# Smooth loop stacks of differentiable stacks and gerbes

David Michael Roberts and Raymond F. Vozzo

(25 November 2016)

Résumé. Nous définissons un groupoïde de Fréchet-Lie
$\operatorname{Map}(S^{1},X)$ d’ana-foncteurs du cercle vers un groupoïde de
Lie $X$. Ceci fournit une présentation du Hom-champ
$\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$, où
$\mathcal{X}$ est le champ différentiable associé à $X$. Nous appliquons cette
construction au groupoïde de Lie sous-jacent au ‘gerbe fibré’ d’une variété
différentiable $M$; le résultat est un gerbe fibré au-dessus de l’espace des
lacets $LM$ de $M$.
Abstract. We define a Fréchet–Lie groupoid $\operatorname{Map}(S^{1},X)$ of
anafunctors from the circle into a Lie groupoid $X$. This provides a
presentation of the Hom-stack
$\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$, where
$\mathcal{X}$ is the differentiable stack associated to $X$. We apply this
construction to the Lie groupoid underlying a bundle gerbe on a manifold $M$;
the result is a bundle gerbe on the loop space $LM$ of $M$.
Keywords. Differentiable stacks, Lie groupoids, Hom-stacks, loop stacks,
gerbes, bundle gerbes
Mathematics Subject Classification (2010). Primary 22A22; Secondary 58B25,
58D15, 14A20, 18F99, 53C08.

## 1\. Introduction

The notion of smooth loop space of a manifold is useful in a variety of areas
of geometry, while at the same time being just outside the usual sphere of
study, namely finite-dimensional manifolds. While it is naturally a
topological space, it carries a very well-behaved smooth structure as an
infinite-dimensional manifold. While recent progress on generalised smooth
spaces means that any mapping space, in particular a loop space, is easily a
smooth space, and the general study of smooth spaces is advancing in leaps and
bounds (see e.g. the lengthy book [IZ13] on diffeological spaces), the fact
that the loop space is a manifold with well-understood charts is extremely
useful.

The area of geometry has in recent years expanded to include what is becoming
known as ‘higher geometry’, where, loosely speaking, the geometric objects of
study have a categorical or higher categorical aspect. One example of such
objects are differentiable, or _Lie_ , groupoids, which are known [Pro96] to
be incarnations of _differentiable stacks_ : stacks that look locally like
manifolds, but with internal symmetries captured by Lie groupoids. A rather
well-known simple case is that of _orbifolds_. Other examples that are still
stacks on manifolds but which are still akin to Lie groupoids, are groupoids
built from infinite-dimensional manifolds, or from smooth spaces. Clearly
these objects can become locally less well-behaved as one becomes more
general; an arbitrary diffeological space, for instance, may have rather
terrible topological and homotopical properties.

The construction that this short paper wishes to address is that of the loop
stack of a differentiable stack. This was introduced in the special case of
orbifolds in [LU02], and then considered in full generality for the purposes
of studying string topology in [BGNX12]. All of these are special cases of the
loop stack of the underlying _topological stack_ , a special case of the
topological mapping stack studied in [Noo10].111The paper [Car12] considers
the more general problem of a cartesian closed bicategory of stacks, whereas
[Noo10] considers the special case with compactness conditions on the domain.
In other words, the end result is only a _topological_ stack, rather than a
differentiable stack.

One (quite reasonable) approach is to consider if we can find a loop stack on
manifolds that arises from a diffeological groupoid (i.e. a diffeological
stack). This is not too difficult, and the parts of our construction that do
not require special handling due to the nature of manifolds are performed for
diffeological stacks. The novelty here is that this construction can be lifted
so that it becomes a stack arising from what we call a _Fréchet–Lie_ groupoid:
a groupoid in the category of Fréchet manifolds. This is the optimal result,
since the construction applied to a manifold returns (a stack equivalent to)
the usual loop space of that manifold, which is an infinite-dimensional
Fréchet manifold in general. This is in contrast to the case of algebraic Hom-
stacks, for instance [Ols06], which are again algebraic stacks.

The benefits of having a smooth version of the loop stack is that one can
start to do actual geometry on it, rather than just topological constructions
(such as the string topology in [BGNX12]). Moreover, while one can perform
smooth geometric constructions on diffeological spaces, as _spaces_ , unlike
manifolds there is little control over the local structures. So, for instance,
our construction provides a _smoothly paracompact_ groupoid, admitting
partitions of unity on object and arrow manifolds.

Another example, which was the original impetus for this article, are the loop
stacks of _bundle gerbes_. Bundle gerbes over manifolds are higher geometric
objects analogous to line bundles, and as such can support structures
analogous to connections. One can form the construction given below to the
groupoid underlying a bundle gerbe and then the resulting groupoid is in fact
still a gerbe, now over a loop space, and this should again carry a connective
structure of the appropriate sort. Of particular interest is the bundle gerbe
underlying the _String 2-group_ , which will be the subject of future work.

$\textstyle{\mathcal{M}=\\{\text{manifolds}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\subset$

$\subset$

$\textstyle{\\{\text{Fr\'{e}chet
manifolds}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\subset$

$\subset$

$\textstyle{\\{\text{diffeological
spaces}\\}=\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\subset$

$\textstyle{\\{\text{Lie
groupoids}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\subset$

$\textstyle{\\{\text{Fr\'{e}chet Lie
groupoids}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\subset$

$\textstyle{\\{\text{diffeological
groupoids}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{\text{Differentiable
stacks}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\subset$

$\subset$

$\textstyle{\\{\text{Fr\'{e}chet differentiable
stacks}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\subset$

$\textstyle{\\{\text{Diffeological
stacks}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

$\subset$

$\textstyle{\operatorname{\bf{Stack}}_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\bf{Stack}}_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{\bf{Stack}}}$
Figure 1: Categories and 2-categories appearing in the paper

We consider in this paper various categories of smooth objects, groupoids in
those categories and corresponding smooth stacks. Figure 1 summarises these,
as well as the relations between them. The first row consists of categories,
the remaining rows consist of 2-categories, and the inclusions denote _full_
subcategories and sub-2-categories. The vertical arrows of type  denote
surjective-on-objects 2-functors. We use $\operatorname{\bf{Stack}}_{X}$ to
denote the 2-category of stacks of groupoids on the site $X$.

The paper outline is as follows:

  * •

Section 2—Gives background on sites, internal groupoids, anafunctors (a type
of generalised morphism between internal groupoids) and stacks presented by
groupoids internal to the base site.

  * •

Section 3—We construct a diffeological groupoid $\operatorname{Map}(S^{1},X)$
of anafunctors and transformations.

  * •

Section 4—Proves that $\operatorname{Map}(S^{1},X)$ is indeed a presentation
over the site of diffeological spacs, making
$\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$ a diffeological
stack.

  * •

Section 5—We show that the construction of $\operatorname{Map}(S^{1},X)$
actually lands in the sub-2-category of Fréchet–Lie groupoids, and that this
gives a (weak) presentation of
$\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$. This is our
first main result.

  * •

Section 6—Gives a treatment of the theory of gerbes on the site of manifolds
presented by (Fréchet–)Lie groupoids, including establishing stability of
various properties under forming the mapping groupoid.

  * •

Section 7—We prove our second main result, namely that given a bundle gerbe (a
special sort of abelian gerbe), the mapping groupoid is again a bundle gerbe.

Acknowledgements. This research was supported Australian Research Council’s
Discovery Projects DP120100106 and DP130102578. This project was born in
‘Coffee Spot’: thanks to the staff for uninterrupted and secluded working
time. A big thanks to Andrew Stacey for writing the paper [Sta13], in order to
prove theorem 5.1, after discussions with the first named author; this theorem
was crucial to the success of the current paper. Thanks also to Alexander
Schmeding for side discussions about possible extensions to the infinite-
dimensional setting. The authors thank the anonymous referee for their careful
reading and helpful suggestions, which helped us find a small error in the
original version of Lemma 6.5.

## 2\. Background and preliminaries

### 2.1 Sites

We will be interested in stacks over sites where the Grothendieck topology,
arises from a _coverage_ (see e.g. [Joh02, Section C.2.1]), rather than the
more familiar data of a pretopology. In this paper we will work only with a
coverage and not the Grothendieck topology generated by it.

######  Definition 2.1.

Let $\mathcal{C}$ be a category. A _coverage_ $J$ on $\mathcal{C}$ is a
collection $J(x)$, for each object $x$, of families of arrows $\\{u_{i}\to
x\mid i\in I\\}$ (called _covering families_) with the property that for each
covering family $\\{u_{i}\to x\mid i\in I\\}\in J(x)$ and $f\colon y\to x$
there is a covering family $\\{v_{k}\to y\mid k\in K\\}\in J(y)$ such that for
all $k$ there is an $i\in I$ and lift as shown

$\textstyle{v_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{x}$

A _site_ $(\mathcal{C},J)$ is then a category $\mathcal{C}$ equipped with a
coverage $J$, and sites with the same underlying category are _equivalent_ if
their coverages generate the same sieves.

It will be the case that the coverages we consider satisfy the saturation
condition that composites of coverages are again coverages, but not still not
necessarily that pullbacks of covering families are covering families.

If we have a pair of covering families $\mathcal{U}=\\{u_{i}\to x\mid i\in
I\\}$ and $\mathcal{V}=\\{v_{j}\to x\mid j\in J\\}$ then we say $\mathcal{V}$
_refines_ $\mathcal{U}$ if for every $j\in J$ there is an $i\in I$ and a lift
of $v_{j}\to x$ through $u_{i}$. We can say that a coverage $J_{1}$ refines
the coverage $J_{2}$ if every covering family in $J_{1}$ refines a covering
family in $J_{2}$. If $J_{1}$ refines $J_{2}$ and $J_{2}$ refines $J_{1}$ then
they give rise to equivalent sites.

A coverage is called a _singleton_ coverage if all covering families consist
of single maps, in which case covering families will be referred to as
_covering maps_. An example of a singleton coverage is a class of maps
containing identity arrows, closed under composition and pullback along
arbitrary maps; such a class will be called a _singleton pretopology_

A _superextensive_ coverage (on an extensive category, see [CLW93]) is one
that is generated by a singleton coverage and the coverage where covering
families are inclusions of summands $\\{u_{i}\to\coprod_{i\in I}u_{i}\mid i\in
I\\}$. For all intents and purposes, a superextensive coverage $J$ can be
reduced to considering just the singleton coverage $\amalg J$ it gives rise
to: $\amalg J$-covering maps are of the form $\coprod_{i}u_{i}\to x$, for
$\\{u_{i}\to x\mid i\in I\\}$ a covering family in the original superextensive
coverage. We shall abuse terminology slightly and say that a superextensive
coverage $J_{1}$ and another singleton coverage $J_{2}$ give rise to
equivalent sites when the _singleton coverage associated to $J_{1}$_ is
equivalent to $J_{2}$. We shall also abuse notation and refer to a covering
map in $\amalg J$ as being in $J$ when no confusion shall arise.

A site is called _subcanonical_ if all representable presheaves are in fact
sheaves. For a singleton coverage this is implied by all covering maps being
regular epimorphisms, and for a subcanonical superextensive coverage $J$, the
singleton coverage $\amalg J$ is subcanonical. In fact all of the coverages we
consider in this paper will be subcanonical.

We will need the following examples over the course of the paper.

######  Example 2.2.

Consider the category $\operatorname{\bf{Cart}}$ with objects $\mathbb{R}^{n}$
for $n=0,1,2,\ldots$ and
$\operatorname{\bf{Cart}}(\mathbb{R}^{n},\mathbb{R}^{m})=C^{\infty}(\mathbb{R}^{n},\mathbb{R}^{m})$.
This has a coverage where a covering family
$\\{\phi_{i}\colon\mathbb{R}^{n}\hookrightarrow\mathbb{R}^{n}\mid i\in I\\}$
is an open cover in the usual sense.

