# Diffeological Principal Bundles and Principal Infinity Bundles

Emilio Minichiello

###### Abstract.

Given a diffeological group $G$ we prove that the nerve of the groupoid of
diffeological principal $G$-bundles over a diffeological space $X$ is weakly
equivalent to the nerve of the category of $G$-principal $\infty$-bundles over
$X$. Using this we show that the Čech cohomology groups of $X$ as defined in
[Igl20] coincides with the corresponding cohomology groups defined using
higher topos theory.

###### Contents

  1. 1 Introduction
  2. 2 Diffeological Spaces
    1. 2.1 Diffeological Spaces as Concrete Sheaves
  3. 3 Stacks
  4. 4 Simplicial Presheaf Theory
  5. 5 Cofibrant Replacement of Diffeological Spaces
  6. 6 Principal Infinity Bundles and Categories of Fibrant Objects
  7. 7 Principal Infinity Bundles and Diffeological Principal Bundles
  8. 8 Comparison to Čech Cohomology
    1. 8.0.1 Comparing Diffeological Čech cohomologies

##  1\. Introduction

Principal bundles and Čech cohomology are important tools in the study of
smooth manifolds. However, in recent years, the desire to expand the typical
objects of study in differential geometry has led to various frameworks in
which one can define a ”generalized smooth space.” In this paper, we draw a
connection between two such frameworks. One of them is diffeology, as
popularized in the textbook [Igl13]. A diffeological space consists of a set
$X$, and a set $\mathcal{D}_{X}$ of functions $\mathbb{R}^{n}\to X$ satisfying
three simple conditions. While the definition of a diffeological space is
simple, a large number of interesting spaces outside the purview of classical
differential geometry can be given a diffeology. Every finite dimensional
smooth manifold inherits a canonical diffeology, as does the set
$C^{\infty}(X,Y)$ of smooth maps between any two diffeological spaces. In fact
the category of diffeological spaces is complete, cocomplete and cartesian
closed. More precisely it is a quasi-topos [BH09]. This is of course not the
case for the category of finite dimensional smooth manifolds, and thus
provides a ”better” category in which to work. Various notions of classical
differential geometry have been defined for diffeological spaces, like
differential forms, deRham cohomology, fiber bundles, tangent spaces [CW15],
and recently Čech cohomology [Igl20] [KWW21].

The second framework is higher topos Theory. Here the objects of interest are
$\infty$-sheaves over the site $\mathsf{Cart}$ of cartesian spaces, which we
call smooth $\infty$-groupoids. A surprising fact is that many constructions
of classical differential geometry can also be extended to smooth
$\infty$-groupoids. One such extension is the notion of a principal
$\infty$-bundle, as defined in [NSS14a] and [NSS14]. Classical principal
$G$-bundles are examples of such principal $\infty$-bundles, as are non-
abelian bundle gerbes, which are precisely $G$-principal $\infty$-bundles,
where $G$ is a Lie $2$-group, which can be presented by a crossed module
$G=[G_{1}\to G_{0}]$. In this framework, Čech cohomology of a manifold $M$
with coefficients in some $\infty$-sheaf $A$ is given by the connected
components of the derived mapping space

$H^{k}(M,A)\coloneqq\pi_{0}\mathbb{R}\text{Hom}(M,B^{k}A),$

where $B^{k}A$ is the $k$-th delooping of $A$, if it exists. This allows for a
robust framework with which one can study more exotic cohomology theories,
such as twisted, equivariant or differential refinements of general cohomology
theories. For more on this theory we recommend the texts [FSS11], [ADH21],
[Sch13], [BNV13]. In this paper we will use the presentation of this theory by
simplicial presheaves. Thus the reader does not need to be comfortable with
$\infty$-categories in order to read this paper.

Our contribution is to give a connection between the above mentioned
frameworks. Baez-Hoffnung proved in [BH09] that the category of diffeological
spaces is equivalent to the category of concrete sheaves on $\mathsf{Cart}$,
which is a full subcategory of the category of sheaves on $\mathsf{Cart}$.
Thus we can think of any diffeological space as a certain kind of sheaf. The
category of sheaves includes into the category of simplicial presheaves
$\mathsf{sPre}(\mathsf{Cart})$, which has a simplicial model category
structure on it, called the local projective model structure. Therefore higher
topos theory tells us that if $A$ is an $\infty$-sheaf, namely a simplicial
presheaf satisfying a condition called descent, then cohomology of a
diffeological space $X$ with coefficients in $A$ is given by derived mapping
space

$\mathbb{R}\text{Hom}(X,A)=\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,A),$

where $Q$ is a cofibrant replacement functor in the local projective model
structure. We are in the lucky situation that this functor has a simple and
explicit description when $X$ is a discrete simplicial presheaf, i.e. a
presheaf, thanks to Dugger [Dug01]. This realization allows us to connect the
theory of diffeological spaces to the theory of smooth $\infty$-groupoids.

Let $X$ be a diffeological space, and let
$B=\coprod_{p\in\mathsf{Plot}(X)}U_{p}$ denote the nebula of $X$, namely the
coproduct of the domains of all of its plots. There is a canonical projection
map $\pi:B\to X$. Let $M$ denote the diffeological monoid whose elements are
smooth maps $f:B\to B$ making the following diagram commute:

${B}$${B}$${X}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$

then $M$ acts on $B$, and we can form the following simplicial diffeological
space:

${{B//M}}$${B}$${{B\times M}}$${{B\times M\times
M}}$${\dots}$$\scriptstyle{\coloneqq}$

This simplicial diffeological space is used in [Igl20] to define his version
of Čech cohomology for difeological spaces. Our first result is:

######  Proposition 5.2.

Given a diffeological space $X$ and its functorial cofibrant replacement $QX$
in the local projective model structure, then there is an isomorphism

$QX\cong B//M$

of simplicial presheaves, equivalently of simplicial diffeological spaces.

