.  In \hypref{s:differentiable stacks}{Section} we apply the above theory to differentiable stacks, proving the three claims above and ending the section with examples.  In \hypref{s:differential forms}{Section} we express the results on differential forms on Lie groupoids in \cite{watts-groupoids} and \cite{KW} in terms of the coarse functor, and extend these results to differential forms on the corresponding discretized sheaves of sets (\hypref{t:differential forms}{Theorem}).

For more details on diffeological spaces, see \cite{iglesias}.  For more details on stacks, see \cite{heinloth} and \cite{vistoli}, as well as \cite{lerman} for a detailed description of the relation between Lie groupoids and differentiable stacks. Coarse moduli spaces have been used in a variety of applications.  For example, for topological stacks, the \emph{topological coarse moduli space} is used in \cite{BGNX} to develop string topology for differentiable stacks over manifolds; in particular, the coarse moduli space is used to find ``hidden loops'' in the loop space of a stack.  In \cite{MT}, the authors use the topological coarse moduli space to extend the Kashiwara index formula from manifolds to orbifolds.  In \cite{hepworth:morse} and \cite{hepworth:vf}, the {\em underlying topological space of a stack} (similar to but fundamentally different from a coarse moduli space) is used to develop morse theory and flows of vector fields for stacks over manifolds.  For more details on the topological coarse moduli space, see \cite{noohi}.  For this paper, we do not focus on topology; however, we do make note that the topology of the underlying topological space of a stack as defined by Hepworth in \cite{hepworth:morse} matches the so-called D-topology of the diffeological space corresponding to a diffeological coarse moduli space (see \hypref{r:topology}{Remark}).

\subsection*{Prerequisites}\labell{ss:prerequisites}
This paper uses a lot of language from category theory.  Besides the basics, the reader is assumed to be familiar with the following.  We give an intuitive idea of what each of these objects are, but pinpoint an actual reference for a definition or statement.  An excellent resource for both intuition and definitions is the $n$-Category Cafe \cite{cafe}.

\begin{itemize}

\item \labell{d:site} Given a category $C$, a \emph{coverage} on $C$ is a generalization of open covers in the topological sense.  Essentially, it is a specification of the open covers of each object of $C$ satisfying some coherence conditions.  See \cite[A2.1.9]{johnstone} for a definition.  A \emph{site} is a category equipped with a coverage.  (Note that we do not require the underlying category of a site to be small, whereas some references do.)  We will be dealing with the site of open subsets of Euclidean spaces with smooth maps as arrows (which is small), as well as the site of smooth manifolds (which includes the former category as a dense ``subsite'' and so is essentially small).  In both of these cases, the coverage is given by usual open covers by open sets.

\item \labell{d:sheaf} Given a category $C$, a \emph{presheaf of sets} on $C$ is a contravariant functor $C^{op}\to\set$.  If $C$ is a site, then a \emph{sheaf of sets} is a presheaf of sets on $C$ that obeys a ``sheaf condition'' with respect to the coverage on $C$.  See \cite[A2.1.9]{johnstone} for a definition.  For example, given the site of manifolds, the sheaf $\Omega^k(\cdot)$ assigns to each manifold $M$ its set of $k$-forms $\Omega^k(M)$.  Presheaves of sets over $C$ form a category (and sheaves a subcategory) with arrows given by natural transformations.

\item \labell{d:sheafification} A presheaf of sets $P$ on a site $C$ has a unique \emph{sheafification}, a sheaf of sets $S$ over $C$ such that there is a natural transformation $\alpha:P\to S$ satisfying the following universal property: if $T$ is any sheaf of sets over $C$ and $\beta:P\to T$ a natural transformation, then $\beta$ factors through $\alpha$.  See \cite[Definition 2.63 and Theorem 2.64]{vistoli}.  Note this means sheafification is the left adjoint to the inclusion of sheaves of sets into presheaves of sets.

\item \labell{l:comparison} \textbf{Comparison Lemma:} Let $C$ be a locally small site, and $D$ a small dense subsite.  Then the restriction functor sending sheaves of sets over $C$ to sheaves of sets over $D$ is an equivalence of categories.  See \cite[C2.2.3]{johnstone}.

\item \labell{d:cfg} Given a category $C$, a \emph{category fibered in groupoids (CFG)} over $C$ is a functor $\pi:D\to C$ whose fibers are groupoids, satisfying some conditions; see \cite[Definition 3.5]{vistoli}.  As an example, let $G$ be a Lie group.  Assign to each manifold $M$ the trivial $G$-bundle $G\times M\to M$.  This induces a CFG whose fiber over $M$ is $\{G\times M\to M\}$.  CFGs over $C$ form a $2$-category, where an arrow $F:(\pi:D\to C)\to(\pi':D'\to C)$ is a functor from $D$ to $D'$ such that $\pi'\circ F=\pi$, and $2$-arrows are given by natural transformations.

\item \labell{d:stack} A \emph{stack} over a site $C$ is a CFG that satisfies a ``descent condition'', which is a $2$-categorical version of the sheaf condition.  See \cite[Definition 4.6]{vistoli}.  An example of a stack is the CFG whose fiber over a manifold $M$ is the groupoid of principal $G$-bundles over $M$, with bundle isomorphisms as arrows, where $G$ is a Lie group.  Stacks over $C$ form a sub-$2$-category of CFGs over $C$.

\item \labell{d:cat of el} Let $P:C^{op}\to\set$ be a presheaf of sets.  Then the \emph{category of elements} of $P$, denoted $\pi:\int P\to C$, is a CFG defined as follows.  The objects of $\int P$ are pairs $(x,c)$ where $c$ is an object in $C$ and $x\in P(c)$, and the arrows $\tilde{f}:(x,c)\to(x',c')$ are arrows $f:c\to c'$ in $C$ such that $P(f)(c')=c$.  Define $\pi$ to be the obvious projection.  Note that in this case, the resulting CFG is \emph{discrete}; that is, the arrows in each fiber are trivial, and so we think of these groupoids as just sets.  See \cite[page 203]{awodey} for more details.  It is not hard to show that $\int$ is a fully faithful functor from sheaves to CFGs, known as the \emph{Grothendieck construction}, sending a map of presheaves to the obvious map of CFGs.  Note that if $C$ is a site and $P$ a sheaf, then $\pi:\int P\to C$ is in fact a stack.

\end{itemize}

\subsection*{Acknowledgments}\labell{ss:acknowledgements}

The authors would like to thank Eugene Lerman for many discussions regarding stacks, and introducing us to various categorical notions required to put the entire puzzle that is this paper together.  The first author would also like to thank Patrick Iglesias-Zemmour and Yael Karshon for many illuminating diffeological examples, and Pierre Albin for discussions on the presentation.