For the purposes of the current paper, we consider manifolds to be finite
dimensional unless otherwise specified.

######  Example 2.3.

The category $\mathcal{M}$ of smooth manifolds has the following coverages:

  * •

the coverage ${O}$ of open covers in the usual sense;

  * •

the coverage $C$, where covering families $C(X)$ are covers of $X$ by regular
closed compact neighbourhoods, such that the interiors also cover;

  * •

the singleton pretopology $\operatorname{Subm}$ where covering maps are
surjective submersions.

All these coverages give equivalent sites, the first two because manifolds are
locally compact and regular222In fact there is a coverage on the category of
locally compact spaces consisting of compact neighbourhoods, and a coverage on
the category of regular spaces consisting of closed neighbourhoods. and the
first and last because surjective submersions have local sections. The first
two coverages are superextensive, and we will be considering their associated
singleton coverages.

Recall that a (smooth) _Fréchet_ manifold is a smooth manifold locally
modelled on Fréchet spaces (a good reference is [Ham82]). The definition does
not assume second-countability, so that the category of Fréchet manifolds
admits small coproducts. A submersion between Fréchet manifolds is a map for
which there are charts on which the map looks locally like a projection out of
a direct sum: $V\oplus W\to V$ (it is not enough to ask that this is
surjective, or even split surjective, on tangent spaces).

######  Example 2.4.

The category $\mathcal{F}$ of Fréchet manifolds has a coverage given by open
covers, and also a singleton pretopology given by surjective submersions. The
first is superextensive, and these give rise to equivalent sites.

Our last example needs some preliminaries. The following definition is quite
different to that which appears in the original article [Sou80], but is in
fact equivalent by work of Baez–Hoffnung [BH11]. An extensive reference is the
book [IZ13].

######  Definition 2.5.

A _diffeological space_ is a sheaf $X$ on $\operatorname{\bf{Cart}}$ that is a
subsheaf of
$\mathbb{R}^{n}\mapsto\operatorname{\bf{Set}}(\mathbb{R}^{n},\underline{X})$,
where $\underline{X}=X(\mathbb{R}^{0})$ is the _set of points of $X$_. A
smooth map of diffeological spaces is just a map between the underlying
sheaves. We denote the category of diffeological spaces by $\mathcal{D}$.

We can think of cartesian spaces $\mathbb{R}^{n}$ as diffeological spaces via
the Yoneda embedding, and for $X$ a diffeological space, the elements of
$X(\mathbb{R}^{n})$ as maps $\mathbb{R}^{n}\to X$ in $\mathcal{D}$. The
category of diffeological spaces is a Grothendieck quasitopos [BH11], in
particular is complete, cocomplete, extensive and cartesian closed.

A map $X\to Y$ of diffeological spaces is a _subduction_ if for every
$f\colon\mathbb{R}^{n}\to Y$ there is a covering family
$\phi_{i}\colon\mathbb{R}^{n}\hookrightarrow\mathbb{R}^{n}$ such that each map
$f\circ\phi_{i}\colon\mathbb{R}^{n}\to Y$ lifts to $X$. Note that there are
fully faithful inclusions
$\mathcal{M}\hookrightarrow\mathcal{F}\hookrightarrow\mathcal{D}$. Surjective
submersions of manifolds and also of Fréchet manifolds are subductions.

######  Example 2.6.

The category of diffeological spaces has a singleton pretopology
$\operatorname{Subd}$ given by subductions.

The following facts about subductions will be useful.

  * •

Every subduction $A\to B$ is refined by a subduction with domain a coproduct
of Euclidean spaces;

  * •

Every subduction $A\to M$ with $M$ a manifold is refined by an open cover of
$M$.

The astute reader will have noticed that almost all of the examples are in
fact pretopologies or singleton pretopologies. The important fact is that we
need to use the singleton coverage $\amalg C$ which is _not_ a pretopology,
but _refines_ a singleton pretopology.

### 2.2 Internal groupoids

We will be dealing with internal groupoids that satisfy extra conditions, due
to the fact that the ambient categories of manifolds are not finitely
complete. To that end, _Lie groupoids_ are groupoids internal to $\mathcal{M}$
where the source and target maps are submersions, and _Fréchet–Lie groupoids_
are groupoids internal to $\mathcal{F}$ where again the source and target maps
are submersions of Fréchet manifolds. We will also consider _diffeological_
groupoids, which are just groupoids internal to ; their source and target maps
are automatically subductions.

Functors between internal groupoids, be they Lie, Fréchet–Lie or diffeological
groupoids, will be assumed to be smooth. The same will be true for natural
transformations between such functors. We denote, for a category
$\mathcal{C}$, the 2-category of groupoids internal to $\mathcal{C}$ by
$\operatorname{\bf{Gpd}}(\mathcal{C})$, with the above caveats for
$\mathcal{C}=\mathcal{M},\ \mathcal{F}$. Since the inclusions
$\mathcal{M}\hookrightarrow\mathcal{F}\hookrightarrow$ are full, we have full
inclusions of 2-categories
$\operatorname{\bf{Gpd}}(\mathcal{M})\hookrightarrow\operatorname{\bf{Gpd}}(\mathcal{F})\hookrightarrow\operatorname{\bf{Gpd}}()$.

It is a well-known problem that there are just not enough morphisms between
internal groupoids, in particular Lie groupoids and their cousins. One
approach to this problem is through the use of _internal anafunctors_. These
were introduced in Bartels’ thesis [Bar06], inspired by work of Makkai on
foundational issues surrounding the Axiom of Choice in category theory. We do
not need the full theory of internal anafunctors, the basic definitions are
enough for the present paper, for the special case where we only consider
internal groupoids. We have also generalised the notion ever so slightly, by
using singleton _coverages_ ; the fragment of the theory we need here does not
lose out by considering this more general setting.

######  Definition 2.7 ([Bar06]).

Let $J$ be a singleton coverage on $\mathcal{C}$ and let $Y$ and $X$ be
groupoids in $\mathcal{C}$. An _anafunctor_ $Y-\\!\\!\\!\mapsto X$ is a span
of internal functors

$Y\xleftarrow{j}Y^{\prime}\xrightarrow{f}X$

where the object component $j_{0}\colon Y^{\prime}_{0}\to Y_{0}$ of $j$ is a
$J$-cover, and the following square is a pullback

$\textstyle{Y^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}_{0}\times
Y^{\prime}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{0}\times
j_{0}}$$\textstyle{Y_{0}\times Y_{0}}$

Of primary interest to us is the case when the groupoid $Y$ has no nontrivial
arrows, that is, it is just an object of $\mathcal{C}$, say $M$. In that case,
any functor $j\colon Y^{\prime}\to M$ satisfying the conditions is determined
by the map on objects and the groupoid $Y$ is what is known as a _Čech
groupoid_ of the covering map $j_{0}$ (or by abuse of notation, of its
domain). If we let $U=Y^{\prime}_{0}$, then $Y^{\prime}_{1}=U\times_{M}U$, and
we denote $Y^{\prime}$ by $\check{C}(U)$. Thus any anafunctor from $M$ to an
internal groupoid $X$ is of the form
$M\xleftarrow{j}\check{C}(U)\xrightarrow{f}X$.

Assume for the moment that $J$ is a singleton pretopology, so that we have
pullbacks of covering maps. Given a pair of anafunctors
$M\leftarrow\check{C}(U_{1})\xrightarrow{f}X$ and
$M\leftarrow\check{C}(U_{2})\xrightarrow{g}X$, we want to define what it means
to have a transformation between them. Let $U_{12}=U_{1}\times_{M}U_{2}$. Then
a transformation is a diagram

---
$\textstyle{\check{C}(U_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$
$\textstyle{M}$$\textstyle{\check{C}(U_{12})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\;,}$$\textstyle{\check{C}(U_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$
$\scriptstyle{\alpha}$

where the two functors $\check{C}(U_{12})\to\check{C}(U_{i})$ are induced by
the projections $U_{12}\to U_{i}$. The picture one should keep in mind here is
a coboundary between $X$-valued Čech cocycles that lives over a common
refinement.

For a singleton _coverage_ , such as the coverage $\amalg C$ of compact
neighbourhoods on manifolds, we can define a transformation to be a diagram as
above, where instead of considering the pullback, which does not necessarily
exist (or if it does, may not be a covering map), one considers a refinement
$U_{12}$, equipped with maps to $U_{1}$ and $U_{2}$. One of the lessons that
can be gleaned from [Rob16] is that when working with anafunctors nothing is
lost by considering a coverage that is cofinal in a pretopology, rather than
the pretopology itself (as in [Bar06]).

Using the notion of anafunctor with respect to a pretopology, internal
groupoids, anafunctors and transformations form a bicategory [Bar06]. We will
not use this bicategory structure directly, but it is relied on implicitly to
take advantage of Theorem 2.10 below.

### 2.3 Stacks

We are considering stacks on the category $\mathcal{M}$ of manifolds using the
coverage $O$ of open covers. A standard reference is [BX11], and we point the
reader to the detailed discussion of stacks in section 2.2 therein. We give
the definition we need and then mention without proof some standard facts.

######  Definition 2.8.

Let $\mathcal{X}\colon\mathcal{M}^{\text{op}}\to\operatorname{\bf{Gpd}}$ be a
weak 2-functor. We say $\mathcal{X}$ is a _stack_ if the following conditions
are satisfied for every covering family $\\{\phi_{i}\colon U_{i}\to M\mid i\in
I\\}$:

  1. 1.

For any pair of objects $x,y$ of $\mathcal{X}(M)$ and any family of
isomorphisms $\sigma_{i}\colon x|_{U_{i}}\to y|_{U_{i}}$ in
$\mathcal{X}(U_{i})$, $i\in I$, there is a _unique_ isomorphism $\sigma\colon
x\to y$ in $\mathcal{X}(M)$ such that $\sigma|_{U_{i}}=\sigma_{i}$.

  2. 2.

For every family of objects $x_{i}\in\mathcal{X}(U_{i})$, $i\in I$, and
collection of isomorphisms $\sigma_{ij}\colon x_{i}|_{U_{ij}}\to
x_{j}|_{U_{ij}}$ in $\mathcal{X}(U_{ij})$, $i,j\in I$ satisfying
$\sigma_{jk}\circ\sigma_{ik}=\sigma_{ik}$ in $\mathcal{X}(U_{ijk})$ (leaving
the restrictions implicit), then there is an object $x$ of $\mathcal{X}(M)$
and isomorphisms $\rho_{i}\colon x|_{U_{i}}\to x_{i}$ for all $i\in I$ such
that $\sigma_{ij}\circ\rho_{i}=\rho_{j}$ (in $\mathcal{X}(U_{ij})$) for all
$i,j\in I$ (where as usual, we write $U_{ij}=U_{i}\cap U_{j}$ and
$U_{ijk}=U_{i}\cap U_{j}\cap U_{k}$).

If only the first point is satisfied, then we say $\mathcal{X}$ is a
_prestack_.

A morphism of stacks is given by a transformation of weak 2-functors, and
there is a 2-category $\operatorname{\bf{Stack}}_{\mathcal{M}}$ of stacks on
$(\mathcal{M},O)$. The relevant points we need are as follows:

  * •

Any manifold $M$ gives rise to a stack, also denoted by $M$ (as $O$ is
subcanonical). Also, any diffeological space is a stack. The Yoneda embedding
ensures that any map of stacks between manifolds or diffeological spaces is
just a smooth map in the usual sense. A stack equivalent to a manifold is
called _representable_.