Conceptually this result shows that Iglesias-Zemmour’s results in [Igl20] are
”homotopically robust.” In more detail, this result allows us to prove:

######  Corollary 8.8.

Iglesias-Zemmour’s cohomology groups agree with $\infty$-sheaf cohomology
groups:

$\check{H}^{k}_{PIZ}(X,\mathbb{R}^{\delta})\cong H^{k}(X,\mathbb{R}^{\delta})$

The results of [NSS14a] tell us that if $G$ is a Lie group, then the nerve of
the category of $G$-principal $\infty$-bundles on a diffeological space $X$ is
weakly equivalent to the derived mapping space $\mathbb{R}\text{Hom}(X,BG)$
where $BG$ is the stack defined by sending a cartesian space $U$ to the nerve
of the groupoid $\left(C^{\infty}(U,G)\rightrightarrows
C^{\infty}(U,*)\right).$ Our main result is:

######  Theorem 7.4.

If $G$ is a diffeological group and $X$ a diffeological space, then there is a
weak equivalence of simplicial sets

$N\mathsf{DiffPrinc}_{G}(X)\simeq N\infty\mathsf{Princ}_{G}(X)$

between the nerve of the groupoid of diffeological principal $G$-bundles as
defined in [Igl13], and the nerve of the category of $G$-principal
$\infty$-bundles as defined in [NSS14a].

This result provides a bridge between two frameworks for higher differential
geometry. In future work we will explore how to move results from higher topos
theory, which have had a long time to be developed, to diffeological spaces,
which is a much newer subject. In particular we plan to investigate how to
define diffeological non-abelian bundle gerbes and diffeological bundles with
connection 111There is a definition of diffeological bundles with connection
in [Igl13], but Iglesias-Zemmour notes that ”This is more a program than a
strict immutable definition.” using this approach. Since we hope that our
audience will include diffeologists, who may not be familiar with higher topos
theory, we take an expository tone in this paper, which accounts for its
length.

######  Remark 1.1.

We will use the homotopical theory of simplicial presheaves on $\mathsf{Cart}$
in this paper, however every simplicial presheaf we shall encounter will
actually be a simplicial sheaf and we easily could have taken the whole theory
to rest on the equivalent homotopical theory of simplicial sheaves. In more
detail, levelwise sheafification defines a Quillen equivalence between the
local projective model structure on simplicial presheaves and on simplicial
sheaves.

##  2\. Diffeological Spaces

In this section we give some background on diffeological spaces, all of which
can be found in [Igl13], and then focus on the perspective of diffeological
spaces as concrete sheaves on $\mathsf{Cart}$.

######  Definition 2.1.

Let $U$ be a smooth finite dimensional manifold diffeomorphic to
$\mathbb{R}^{n}$ for some $n\geq 0$. We call $U$ a Cartesian space. We say a
collection of inclusion maps $\\{U_{i}\hookrightarrow U\\}$ is a covering
family if it is a differentiably good open cover. Namely every finite
intersection $U_{i_{0}}\cap\dots\cap U_{i_{n}}$ is a cartesian space and the
canonical map $\coprod_{i}U_{i}\to U$ is surjective. Let $\mathsf{Cart}$
denote the category of Cartesian spaces with smooth maps as morphisms. Given a
set $X$, let $\mathsf{Param}(X)$ denote the set of parametrizations of $X$,
namely the set of set functions $U\xrightarrow{p}X$, where
$U\in\mathsf{Cart}$.

######  Definition 2.2.

Given a set $X$, a diffeology222This is not the exact definition of
diffeological spaces as given in the literature. Usually parametrizations are
set functions whose domains are open subsets of cartesian spaces. However
Remark 2.38 shows that the category of diffeological spaces given by that
definition and that of Definition 2.2 are equivalent. on $X$ is a subset
$\mathcal{D}_{X}\subseteq\mathsf{Param}(X)$ such that:

  * •

(D1) all constant maps $U\to*\to X$ belong to $\mathcal{D}_{X}$,

  * •

(D2) if $U\xrightarrow{p}X\in\mathcal{D}_{X}$, and $V\xrightarrow{f}U$ is a
smooth map, then the composition $V\xrightarrow{f}U\xrightarrow{p}X$ also
belongs to $\mathcal{D}_{X}$, and

  * •

(D3) if $U\xrightarrow{p}X$ is a parametrization, and $\\{U_{i}\hookrightarrow
U\\}$ is a covering family for $U$ such that the restriction
$U_{i}\hookrightarrow U\xrightarrow{p}X$ belongs to $\mathcal{D}_{X}$ for
every $i$, then $p\in\mathcal{D}_{X}$.

A set $X$ equipped with a diffeology $\mathcal{D}_{X}$ is called a
diffeological space. Parametrizations that belong to a diffeology are called
plots. We say a set function $X\xrightarrow{f}Y$ between diffeological spaces
is smooth if for every plot $U\xrightarrow{p}X$ in $\mathcal{D}_{X}$, the
composition $U\xrightarrow{p}X\xrightarrow{f}Y$ belongs to $\mathcal{D}_{Y}$.

Denote the category whose objects are diffeological spaces and morphisms are
smooth maps between them by $\mathsf{Diff}$. An isomorphism in this category
is called a diffeomorphism.

Let $\mathcal{D}(X)$ denote the poset of all diffeologies on $X$ ordered by
inclusion. Let $\mathcal{D}_{X},\mathcal{D}^{\prime}_{X}$ be two diffeologies
on $X$. We say that $\mathcal{D}_{X}$ is smaller than
$\mathcal{D}^{\prime}_{X}$ if
$\mathcal{D}_{X}\subseteq\mathcal{D}^{\prime}_{X}$, in which case we also say
that $\mathcal{D}^{\prime}_{X}$ is larger than $\mathcal{D}_{X}$.

######  Remark 2.3.

In the literature we typically replace the word smaller with coarser or weaker
and larger with finer or stronger.