\section{From Diffeology to Concrete Sheaves}\labell{s:diffeology}

In this section we review the definition of a diffeology, a concrete sheaf, and establish an equivalence of categories between diffeological spaces and concrete sheaves over manifolds \hypref{c:diffeol equiv csheaves}{Corollary} (see also \hypref{r:diffeol equiv csheaves}{Remark}).  Our main resource for diffeology is the book by Iglesias-Zemmour \cite{iglesias}.  For more details on concrete sheaves, see \cite{BH}.

\begin{definition}
Let $\open$ be the category of open subsets of $\R^n$ for any $n$ and their smooth maps.  Let $\mfld$ be the category of manifolds.  We implicitly realize these categories as sites with coverages generated by standard open covers.
\end{definition}

\begin{definition}[Diffeological Space]\labell{d:diffeology}
Let $X$ be a set.  A \emph{parametrization} $p:U\to X$ is a function from an open subset $U$ of $\open$.  A \emph{diffeology} $\mathcal{D}$ on $X$ is a family of parametrizations satisfying the following three axioms:

\begin{itemize}
\item \textbf{(Covering Axiom)} All constant maps $p:U\to X$ are contained in $\mathcal{D}$.
\item \textbf{(Gluing Axiom)} If $p:U\to X$ is a parametrization such that there exist an open cover $\{U_\alpha\}_{\alpha\in A}$ of $U$ and a family $\{p_\alpha:U_\alpha\to X\}_{\alpha\in A}\subseteq\mathcal{D}$ of parametrizations that satisfy $p_\alpha=p|_{U_\alpha}$, then $p$ is contained in $\mathcal{D}$.
\item \textbf{(Smooth Compatibility Axiom)} If $p:U\to X$ is in $\mathcal{D}$ and $f:V\to U$ is a smooth map between open sets of Euclidean spaces, then $f^*p=p\circ f$ is also contained in $\mathcal{D}$.
\end{itemize}
The pair $(X,\mathcal{D})$ is called a \emph{diffeological space}, and the elements of $\mathcal{D}$ are called \emph{plots}.
\end{definition}

\begin{definition}[Smooth Map]\labell{d:smoothmap}
Let $(X,\mathcal{D}_X)$ and $(Y,\mathcal{D}_Y)$ be diffeological spaces, and let $F:X\to Y$ be a map.  Then $F$ is \emph{(diffeologically) smooth} if for every plot $p\in\mathcal{D}_X$, the composition $F\circ p$ is a plot in $\mathcal{D}_Y$.
\end{definition}

\begin{remark}\labell{r:diffeol is a cat}
Diffeological spaces with smooth maps between them form a category, denoted by $\diffeol$.  This category is complete and cocomplete \cite{BH}, and contains the category of smooth manifolds with smooth maps $\mfld$ as a full subcategory.
\end{remark}

\begin{notation}
For $C$ a site, we will use the symbols $\mathbf{PShf}(C)$, $\mathbf{Shf}(C)$, $\mathbf{CFG}(C)$, and $\mathbf{St}(C)$ to denote the categories of presheaves of sets, sheaves of sets, categories fibered in groupoids, and stacks over $C$.
\end{notation}

\begin{definition}[Concrete Sheaves, Definition 4.10 of \cite{BH}]\labell{d:concrete sheaf}
Let $C$ be the site $\open$ or $\mfld$.  A presheaf $P:C^{op}\to\set$ is {\em concrete at $U\in C_0$} if the function sending each element $p\in P(U)$ to the function
\begin{equation}\labell{e:concrete}
\underline{p}:\hom(*,U)\to P(*):u\mapsto u^*p
\end{equation}
is injective.  We say that $P$ is a {\em concrete presheaf} if it is concrete at each object $U$.  If in addition $P$ satisfies the sheaf axiom, then we call $P$ a \emph{concrete sheaf}.  We will refer to the category of concrete sheaves over $\open$ and $\mfld$ (with natural transformations as arrows) by $\mathbf{CShf}(\open)$ and $\csheaves$, respectively.
\end{definition}

\begin{lemma}[Baez-Hoffnung \cite{BH}, Lemma 4.14]\labell{l:baez-hoffnung}\labell{l:BH1}
There is an equivalence of categories from $\diffeol$ to $\mathbf{CShf}(\open)$.
\end{lemma}

The proof of the following lemma is a straightforward calculation, and so the proof is omitted.

\begin{lemma}\labell{l:local lemma}
Let $C$ be the site $\open$ or $\mfld$ and $P:C^{op} \to \set$ a presheaf.  Let $U\in C_0$ be an object of $C$ and $\{U_i\to U\}$ a cover of $U$.  Then if $P$ is concrete at each $U_i$, it is concrete at $U$.
\end{lemma}

\begin{lemma}[Concrete Comparison Lemma]\labell{l:concrete comparison}
There is an equivalence of categories from $\mathbf{CShf}(\open)$ to $\csheaves$.
\end{lemma}

\begin{proof}
Recall that the Comparison Lemma tells us that there exists an equivalence of categories:
\begin{equation}\labell{comparisoneq}
  \xymatrix{\mathbf{Shf(\open)} \ar@<1ex>[r]^{\ext} & \sheaves \ar@<1ex>[l]^{\res}}
\end{equation}
where $\ext(S)$ is the Kan extension of $S:\open^{op}\to \set$ against the inclusion of $\open$ into $\mfld$ and $\res(S')$ is the restriction of $S':\mfld^{op}\to \set$ to elements of $\open$ in $\mfld$.

Clearly, $\res$ takes $\csheaves$ to $\mathbf{CShf}(\open)$.  Since any $n$-dimensional manifold $M$ has an open cover by open subsets of $\RR^n$, it follows from \hypref{l:local lemma}{Lemma} that the sheaf $S:\mfld^{op}\to \set$ must be concrete if $\res(S)$ is concrete.  So $\res$ takes {\em only} concrete sheaves of $\sheaves$ to concrete sheaves of $\mathbf{Shf(\open)}$.  Therefore, since $\res\circ\ext$ must take $\mathbf{CShf}(\open)$ to $\mathbf{CShf}(\open)$ (concreteness is preserved under isomorphism), it follows that $\ext$ takes $\mathbf{CShf}(\open)$ to $\csheaves$.  Thus the equivalence of categories (\hypref{comparisoneq}{Diagram}) descends to our desired equivalence between
$\mathbf{CShf}(\open)$ and $\csheaves$.
\end{proof}

\begin{corollary}\labell{c:diffeol equiv csheaves}
There is an equivalence of categories from $\diffeol$ to $\csheaves$.
\end{corollary}