  * •

Any Lie groupoid gives rise to a _prestack_ , by sending the groupoid $X$ to
the presheaf of groupoids
$\mathcal{M}(-,X)\colon\mathcal{M}^{\text{op}}\to\operatorname{\bf{Gpd}}$, and
this prestack can be ‘stackified’. More generally, any Fréchet–Lie or
diffeological groupoid gives rise to a prestack and hence a stack.

  * •

The 2-category of stacks $\operatorname{\bf{Stack}}$ on
$(\operatorname{Subd})$ is equivalent to
$\operatorname{\bf{Stack}}_{\mathcal{M}}$. This follows from a stack version
of the ``lemme de comparaison'' [SGA4.1, Esposé III, Théorème 4.1]; see
discussion at [Car13].

The correct notion of ‘pullback’ for stacks is a _comma object_.333This is
sometimes called a weak pullback, or even just a pullback, in the stack
literature. However the definition usually given is clearly that of a comma
object. For a cospan $G\xrightarrow{f}H\xleftarrow{g}K$ of groupoids, the
comma object $G\downarrow_{H}K$ (or sometimes $f\downarrow g$) can be computed
as the _strict_ limit $G\times_{H}H^{\mathbbm{2}}\times_{H}K$ where
$H^{\mathbbm{2}}$ is the arrow groupoid of $H$. The comma object of a cospan
of stacks is calculated pointwise, that is,
$\left(\mathcal{X}\downarrow_{\mathcal{Z}}\mathcal{Y}\right)(M)=\mathcal{X}(M)\downarrow_{\mathcal{Z}(M)}\mathcal{Y}(M)$.
The comma object fits into a 2-commuting square called a _comma square_ ,

$\textstyle{\mathcal{X}\downarrow_{\mathcal{Z}}\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\mathcal{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which is universal among such 2-commuting squares.

A stack is said to be _presentable_ if it is the stackification of an internal
groupoid. In this case, there is extra structure that the stack admits, from
which we can recover the groupoid up to weak equivalence [Pro96, BX11].

First, we say a map of stacks $\mathcal{Y}\to\mathcal{X}$ is _representable_
(resp. representable by diffeological spaces) if for every manifold $M$ and
map $M\to\mathcal{X}$, the comma object $M\downarrow_{\mathcal{X}}\mathcal{Y}$
is representable by a manifold (resp. a diffeological space). We can talk
about properties of representable maps arising from properties of maps in
$\mathcal{M}$ or ; if $P$ is a property of maps of manifolds (or diffeological
spaces) that is stable under pullback and local on the target in a given
coverage $J$, then we say a representable map of stacks
$\mathcal{Y}\to\mathcal{X}$ has property $P$ if for every $M\to\mathcal{X}$
the projection $M\downarrow_{\mathcal{X}}\mathcal{Y}\to M$ has property $P$.

######  Definition 2.9.

A stack $\mathcal{X}$ on $(\mathcal{M},O)$ is _presentable_ (resp. presentable
by a diffeological groupoid) if there is a manifold (resp. diffeological
space) $X_{0}$ and a representable epimorphism $p\colon X_{0}\to\mathcal{X}$
that is a submersion (resp. subduction).

It follows from the definition that the comma object
$X_{1}:=X_{0}\downarrow_{\mathcal{X}}X_{0}$ is representable (by a manifold or
diffeological space), the two projection maps $X_{1}\to X_{0}$ are submersions
(or subductions) and $X_{1}\rightrightarrows X_{0}$ is an internal groupoid.
This internal groupoid is said to _present_ the stack $\mathcal{X}$. Then
$\mathcal{X}$ is the stackification of the prestack arising from this internal
groupoid. Note that this definition also works if we ask for presentability by
a Fréchet–Lie groupoid: one asks for a representable submersion from a Fréchet
manifold.

The usual name for a stack presentable by a Lie groupoid is _differentiable
stack_ , and we will call stacks presentable by diffeological groupoids,
_diffeological stacks_. Stacks presented by a Fréchet–Lie groupoid shall be
called _Fréchet differentiable stacks_.

The main result we need here is the following, and follows from the
combination of the general theory of [Pro96] and [Rob12, Theorem 7.2] in the
case of Lie groupoids, and uses an adaptation of Pronk’s argument for the case
of diffeological groupoids.

######  Theorem 2.10.

The 2-category of differentiable stacks (resp. diffeological stacks) is
equivalent to the bicategory of Lie groupoids (resp. diffeological groupoids),
anafunctors and transformations.

What this means in practice is that we can pass between maps between
presentable stacks and anafunctors between the presenting groupoids, and we
shall use this below.

If we have an epimorphism $p\colon X_{0}\to\mathcal{X}$ from a representable
stack $X_{0}$ such that merely the comma object
$X_{0}\downarrow_{\mathcal{X}}X_{0}$ is representable and the projections are
surjective submersions, then we call $p$ a _weak presentation_. For certain
sites a weak presentation gives a strong presentation: this is true for
instance for presentations by diffeological spaces. This relies on the
following lemma adapted from [BX11, Lemma 2.2], which works in the framework
of stacks on the site of _not-necessarily-Hausdorff_ (finite-dimensional)
manifolds.

######  Lemma 2.11.

Let $f\colon\mathcal{Y}\to\mathcal{X}$ be a morphism in
$\operatorname{\bf{Stack}}_{\mathcal{M}}$. If $M$ is a diffeological space,
$M\to\mathcal{X}$ an epimorphism of stacks, and the comma object
$M\downarrow_{\mathcal{X}}\mathcal{Y}$ is a diffeological space, then $f$ is
representable as a map of stacks considered in the equivalent 2-category
$\operatorname{\bf{Stack}}$.

The analogous result is not true for stacks on the category $\mathcal{M}$, but
it _is_ true (following [BX11]) if we allow ourselves to use possibly non-
Hausdorff manifolds. In practice, one often finds that the stack is weakly
presented by a Lie groupoid, which is made up of (Hausdorff) manifolds, which
then can be used without reference to non-Hausdorff manifolds. The same can be
said for weak presentations by Fréchet–Lie groupoids, an example of which will
arise in our main construction.

While it is not always the case that the 2-category of internal groupoids has
internal homs, the 2-category of stacks _does_ have internal homs, namely for
a pair of stacks $\mathcal{X},\mathcal{Y}$, there is a stack
$\operatorname{\underline{\mathcal{H}om}}(\mathcal{Y},\mathcal{X})$ and an
evaluation map
$\mathcal{Y}\times\operatorname{\underline{\mathcal{H}om}}(\mathcal{Y},\mathcal{X})\to\mathcal{X}$
with the necessary properties.

######  Definition 2.12.

The _Hom-stack_
$\operatorname{\underline{\mathcal{H}om}}(\mathcal{Y},\mathcal{X})$ is defined
by taking the value on the object $M$ to be the groupoid
$\operatorname{\bf{Stack}}_{\mathcal{M}}(\mathcal{Y}\times M,\mathcal{X})$.

Thus we have a Hom-stack for any pair of stacks on $\mathcal{M}$. The case we
are interested in is where we have a stack $\mathcal{X}$ associated to an
internal groupoid $X$ in $\mathcal{M}$ or , and the Hom-stack
$\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$.

## 3\. Construction of the diffeological loop groupoid

We will now describe the construction of the loop groupoid of a diffeological
groupoid $X$. This will naturally be a groupoid also internal to , and we
shall show in the next section that it in fact presents the Hom-stack
$\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$, for
$\mathcal{X}$ the stack associated to $X$.

The objects of the diffeological mapping groupoid are anafunctors
$S^{1}-\\!\\!\\!\mapsto X$, using the compact neighbourhood coverage $C$ of
Example 2.3.

As the category of diffeological spaces is cartesian closed and finitely
complete, results of Bastiani–Ehresmann [BE72] imply that the category
$\operatorname{\bf{Gpd}}()$ of diffeological groupoids is also cartesian
closed and finitely complete. Therefore the set
$\operatorname{\bf{Gpd}}()(\check{C}(V),X)_{0}$ of objects of the internal
hom—a groupoid—is in fact a diffeological space. We shall, for the sake of
saving space, write
${X}^{\check{C}(V)}:=\operatorname{\bf{Gpd}}()(\check{C}(V),X)_{0}$. The
category is also cocomplete (in fact extensive) and so we define the object
space $\operatorname{Map}(S^{1},X)_{0}$ to be the diffeological space

$\coprod_{V\in C(S^{1})}{X}^{\check{C}(V)}.$

######  Remark 3.1.

This mirrors the construction of the topological loop groupoid as in [LU02,
BGNX12], even though for the purposes of diffeological groupoids it is not
necessary to focus only on _compact_ neighbourhoods; the diffeological
groupoid of functors would exist using ordinary open covers. This would even
give an equivalent mapping diffeological groupoid in the end. However, for the
Lie groupoids in section 5 we _do_ need to use compact covers to get the
appropriate Fréchet topology on mapping spaces.

The picture we keep in mind for the elements of the object space is a sequence
like:

where the horizontal lines are paths in $X_{0}$ and the vertical arrows,
varying smoothly, are given by a path in $X_{1}$.

Next we move on to the arrow space of $\operatorname{Map}(S^{1},X)$. Recall
that a _transformation_ $t\colon f\to g$ of anafunctors $f,g\colon
S^{1}-\\!\\!\\!\mapsto X$ is a diagram

$\textstyle{\check{C}(V_{12})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{C}(V_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$
$\textstyle{\check{C}(V_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $V_{12}$ is the chosen refinement of $V_{1}\times_{S^{1}}V_{2}$ as
discussed in section 2.2. Note that $t$ is necessarily a natural isomorphism
as $X$ is a groupoid.

For arbitrary $f$ and $g$ with domains $\check{C}(V_{1})$ and
$\check{C}(V_{2})$, respectively, the diffeological space of all
transformations is

${X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}$

where the two maps

${X}^{\check{C}(V_{i})}\to{X}^{\check{C}(V_{12})}$

are given by precomposition with the canonical functors
$\check{C}(V_{12})\to\check{C}(V_{i})$. Here the groupoid $X^{\mathbbm{2}}$ is
the arrow groupoid of $X$ and we are pulling back along the maps

${(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\to{X}^{\check{C}(V_{12})}$

which are given by postcomposition with the functors $S,T\colon
X^{\mathbbm{2}}\to X$ (on objects these are the usual source and target maps).

The space of arrows $\operatorname{Map}(S^{1},X)_{1}$ is then

$\coprod_{V_{1},V_{2}\in
C(S^{1})}{X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}$

The source and target maps are projections on to the first and last factors.
These are automatically smooth maps, and are both split by the unit map and
hence are quotient maps; quotient maps in are subductions and hence the source
and target maps are subductions.

Composition of transformations of anafunctors [Bar06, Proposition 12] (or
[Rob12, Section 5] for a description closer to what is given here) is a little
involved, but is essentially induced by the composition in $X^{\mathbbm{2}}$,
which is smooth. This implies that composition in
$\operatorname{Map}(S^{1},X)$ is smooth, and hence that
$\operatorname{Map}(S^{1},X)$ is a diffeological groupoid.

## 4\. Presentation by a diffeological groupoid

For $X$ a diffeological groupoid, to give a presentation over of the Hom-stack
$\operatorname{\underline{\mathcal{H}om}}(S^{1},X)$ we will need to define a
map from some diffeological space $A$, considered as a stack, to
$\operatorname{\underline{\mathcal{H}om}}(S^{1},X)$. Such a map is determined
by a map of stacks $A\times S^{1}\to\mathcal{X}$. Since these are all stacks
arising from diffeological groupoids, this a map can be specified by
constructing an anafunctor $A\times S^{1}-\\!\\!\\!\mapsto X$ in the category
of diffeological spaces as per Theorem 2.10.