######  Definition 2.4.

Given a diffeological space $(X,\mathcal{D}_{X})$, a generating family
$\mathcal{G}$ for $\mathcal{D}_{X}$ is a collection
$\\{V_{i}\xrightarrow{q_{i}}X\\}$ of parametrizations such that for every plot
$U\xrightarrow{p}X$ there exists a covering family $\\{U_{i}\hookrightarrow
U\\}$ of $U$ and a smooth map $f_{i}:U_{i}\to V_{i}$ such that the following
diagram commutes:

(1)
${U}$${X}$${{U_{i}}}$${{V_{i}}}$$\scriptstyle{f_{i}}$$\scriptstyle{p}$$\scriptstyle{q_{i}}$

If $\mathcal{G}$ is a generating family for $\mathcal{D}_{X}$, we write
$\mathcal{D}_{X}=\langle\mathcal{G}\rangle$. Conversely, given any set of
parametrizations $\mathcal{G}\subseteq\mathsf{Param}(X)$, such that the images
of the parametrizations in $\mathcal{G}$ cover $X$, then let
$\langle\mathcal{G}\rangle$ denote the diffeology it generates, namely plots
of $\langle\mathcal{G}\rangle$ are plots $p$ that satisfy (1).

######  Example 2.5.

The diffeology on a set $X$ generated by the empty set
$\langle\varemptyset\rangle$ is precisely the discrete diffeology
$\mathcal{D}^{\circ}_{X}$, since it is the smallest possible diffeology on
$X$. Similarly $\langle\mathsf{Param}(X)\rangle=\mathcal{D}^{\bullet}_{X}$ is
the codiscrete diffeology.

######  Definition 2.6.

Given a set $X$, a diffeological space $(Y,\mathcal{D}_{Y})$ and a set
function $X\xrightarrow{f}Y$, there exists a diffeology:

(2)
$f^{*}(\mathcal{D}_{Y})=\\{p\in\mathsf{Param}(X)\,:\,fp\in\mathcal{D}_{Y}\\}$

called the pullback diffeology on $X$.

Similarly if $(X,\mathcal{D}_{X})$ is a diffeological space, $Y$ is a set, and
$X\xrightarrow{f}Y$ is a set function, then there exists a diffeology:

(3) $f_{*}(\mathcal{D}_{X})=\langle f\circ\mathcal{D}_{X}\rangle$

called the pushforward diffeology on $Y$, and where

$f\circ\mathcal{D}_{X}=\\{V\xrightarrow{p}Y\in\mathsf{Param}(Y)\,:\,p=f\varphi,\text{
where }\varphi\in\mathcal{D}_{X}\\}$

######  Remark 2.7.

The pushforward diffeology is a little less intuitive than the pullback so
let’s discuss it in more detail. Suppose $X\xrightarrow{f}Y$ is a set function
and $X$ has a diffeology $\mathcal{D}_{X}$. Then what is a plot of
$f_{*}(\mathcal{D}_{X})$? A parametrization $U\xrightarrow{p}Y$ is a plot of
$f_{*}(\mathcal{D}_{X})$ if it is constant or locally factors through $f$:

(4)
${{U_{i}}}$${U}$${X}$${Y}$$\scriptstyle{p_{i}}$$\scriptstyle{f}$$\scriptstyle{p}$

where $p_{i}\in\mathcal{D}_{X}$ for all $i\in I$.

######  Definition 2.8.

A map $(X,\mathcal{D}_{X})\xrightarrow{f}(Y,\mathcal{D}_{Y})$ between
diffeological spaces is called an:

  * •

induction if it is injective and $f^{*}(\mathcal{D}_{Y})=\mathcal{D}_{X}$

  * •

subduction if it is surjective and $f_{*}(\mathcal{D}_{X})=\mathcal{D}_{Y}$.

######  Definition 2.9.

Let $(X,\mathcal{D}_{X})$ be a diffeological space and
$A\xhookrightarrow{\iota}X$ a subset. Then consider:

(5)
$\mathcal{D}^{X}_{A}=\iota^{*}(\mathcal{D}_{X})=\\{U\xrightarrow{p}A\,:\,\iota
p\in\mathcal{D}_{X}\\}$

the pullback diffeology on $A$ by the inclusion map. We call this the subspace
diffeology on $A$. Recall that this is the largest diffeology on $A$ that
makes the inclusion smooth. $\iota$ is always an induction when $A$ is
equipped with the subspace diffeology.

######  Definition 2.10.

Let $(X,\mathcal{D}_{X})$ be a diffeological space and
$X\xrightarrow{\pi}X/\sim$ the projection map onto the quotient by some
equivalence relation taking a point $x\in X$ to its equivalence class $[x]\in
X/\sim$. Then consider:

(6) $D_{\sim}^{X}=\pi_{*}(\mathcal{D}_{X})$

the pushforward diffeology on $X/\sim$ by the projection map. We call this the
quotient diffeology. Recall that this is the smallest diffeology on $X/\sim$
that makes the projection smooth. $\pi$ is always a subduction when $X/\sim$
is equipped with the quotient diffeology.

######  Remark 2.11.

A parametrization $U\xrightarrow{p}X/\sim$ is exactly a plot if it is locally
a composition of $\pi$ and a plot in $X$. In other words, there exists a
covering $\\{U_{i}\hookrightarrow U\\}$, and plots $U_{i}\to X$ for every $i$,
such that the following diagram commutes:

${U}$${{X/\sim}}$${{U_{i}}}$${X}$$\scriptstyle{\pi}$$\scriptstyle{p_{i}}$$\scriptstyle{p}$

In fact, the category of diffeological spaces is complete and cocomplete.
Suppose $F:\mathsf{J}\to\mathsf{Diff}$ is a diagram of diffeological spaces.
Then a parametrization $U\xrightarrow{p}\lim F$, (where we are taking $\lim F$
to be the limit of the underlying sets of the $F_{j}$) is a plot iff the
composite $U\xrightarrow{p}\lim F\to F_{j}$ is a plot for every
$j\in\mathsf{J}$. Similarly a parametrization
$U\xrightarrow{p}\text{colim}\,F$ is a plot iff there exists a covering
$\\{U_{i}\to U\\}$ and plots $U_{i}\xrightarrow{p_{i}}F_{j_{i}}$ for each $i$,
such that the following diagram commutes:

${U}$${{\text{colim}\,F}}$${{U_{i}}}$${{F_{j_{i}}}}$$\scriptstyle{p_{i}}$$\scriptstyle{p}$

######  Lemma 2.12.

The category $\mathsf{Set}$ is Cartesian closed, in other words for any three
sets $A,B,C\in\mathsf{Set}$ we have the following isomorphism, natural in all
three variables:

(7) $\mathsf{Set}(A,\mathsf{Set}(B,C))\xrightarrow{\\#}\mathsf{Set}(A\times
B,C)$

where $\\#$ is called the transpose, defined by $f^{\\#}(a,b)=[f(a)](b)$, with
inverse $(\\#)^{-1}=\flat$.

######  Definition 2.13.

Given any two diffeological spaces $X,Y$, consider the set
$\mathsf{Diff}(X,Y)$ of smooth maps $X\xrightarrow{f}Y$. Equip it with the
functional diffeology:

(8) $\mathcal{D}_{X\to Y}=\\{U\xrightarrow{p}\mathsf{Diff}(X,Y)\,:U\times
X\xrightarrow{p^{\\#}}Y\text{ is smooth}\\}$

and denote the diffeological space by
$C^{\infty}(X,Y)\coloneqq(\mathsf{Diff}(X,Y),\mathcal{D}_{X\to Y})$.

######  Remark 2.14.

We will also occasionally denote $C^{\infty}(X,Y)$ by the notation $Y^{X}$.

######  Definition 2.15.

Given two diffeological spaces $X$ and $Y$, define the set function:

(9) $C^{\infty}(X,Y)\times X\xrightarrow{\text{ev}}Y$

by $\text{ev}(f,x)=f(x)$.

######  Lemma 2.16.

$\mathcal{D}_{X\to Y}$ is the largest diffeology on $C^{\infty}(X,Y)$ such
that ev is smooth.

######  Theorem 2.17 ([Lau06, Theorem 3.2]).

The transpose map $\\#$ is a diffeomorphism, with inverse $\flat$:

(10) ${{C^{\infty}(X,C^{\infty}(Y,Z))}}$${{C^{\infty}(X\times
Y,Z)}}$$\scriptstyle{\\#}$$\scriptstyle{\flat}$

######  Definition 2.18.

Let $X$ be a diffeological space. A differential $k$-form $\omega$ on $X$ is
defined to be a collection $\\{\omega_{p}\\}$, where
$\omega_{p}\in\Omega^{k}(U)$ for every plot $U\xrightarrow{p}X$, such that if

${U}$${V}$${X}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{f}$

is a map of plots, then $f^{*}\omega_{q}=\omega_{p}$.

Using Definition 2.18, one can also define exterior derivatives, wedge
products, and deRham cohomology on arbitrary diffeological spaces.

######  Definition 2.19.

A diffeological group is a group $G$ equipped with a diffeology
$\mathcal{D}_{G}$ such that the multiplication map $m:G\times G\to G$, and
inverse map $i:G\to G$ are smooth.

######  Definition 2.20.

A (left) diffeological group action of a diffeological group $G$ on a
diffeological space $X$, denoted $\rho:G\curvearrowright X$, is a smooth map

$\rho:G\times X\to X$

such that

  1. (1)

$\rho(e_{G},x)=x$, and

  2. (2)

$\rho(h,\rho(g,x))=\rho(hg,x)$,

where $e_{G}$ denotes the identity element of $G$.

######  Definition 2.21.

Given a group $G$, we can always equip it with a diffeology such that $m$ and
$i$ are smooth. We call this a group diffeology on $G$. We can always choose
the largest of these diffeologies, which we will often refer to as the group
diffeology on $G$.

######  Definition 2.22.

Let $G$ be a diffeological group, and $P$ be a right $G$-space. A map
$P\xrightarrow{\pi}X$ of diffeological spaces is a diffeological principal
$G$-bundle if:

  1. (1)

the map $P\xrightarrow{\pi}X$ is a subduction, and

  2. (2)

the map $P\times G\to P\times_{X}P$ defined by $(p,g)\mapsto(p,p\cdot g)$ is a
diffeomorphism, and

A map of diffeological principal $G$-bundles $P\to P^{\prime}$ over $X$ is a
diagram

${P}$${{P^{\prime}}}$${X}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{f}$

where $f$ is a $G$-equivariant smooth map. Notice that such a map is always a
diffeomorphism. A diffeological principal $G$-bundle is said to be trivial if
it is isomorphic to the diffeological principal $G$-bundle $U\times
G\xrightarrow{\text{pr}_{1}}U$.

######  Remark 2.23.

Condition (1) of Definition 2.22 is equivalent to saying that $G$ acts on the
fibers of $\pi$ freely and transitively.

######  Lemma 2.24.

Given a diffeological group $G$ acting on a diffeological space $P$, a map
$P\xrightarrow{\pi}X$ is a diffeological principal $G$-bundle iff for every
plot $U\xrightarrow{q}X$, the pullback $q^{*}P$ is trivial, and the map
$P\times G\to P\times_{X}P$ defined by $(p,g)\mapsto(p,p\cdot g)$ is a
diffeomorphism.

###### Proof.

$(\Rightarrow)$ Proposition 5.3.(4) of [KWW21] proves that $q^{*}P\to U$ is a
diffeological principal $G$-bundle over a cartesian space, thus it is trivial.
$(\Leftarrow)$ If $q^{*}P\cong U\times G$, then the differentiably good cover
$\\{U\\}$ shows that $\pi$ is a subduction, namely the following diagram
commutes:

${U}$${P}$${U}$${X}$$\scriptstyle{q}$$\scriptstyle{\pi}$$\scriptstyle{s}$

where $s$ is the composition $U\to U\times G\cong q^{*}P\to P$, where the
first map is the section $x\mapsto(x,e_{G})$. ∎

######  Lemma 2.25.

If $G$ is a diffeological group and $P\xrightarrow{\pi}U$ is a diffeological
principal $G$-bundle over a cartesian space $U$, then $\pi$ is trivial.