\begin{remark}\labell{r:diffeol equiv csheaves}
The construction of a functor $I$ from $\diffeol$ to $\csheaves$ that is part of the equivalence in \hypref{c:diffeol equiv csheaves}{Corollary} is given as follows.  Given a diffeological space $(X,\mathcal{D})$ and a manifold $M$ of dimension $m$, define a sheaf $I(\mathcal{D})$ over $\mfld$ by setting $I(\mathcal{D})(M)$ to be the set of all maps $p:M\to X$ such that there exist an open cover $\{U_\alpha\}$ of $M$ in which $U_\alpha\subseteq\RR^m$ for each $\alpha$ and plots $q_\alpha:U_\alpha\to X$ in $\mathcal{D}$ satisfying $p|_{U_\alpha}=q_\alpha$.  This is the concrete sheaf of manifolds associated to $\mathcal{D}$.  A diffeologically smooth map $F$ is sent via $I$ to a map of sheaves sending a plot $p$ to $F\circ p$.
\end{remark}

\begin{remark}\labell{r:YonedaExt}
The fully faithful embedding of manifolds into diffeologies takes any manifold $M$ to the diffeological space built on the underlying set of $M$ with plots all smooth maps from open subsets of Euclidean space to $M$.  So the concrete sheaf $I(M)$ (for $I$ the functor of the previous remark) is just the sheaf of smooth functions to $M$: $I(M)(N)=C^\infty(N,M)$.  Write $\iota$ for the composition of the inclusion of the embedding of manifolds into diffeologies with $I$.  Then, from the definition of the Yoneda embedding $y:\mfld \to \stacks$, it is obvious that the diagram:
\[\xymatrix{& \csheaves \ar[dr]^{\int} & \\ \mfld \ar[ur]^{\iota} \ar[rr]_y& & \stacks}\]
commutes, where $\int:\csheaves\to \stacks$ is the Grothendieck construction.  Thus, we may think of $\int$ as an extension of the Yoneda embedding.
\end{remark}

\section{From Concrete Sheaves to Sheaves to Stacks}\labell{s:concrete}

In this section we begin by reviewing concretization.  This is a functor introduced by Baez and Hoffnung in \cite{BH}, where they prove that it is left adjoint functor to the inclusion $\csheaves\to\sheaves$.  We then define discretization, and show that it is left adjoint to the Grothendieck construction.  These two functors compose to form the functor $\coarse$.  It follows that $\coarse$ is left adjoint to the restricted Grothendieck construction (\hypref{t:coarse adj}{Theorem}).  Moreover, $\coarse$ is co-continuous, taking 2-colimits to colimits (\hypref{t:coarse co-continuous}{Theorem}).

\begin{definition}[Concretization]\labell{d:concretization}
Let $S$ be a sheaf over $\mfld$.  Define $\kappa(S):\mfld^{op}\to \set$ to be the presheaf assigning to each manifold $M$ the quotient set $S(M)/\!\sim$, where $p_1\sim p_2$ if $\underline{p_1}=\underline{p_2}$ (see \hypref{e:concrete}{Equation} for a definition of $\underline{p_i}$).  Define the \emph{concretization} of $S$, denoted by $\concr(S):\mfld^{op} \to \set$, to be the sheafification of $\kappa(S)$.
\end{definition}

\begin{lemma}[Baez-Hoffnung, Lemma 5.20 of \cite{BH}]\labell{l:concr adj}
$\concr:\sheaves\to\csheaves$ is a left adjoint functor to the inclusion functor $\csheaves\hookrightarrow\sheaves$.
\end{lemma}

We'll see below that concretization can have interesting effects, especially on sheaves coming from differentiable stacks.  However, some sheaves of sets are practically destroyed by the process:
\begin{example}[$\concr$ Applied to $\Omega^k$]\labell{x:differential forms}
Let $\Omega^k:\mfld^{op}\to \set$ be the sheaf of differential $k$-forms.  If $k=0$, then $\concr(\Omega^k)$ is precisely equal to $\CIN(-,\RR)$, the Yoneda embedding of $\RR$ in $\sheaves$, or equivalently, $I(\mathcal{D})$ where $\mathcal{D}$ is the standard diffeology on $\RR$ and $I$ is as in \hypref{r:diffeol equiv csheaves}{Remark}.  If $k>0$, then for every manifold $M$, we have $\concr(\Omega^k)(M)=\{0\}$. \eoe
\end{example}

\begin{definition}[Discretization]\labell{d:discretization}
Let $\mathcal{X}$ be a stack over $\mfld$.  Define $\pizero(\mathcal{X})$ to be the assignment to each manifold $M$ the set of isomorphism classes in $\mathcal{X}(M)$.  This is a presheaf over $\mfld$.  Define $\discr(\mathcal{X})$ to be the sheafification of $\pizero(\mathcal{X})$. For any manifold $M$ and $\xi\in\mathcal{X}(M)$, denote by $[\xi]$ the image of the isomorphism class of $\xi$ in $\discr(\mathcal{X})(M)$ via the natural map $\pizero(\mathcal{X})\to\discr(\mathcal{X})$.
\end{definition}

\begin{remark}
In general, sheafification is necessary in the definition of discretization; see \hypref{x:BZ2}{Example}.
\end{remark}

\begin{lemma}\labell{l:discr adj}
$\discr:\stacks\to\sheaves$ is left adjoint to the Grothendieck construction $\int:\sheaves\to\stacks$.
\end{lemma}

\begin{remark}
It is already known that $\int$ has a left adjoint functor, but this is typically described in much more generality (see \cite{cafe} or \cite{awodey}).
\end{remark}

\begin{proof}
We will show $\int:\sheaves\to\stacks$ has the universal property of a right adjoint.  Fix a stack $\mathcal{X}\in\stacks$.  Then there exists a natural map of CFGs $\phi_\mathcal{X}: \mathcal{X} \to \int \pizero \mathcal{X}$ with $\phi_\mathcal{X}(\xi)=[\xi]$; that is, the groupoid fibers of $\mathcal{X}$ are replaced with the sets of their equivalences classes, and above each fiber $\phi_\mathcal{X}$ is the quotient map.

Now, suppose $S:\mfld^{op} \to \set$ is a sheaf of sets and $\psi: \mathcal{X} \to \int S$ is any map of stacks.  Then, since the fibers of $\int S$ are sets, $\psi$ must be constant on isomorphism classes of the fibers of $\mathcal{X}$.  Thus, it readily follows that $\psi$ factors uniquely through $\phi_\mathcal{X}$.