Consider then the covers $V\to S^{1}$ used in the construction of the
diffeological mapping groupoid, which are subductions since they admit local
sections over open sets. The product of subductions is again a subduction, so
we can, for each $V\in C(S^{1})$ define the anafunctor

$S^{1}\times{X}^{\check{C}(V)}\leftarrow\check{C}(V)\times{X}^{\check{C}(V)}\xrightarrow{\operatorname{ev}}X$

where the right-pointing arrow is just the evaluation map for diffeological
groupoids. This gives us, via the preceeding argument, a map

${X}^{\check{C}(V)}\to\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$

of stacks, and hence a map
$q\colon\operatorname{Map}(S^{1},X)_{0}\to\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$.

######  Proposition 4.1.

For $X$ a diffeological groupoid, the map $q$ is an epimorphism of stacks.

###### Proof.

It is enough to show that for any
$f\colon\mathbb{R}^{n}\to\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$
there is an open cover of $\mathbb{R}^{n}$ and local sections of $q$ over it.
The map $f$ is determined by a map $\mathbb{R}^{n}\times S^{1}\to\mathcal{X}$
of stacks, and hence an anafunctor ${F}\colon{\mathbb{R}^{n}\times
S^{1}-\\!\\!\\!\mapsto}X$ of diffeological groupoids. Any subduction with
codomain a manifold is refined by an open cover of the manifold, so we can
replace the anafunctor by an isomorphic one of the form $\mathbb{R}^{n}\times
S^{1}\leftarrow\check{C}(U)\to X$, where
$U=\coprod_{i}U_{i}\to\mathbb{R}^{n}\times S^{1}$ is an open cover. We can
further repeat the argument from [Noo10, Proof of Theorem 4.2] to construct an
open cover $\coprod_{j}W_{j}\to\mathbb{R}^{n}$ and for each $j$ an open cover
$V^{o}_{j}\to S^{1}$ with closure $V_{j}\to S^{1}$ an element of $C(S^{1})$.
Then the restrictions $W_{j}\times\check{C}(V_{j})\to X$ of $F$ give maps of
diffeological spaces $W_{j}\to{X}^{\check{C}(V_{j})}$, i.e. local sections of
$q$ over the open cover $\\{W_{j}\\}$. ∎

To show that the groupoid $\operatorname{Map}(S^{1},X)$ presents the Hom-
stack, we need to show that the comma object of the map $q$ with itself is the
arrow space $\operatorname{Map}(S^{1},X)_{1}$ of our diffeological groupoid.

First, we do indeed have a 2-commuting square

|
---|---
$\textstyle{\operatorname{Map}(S^{1},X)_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{t}$$\textstyle{\operatorname{Map}(S^{1},X)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$
$\textstyle{\operatorname{Map}(S^{1},X)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$
$\textstyle{\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which we can see by considering a component of the top left corner labelled by
$V_{1},V_{2}\in C(S^{1})$. The projection

$X^{\check{C}(V_{1})}\times_{X^{\check{C}(V_{12})}}(X^{\mathbbm{2}})^{\check{C}(V_{12})}\times_{X^{\check{C}(V_{12})}}X^{\check{C}(V_{2})}\to(X^{\mathbbm{2}})^{\check{C}(V_{12})}$

can be unwound to give a natural transformation between the maps $q\circ s$
and $q\circ t$.

To show the above diagram is indeed a comma square, we shall show that for any
Euclidean space $\mathbb{R}^{n}$ the diagram of groupoids

|
---|---
$\textstyle{X^{\check{C}(V_{1})}\times_{X^{\check{C}(V_{12})}}(X^{\mathbbm{2}})^{\check{C}(V_{12})}\times_{X^{\check{C}(V_{12})}}X^{\check{C}(V_{2})}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\check{C}(V_{1})}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$
$\textstyle{X^{\check{C}(V_{2})}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$
$\textstyle{\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

is a comma square. It is immediate that all but the bottom right corner are
sets, so we need to show the canonical map

$c\colon
X^{\check{C}(V_{1})}\times_{X^{\check{C}(V_{12})}}(X^{\mathbbm{2}})^{\check{C}(V_{12})}\times_{X^{\check{C}(V_{12})}}X^{\check{C}(V_{2})}(\mathbb{R}^{n})\\\
\longrightarrow
X^{\check{C}(V_{1})}\downarrow_{\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})(\mathbb{R}^{n})}X^{\check{C}(V_{2})}(\mathbb{R}^{n})$

is a bijection. These sets are as follows:

$X^{\check{C}(V_{1})}\times_{X^{\check{C}(V_{12})}}(X^{\mathbbm{2}})^{\check{C}(V_{12})}\times_{X^{\check{C}(V_{12})}}X^{\check{C}(V_{2})}(\mathbb{R}^{n})\\\
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48.77501pt\raise-35.25931pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.4416pt\raise-0.89722pt\hbox{\lx@xy<EMAIL_ADDRESS>0.89722pt\hbox{\lx@xy@drawline@}}}}\ignorespaces{\hbox{\hbox{\kern
0.4416pt\raise-0.89722pt\hbox{\lx@xy<EMAIL_ADDRESS>0.89722pt\hbox{\lx@xy<EMAIL_ADDRESS>
It is not difficult to see that $c$ must be injective; in particular $f$ and
$g$ correspond to $\tilde{f}$ and $\tilde{g}$, respectively. Unravelling the
description of $\tilde{\alpha}$ we can see it must arise from some $\alpha$ as
in the first set, and so the map is bijective, and hence
$X^{\check{C}(V_{1})}\downarrow_{\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})(\mathbb{R}^{n})}X^{\check{C}(V_{2})}$
is representable, by the component of $\operatorname{Map}(S^{1},X)_{1}$
labelled by $V_{1},V_{2}$.

Since $q$ is an epimorphism, we then see that $q$ is representable (by Lemma
2.11) and hence

######  Theorem 4.2.

For $X$ a diffeological groupoid, the Hom-stack
$\operatorname{\underline{\mathcal{H}om}}(S^{1},X)$ is presented by the
diffeological groupoid $\operatorname{Map}(S^{1},X)$.

Note that there was nothing special about $S^{1}$ in this argument: we only
required $S^{1}$ to be a manifold in order for the proof of Proposition 4.1 to
work. However for the next section the analysis is more delicate and so we
have only treated the case of $S^{1}$.

## 5\. Presentation by a Fréchet–Lie groupoid

The diffeological groupoid $\operatorname{Map}(S^{1},X)$ can also be
considered in the case that $X$ is a Lie groupoid. In this section we will
show that whenever $X$ is a Lie groupoid, the diffeological groupoid
$\operatorname{Map}(S^{1},X)$ defined in section 3 is in fact a Fréchet–Lie
groupoid (Theorem 5.9) and that it also weakly presents the Hom-stack
$\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$ over the site of
manifolds (Theorem 5.12).

From now on we will work with the coverage $C$ as in section 3 but we will
always use _minimal_ covers of $S^{1}$ (those such that triple intersections
are empty), which are cofinal in $C(S^{1})$. We denote the set of these
minimal covers by $C(S^{1})_{\text{min}}$ The object space is then

$\coprod_{V\in C(S^{1})_{\text{min}}}{X}^{\check{C}(V)},$

where again each component ${X}^{\check{C}(V)}$ is the space of (smooth)
functors $\check{C}(V)\to X$. This is naturally described as the iterated
pullback

$X_{0}^{I_{1}}\times_{X_{0}^{J_{1}}}X_{1}^{J_{1}}\times_{X_{0}^{J_{1}}}X_{0}^{I_{2}}\times_{X_{0}^{J_{2}}}X_{1}^{J_{2}}\times_{X_{0}^{J_{2}}}\ldots\times_{X_{0}^{J_{n-1}}}X_{0}^{I_{n}}$

where $I_{i}$ are closed subintervals of $S^{1}$, $V=\coprod_{i=1}^{n}I_{i}$
and $J_{i}=I_{i}\cap I_{i+1}$, the maps

$X_{0}^{I_{i}}\to X_{0}^{J_{i}}\leftarrow X_{0}^{I_{i+1}}$

are given by restriction, and the maps $X_{1}^{J_{i}}\to X_{0}^{J_{i}}$ are
induced by the source and target maps alternately. Here the $I_{i}$ and
$J_{i}$ are intervals so that a functor $\check{C}(V)\to X$ consists of a
series of paths $I_{i}\to X_{0}$ and a series of paths $J_{i}\to X_{1}$ that
“patch together using source and target”.

Recall that a pullback of a submersion in the category of Fréchet manifolds
exists, and is again a submersion. Our strategy is to show that the maps above
are all submersions, which will imply that the object space is a Fréchet
manifold.

The following result of Stacey [Sta13] guarantees that the maps
$X_{1}^{J_{i}}\to X_{0}^{J_{i}}$ are submersions; see also [AS17, Lemma 2.4].

######  Theorem 5.1 (Stacey).

Let $M\to N$ be a submersion of finite-dimensional manifolds and $K$ a compact
manifold. Then the induced map of Fréchet manifolds $M^{K}\to N^{K}$ is a
submersion.

For the maps $X_{0}^{I_{i}}\to X_{0}^{J_{i}}\leftarrow X_{0}^{I_{i+1}}$ we
will need the following theorem, which may be derived from the result in
[See64] (see also [Mit61, §7], which essentially proves Corollary 5.3
directly).

######  Theorem 5.2 (Seely).

The Fréchet space $(\mathbb{R}^{n})^{\mathbb{R}_{+}}$ is a direct summand of
$(\mathbb{R}^{n})^{\mathbb{R}}$, where we take the topology of uniform
convergence of all derivatives on compact subsets.

######  Corollary 5.3.

The Fréchet space $(\mathbb{R}^{n})^{[0,1]}$ is a direct summand of
$(\mathbb{R}^{n})^{[-1,1]}$, hence the restriction map
$(\mathbb{R}^{n})^{[-1,1]}\to(\mathbb{R}^{n})^{[0,1]}$ is a submersion of
Fréchet spaces. The same is true with $[0,1]\subset[-1,1]$ replaced with any
inclusion $J\subset I$ of compact intervals.

This allows us to prove

######  Proposition 5.4.

Let $M$ be an $n$-dimensional manifold and $J\subset I$ two compact intervals.
Then the restriction map $M^{I}\to M^{J}$ is a submersion of Fréchet
manifolds.

###### Proof.

Let $f\colon I\to M$ be a smooth function, and denote by $f_{J}\colon J\to M$
its restriction along the inclusion. To show that $M^{I}\to M^{J}$ is a
submersion, we need to find charts around $f$ and $f_{J}$ such that the map is
a submersion of Fréchet spaces on those charts. Recall [Ham82, §I.4.1] that a
chart around $f$ is a neighbourhood of the zero section in
$\Gamma(I,I\times_{f,M}TM)$, and similarly for $f_{J}$. Clearly
$I\times_{f,M}TM\simeq I\times\mathbb{R}^{n}$, and given such an isomorphism
we get an induced isomorphism $J\times_{f_{J},M}TM\simeq
J\times\mathbb{R}^{n}$ that is compatible with the restriction map. The
induced map on spaces of sections,

$(\mathbb{R}^{n})^{I}=\Gamma(I,I\times\mathbb{R}^{n})\to\Gamma(J,J\times\mathbb{R}^{n})=(\mathbb{R}^{n})^{J},$

is just the obvious restriction map, and this map is locally the same, after
unwinding the isomorphisms just given, to the restriction map. But Corollary
5.3 says that this map is a submersion, as we needed. ∎

Proposition 5.4 implies that the maps $X_{0}^{I_{i}}\to
X_{0}^{J_{i}}\leftarrow X_{0}^{I_{i+1}}$ are submersions and hence we have

######  Proposition 5.5.

For $X$ a Lie groupoid, the object space $\operatorname{Map}(S^{1},X)_{0}$ is
a Fréchet manifold.