###### Proof.

Notice that $U\xrightarrow{1_{U}}U$ is a plot of $U$, thus the pullback
$1_{U}^{*}P\cong P\cong U\times G$ by definition. ∎

###  2.1. Diffeological Spaces as Concrete Sheaves

A major development in the theory of diffeological spaces was made in [BH09],
which showed that the category of diffeological spaces can be identified with
a full subcategory of the category of sheaves on a site $\mathsf{Cart}$. Here
we recall this theorem and its setup.

######  Definition 2.26.

Let $\mathsf{C}$ be a (small) category. A coverage $J$ on $\mathsf{C}$ is a
function that assigns to every object $U\in\mathsf{C}$ a set $J(U)$ of
families of morphisms $\\{U_{i}\xrightarrow{r_{i}}U\\}$ with the following
property: if $\\{U_{i}\xrightarrow{r_{i}}U\\}\in J(U)$, and
$V\xrightarrow{g}U$ is a map in $\mathsf{C}$, then there exists a family
$\\{V_{j}\xrightarrow{t_{j}}V\\}\in J(U)$ such that $gt_{j}$ factors through
some $r_{i}$. Namely for every $j$ there exists a map $s_{ji}$ making the
following diagram commute:

(11)
${{V_{j}}}$${{U_{i}}}$${V}$${U}$$\scriptstyle{t_{j}}$$\scriptstyle{s_{ji}}$$\scriptstyle{r_{i}}$$\scriptstyle{g}$

A category equipped with a coverage is called a site. The families
$\\{U_{i}\xrightarrow{r_{i}}U\\}\in J(U)$ are called covering families over
$U$.

######  Example 2.27.

The main example of a site we will use in this paper is $\mathsf{Cart}$.
Recall this is the category whose objects are smooth manifolds diffeomorphic
to $\mathbb{R}^{n}$ for some $n\in\mathbb{N}$ (including the empty set
$\varnothing$), and whose morphisms are smooth maps between them. We call the
objects of this category cartesian spaces. Define a coverage on
$\mathsf{Cart}$ as follows: For $U\in\mathsf{Cart}$, a covering family is a
collection of inclusions of open sets $\\{U_{i}\hookrightarrow U\\}$ such that
every finite intersection $U_{i_{0}}\cap\dots\cap U_{i_{n}}$ is a cartesian
space. This is called a differentiably good cover of $U$. Let us show that the
differentiably good covers form a coverage. If $\\{U_{i}\hookrightarrow U\\}$
is a differentiably good cover and $V\xrightarrow{g}U$ a smooth map, then
$\\{g^{-1}(U_{i})\hookrightarrow V\\}$ is an open cover, but not necessarily
differentiably good. However by [FSS11, Proposition A.1], differentiably good
covers always exist on paracompact smooth manifolds, and cartesian spaces are
paracompact, thus there is a differentiably good cover
$\\{W_{ij}\hookrightarrow g^{-1}(U_{i})\\}$ for each $i$, and thus the
composite collection $\\{W_{ij}\hookrightarrow V\\}$ is a differentiably good
cover of $V$ for every $i$, and we can consider the map $g|_{W_{ij}}\to
U_{i}$, which will make the diagram (11) commute.

######  Remark 2.28.

In the above example, the pullback of two inclusions $U_{i}\hookrightarrow U$
and $U_{j}\hookrightarrow U$ is precisely the inclusion of the intersection
$U_{i}\cap U_{j}\hookrightarrow U$. We will commonly denote such intersections
by $U_{ij}\coloneqq U_{i}\cap U_{j}$, and we will sometimes abuse this
notation for other sites such that $U_{ij}\coloneqq U_{i}\times_{U}U_{j}$.

######  Definition 2.29.

A presheaf on a category $\mathsf{C}$ is a functor
$F:\mathsf{C}^{op}\to\mathsf{Set}$. An element $x\in F(U)$ for an object
$U\in\mathsf{C}$ is called a section over $U$. If $U\xrightarrow{f}V$ is a map
in $\mathsf{C}$, and $x\in F(V)$, then we sometimes denote $F(f)(x)$ by
$x|_{U}$. If $\\{U_{i}\xrightarrow{r_{i}}U\\}_{i\in I}$ is a covering family,
then a matching family is a collection $\\{x_{i}\\}_{i\in I}$, $x_{i}\in
F(U_{i})$, such that given a diagram in $\mathsf{C}$ of the form

${V}$${{U_{j}}}$${{U_{i}}}$${U}$$\scriptstyle{r_{i}}$$\scriptstyle{r_{j}}$$\scriptstyle{f}$$\scriptstyle{g}$

then $F(f)(x_{i})=F(g)(x_{j})$ for all $i,j\in I$. An amalgamation $x$ for a
matching family $\\{x_{i}\\}$ is a section $x\in F(U)$ such that
$x_{i}|_{U}=x$ for all $i$.

######  Definition 2.30.

A presheaf on a site $(\mathsf{C},J)$ is a sheaf if for every object
$U\in\mathsf{C}$, every covering family $\\{U_{i}\to U\\}\in J(U)$ and every
matching family $\\{x_{i}\\}$, there exists a unique amalgamation $x\in F(U)$.

######  Remark 2.31.

If $(\mathsf{C},J)$ is a site that has pullbacks, then we can equivalently
express the condition for $F$ being a sheaf as requiring that for every
$U\in\mathsf{C}$ and every covering family $\\{U_{i}\to U\\}\in J(U)$, the
diagram:

(12) ${{F(U)}}$${{\prod_{i}F(U_{i})}}$${{\prod_{i,j}F(U_{i}\times_{U}U_{j})}}$

is an equalizer.