So, since $\int$ is fully faithful, the unique map of categories fibered in groupoids $\bar{\psi}:\int \pizero \mathcal{X} \to \int S$ satisfying $\psi=\bar{\psi}\circ\phi_\mathcal{X}$ is equal to $\int$ applied to a unique map of presheaves $f:\pizero \mathcal{X} \to S$.  Furthermore, the map $f$ must factor uniquely through the sheafification $\eta_\mathcal{X}:\pizero \mathcal{X} \to \discr \mathcal{X}$.  In other words, there exists a unique map of sheaves $g:\discr \mathcal{X} \to S$ with $f=g\circ \eta$.  Thus, $\int(g):\int\discr\mathcal{X}\to \int S$ is the unique map of stacks such that $g\circ (\int(\eta_{\mathcal{X}})\circ\phi_\mathcal{X}) = \psi$.

It follows $\int:\sheaves \to \stacks$ satisfies the universal property of a right adjoint with unit of adjunction $\mathcal{X}\mapsto\int(\eta_\mathcal{X})\circ \phi_\mathcal{X}$ and left adjoint $\discr$.

\end{proof}

\begin{remark}
To prove \hypref{l:discr adj}{Lemma}, we could simply show the standard adjunction hom-set isomorphism exists and, while showing that there is a bijection is a bit simpler than what appears above, showing that there is a natural choice is quite a bit more difficult, and so we prefer the above method.
\end{remark}

\begin{remark}\labell{r:2-functor}
 It is easy to check that $\pizero: \stacks\to\presheaves$ takes $2$-commuting diagrams in $\stacks$ to strictly commuting diagrams in $\presheaves$.  It follows that, by thinking of $\sheaves$ as a 2-category with trivial 2-arrows, we may call $\discr$ a 2-functor.
\end{remark}

\begin{definition}[The Functor $\coarse$]\labell{d:coarse}
Define the \emph{coarse (2-)functor} $\coarse:\stacks\to\csheaves$ as the composition $\concr\circ\discr$.  For any stack $\mathcal{X}$, we will call the concrete sheaf $\coarse(\mathcal{X})$ {\em the diffeological coarse moduli space of $\mathcal{X}$}.  This nomenclature is justified in \hypref{r:coarse moduli space}{Remark}.
\end{definition}

\begin{remark}\labell{r:coarse}
We could have defined $\coarse$ more directly without using the composition of $\concr$ and $\discr$.  Indeed, the sheafification of the presheaf which assigns to a manifold $M$ the quotient set $\pizero(\mathcal{X})(M)/\!\sim$, where $\sim$ is as defined in \hypref{d:concretization}{Definition}, yields a sheaf isomorphic to $\coarse(\mathcal{X})$ for any stack $\mathcal{X}$.  The point is that you do not need to sheafify twice.  This said, $\concr$ and $\discr$ are interesting functors on their own.
\end{remark}

\begin{theorem}[$\coarse$ is a Left Adjunction]\labell{t:coarse adj}
The 2-functor $\coarse$ is left adjoint to the Grothendieck construction restricted to $\csheaves$.
\end{theorem}

\begin{proof}
This follows directly from \hypref{l:concr adj}{Lemma} and \hypref{l:discr adj}{Lemma}.
\end{proof}

\begin{remark}[Diffeological Coarse Moduli Space]\labell{r:coarse moduli space}
As noted in \hypref{r:YonedaExt}{Remark}, the functor $\int:\csheaves\to\stacks$ is an extension of the Yoneda embedding of manifolds into stacks and, as shown in \hypref{t:coarse adj}{Theorem}, the functor $\coarse$ is a left adjoint to this extension.  Hence, following the nomenclature of Hepworth (see \cite[Section 2]{hepworth:morse}), we will call the image of a stack $\mathcal{X}$ under $\coarse$ {\em the diffeological coarse moduli space of $\mathcal{X}$}.  We avoid the term {\em coarse moduli space} simply because this term should be reserved for objects in the underlying site of a stack, while $\coarse(\mathcal{X})$ need not be a manifold in general.
\end{remark}

\begin{theorem}[Coarse is Co-continuous]\labell{t:coarse co-continuous}
If a stack $\mathcal{X}$ is equal to a 2-colimit $\operatorname{colim}F$ in the 2-category of stacks, then $\coarse(\mathcal{X})$ is equal to the 1-colimit $\operatorname{colim}(\coarse\circ F)$.
\end{theorem}

\begin{proof}
As explained in \hypref{r:2-functor}{Remark}, $\coarse$ takes $2$-commutative diagrams in $\stacks$ to strictly commuting diagrams in $\csheaves$.  Therefore, the $2$-cocones used to define colimits in $\stacks$ are sent to cocones in $\csheaves$ and a standard proof for co-continuity of left adjoints may be used (the dual argument can be found here \cite[Theorem 1, page 118]{MacLane}, for instance).

\end{proof}

\begin{remark}[Topological Considerations]\labell{r:topology}
Before moving on to differentiable stacks, we make the following note about the topology of the diffeological coarse moduli space.  Given a diffeological space $(X,\mathcal{D})$ the \emph{D-topology} of $X$ is defined to be the strongest topology on $X$ such that all plots in $\mathcal{D}$ are continuous.  That is, $A\subseteq X$ is open if and only if for every $p\in\mathcal{D}$, the set $p^{-1}(A)$ is open in the domain of $p$.

Now, assume that $\mathcal{X}$ is a stack whose diffeological coarse moduli space can be identified with $(X,\mathcal{D})$.  In particular, $\pizero(\mathcal{X}(*))$ is identified with the set $X$.  Then $A\subseteq \pizero(\mathcal{X}(*))=X$ is open if and only if for every manifold $M$ and object $\xi\in\mathcal{X}(M)$, we have that $\underline{[\xi]}^{-1}(A)\subseteq M$ is open.  Applying the Yoneda embedding $y$, we have that $A$ is open if and only if for every manifold $M$ and every map of stacks $\varphi:y(M)\to\mathcal{X}$, we have $\underline{[\varphi]}^{-1}(A)\subseteq M$ is open.  This matches the definition of {\em the underlying space of a stack} used by Hepworth in \cite[Definition 2.11]{hepworth:morse}.