To see that the set of arrows has a manifold structure as well, recall that
this set is given by

$\coprod_{V_{1},V_{2}\in
C(S^{1})_{\text{min}}}{X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}$

where the chosen refinement $V_{12}$ is also a minimal cover of $S^{1}$. To
use the same reasoning as above we need to know that the maps

${(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\to{X}^{\check{C}(V_{12})},$

induced by $S,T\colon X^{\mathbbm{2}}\to X$, and

${X}^{\check{C}(V_{i})}\to{X}^{\check{C}(V_{12})}$

($i=1,2$) are submersions.

Now for $M\to N$ a map of finite-dimensional manifolds, and $C\to D$ a map of
compact manifolds with boundary, the two induced maps

$M^{C}\to N^{C},\quad\text{and}\quad M^{D}\to M^{D}$

have a rather nice property in that on certain canonical charts they are
actually _linear_ maps (recall that these maps above look locally like maps
between spaces of sections induced by vector bundle maps). More generally one
can consider larger diagrams, all of whose maps have this local linearity, and
further the charts exhibiting this local behaviour can all be chosen
compatibly. Such a diagram will be called be called _locally linear_.

An example of such a diagram is one where all the objects are mapping spaces
as above, and all arrows are induced by pre- or post-composition as above. A
much simpler and familiar example would be in the finite-dimensional setting,
where the exponential map is a local diffeomorphism. The induced diagram on
tangent spaces, for any compatible system of basepoints, is then a diagram of
vector spaces.

We have the following Lemma:

######  Lemma 5.6.

Let

$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
A_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
A_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
B_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
B_{n}}$

be a diagram of submersions that is locally linear. Then the natural map

$\lim A_{i}\to\lim B_{i},$

where the limits are iterated fibre products, is also a submersion.

###### Proof.

The local linearity of the diagram means that we can find a diagram of the
same shape in the category of Fréchet _spaces_ and linear maps, and in fact
_split_ linear maps, since all of the maps are submersions, hence locally
split. Then the proof that the induced map is a split submersion of Fréchet
spaces proceeds exactly as one would in the finite-dimensional case. One can
induct on the length of the zig-zags and so reduce to the case of a diagram

$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
A_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
B_{3}}$

in the category of Fréchet spaces and linear maps and then show that one can
find a section of the linear map $A_{1}\times_{A_{2}}A_{3}\to
B_{1}\times_{B_{2}}B_{3}$. ∎

Let $X\to Y$ be a functor between Lie groupoids such that the object and arrow
components are submersions. We call such a functor _submersive_. We have the
following result.

######  Lemma 5.7.

  1. 1.

Let $X\to Y$ be a submersive functor between Lie groupoids. Then the induced
map

${X}^{\check{C}(V)}\to{Y}^{\check{C}(V)}$

is a submersion.

  2. 2.

Let $X$ be a Lie groupoid and $V_{1}\to V_{2}$ be a refinement of minimal
covers. Then the induced map

${X}^{\check{C}(V_{2})}\to{X}^{\check{C}(V_{1})}$

is a submersion.

###### Proof.

The first part follows from Theorem 5.1 and Lemma 5.6 and the second follows
from Proposition 5.4 and Lemma 5.6. ∎

Lemma 5.7 implies that the maps above are submersions and so we have

######  Proposition 5.8.

For $X$ a Lie groupoid, the arrow space $\operatorname{Map}(S^{1},X)_{1}$ is a
Fréchet manifold.

Now happily, the source and target map for our Fréchet–Lie groupoid are given,
on each component of the arrow Fréchet manifold, by the two projections

${X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\to{X}^{\check{C}(V_{i})}$

where $i=1,2$, which are submersions. Therefore

######  Theorem 5.9.

For $X$ a Lie groupoid, $\operatorname{Map}(S^{1},X)$ is a Fréchet–Lie
groupoid.

Observe that $\mathbb{L}X$ is built by taking disjoint unions of pullbacks of
smooth path spaces, and smooth path spaces are metrisable and smoothly
paracompact (as they are nuclear Fréchet spaces). By a combination of Lemma
27.9 and the comments in §27.11 of [KM97], the pullback
$M_{1}\times_{N}M_{2}$, where $M_{1},M_{2}$ are metrisable smoothly
paracompect and where at least one of $M_{i}\to N$ is a submersion, is
smoothly paracompact. Thus by induction the iterated pullback that defines
${X}^{\check{C}(V)}$ is a smoothly paracompact manifold, and so the object and
arrow manifolds of $\mathbb{L}X$ are smoothly paracompact. This means that
every open cover admits subordinate _smooth_ partitions of unity, and so any
geometric constructions with smooth objects (differential forms and so on) can
be built locally.

In fact the spaces $\mathbb{L}X_{n}$ of sequences of $n$ composable arrows are
also paracompact, so that $\mathbb{L}X$ is a paracompact groupoid in the
terminology of Gepner–Henriques. As a result we know that the fat geometric
realisation $||\mathbb{L}X||$ of the nerve of $\mathbb{L}X$ is a paracompact
space [GH07, Lemma 2.25].

The following Proposition means that the endo-2-functor on stacks on
$\mathcal{M}$ lifts to a 2-functor on _presentations_ of stacks. It is thus a
kind of rigidification of the loop stack functor.

######  Proposition 5.10.

The assignment $X\mapsto\mathbb{L}X$ extends to a 2-functor

$\mathbb{L}\colon\operatorname{\bf{Gpd}}(\mathcal{M})\to\operatorname{\bf{Gpd}}(\mathcal{F}).$

###### Proof.

Given a functor $f\colon X\to Y$ between Lie groupoids, we clearly get a
functor $\mathbb{L}f\colon\mathbb{L}X\to\mathbb{L}Y$ between Fréchet–Lie
groupoids, by composing everything in sight with $f$. Moreover, given a second
functor $k\colon Y\to Z$, we clearly have
$\mathbb{L}(kf)=\mathbb{L}k\,\mathbb{L}f$.

Assume now that we have a natural transformation $\alpha\colon f\Rightarrow
g\colon X\to Y$, or in other words a functor $X\to Y^{\mathbbm{2}}$. We need
to show that this induces a natural transformation
$\mathbb{L}f\Rightarrow\mathbb{L}g$, which is determined by the data of a
smooth map

$\operatorname{Map}(S^{1},X)_{0}\to\operatorname{Map}(S^{1},Y)_{1}.$

We first need to describe this map on the level of underlying sets. Let
$S^{1}\leftarrow\check{C}(V)\xrightarrow{h}X$ be an anafunctor. The value of
the natural transformation
$\mathbb{L}\alpha\colon\mathbb{L}f\Rightarrow\mathbb{L}g$ at $h$ is a
transformation of anafunctors

$\mathbb{L}\alpha(h)\colon(\check{C}(V)\xrightarrow{fh}Y)\Rightarrow(\check{C}(V)\xrightarrow{gh}Y)$

and so lives in the component

${Y}^{\check{C}(V)}\times_{{Y}^{\check{C}(V)}}{(Y^{\mathbbm{2}})}^{\check{C}(V)}\times_{{Y}^{\check{C}(V)}}{Y}^{\check{C}(V)}\simeq{(Y^{\mathbbm{2}})}^{\check{C}(V)}$

Moreover, the transformation $\mathbb{L}\alpha(h)$ is simply the left
whiskering of $\alpha$ by the functor $h$. Thus $\mathbb{L}\alpha$ is given
(on one component) by the map

${X}^{\check{C}(V)}\to{(Y^{\mathbbm{2}})}^{\check{C}(V)},$

induced by composition with the given $X\to Y^{\mathbbm{2}}$, hence the
component map of the natural transformation
$\mathbb{L}f\Rightarrow\mathbb{L}g$ is smooth.

Now it remains to show firstly that $\mathbb{L}\alpha$ is natural, and
secondly that this is functorial for both compositions of 2-cells. Naturality
follows from the proof that anafunctors are 1-cells in a bicategory, and that
functors are 1-cells in the locally full sub-bicategory
$\operatorname{\bf{Gpd}}(\mathcal{M})$. Functoriality follows from the fact
whiskering is a functorial process. ∎

######  Remark 5.11.

The 2-functor
$\mathbb{L}\colon\operatorname{\bf{Gpd}}(\mathcal{M})\to\operatorname{\bf{Gpd}}(\mathcal{F})$
preserves products up to weak equivalence. This follows formally using the
equivalence between differentiable stacks and Lie groupoids and anafunctors,
and the fact that the product of differentiable stacks is presented by the
product of Lie groupods. However we actually have a slightly more rigid
result, with the coherence functor (in one direction) being the canonical
inclusion

$\mathbb{L}(X\times Y)\hookrightarrow\mathbb{L}X\times\mathbb{L}Y,$

rather than some comparison anafunctor. This has a quasi-inverse _functor_
that takes a pair of objects, in summands indexed by the covers $V_{1}$ and
$V_{2}$ respectively, to the isomorphic pair indexed by the same cover
$V_{12}$, the chosen common refinement of $V_{1}$ and $V_{2}$.

Now the construction of the map $q$ from section 4 is identical, we need to
additionally show that it is a submersion. There is a small subtlety here, in
that we haven’t been able to show directly that $q$ is a representable map of
stacks, rather we will rely on (a submersion variant of) the weaker notion of
presentation from [Pro96, §6.2.0.1], which only requires that the comma object
of $q$ with itself gives a submersion between manifolds. Since we know the
comma object $q\downarrow q$ is already a manifold, namely the arrow space
$\operatorname{Map}(S^{1},X)_{1}$, and the projections are the source and
target maps, which are submersions, then we have our first main result.

######  Theorem 5.12.

For $X$ a Lie groupoid, the Hom-stack
$\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$ is weakly
presented by the Fréchet–Lie groupoid $\operatorname{Map}(S^{1},X)$.

As the stack $\operatorname{\underline{\mathcal{H}om}}(M,\mathcal{X})$ is
presented by a paracompact groupoid it is well-behaved homotopically.
Proposition 8.5 in [Noo12] ensures that since $\mathbb{L}X$ has object and
arrow manifolds metrisable,
$\operatorname{\underline{\mathcal{H}om}}(M,\mathcal{X})$ has a
_hoparacompact_ underlying topological stack. Then the classifying space of
$\operatorname{\underline{\mathcal{H}om}}(M,\mathcal{X})$ (as defined in
[Noo12]) is well-defined up to homotopy equivalence, rather than weak homotopy
equivalence.

We note that with minor modifications, one can repeat the above analysis for
the case of the mapping stack
$\operatorname{\underline{\mathcal{H}om}}([0,1],\mathcal{X})$, but we leave
that to the interested reader.

## 6\. Recap on differentiable gerbes

######  Definition 6.1.

A (Fréchet-)Lie groupoid $X\to M$ is a _gerbe_ if $\pi\colon X_{0}\to M$ and
$(s,t)\colon X_{1}\to X_{0}^{[2]}$ are surjective submersions. The stack on
$\mathcal{M}$ that such a groupoid (weakly) presents will be called a
(Fréchet-)differentiable gerbe.

Equivalently, we can require that $X\to M$ and $X\to\check{C}(X_{0})$ are
submersive functors that are surjective on objects and arrows. We rephrase
these properties in terms of functors rather than component maps because later
we wish to prove stability of these properties under forming mapping
groupoids.

######  Remark 6.2.

In this section the results also apply to general Fréchet–Lie groupoids, even
though we have only stated them for Lie groupoids for brevity.