######  Example 2.32.

Given a smooth manifold $M$, the presheaf

$U\mapsto C^{\infty}(U,M)$

which we denote by either $\underline{M}$ or just $M$, is a sheaf on
$\mathsf{Cart}$. Another important example of a sheaf is

$U\mapsto\Omega^{k}(U)$

for any $k\geq 0$. If $V$ is a cartesian space, we will denote its image under
the yoneda embedding by $yV$. This is the presheaf

$U\mapsto C^{\infty}(U,V)$

and by the above example is a sheaf. We call a site $(\mathsf{C},J)$ where
every representable is a sheaf subcanonical. Thus we see that $\mathsf{Cart}$
is a subcanonical site.

######  Definition 2.33.

A site $(\mathsf{C},J)$ is concrete if:

  1. (1)

it is subcanonical,

  2. (2)

it has a terminal object $*$,

  3. (3)

the functor $\mathsf{C}(*,-):\mathsf{C}\to\mathsf{Set}$ is faithful, and

  4. (4)

for every covering family $\\{U_{i}\to U_{i}\\}$, the family of maps
$\mathsf{C}(*,U_{i})\to\mathsf{C}(*,U)$ is jointly surjective, namely the map
$\prod_{i}\mathsf{C}(*,U_{i})\to\mathsf{C}(*,U)$ is surjective.

######  Lemma 2.34.

$\mathsf{Cart}$ is a concrete site.

######  Definition 2.35.

If $(\mathsf{C},J)$ is a concrete site and $F$ is a presheaf, then we call
$F(*)$ its underlying set, and for any $U\in\mathsf{C}$ there always exists a
map

$\phi_{U}:F(U)\to\mathsf{Set}(\mathsf{C}(*,U),F(*))$

defined by $\phi_{U}(x)=\left(u\mapsto F(u)(x)\right)$. We say a sheaf $F$ is
concrete if for every object $U\in\mathsf{C}$, the function $\phi_{U}$ is
injective. Let $\mathsf{ConSh}(\mathsf{C})$ denote the full subcategory of
concrete sheaves on a concrete site $(\mathsf{C},J)$.

######  Example 2.36.

For any smooth manifold $M$, the sheaf $\underline{M}$ is concrete. This is
equivalent to saying that for every cartesian space $U$, the function

$\phi_{U}:C^{\infty}(U,M)\to\mathsf{Set}(U(*),M(*))$

is injective, which is the same thing as saying that the set of smooth maps
from $U$ to $M$ is a subset of all set functions from $U$ to $M$.

Note that the sheaf $\Omega^{k}$ is not concrete. Indeed if $U$ is a cartesian
space, then $\phi_{U}$ takes the form

$\phi_{U}:\Omega^{k}(U)\to\mathsf{Set}(U(*),\Omega^{k}(*))$

but $\Omega^{k}(*)=\\{0\\}$ is the zero vector space, thus
$\mathsf{Set}(U(*),\\{0\\})=*$ is the singleton set. It is well known that
unless $U=*$, $\Omega^{k}(U)$ is not zero dimensional for any $k$.

######  Theorem 2.37 ([BH09, Proposition 24]).

The category $\mathsf{Diff}$ of diffeological spaces is equivalent to the
category of concrete sheaves $\mathsf{ConSh}(\mathsf{Cart})$.

######  Remark 2.38.

The above theorem is stated differently in [BH09], namely they use a site
whose objects are open subsets of cartesian spaces. However this site is a
dense-subsite of $\mathsf{Cart}$, and thus their categories of sheaves are
equivalent. For more details see [nLa21, Proposition 2.31].

Theorem 2.37 allows us to make a perspective shift. Constructions made in
$\mathsf{Diff}$ can be compared with already defined notions of sheaves. For
example a differential $k$-form $\omega$ on a diffeological space $X$, given
in Definition 2.18, is equivalent to a map

$X\xrightarrow{\omega}\Omega^{k}$

of sheaves on $\mathsf{Cart}$.

##  3\. Stacks

######  Definition 3.1.

Given a category $\mathsf{C}$, a groupoidal presheaf on $\mathsf{C}$ is any of
the following objects:

  1. (1)

a category fibered in groupoids (CFG) $\mathsf{E}\to\mathsf{C}$,

  2. (2)

a pseudofunctor $F:\mathsf{C}^{op}\to\mathsf{Gpd}$, or

  3. (3)

a strict functor $F:\mathsf{C}^{op}\to\tau_{0}(\mathsf{Gpd})$, where
$\tau_{0}$ denotes the underlying $1$-category of $\mathsf{Gpd}$.

The $2$-category of each of the above three objects with their corresponding
morphisms are equivalent. For concreteness we will assume our groupoidal
presheaves are strict functors, but all statements we make can be done using
any of the three models. Let $\mathsf{GpdPre}(\mathsf{C})$ denote the
$2$-category of groupoidal presheaves on a category $\mathsf{C}$. For more
details on these definitions and their equivalence, see [JY20] and [Hol07].

######  Definition 3.2.

A groupoidal presheaf $F$ is a stack on a site $\mathsf{C}$ if for all
$U\in\mathsf{C}$, and for any covering $\\{U_{i}\to U\\}$, the following hold:

  1. (1)

(Gluing on Objects) for every $x_{i}\in F(U_{i})$, and map
$g_{ij}:x_{i}|_{U_{ij}}\to x_{j}|_{U_{ij}}$ such that
$g_{ik}|_{U_{ijk}}=g_{jk}|_{U_{ijk}}g_{ij}|_{U_{ijk}}$, there exists a unique
$x\in F(U)$ and an isomorphism $\varphi_{i}:x|_{U_{i}}\to x_{i}$ such that
$g_{ij}\varphi_{i}|_{U_{ij}}=\varphi_{j}|_{U_{ij}}$, and

  2. (2)

(Gluing on Morphisms) for every $x,y\in F(U)$, if for every $i$, there is a
map $x|_{U_{i}}\xrightarrow{f_{i}}y|_{U_{i}}$ such that
$f_{i}|_{U_{ij}}=f_{j}|_{U_{ij}}$, then there exists a unique map $f:x\to y$
such that $f|_{U_{i}}=f_{i}$.