\end{remark}

\section{Differentiable Stacks}\labell{s:differentiable stacks}

This section assumes that the reader is familiar with Lie groupoids, their actions and principal bundles, the localization of the category of Lie groupoids and its equivalence to differentiable stacks over manifolds.  Our main resource for this theory is Lerman's survey on the topic, \cite{lerman}.  In this section, we show that the quotient diffeology on the orbit space $G_0/G_1$ of a Lie groupoid $G$ matches the diffeology associated to $\coarse(BG)$ (\hypref{t:orbitspace}{Theorem}).  Then in the remainder of the section, we discuss the relationship between principal $G$-bundles and plots of the orbit space $G_0/G_1$.

\begin{theorem}[The Orbit Space of a Lie Groupoid]\labell{t:orbitspace}
Let $\mathcal{X}$ be a differentiable stack.  Let $G=(G_1\toto G_0)$ be a Lie groupoid representing $\mathcal{X}$ (that is, $\mathcal{X}\cong BG$).  Then $\coarse(\mathcal{X})$ is isomorphic to the concrete sheaf associated to the quotient diffeology on the orbit space $G_0/G_1$ of $G$.
\end{theorem}

\begin{remark}
In particular, the above theorem shows that the diffeology associated to the diffeological coarse moduli space $\coarse(\mathcal{X})$ is more than just independent of atlas; it is only dependent on the isomorphism class of stacks in the category of \emph{all} stacks over manifolds.
\end{remark}

\begin{proof}
Since $BG$ is the colimit of the diagram $y(G_1\toto G_0)$ in $\stacks$ (where $y$ is the Yoneda embedding) it follows from \hypref{t:coarse co-continuous}{Theorem} and \hypref{c:diffeol equiv csheaves}{Corollary} that the diffeological space associated to $\coarse(BG)$ is isomorphic to $G_0/G_1$ equipped with the quotient diffeology, since this is the colimit of the diagram $G_1\toto G_0$ in the category $\diffeol$.
\end{proof}

\begin{remark}
If $\mathcal{X}$ and $\mathcal{Y}$ are differentiable stacks, and $F:\mathcal{X}\to\mathcal{Y}$ is a map of stacks between them, then $\coarse(F)$ corresponds to a smooth map between the corresponding quotient diffeologies of representative Lie groupoid orbit spaces.  In particular, if $\mathcal{X}\cong BG$ and $\mathcal{Y}\cong BH$ for Lie groupoids $G$ and $H$, then $\coarse(F)$ corresponds to a smooth map between $G_0/G_1$ and $H_0/H_1$, both equipped with their quotient diffeologies. If $G$ and $H$ are Morita equivalent, then $F$ will be an isomorphism of stacks, and $\coarse(F)$ will correspond to a diffeomorphism of quotient diffeologies.  Similar results appear in \cite{watts-groupoids} and \cite{KZ}, but purely in the context of groupoids and diffeology.
\end{remark}

Let $G$ be a Lie groupoid, and let $BG$ be its corresponding differentiable stack.  Given principal $G$-bundles $P$ and $Q$ over a manifold $B$, we have that $P$ and $Q$ are isomorphic (\emph{i.e.} $G$-equivariantly bundle-diffeomorphic) if and only if they represent the same element in $\pizero(BG)(B)$.  This equivalence no longer holds upon sheafification of $\pizero(BG)$ to $\discr(BG)$ (see \hypref{x:BZ2}{Example}).

\begin{definition}[Locally Isomorphic Bundles]\labell{d:loc eqvt diffeo}
Let $G$ be a Lie groupoid, and let $P$ and $Q$ be principal $G$-bundles over a manifold $B$.  Then $P$ and $Q$ are \emph{locally isomorphic} if there exists an open cover $\{i_\alpha:B_\alpha\to B\}$ of $B$ such that for each $\alpha$, the principal $G$-bundles $i_\alpha^*P$ and $i_\alpha^*Q$ are isomorphic.
\end{definition}

\begin{proposition}[Discretization for Differentiable Stacks]\labell{p:loc eqvt diffeo}
Let $G$ be a Lie groupoid, and let $P$ and $Q$ be principal $G$-bundles over a manifold $B$.  Then, $P$ and $Q$ are locally isomorphic if and only if $[P]=[Q]$ in $\discr(BG)(B)$.
\end{proposition}

\begin{proof}
$P$ and $Q$ are locally isomorphic if and only if there exists an open cover $\{i_\alpha:B_\alpha\to B\}$ of $B$ such that for each $\alpha$, there is an isomorphism $i_\alpha^*P \cong i_\alpha^*Q$.  This is equivalent to $\pizero(BG)(i_\alpha^*P)=\pizero(BG)(i_\alpha^*Q)$ for each $\alpha$.  By definition of sheafification, this is equivalent to $\discr(BG)(P)=\discr(BG)(Q)$.
\end{proof}

The following example illustrates the difference between $\pizero(BG)$ and $\discr(BG)$.

\begin{example}[$B\ZZ_2$]\labell{x:BZ2}
Let $\mathcal{X}$ be the differentiable stack $B\ZZ_2$, where $\ZZ_2=\ZZ/2\ZZ$.  Then $B\ZZ_2(\SS^1)$ contains two isomorphism classes of bundles: one represented by the trivial $\ZZ_2$-bundle which we denote by $P$, and the other represented by the boundary of the M\"obius band which we denote by $Q$.  While $P$ and $Q$ do not represent the same elements in $\pizero(BG)$, by \hypref{p:loc eqvt diffeo}{Proposition} we know that $[P]=[Q]$ in $\discr(BG)$.
\eoe
\end{example}

\begin{remark}
The above example is not unique; given {\em any} Lie group $\Gamma$, since the stack $B\Gamma$ is a gerbe, it follows that the fibers of $\discr(B\Gamma)$ are all singleton sets.
\end{remark}

Now, given a Lie groupoid $G$, and two principal $G$-bundles $P$ and $Q$, we give conditions under which $[P]$ and $[Q]$ yield the same objects after applying the functor $\kappa$ to $\discr (BG)$.  Of course, $\underline{[P]}=\underline{[Q]}$ does not imply $[P]=[Q]$, as illustrated in \hypref{x:fiber eqvt diffeo}{Example}.

\begin{definition}[Fiberwise Isomorphic Bundles]\labell{d:fiber eqvt diffeo}
Let $G$ be a Lie groupoid, and let $\rho_P:P\to B$ and $\rho_Q:Q\to B$ be principal $G$-bundles over a manifold $B$.  Then $P$ and $Q$ are \emph{fiberwise isomorphic} if for every $b\in B$, there exists a $G$-equivariant diffeomorphism from $\rho_P^{-1}(b)$ to $\rho_Q^{-1}(b)$.
\end{definition}

\begin{proposition}[$\kappa$ and Differentiable Stacks]\labell{p:fiber eqvt diffeo}
Let $G$ be a Lie groupoid, and let $B$ be a manifold.  Let $\rho_P:P\to B$ and $\rho_Q:Q\to B$ be principal $G$-bundles over $B$.  Then, $P$ and $Q$ are fiberwise isomorphic if and only if $P$ and $Q$ descend to the same elements in $\kappa(\discr(BG))(B)$.
\end{proposition}