Because $(s,t)$ is a submersion the pullback $\Lambda X:=\Delta^{*}X_{1}\to
X_{0}$, for $\Delta\colon X_{0}\to X_{0}^{[2]}$ the diagonal, is a bundle of
Lie groups. We shall call $\Lambda X$ the _inertia bundle_. If the fibre
$\Lambda X_{x}\simeq G$ for every $x\in X_{0}$ then this gives a _$G$ -gerbe_
in the sense of [LGSX09], that is, an extension of groupoids

$\Lambda X\to X\to\check{C}(X_{0}).$

However, we wish to use a mental picture as close to bundle gerbes [Mur96] as
possible, so offer the following diagram encoding a gerbe $X\to M$:

$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}^{[2]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

We have left and right actions of $\Lambda X$ on $X_{1}$, or rather a left
action of $\Lambda X_{L}:=\operatorname{pr}_{1}^{*}\Lambda X$ and a right
action of $\Lambda X_{R}:=\operatorname{pr}_{2}^{*}\Lambda X$ on $X_{1}$,
preserving the fibres of $(s,t)$, by composition in the groupoid $X$. This
makes $X_{1}\to X_{0}^{[2]}$ a principal $\Lambda X_{L}$-$\Lambda
X_{R}$-bibundle. Notice that $X_{1}$ is locally isomorphic to $\Lambda X_{L}$
and to $\Lambda X_{R}$ (as spaces over $X_{0}^{[2]}$) using local sections of
$(s,t)$.

There is also an action of $X$ as a groupoid on the family $\Lambda X\to
X_{0}$, covering the action of $X$ on $X_{0}$. This is by conjugation in the
groupoid: if $f\colon x\to y\in X_{1}$ and $\alpha\in\Lambda X_{x}$, then
$f^{-1}\alpha f\in\Lambda X_{y}$, where we are using the diagrammatic (or
algebraic) composition order. This defines a smooth map

$\Lambda X\times_{X_{0},s}X_{1}\to\Lambda X$

over $X_{0}$, using the target map composed with the second projection on the
domain. We also want to think of this in the equivalent form of

$\Lambda X_{L}\times_{X_{0}^{[2]}}X_{1}\to\Lambda X_{R},$

a map over $X_{0}^{[2]}$. This action defines an action groupoid $\Lambda
X/\\!/X$ with objects $\Lambda X$ and morphisms $\Lambda
X_{L}\times_{X_{0}^{[2]}}X_{1}$. This groupoid will become important for
calculations in the next section. We will denote an object and an arrow of
$\Lambda X/\\!/X$ by

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29.85764pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{y}$}}}}}}}\ignorespaces}}}}\ignorespaces\;,$

respectively. The action of $X$ on $\Lambda X$, that is, the target map of
$\Lambda X/\\!/X$, is

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}}\ignorespaces\ignorespaces{\hbox{\kern-29.56981pt\raise
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0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\kern
41.26428pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}}\ignorespaces\ignorespaces{\hbox{\kern-41.26428pt\raise
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0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces$
(1)

For the purposes of being confident that various pullbacks exist in what
follows, we record some trivial consequences of the conditions on the
definition of a gerbe. Note that the surjectivity requirements are superfluous
at this point, but will become important later.

######  Lemma 6.3.

For a Lie groupoid $X$ with submersive functors $X\to M$ and
$X\to\check{C}(X_{0})$, the following functors are also submersive:

  1. 1.

$(S,T)\colon X^{\mathbbm{2}}\to X\times_{M}X$

  2. 2.

$\operatorname{pr}_{i}\colon X\times_{M}X\to X$ for $i=1,2$

  3. 3.

$S,T\colon X^{\mathbbm{2}}\to X$

While there may be some utility in maintaining extra generality at this point,
our results will ultimately be applied in the case that $\Lambda X$ is a
bundle of _abelian_ Lie groups. Thus from now on we make this assumption. Note
however that an _abelian gerbe_ in the sense of [Bre94, Definition 2.9] is
more restrictive than simply demanding $\Lambda X_{x}$ is abelian for every
$x\in X_{0}$. We will get to this type of gerbe soon (see Definition 6.6
below)

We are also interested primarily in the case that $\Lambda X\to X_{0}$
descends to $M$. This means that there is an isomorphism

$\phi\colon\Lambda X_{L}\xrightarrow{\sim}\Lambda X_{R}$

over $X_{0}^{[2]}$ which satisfies the cocycle condition over $X_{0}^{[3]}$.
We will refer to $\phi$ as the _descent isomorphism_ for $\Lambda X$. We can
denote this isomorphism by

$\raisebox{17.22217pt}{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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where $(x,y)\in X_{0}\times_{M}X_{0}$. There is then a bundle of groups
$\mathcal{A}\to M$ such that $\pi^{*}\mathcal{A}\simeq\Lambda X$. Another way
to phrase this is that there is an action $\Lambda
X\times_{X_{0},\operatorname{pr}_{1}}X_{0}^{[2]}=\Lambda X_{L}\to\Lambda X$ of
the groupoid $\check{C}(X_{0})$ on $\Lambda X$, and hence we have an action
groupoid $\Lambda X/\\!/\check{C}(X_{0})$ with arrows $\Lambda X_{L}$. This
has a projection map to $\check{C}(X_{0})$ making $\Lambda
X/\\!/\check{C}(X_{0})\to\check{C}(X_{0})$ a bundle of groups object in the
category of Lie groupoids.

######  Lemma 6.4.

If $\Lambda X$ descends to $\mathcal{A}$ on $M$, the following square is a
pullback of Lie groupoids

$\textstyle{\Lambda
X/\\!/\check{C}(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{C}(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\,.}$

###### Proof.

We can verify this by looking at the level of objects and arrows,
individually. The object manifold of $\Lambda X/\\!/\check{C}(X_{0})$ is
$\Lambda X$, and by assumption this is isomorphic to
$X_{0}\times_{M}\mathcal{A}$, as needed. The square at the level of arrow
manifolds is

$\textstyle{\Lambda
X\times_{X_{0},\operatorname{pr}_{1}}(X_{0}\times_{M}X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\times_{M}X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$

and so we need to show that the induced map

$\Lambda
X\times_{X_{0},\operatorname{pr}_{1}}(X_{0}\times_{M}X_{0})\to(X_{0}\times_{M}X_{0})\times_{M}\mathcal{A}$
(2)

is an isomorphism. But $\Lambda X\simeq X_{0}\times_{M}\mathcal{A}$, so (2) is
just the canonical isomorphism rearranging the factors of a iterated pullback.
∎

We can hence talk about _$\mathcal{A}$ -gerbes on $M$_ for a fixed bundle of
abelian groups $\mathcal{A}\to M$, and we will restrict attention to this case
from now on. The bundle $\mathcal{A}$ will be referred to as the _structure
group bundle_.

To go further and talk about _abelian_ $\mathcal{A}$-gerbes we need to say
what it means for the left and right actions of $\Lambda X_{L}$ and $\Lambda
X_{R}$ on $X_{1}$ to agree. In the special case that $\Lambda X=X_{0}\times
A$, then $\Lambda X_{L}=X_{0}^{[2]}\times A=\Lambda X_{R}$, and we could ask
that the $A$-$A$-bibundle $X_{1}$ is in fact just an $A$-bundle, with the
right action equal to the left action.

In the case that $\Lambda X=\pi^{*}\mathcal{A}$ is non-trivial, the best we
can do is identify $\Lambda X_{L}$ with $\Lambda X_{R}$ via the given descent
isomorphism, and ask that _relative to this identification_ , the left and
right actions agree. There are two ways to look at this agreement, from the
point of view of the actions of $\Lambda X_{L}$, $\Lambda X_{R}$ on $X_{1}$,
or the action of $X$ on $\Lambda X$. However, we want to also introduce a
third way, that uses a more global, groupoid-based approach to be used in the
next section.

Recall that $X^{\mathbbm{2}}$ is the arrows of a groupoid object in Lie
groupoids—that is, a double groupoid. There is a groupoid action _in the
category of Lie groupoids_

$\Lambda X/\\!/X\times_{X,S}X^{\mathbbm{2}}\to\Lambda X/\\!/X$

with the object component of this functor given by equation (1). The arrow
component is given by

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We remind the reader that here notation for the conjugation action is using
the diagrammatic order for composition.

While this seems to iterate our data to another level of complexity, this
allows us to consider stability of structures under the functor

$(-)^{\check{C}(V)}\colon\operatorname{\bf{Gpd}}(\mathcal{M})\to\mathcal{F}.$

In particular, since $(-)^{\check{C}(V)}$ preserves products and even
pullbacks of submersive functors, for a bundle of groups $\mathcal{G}\to X$ in
$\operatorname{\bf{Gpd}}(\mathcal{M})$ (considered as a 1-category),
$\mathcal{G}^{\check{C}(V)}\to X^{\check{C}(V)}$ is a bundle of Fréchet–Lie
groups. This will allow a calculation of the structure group bundle of the
(putative) gerbe $\mathbb{L}X$, once we prove that it is in fact a gerbe.

######  Lemma 6.5.

For $X$ an $\mathcal{A}$-gerbe, the following are equivalent:

  1. 1.

The diagram

---
$\textstyle{\Lambda
X_{L}\times_{X_{0}^{[2]}}X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\phi\times\operatorname{id}_{X_{1}}}$$\textstyle{\Lambda
X\times_{X_{0},s}X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}}$$\textstyle{\Lambda
X_{R}\times_{X_{0}^{[2]}}X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{X_{1}\times_{t,X_{0}}\Lambda
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

sitting over $X_{0}^{[2]}$ commutes (“the right and left actions of $\Lambda
X$ on $X_{1}$ agree”);

  2. 2.

The conjugation action of $X$ on $\Lambda X$ factors through the action of
$\check{C}(X_{0})$ on $\Lambda X$, via the projection $X\to\check{C}(X_{0})$;

  3. 3.

The action of $X^{\mathbbm{2}}$ on $\Lambda X/\\!/X$ factors through an action

$\Lambda X/\\!/X\times_{X,\operatorname{pr}_{1}}(X\times_{M}X)\to\Lambda
X/\\!/X,$ (3)

of the double groupoid $X\times_{M}X\rightrightarrows X$ on $\Lambda X/\\!/X$,
in the category of Lie groupoids, whose object component is the descent
isomorphism $\phi$ for $\Lambda X$, via the functor $(S,T)\colon
X^{\mathbbm{2}}\to X\times_{M}X$.

###### Proof.

We will first prove that 1. and 2. are equivalent. The implication
3.$\Rightarrow$2\. is immediate because 2. is merely the object component of
3. We will then show how 3. follows from 2.

The diagram in 1. commuting means that for all $(\alpha,f)\in\Lambda
X_{L}\times_{X_{0}^{[2]}}X_{1}$, $\alpha f=f\phi(\alpha)$. In other words,
that $\phi(\alpha)=f^{-1}\alpha f$, but this is precisely what it means for
the action of $X$ on $\Lambda X$ to factor through the action of
$\check{C}(X_{0})$ on $\Lambda X$, and so 1.$\Leftrightarrow$2.

To prove that 2. implies 3., we need first to describe an action as in (3)
with object component $\Lambda X_{L}=\Lambda
X\times_{X_{0},\operatorname{pr}_{2}}X_{0}^{[2]}\xrightarrow{\phi}\Lambda
X_{R}\xrightarrow{\operatorname{pr}}\Lambda X$. If 2. holds then

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(4)

for any $f\colon x\to y\in X_{1}$. Hence we can define the arrow component of
(3) by

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which is indeed a functor by virtue of (4), and the fact it is an action
follows from the cocycle identity for $\phi$. The action of $X^{\mathbbm{2}}$
on $\Lambda X/\\!/X$ factors through this action by construction.