We call the requirement that $g_{ik}=g_{jk}g_{ij}$ and $g_{ii}=e_{G}$ the
cocycle conditions.

We call a map $f:X\to Y$ of groupoidal presheaves such that $f_{U}:X(U)\to
Y(U)$ is an equivalence of groupoids for every $U\in\mathsf{C}$ an objectwise
equivalence. It is not immediately clear from the definition that being a
stack is preserved under objectwise equivalence, but it is true.

######  Lemma 3.3 ([Vis07, Propositions 3.36 and 4.12]).

Let $\mathsf{C}$ be a site, and $f:X\to Y$ is an objectwise equivalence of
groupoidal presheaves, then $X$ is a stack iff $Y$ is a stack.

Let $G$ be a diffeological group, and let $\mathsf{DiffPrinc}_{G}$ denote the
groupoidal presheaf on $\mathsf{Cart}$ that sends a cartesian space to its
groupoid of diffeological principal $G$-bundles. We will show that
$\mathsf{DiffPrinc}_{G}$ is a stack on $\mathsf{Cart}$.

######  Remark 3.4.

Even though the proof of this is quite standard, and a similar theorem is
proven in [Bun21, Section 4.1], using different methods, and only for
diffeological vector bundles, we haven’t seen any full proofs in the
literature and thus record it here. This proof can easily be generalized to
show that $\mathsf{DiffPrinc}_{G}$ extends to a stack on the category of all
diffeological spaces with the subduction Grothendieck topology [Bun21, Remark
4.6].

######  Remark 3.5.

If $G$ were a Lie group, then a diffeological principal $G$-bundle on a smooth
finite dimensional manifold is precisely a classical principal $G$-bundle, and
it is well known that in this case that $\mathsf{DiffPrinc}_{G}$ is a stack
not only on $\mathsf{Cart}$ but also on the site of finite dimensional smooth
manifolds.

To prove that $\mathsf{DiffPrinc}_{G}$ is a stack on $\mathsf{Cart}$ we need a
diffeological version of the bundle construction theorem in classical smooth
manifold theory. Let $U$ be a cartesian space equipped with a differentiably
good open cover $\mathcal{U}=\\{U_{i}\hookrightarrow U\\}$, and suppose there
are maps $g_{ij}:U_{ij}\to G$ for each pair $U_{i},U_{j}\in\mathcal{U}$ such
that $g_{ik}|_{U_{ijk}}=g_{jk}|_{U_{ijk}}g_{ij}|_{U_{ijk}}$ and
$g_{ii}=e_{G}$. We call such a collection $\\{g_{ij}\\}$ a cocycle. Given two
cocycles $\\{g_{ij}\\},\\{g^{\prime}_{ij}\\}$, we say a collection
$\\{h_{i}:U_{i}\to G\\}$ is a map of cocycles from $\\{g_{ij}\\}$ to
$\\{g^{\prime}_{ij}\\}$ if $g^{\prime}_{ij}h_{i}=h_{j}g_{ij}$, for every
$i,j$.

######  Proposition 3.6.

Given a diffeological group $G$, a cartesian space $U$, a differentiably good
open cover $\mathcal{U}=\\{U_{i}\hookrightarrow U\\}$, and a cocycle
$\\{g_{ij}\\}$, form the quotient

$P\coloneqq\coprod U_{i}\times G/\sim$

where $(x,i,h)\sim(x,j,k)$ if $k=g_{ij}(x)h$. Then the map
$P\xrightarrow{\pi}U$ defined by $\pi([x,i,h])=x$ is a diffeological principal
$G$-bundle.

Further, every map of cocycles $\\{h_{i}\\}$ gives rise to a $G$-equivariant
map of diffeological principal $G$-bundles.

###### Proof.

According to Definition 2.22, to show that $P\xrightarrow{\pi}U$ is a
diffeological principal $G$-bundle, we must first show that $G$ acts freely
and transitively on the fibers of $\pi$. This will follow from the standard
proof, which we reproduce for convenience. Namely, define a (right) action of
$G$ on $P$ by $\rho([x,i,h],g)=[x,i,hg]$. This is easily seen to be well
defined and smooth. Now suppose that $[x,i,hg]=[x,i,h]$. Then $hg=g_{ii}(x)h$,
but $g_{ii}(x)=e_{G}$ by assumption, so $g=e_{G}$, thus the action is free.
Let us show it is transitive on fibers of $\pi$. Consider two points $[x,i,h]$
and $[x,j,k]$ in the same fiber of $\pi$. This means that $x\in U_{ij}$, so
then $[x,i,h]=[x,j,g_{ij}(x)h]$, thus set $g=h^{-1}g_{ij}^{-1}(x)k$, so then
$[x,i,h]\cdot g=[x,j,g_{ij}(x)h]\cdot g=[x,j,k]$. Thus $G$ acts transitively
on fibers.

Now let us show that $P\xrightarrow{\pi}X$ is a subduction. Suppose that
$V\xrightarrow{p}U$ is a plot, namely a smooth map of cartesian spaces. Then
as in the proof of Example 2.27, there exists a differentiably good cover
$\\{V_{j}\\}$ of $V$ such that the following diagram commutes:

${{V_{j}}}$${{U_{i}}}$${P}$${V}$${U}$${U}$$\scriptstyle{\pi}$$\scriptstyle{s_{ij}}$$\scriptstyle{\sigma_{i}}$$\scriptstyle{p}$

where $\sigma_{i}$ is defined as $\sigma_{i}(x)=[x,i,e_{G}]$. Thus $\pi$ is a
subduction.