\begin{remark}\labell{r:fiber eqvt diffeo}
Unlike the case of discetization, we do not need to sheafify in order to obtain this result,  though the proposition applies equally well to elements of the sheafification $\concr(\discr(BG))$.  Indeed, it follows from \cite[Lemma 5.21]{BH} (which states that concrete presheaves are \emph{separated}) that for any presheaf $P$ and manifold $M$, sheafification induces an injection $\kappa(P)(M)\to\concr(P)(M)$.  We do not know, however, if this is a bijection in general.
\end{remark}

\begin{proof}
$P$ and $Q$ descend to the same elements of $\kappa(\discr(BG))(B)$ if and only if $\underline{[P]}$ and $\underline{[Q]}$ are the same.  But this is equivalent to $[b^*P]=[b^*Q]$ for each $b\in B$. That is, for each $b\in B$, there is a $G$-equivariant diffeomorphism between $b^*P$ and $b^*Q$.  But these two objects are exactly the fibers $\rho_P^{-1}(b)$ and $\rho_Q^{-1}(b)$, respectively.
\end{proof}

The following corollary follows from the definition of $\coarse$, Propositions \ref{p:loc eqvt diffeo} and \ref{p:fiber eqvt diffeo}, and \hypref{r:fiber eqvt diffeo}{Remark}.

\begin{corollary}[Coarse Applied to a Differentiable Stack]\labell{c:fiber eqvt diffeo}
Let $G$ be a Lie groupoid, and let $B$ be a manifold.  Let $P$ and $Q$ be principal $G$-bundles over $B$.  Then $P$ and $Q$ are fiberwise isomorphic if and only if they represent the same object in $\coarse(BG)(B)$.
\end{corollary}

\begin{example}[Reflection in $\RR$]\labell{x:fiber eqvt diffeo}
Let $G$ be the action groupoid $(\ZZ_2\times\RR)\toto\RR$ obtained from the action of $\ZZ_2$ on $\RR$ sending $(\pm1,x)$ to $\pm x$.  Consider the following two plots $p_1:\RR\to\RR$ and $p_2:\RR\to\RR$.

$$ p_1(\tau)=\begin{cases}
-e^{-1/\tau^2} & \text{ if $\tau<0$,}\\
0 & \text{ if $\tau=0$,}\\
e^{-1/\tau^2} & \text{ if $\tau>0$,}\\
\end{cases}$$
$$ p_2(\tau)=\begin{cases}
-e^{-1/\tau^2} & \text{if $\tau\neq 0$,}\\
0 & \text{ if $\tau=0$.}\\
\end{cases}$$

Then, calculating the pullback bundles $P_i:=p_i^*t$ ($i=1,2$), where $t:G_1\to G_0$ is the unit bundle, we have
\begin{align*}
P_1=&~\{(\tau,a,x)\in\RR\times\ZZ_2\times\RR~|~p_1(\tau)=a\cdot x\}\\
=&~\{(\tau,1,-e^{-1/\tau^2})~|~\tau<0\}\cup\{(0,\pm1,0)\}\cup\{(\tau,1,e^{-1/\tau^2})~|~\tau>0\}\\
&~\cup\{(\tau,-1,e^{-1/\tau^2})~|~\tau<0\}\cup\{(\tau,-1,-e^{-1/\tau^2})~|~\tau>0\},
\end{align*}
and similarly,
\begin{align*}
P_2=&~\{(\tau,1,-e^{-1/\tau^2})~|~\tau<0\}\cup\{(0,\pm1,0)\}\cup\{(\tau,1,-e^{-1/\tau^2})~|~\tau>0\}\\
&~\cup\{(\tau,-1,e^{-1/\tau^2})~|~\tau<0\}\cup\{(\tau,-1,e^{-1/\tau^2})~|~\tau>0\}.
\end{align*}

If $\phi:P_1\to P_2$ is a bundle-diffeomorphism, then in particular it must send $\tau$ to $\tau$; that is, $\phi$ must fix the first argument of each triple in $P_1$.  Now, the anchor map $\alpha_i:P_i\to \RR$ is just the projection to the third argument of each triple, and so if $\phi$ is also equivariant, then it must fix the third argument of each triple in $P_1$.  Thus, $\phi$ must be the identity map when restricted to fibers for which $t<0$, and it is switches connected components of the fibers for $t>0$.  Hence, $\phi$ cannot be continuous at $t=0$, and so we conclude that there is no equivariant diffeomorphism of bundles; in fact, not even a local one exists.  However, it is clear that the fibers of $P_1$ and $P_2$ are equivariantly diffeomorphic, and so by \hypref{p:fiber eqvt diffeo}{Proposition} we have $\underline{[P_1]}=\underline{[P_2]}$.  This corresponds to the fact that $\pi_G\circ p_1=\pi_G\circ p_2$ where $\pi_G:G_0\to G_0/G_1$ is the quotient map (see \hypref{p:plots}{Proposition}). \eoe
\end{example}

In light of \hypref{r:diffeol equiv csheaves}{Remark}, we show in the following proposition how elements of $\coarse(BG)$ and plots of the corresponding diffeology are related.

\begin{proposition}[From Principal Bundles to Plots]\labell{p:plots}
Let $G$ be a Lie groupoid, and let $\rho:P\to B$ be a principal $G$-bundle, where $B$ is an open subset of some Euclidean space.  Then, $P$ induces a plot $p:B\to G_0/G_1$ defined as $$p(b)=\pi_G\circ\alpha\circ\sigma(b)$$ where $\pi_G:G_0\to G_0/G_1$ is the quotient map, $\alpha:P\to G_0$ is the anchor map, and $\sigma$ is a local section of $\rho$ about $b$.  Moreover, every plot in the quotient diffeology locally arises in this way.  In particular, $\pizero(BG(*))$ is in bijection with the underlying set of $G_0/G_1$.
\end{proposition}

\begin{remark}\labell{r:plots}
The last statement is already known, but is crucial to understanding the diffeological coarse moduli space of a differentiable stack from the standpoint of diffeology, and so is worth mentioning.  Also, if $B$ is a manifold, but not an open subset of some Euclidean space, then the proposition still holds where $p$ is no longer a plot, but is contained in $I(\mathcal{D})(B)$ where $\mathcal{D}$ is the quotient diffeology on $G_0/G_1$ and $I$ is as given in \hypref{r:diffeol equiv csheaves}{Remark}.
\end{remark}

\begin{proof}
Since $\rho$ is a surjective submersion, local sections of $\rho$ exist.  Assume that $\sigma$ and $\sigma'$ are two local sections of $\rho$ about $b$.  Then, there exists $g\in G_1$ such that $\sigma(b)\cdot g=\sigma'(b)$.  Thus, $\alpha(\sigma(b))$ and $\alpha(\sigma'(b))$ lie in the same $G$-orbit, and so descend to the same point via $\pi_G$.  Thus, $p$ is independent of the local section chosen, and so is well-defined.  Moreover, $p$ is diffeologically smooth, and so is a plot.  To show that $p$ and $\underline{[P]}$ are the same, consider the bundle $b^*P$, which is the fiber of $P$ over $b$.  The anchor map restricted to $b^*P$ has an image contained in exactly one $G$-orbit of $G_0$, which by definition of $G_0/G_1$ is exactly $p(b)$.  But $[b^*P]=\underline{[P]}(b)$, and this proves the claim.