The arrow component of the action in 3. is in fact determined uniquely, rather
than merely being ‘an’ action, since 3. implies 2. and then one can construct
the required arrow component of the action functor. ∎

Note that in particular that if the conditions of the lemma are satsified
there is a functor $\Lambda X/\\!/X\to\Lambda X/\\!/\check{C}(X_{0})$ sitting
over $X\to\check{C}(X_{0})$.

######  Definition 6.6.

We call an $\mathcal{A}$-gerbe $X\to M$ _abelian_ if the equivalent conditions
of Lemma 6.5 hold.

######  Lemma 6.7.

For an abelian $\mathcal{A}$-gerbe $X\to M$, the left and hence all squares
below are pullbacks of Lie groupoids

$\textstyle{\Lambda
X/\\!/X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda
X/\\!/\check{C}(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{C}(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\,.}$

In particular, $\Lambda X/\\!/X\to X$ is a bundle of groups object in the
category of Lie groupoids.

###### Proof.

Since $X$ is an abelian $\mathcal{A}$-gerbe, and hence an $\mathcal{A}$-gerbe,
the right square is a pullback by Lemma 6.4. By the pullback pasting lemma,
the left square is a pullback if and only if the outer rectangle is a
pullback; we shall prove the former. The object components of the top and
bottom horizontal functors in the left square are identity maps
$\operatorname{id}_{\Lambda X}$ and $\operatorname{id}_{X_{0}}$ respectively,
and the left and right vertical maps are both $\Lambda X\to X_{0}$, hence on
objects the left square is a pullback. The morphism components of the left
square give the square

$\textstyle{\Lambda
X_{L}\times_{X_{0}^{[2]}}X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\times(s,t)}$$\scriptstyle{\operatorname{pr}_{2}}$$\textstyle{\Lambda
X_{L}\times_{X_{0}^{[2]}}X_{0}^{[2]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{2}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(s,t)}$$\textstyle{X_{0}^{[2]}\,,}$

which is manifestly a pullback. ∎

######  Example 6.8.

Let $A$ be an abelian Lie group. An $A$-bundle gerbe on $M$ in the sense of
[Mur96] is an abelian $M\times A$ gerbe $X\to M$. Most often one just
considers the case444There is also a version of bundle gerbes where $X_{1}\to
X_{0}^{[2]}$ is a _line bundle_ , rather than a principal bundle. This is
captured in our framework if we allow for Lie groupoids that are enriched over
a monoidal category of smooth objects, in this case the category
$\operatorname{Lines}_{\mathbb{C}}$ of complex lines with the usual tensor
product. Asking that $X$ is enriched over $\operatorname{Lines}_{\mathbb{C}}$
in this internal setting is nothing other than asking that $X_{1}\to
X_{0}\times X_{0}$ is a line bundle over its image.  of $A=U(1)$ or
$\mathbb{C}^{\times}$. Note that the local triviality of the $A$-bundle
$X_{1}\to X_{0}\times_{M}X_{0}$ follows from the rest of the definition, as it
is a surjective submersion, hence has local sections, and has an action by $A$
that is free and transitive on fibres.

######  Example 6.9.

In [HMSV13] the second-named author and collaborators considered ‘bundle
gerbes with non-constant structure group bundle’. This is a case intermediate
between gerbes as defined here and ordinary bundle gerbes as in [Mur96],
requiring that $\mathcal{A}$ is a locally trivial bundle of groups, and
$X_{1}\to X_{0}^{[2]}$ is locally trivial in a way compatible with the induced
local trivialisations of $\pi^{*}\mathcal{A}$. The main nontrivial example of
[HMSV13] is however infinite-dimensional, meaning the results of the present
paper can only be applied if we consider it as a diffeological groupoid.

In fact, assuming $\mathcal{A}$ is a locally trivial bundle of groups has
consequences for the structure of abelian $\mathcal{A}$-gerbes.

######  Lemma 6.10.

Let $\mathcal{A}\to M$ be a locally trivial bundle of abelian groups. Then for
any abelian $\mathcal{A}$-gerbe $X\to M$, the map $(s,t)\colon X_{1}\to
X_{0}\times_{M}X_{0}$ is a locally trivial bundle.

###### Proof.

The locally trivial bundles of groups $\Lambda X_{L}$ and $\Lambda
X_{R}$—pullbacks of $\mathcal{A}$—act principally on $X_{1}$ (that is: freely,
and transitively on the fibres of $(s,t)$), and $(s,t)$ admits local sections
as it is a submersion. From these local sections and local trivialisations of,
say $\Lambda X_{L}$, we can construct local trivialisations of $X_{1}\to
X_{0}\times_{M}X_{0}$ ∎

We end with a final technical lemma used in the next section, but of
independent interest.

######  Lemma 6.11.

The bundle of groups $\Lambda X/\\!/X\to X$ (internal to Lie groupoids) is the
pullback of $(S,T)\colon X^{\mathbbm{2}}\to X\times X$ along the diagonal
$\Delta\colon X\to X\times X$.

###### Proof.

On the level of objects this says that $\Lambda X$ is the pullback of $X_{1}$
along $X_{0}\to X_{0}\times X_{0}$, which is true by definition. The arrow
manifold of $X^{\mathbbm{2}}$ can be described as the pullback of $(s,t)\colon
X_{1}\to X_{0}^{2}$ along $(s,s)\colon X_{1}^{2}\to X_{0}^{2}$. In this
description, the (arrow component of the) source functor $S$ projects on the
first factor of $X_{1}^{2}$, and the (arrow component of the) target functor
$T$ projects on the other factor. Thus the pullback of
$X_{1}\times_{X_{0}^{2}}X_{1}^{2}$ along the diagonal $X_{1}\to X_{1}\times
X_{1}$ forces the last two components to be equal, and hence that the middle
factor must be $\Lambda X$, and the pullback is $X_{1}\times_{s,X_{0}}\Lambda
X$ which is the arrow manifold of $\Lambda X/\\!/X$. ∎

## 7\. The loop stack of a gerbe

This section shows that given a differentiable gerbe $\mathcal{X}$ on a
manifold $M$ presented by a Lie groupoid $X$ satisfying (a) a connectedness
property for its automorphism groups $X(x,x)$ and (b) a weak form of local
triviality of $X_{1}\to X_{0}\times_{M}X_{0}$; then the loop stack is again a
(Fréchet) differentiable gerbe. An example of such a groupoid is a bundle
gerbe (see below), in which case $(s,t)$ is the projection map for a principal
bundle.

In the following, denote $\operatorname{Map}(S^{1},X)$ by $\mathbb{L}X$. We
will also denote $(\mathbb{L}X)_{i}$, i.e. the object and arrow manifolds,
simply by $\mathbb{L}X_{i}$.

######  Proposition 7.1.

Let $X$ be a Lie groupoid with a submersive functor
$X\to\operatorname{disc}(M)$ such that the resulting map $X_{1}\to
X_{0}\times_{M}X_{0}$ is a submersion. Then
$\mathbb{L}X\to\operatorname{disc}(LM)$ is submersive and
$(s,t)^{\mathbb{L}X}\colon\mathbb{L}X_{1}\to\mathbb{L}X_{0}\times_{LM}\mathbb{L}X_{0}$
is a submersion.

Note that we do not need to assume that $X$ presents a gerbe on $M$, so that
the result will be applicable to more general bundles of groupoids, in
particular those whose fibres are not necessarily transitive.

###### Proof.

Firstly, as $X\to\operatorname{disc}(M)$ is submersive we have the composite
map ${X}^{\check{C}(V)}\to{\operatorname{disc}(M)}^{\check{C}(V)}\to LM$ a
submersion (Lemma 5.7 parts 1 and 3), and so applying Lemma 5.6 we get that
$\mathbb{L}X_{0}\to LM$ is a submersion. Thus we know
$\mathbb{L}X_{0}\times_{LM}\mathbb{L}X_{0}$ is a Fréchet manifold.

The crux of the proof to show $(s,t)^{\mathbb{L}X}$ is a submersion is in
finding an isomorph555An _isomorph_ of a map $f\colon A\to B$ is a map
$g\colon A^{\prime}\to B^{\prime}$ such that there are isomorphisms $A\simeq
A^{\prime}$ and $B\simeq B^{\prime}$ making the resulting square commute. It
is obvious that the isomorph of a submersion is a submersion. of the map
$(s,t)^{\mathbb{L}X}$ in such a way that Lemmata 5.6 and 5.7 can be applied.

Firstly notice that we can work with $(s,t)^{\mathbb{L}X}$ over each component
of its domain and codomain, which are indexed by pairs $V_{1},V_{2}$ of covers
of $S^{1}$. This is because the disjoint union of submersions is again a
submersion. Hence we are only dealing with the map

${X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\longrightarrow{X}^{\check{C}(V_{1})}\times_{LM}{X}^{\check{C}(V_{2})}$
(5)

which is projection on the first and third factors of the domain. There are
isomorphisms

$\displaystyle{X}^{\check{C}(V_{1})}\times_{LM}{X}^{\check{C}(V_{2})}$
$\displaystyle\simeq{X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}\left({X}^{\check{C}(V_{12})}\times_{LM}{X}^{\check{C}(V_{12})}\right)\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}$
$\displaystyle\simeq{X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X\times_{M}X)}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}$

which arise from the isomorphisms

${X}^{\check{C}(V_{12})}\times_{LM}{X}^{\check{C}(V_{12})}\simeq{(X\times_{M}X)}^{\check{C}(V_{12})}$

and ${\operatorname{disc}(M)}^{\check{C}(V_{12})}\simeq LM.$

Now we have the following isomorph of (5):

$\textstyle{{X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X\times_{M}X)}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}}$
(6)

which is the map induced from the map

${(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\longrightarrow{(X\times_{M}X)}^{\check{C}(V_{12})}$
(7)

by interated pullback. This is, in turn, induced by applying the functor
${(-)}^{\check{C}(V_{12})}$ to the internal functor

$(S,T)\colon X^{\mathbbm{2}}\to X\times_{M}X,$

which is submersive by Lemma 6.3. We can then apply Lemma 5.7.1 to see that
the map (7) is a submersion.

Now notice that we can apply Lemma 5.7.2 to the maps
${X}^{\check{C}(V_{i})}\to{X}^{\check{C}(V_{12})}$ ($i=1,2$) to see they are
submersions. It also follows from Lemma 6.3 together with Lemma 5.7 that the
two maps ${(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\to{X}^{\check{C}(V_{12})}$
induced by $S,T\colon X^{\mathbbm{2}}\to X$, and the two maps
${(X\times_{M}X)}^{\check{C}(V_{12})}\to{X}^{\check{C}(V_{12})}$ induced by
the two projections are submersions. Now we can apply Lemma 5.6, as the
diagram giving the iterated pullback defining the map (6) to get the desired
result, namely that (6) is a submersion. ∎

Let us say a gerbe has _connected stabilisers_ if $\Lambda X\to X_{0}$ is a
bundle of connected groups. It then follows that $X_{1}\to
X_{0}\times_{M}X_{0}$ has connected fibres, and the group $\Lambda X_{x}$ acts
simply transitively on all fibres $(s,t)^{-1}(x,y)$.

Call a submersion $E\to B$ _curvewise trivial_ if for every map
$\eta\colon[a,b]\to B$, the projection $\eta^{*}E\to[a,b]$ is isomorphic to a
trivial bundle $[a,b]\times F\to[a,b]$. For a gerbe $X$ that has connected
stabilisers, if $(s,t)$ is curvewise trivial then the manifold $F$ is
connected.

A gerbe $X$ that has $(s,t)$ curvewise trivial satisfies the property that a
lift, as shown in the diagram

$\textstyle{[a,c]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(s,t)}$$\textstyle{[a,b]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\times_{M}X_{0}}$

always exists, for $c\in[a,b)$. If the gerbe additionally has connected
stabilisers, then there is always a lift as in this diagram:

$\textstyle{[a,c]\coprod[d,b]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(s,t)}$$\textstyle{[a,b]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\times_{M}X_{0}}$

for $a<c<d<b$. Both of these follow from the ability to extend functions
$[a,c]\to F$ (respectively $[a,c]\coprod[d,b]\to F$) to $[a,b]$, using
Corollary 5.3 (and the fact $F$ is connected in the latter case).

######  Lemma 7.2.

Let $M$ be a finite-dimensional manifold and $X$ a Lie groupoid that is a
gerbe on $M$ with $(s,t)$ curvewise trivial, then $\mathbb{L}X_{0}\to LM$ is a
surjective submersion. If additionally $X$ has connected stabilisers then
$(s,t)^{\mathbb{L}X}\colon\mathbb{L}X_{1}\to\mathbb{L}X_{0}\times_{LM}\mathbb{L}X_{0}$
is a surjective submersion.

###### Proof.

For the first statement, note that it is immediate that
$\mathbb{L}\check{C}(X_{0})\to LM$ is surjective, because since $X_{0}\to M$
is a surjective submersion, it has local sections which can be used to lift
locally any loop $\gamma\colon S^{1}\to M$. Then to show that
$\mathbb{L}X\to\mathbb{L}\check{C}(X_{0})$ is surjective, we need to use the
first assumption on $(s,t)$.

Note that we only need to show we can lift paths $[a,b]\to
X_{0}\times_{M}X_{0}$ through $(s,t)\colon X_{1}\to X_{0}\times_{m}X_{0}$,
where $[a,b]\subset\check{C}(V)_{1}$; there are no compatibility conditions.
But note that since $X_{1}$ trivialises after pulling back to $[a,b]$, one can
just use a section to lift paths as needed. Thus
$\mathbb{L}X\to\mathbb{L}\check{C}(X_{0})$ is surjective, and so the first
claim follows.

For the second claim we only need to prove that $(s,t)^{\mathbb{L}X}$ is
surjective (it is already a submersion), so consider a single component
${X}^{\check{C}(V_{1})}\times_{LM}{X}^{\check{C}(V_{2})}\subset\mathbb{L}X_{0}\times_{LM}\mathbb{L}X_{0}$.
It suffices to prove that
${(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\to{X}^{\check{C}(V_{12})}\times_{LM}{X}^{\check{C}(V_{12})}\simeq{(X\times_{M}X)}^{\check{C}(V_{12})}$
is surjective, since $(s,t)^{\mathbb{L}X}$ is a disjoint union of pullbacks of
such maps. Write $V=V_{12}$, and consider
$\gamma=(\gamma_{1},\gamma_{2})\colon\check{C}(V)\to X\times_{M}X$. We need to
find a lift $\widehat{\gamma}$ as in the diagram:

$\textstyle{X^{\mathbbm{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{C}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\widehat{\gamma}}$$\textstyle{X\times_{M}X}$

We will iterate through the connected components of $V$ to define
$\widehat{\gamma}$ on both objects and arrows. The functor $\gamma$ has an
underlying object component a map $\coprod_{i=0}^{n}J_{i}\to X_{0}^{[2]}$, and
starting with $J_{0}=[a_{0},b_{0}]$ we can find an arrow
$b\colon\gamma_{1}(a_{0})\to\gamma_{2}(a_{0})\in X_{1}$. This uses the fact
$X_{1}\to X_{0}^{[2]}$ is surjective. Since $(s,t)$ is curvewise trivial, we
can find a section over $J_{0}$, and hence a map $J_{0}\to
X_{1}=(X^{\mathbbm{2}})_{0}$.

If we denote $J_{i-1}\cap J_{i}$ by $J^{i-1}_{i}$ (working mod $n+1$), then by
naturality the object component $\widehat{\gamma}_{0}$ of the lift $\gamma$ on
$J^{i-1}_{i}\subset J_{i}$ is determined by its value on $J^{i-1}_{i}\subset
J_{i-1}$. By this we mean that for $\widehat{\gamma}$ to be a functor to
$X^{\mathbbm{2}}$, or in other words a natural transformation
$\gamma_{1}\Rightarrow\gamma_{2}$, it must for every point in
$J^{i-1}_{i}\subset\check{C}(V)_{1}$ satistfy naturality. Thus from the lift
on $J_{0}$ we can define the lift on $J^{0}_{1}\subset J_{1}$, and then again
use the fact $(s,t)$ is curvewise trivial to continue the lift on the rest of
$J_{1}$.

So starting from $J_{0}$ we can work through the indexing set for the cover
until we have defined $\widehat{\gamma}_{0}$ on $J_{n-1}$, and hence on
$J^{n-1}_{n}\subset J_{n}$. The first lift, on $J_{0}$ defines
$\widehat{\gamma}_{0}$ on $J^{n}_{0}\subset J_{n}$, and so we need to be able
to define a map $J_{n}\to X_{1}$ extending both of these partial maps. This is
the situation as in Figure 2, where we need to define the dotted portion of
upper central arc.

$\widehat{\gamma}_{0}(J_{0})$$\widehat{\gamma}_{1}(J_{0}^{n})$$\widehat{\gamma}_{0}(J_{0}^{n})$$\widehat{\gamma}_{0}(J_{n}^{n-1})$$\widehat{\gamma}_{1}(J_{n}^{n-1})$$\widehat{\gamma}_{0}(J_{n-1})$
Figure 2: Defining a lift to $X^{\mathbbm{2}}$

It is here we use the hypothesis that $X$ has connected stabilisers, since if
we pull back $X_{1}\to X_{0}^{[2]}$ along $\gamma\big{|}_{J_{n}}$, we can
trivialise to $J_{n}\times A$. Then $A$ is necessarily a connected Lie group,
so we can extend the map $J_{n}\supset J^{n-1}_{n}\coprod J^{n}_{0}\to A$ to
all of $J_{n}$, completing the lift $\widehat{\gamma}\colon\check{C}(V)\to
X^{\mathbbm{2}}$, and the proof. ∎

Thus we get the first main result of this section.

######  Theorem 7.3.

For a differentiable gerbe $\mathcal{X}$ presented by a Lie groupoid $X$ with
connected stabilisers such that $(s,t)$ is curvewise trivial, then
$\operatorname{\underline{\mathcal{H}om}}(S^{1},\mathcal{X})$ is a Fréchet
differentiable gerbe.

We have the following additional result if we know some more about the gerbe
$X$.

######  Proposition 7.4.

Let $X\to M$ be an abelian $\mathcal{A}$-gerbe where $\mathcal{A}$ is a
locally trivial bundle of connected (abelian Lie) groups. Then
$\Lambda\mathbb{L}X\simeq\mathbb{L}X_{0}\times_{LM}L\mathcal{A}$.

###### Proof.

The assumptions on $X$ mean that $\mathbb{L}X$ is a gerbe. The definition of
$\Lambda\mathbb{L}X$ is that it is the pullback

$\textstyle{\Lambda\mathbb{L}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(s,t)^{\mathbb{L}X}}$$\textstyle{\mathbb{L}X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{(\mathbb{L}X_{0})^{[2]}}$

and on the component ${X}^{\check{C}(V)}\subset\mathbb{L}X_{0}$ this is
precisely the pullback

$\textstyle{(\Lambda\mathbb{L}X)_{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{(X^{\mathbbm{2}})}^{\check{C}(V)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{X}^{\check{C}(V)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{X}^{\check{C}(V)}\times_{LM}{X}^{\check{C}(V)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{(X\times_{M}X)}^{\check{C}(V)}\,.}$

But since ${(-)}^{\check{C}(V)}$ preserves strict pullbacks of submersive
functors, we have
$(\Lambda\mathbb{L}X)_{V}\simeq{(X\times_{X\times_{M}X}X^{\mathbbm{2}})}^{\check{C}(V)}$.
By Lemma 6.11, $X\times_{X\times_{M}X}X^{\mathbbm{2}}\simeq\Lambda X/\\!/X$,
and since $X$ presents an abelian $\mathcal{A}$-gerbe, $\Lambda
X/\\!/X\simeq\mathcal{A}\times_{M}X$, by Lemma 6.7. Thus the summand
$(\Lambda\mathbb{L}X)_{V}$ of $\Lambda\mathbb{L}X$ over ${X}^{\check{C}(V)}$
is isomorphic to ${(\mathcal{A}\times_{M}X)}^{\check{C}(V)}\simeq
L\mathcal{A}\times_{LM}{X}^{\check{C}(V)}$ (where we have implicitly
identified ${M}^{\check{C}(V)}$ with $LM$ and similarly for $L\mathcal{A}$). ∎

This gives us the final main result, and in fact the original motivation for
this paper.

######  Theorem 7.5.

Let $M$ be a finite-dimensional smooth manifold, $\mathcal{A}$ be a locally
trivial bundle of connected abelian Lie groups on $M$ and $X\to M$ a finite-
dimensional abelian $\mathcal{A}$-gerbe. Then $\mathbb{L}X$ is an abelian
$L\mathcal{A}$-gerbe on $LM$.

###### Proof.

Theorem 7.3 ensures that $\mathbb{L}X$ is again a gerbe, and from Proposition
7.4 we know that $\Lambda\mathbb{L}X$ descends to a bundle of groups
$L\mathcal{A}\to LM$. Hence we know $\mathbb{L}X$ is an $L\mathcal{A}$-gerbe,
and we thus need to show that $\mathbb{L}X$ is an _abelian_ gerbe. This will
be done by showing condition 2 of Lemma 6.5 holds for $\mathbb{L}X$, given
that condition 3 of Lemma 6.5 holds for $X$. That is, we need to show the
diagram

$\textstyle{\Lambda\mathbb{L}X\times_{\mathbb{L}X_{0}}\mathbb{L}X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{\Lambda\mathbb{L}X}\times(s,t)^{\mathbb{L}X}}$$\textstyle{\Lambda\mathbb{L}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda\mathbb{L}X\times_{\mathbb{L}X_{0}}\mathbb{L}X_{0}\times_{LM}\mathbb{L}X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda\mathbb{L}X}$

commutes. This reduces (using Lemma 6.11) to showing the following diagram
commutes, for all $V_{1},V_{2}$:

$\textstyle{{(\Lambda
X/\\!/X)}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{act}_{\mathbb{L}}}$$\scriptstyle{\operatorname{pr}_{124}}$$\scriptstyle{?}$$\textstyle{{(\Lambda
X/\\!/X)}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{(\Lambda
X/\\!/X)}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{1})}}\left({X}^{\check{C}(V_{1})}\times_{LM}{X}^{\check{C}(V_{2})}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{(\Lambda
X/\\!/X)}^{\check{C}(V_{2})}}$

Using the isomorphism ${(\Lambda
X/\\!/X)}^{\check{C}(V)}\simeq{(\mathcal{A}\times_{M}X)}^{\check{C}(V)}\simeq
L\mathcal{A}\times_{LM}{X}^{\check{C}(V)}$, we can rewrite the desired diagram
as

$\textstyle{L\mathcal{A}\times_{LM}{X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{14}}$$\scriptstyle{\operatorname{pr}_{124}}$$\textstyle{L\mathcal{A}\times_{LM}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\mathcal{A}\times_{LM}{X}^{\check{C}(V_{1})}\times_{LM}{X}^{\check{C}(V_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{13}}$$\textstyle{L\mathcal{A}\times_{LM}{X}^{\check{C}(V_{2})}}$

in other words, we need to prove that the action map
$\operatorname{act}_{\mathbb{L}}$ above (defined using conjugation of
transformations of anafunctors) is, up to isomorphism, the projection