On the other hand, given any plot $p:U\to G_0/G_1$ in the quotient diffeology, there exist an open cover $\{U_\alpha\}$ of $U$ and a family $\{q_\alpha:U_\alpha\to G_0\}$ of smooth maps such that $p|_{U_\alpha}=\pi_G\circ q_\alpha$.  Taking the pullback bundles $q_\alpha^*(t:G_1\to G_0)$ of the unit bundle proves the claim.
\end{proof}

\section{Differential Forms}\labell{s:differential forms}

In this section we give a notion of differential form on a stack.  In the case of a differentiable stack, it coincides with basic forms of a Lie groupoid representative (see \hypref{p:differential forms}{Proposition}).  In the case of a stack equal to $\int$ applied to a concrete sheaf, the definition matches that for a corresponding diffeological space (see \hypref{r:differential forms}{Remark}).  It turns out that this definition only depends on the discretization of the stack, and in the case that the stack is differentiable and represented by a specific type of Lie groupoid, it only depends on the underlying diffeological coarse moduli space (see \hypref{t:differential forms}{Theorem}).  It remains an open question whether this dependence on the diffeological coarse moduli space holds for all stacks.

\begin{definition}[Differential Forms]\labell{d:differential forms1}
Let $\Omega^k:\mfld^{op} \to \set$ be the sheaf of differential $k$-forms over manifolds.  Then, for a sheaf $S:\mfld^{op}\to \set$, we will define a \emph{differential $k$-form} as a map of sheaves $\alpha:S\to \Omega^k$.  Define $\Omega^k(S):=\hom_\sheaves(S,\Omega^k)$.  Similarly, for a stack $\mathcal{X}$, define a $k$-form to be a map $\beta:\mathcal{X}\to \int \Omega^k$ and let $\Omega^k(\mathcal{X}):=\hom_{\stacks}(\mathcal{X},\int \Omega^k)$.  Note that $\Omega^k(\mathcal{X})$ is discrete, and thus we may think of $\Omega^k(\mathcal{X})(M)$ as a set for any manifold $M$.
\end{definition}

\begin{remark}
To conveniently tie forms in with our discussion in this paper, we will suppress the vector space structure of each $\Omega^k(M)$ and regard them only as sets.  In general, of course, $\Omega^k$ is a sheaf of vector spaces and the sets of morphisms $\Omega^k(S)$ and $\Omega^k(\mathcal{X})$ both inherit the structure of vector spaces as well.
\end{remark}

\begin{definition}[Differential Forms in Diffeology, see Article 6.28 of \cite{iglesias}]\labell{d:differential forms2}
A \emph{differential $k$-form} $\alpha$ on a diffeological space $(X,\mathcal{D}_X)$ is an assignment to each plot $p:U\to X$ in $\mathcal{D}_X$ a differential form $\alpha(p)\in\Omega^k(U)$ satisfying the following compatibility condition: if $f:V\to U$ is a smooth map between open subsets of Euclidean spaces, then $\alpha(p\circ f)=f^*\alpha(p)$.
\end{definition}

\begin{remark}\labell{r:differential forms}
When considering a diffeological space $(X,\mathcal{D})$ as a concrete sheaf $I(\mathcal{D}):\mfld^{op} \to \set$, this definition says precisely that a differential $k$-form for a diffeological space is a map of sheaves $\alpha: I(\mathcal{D}) \to \Omega^k$.  So our definitions for differential forms are coherent.
\end{remark}

\begin{proposition}[Basic Forms of a Lie Groupoid]\labell{p:differential forms}
Let $G=(G_1\toto G_0)$ be a Lie groupoid.  Then an atlas $\theta: y(G_0)\to BG$ (where $y$ is the Yoneda embedding of $\mfld$ into $\stacks$) induces a bijection from $\Omega^k(BG)$ to basic $k$-forms on $G_0$; that is, differential $k$-forms $\mu$ such that $s^*\mu=t^*\mu$ where $s$ and $t$ are the source and target maps of $G$, respectively.
\end{proposition}

\begin{proof}
Let $\alpha\in\Omega^k(BG)$, and let $\theta:y(G_0)\to BG$ be an atlas.  Then since $BG$ is the colimit of the diagram $y(G_1\toto G_0)$ we have $$\alpha\circ\theta\circ y(s)=\alpha\circ\theta\circ y(t).$$  Thus, $\alpha\circ\theta$ corresponds to a basic differential form on $G_0$.

In the other direction, any basic form $\mu$ on $G_0$ induces a map of stacks $\tilde{\mu}:y(G_0)\to\int\Omega^k$ such that $$\tilde{\mu}\circ y(s)=\tilde{\mu}\circ y(t).$$  By the universal property of colimits, we have a unique map $\mu':BG\to\int\Omega^k$ such that $\tilde{\mu}=\mu'\circ\theta$.
\end{proof}

\begin{theorem}[Differential Forms and $\coarse$]\labell{t:differential forms}
Let $\mathcal{X}$ be a stack.  Then, pre-composition with $\discr$ induces a bijection between $\Omega^k(\mathcal{X})$ and $\Omega^k(\int\discr(\mathcal{X}))$, and pre-composition with $\coarse$ induces an injection from $\Omega^k(\int\coarse(\mathcal{X}))$ to $\Omega^k(\mathcal{X})$.  Moreover, if $G$ is
\begin{enumerate}
\item \labell{i:groupoid} a finite-dimensional, Hausdorff, paracompact, proper Lie groupoid, or
\item \labell{i:gp action} the action groupoid of a Lie group on a manifold whose identity component acts properly,
\end{enumerate}
then the injection from $\Omega^k(\int\coarse(BG))$ to $\Omega^k(BG)$ is in fact a bijection.
\end{theorem}

\begin{proof}
The first claim is immediate from the fact that the Grothendieck construction is a fully faithful functor, along with \hypref{l:discr adj}{Lemma}.  The second claim follows from the fact that any differential form $\alpha:\coarse(\mathcal{X})\to\int\Omega^k$ is completely determined by it values on elements of the form $\underline{[\xi]}$ where $\xi$ are objects in $\mathcal{X}$.  Finally, the last claim follows from \hypref{l:BH1}{Lemma}, \hypref{r:differential forms}{Remark}, and \cite{watts-groupoids} in the case of \hypref{i:groupoid}{Item} or \cite{KW} in the case of \hypref{i:gp action}{Item}.
\end{proof}

config: arxiv


TITLE: Why is the product of two consecutive integers $n \cdot( n+1) \forall n > 2$ guaranteed to have at least two prime factors?
QUESTION [2 upvotes]: I was reading this paper: http://fermatslibrary.com/s/a-new-proof-of-euclids-theorem and became confused when reading this line:

Since $n$ and $n + 1$ are consecutive integers, they must be coprime.
  Hence the number $N_2 = n(n + 1)$ must have at least two diﬀerent prime
  factors

I see that this is true in practice when writing out several examples, but was hoping for an explanation of why? (which the paper left out). I understand why $n$ and $n+1$ are co-prime, but not why that implies that $n\cdot(n+1)$ has at least two different prime factors. What does being co-prime have to do with this?
Any insight would be really helpful.
EDIT:
I believe I understand now:
1) $n$, $n+1$ have no factors in common except $1$
2) If $n$, $n+1 \ge 2$ they have at least one prime factor
3) From (1) these factors must be different, so there must be at least two different prime factors

REPLY [1 votes]: Assume $n>1$.
$n,n+1$ are adjacent integers, so at least one of them is even, meaning that $n(n+1)$ is even.
$n(n+1)$ will also have a factor other than $2$, as otherwise it would have to be a power of $2$. But $n(n+1)=2^k$ means that $n,n+1$ must both be powers of $2$. One of them is odd, meaning that either $n$ or $n+1$ is $1$ (the only odd power of 2). Given that $n>1$, this can't be true.
So, $n(n+1)$ has a factor of $2$, but can't be factored as $2^k$. So it must have another prime factor somewhere.


TITLE: Do every two orthogonal matrices in $\text{SO}(n)$ lie in the same coset of $\text{SO}(2)$?
QUESTION [2 upvotes]: Let $A,B  \in \text{SO}(n)$.
Does there exist a homomorphism of Lie groups $\phi:\text{SO}(2) \to \text{SO}(n)$, such that $A,B$ lie in the same coset of $\phi(\text{SO}(2))\le \text{SO}(n)$?

REPLY [1 votes]: The answer is yes if $n\leq 3$ and is no for $n\geq 4$.
As freakish says, you question is equivalent to the following:

Given any $p\in SO(n)$, is there a circle subgroup of $SO(n)$ containing $p$?

We will actually address this question for all compact Lie groups $G$ simultaneously.
Proposition 1:  Suppose $G$ is a compact Lie group of rank $1$.  Then any $p\in G$ lies on an $SO(2)\subseteq G$.
Proof:  Recall that in any compact Lie group $G$, any $p\in G$ lies on a maximal torus $T$.  If the rank of $G$ is $1$, then $T = S^1 = SO(2)$.  $\square$
And now the more fun direction:
Proposition: If every $p\in G$ lies in an $SO(2)$ subgroup, then $G$ has rank $1$.
Proof: Let $p \in G$ with the property that the closure of the subgroup generated by $p$, $\overline{\langle p\rangle}$ is a maximal torus $T$.  Such a $p$ always exists: beginning with any maximal torus $T'$, let $p = (\theta_1,...,\theta_k)\in T'$ with the $\{\theta_i\}$ linearly independent over $2\pi \mathbb{Q}$.
By assumption, $p\in SO(2)$ for some $SO(2)\subseteq G$.  Because $SO(2)$ is a subgroup, $\langle p\rangle\subseteq SO(2)$.  Because $SO(2)$ is closed (because it is compact), $\overline{\langle p \rangle}\subseteq SO(2)$.  By construction, $T = \overline{\langle p \rangle}$, so $T\subseteq SO(2)$, so a maximal torus lies in $SO(2)$.  That is, $G$ has rank $1$. $\square$


TITLE: Random permuted blocks - Number of unique blocks
QUESTION [1 upvotes]: When using random permuted blocks to produce a randomisation list for treatment groups in a clinical trial. I know that the formula for the number of unique blocks (assuming equal allocation to each groups) is
$\frac{L!}{(M!)^N} $, where L= Block length, N= Number of treatment groups and M=L/N.
How would I got about generalising this to the case where the ratio isn't 1:1? This is because I wish to work out the number of unique blocks when there is 3 treatment groups with an allocation ratio 2:1:1.

REPLY [0 votes]: This is effectively a multinomial coefficient. For the case when one group is size $2M$ and the other two are size $M$, you'd have
$$\frac{L!}{(2M)!(M!)^2}\text{ possible orderings}$$


TITLE: solving biquadratic equations by ferrari method
QUESTION [0 upvotes]: I know how to solve ax^4+bx^3+cx^2+dx+e by ferrari method but can someone please explain how the below equation will bo solved.
x^4+x^2+1=0

REPLY [1 votes]: Welcome to MSE! Your question could use a bit more context and/or explanation: What have you tried so far? Are you specifically meant to use the Ferrari method? That sort of information will help us get you the sort of answer or help you need!

That said, an alternative to what others have mentioned: you may know the factorization:
$$x^4+x^2+1=(x^2+x+1)(x^2-x+1)$$
(I'd you don't, it's a good one to try to remember.)
That factorization is also equal to $\Phi_3(x)\cdot \Phi_6(x)$, where $\Phi_n(x)$ is the $n$th cyclotomic polynomial. That's useful because the roots of those polynomials are the $n$th roots of unity, to the powers of integers coprime to $n$. Hence the four roots of the equation are:
$$x=(\zeta_3, \zeta_3^2, \zeta_6, \zeta_6^5) = (\zeta_6^2, \zeta_6^4, \zeta_6, \zeta_6^5)$$
where $\zeta_n = e^{i\tau/n}$ is the first of the $n$ roots of unity.


TITLE: Riemann integration and the Fundamental Theorem of Calculus
QUESTION [1 upvotes]: Hello, I am unsure of how to define the partition to find a suitable result for the Riemann integrable, how do I do this for a discontinuous function?

REPLY [2 votes]: You  may omit finitely many points from the integration without changing the integral. This allows you to compute $$
  \int_{-1}^1 s(x)\,dx = \int_{-1}^0 -1 \,dx + \int_0^1 1 \,dx = -1 + 1 = 0
$$
without worrying about how often this includes the point $0$, and about whether $s$ is continuous there or not.