Now suppose that $\\{g_{ij}\\}$ and $\\{g^{\prime}_{ij}\\}$ are cocycles for
the same differentiably good cover $\mathcal{U}=\\{U_{i}\hookrightarrow U\\}$,
and $\\{h_{i}\\}$ is a map of cocycles. We shall define a map
$P\xrightarrow{\varphi}P^{\prime}$ between their respective bundles. Given a
point $[x,i,h]\in P$, define $\varphi([x,i,h])=[x,i,h_{i}h]$. Let us show this
is well defined. Suppose that $[x,i,h]=[x,j,k]$, so that $k=g_{ij}(x)h$. Then

$\varphi([x,j,k])=[x,j,h_{j}k]=[x,j,h_{j}g_{ij}(x)h]=[x,j,g^{\prime}_{ij}(x)h_{i}h]=[x,i,h_{i}h]=\varphi([x,i,h])$

It is straightforward to check that this is a smooth, $G$-equivariant map of
diffeological principal $G$-bundles. ∎

######  Proposition 3.7.

Given a diffeological group $G$, the groupoidal presheaf
$\mathsf{DiffPrinc}_{G}$ is a stack on $\mathsf{Cart}$.

###### Proof.

Let us fix a cartesian space $U$ and a differentiably good cover
$\mathcal{U}=\\{U_{i}\hookrightarrow U\\}$. Suppose we are given diffeological
principal $G$-bundles $P_{i}$ on each $U_{i}$, with isomorphisms
$g_{ij}:P_{i}|_{U_{ij}}\to P_{j}|_{U_{ij}}$ satisfying the cocycle conditions.
By Lemma 2.25, we can identify $P_{i}$ with $U_{i}\times G$, and since
$g_{ij}$ is a $G$-equivariant bundle map, we can identify it with a map
$g_{ij}:U_{ij}\to G$. The cocycle conditions of Definition 3.2.(1) imply that
the collection $\\{g_{ij}\\}$ is a cocycle, and thus applying Proposition 3.6,
there exists a diffeological principal $G$-bundle $P$ on $U$. Let us define
isomorphisms $P|_{U_{i}}\xrightarrow{\varphi_{i}}P_{i}\cong U_{i}\times G$ by
$\varphi_{i}([x,i,h])=(x,h)$. Then if we consider the composition

$P|_{U_{ij}}\xrightarrow{\varphi_{i}|_{U_{ij}}}U_{ij}\times
G\xrightarrow{g_{ij}}U_{ij}\times G$

this is the map $[x,i,h]\mapsto(x,h)\mapsto(x,g_{ij}(x)h)$, and if $x\in
U_{ij}$ so that $[x,i,h]=[x,j,k]$, with $k=g_{ij}(x)h$, then this map
coincides with
$\varphi_{j}|_{U_{ij}}([x,i,h])=\varphi_{j}|_{U_{ij}}([x,j,k])=(x,k)=(x,g_{ij}(x)h)$.
Thus condition (1) of the stack conditions are satisfied.

Now let us show that condition (2) is satisfied. Namely suppose we are given
two diffeological principal $G$-bundles $P,P^{\prime}$ over $U$, and maps
$h_{i}:P|_{U_{i}}\to P^{\prime}|_{U_{i}}$ such that
$h_{i}|_{U_{ij}}=h_{j}|_{U_{ij}}$. The maps $h_{i}$ are equivalent to
$G$-equivariant bundle maps $U_{i}\times G\to U_{i}\times G$, and thus
equivalent to maps $h_{i}:U_{i}\to G$. These are clearly maps of cocycles, and
thus by Proposition 3.6 give rise to a map $P\xrightarrow{h}P^{\prime}$. ∎

Now given a diffeological group $G$, consider the diffeological groupoid

$G\rightrightarrows*$

this defines a groupoidal presheaf on $\mathsf{Cart}$ by

(13) $U\mapsto\left(C^{\infty}(U,G)\rightrightarrows C^{\infty}(U,*)\right)$

For every $U\in\mathsf{Cart}$, there is a canonical map
$BG(U)\to\mathsf{DiffPrinc}_{G}(U)$, natural in $U$, defined on objects by
sending the unique object to the diffeological principal $G$-bundle $U\times
G$. Since diffeological principal $G$-bundles are trivial on cartesian spaces
by Lemma 2.25, this map is essentially surjective. We claim that
$\mathsf{DiffPrinc}_{G}(P,P^{\prime})\cong C^{\infty}(U,G)$, because $P\cong
U\times G\to U\times G\cong P^{\prime}$ is a $G$-equivariant bundle map and
therefore equivalent to a map $U\to G$. Thus the map is also fully faithful,
and therefore an equivalence of groupoids. So by Lemma 3.3 we have proven:

######  Proposition 3.8.

The canonical map $BG\to\mathsf{DiffPrinc}_{G}$ is an objectwise equivalence
and thus $BG$ is a stack on $\mathsf{Cart}$.

######  Remark 3.9.

The groupoidal presheaf $BG$ is not a stack on the site of finite dimensional
smooth manifolds.

##  4\. Simplicial Presheaf Theory

In this section we develop the tools needed for higher topos theory, namely
that of simplicial presheaves and their model categories. For a nice
introduction to simplicial sets, we recommend [Fri21] and [Rie11]. For model
categories we recommend [Hir09] and [Hov07].

######  Definition 4.1.

A simplicial presheaf on $\mathsf{Cart}$ is a simplicial object in the
category $\mathsf{Pre}(\mathsf{Cart})$, namely a functor
$\Delta^{op}\to\mathsf{Pre}(\mathsf{Cart})$. Morphisms of simplicial
presheaves are then natural transformations of the corresponding functors. Let
$\mathsf{sPre}(\mathsf{Cart})$ denote the category of simplicial presheaves on
$\mathsf{Cart}$.

This category is simplicially enriched: If $X,Y$ are simplicial presheaves and
$K$ a simplicial set then we define:

  1. (1)

the tensor product $(X\otimes K)(U)=X(U)\times K$,

  2. (2)

the cotensor $(X^{K})(U)=X(U)^{K}$, where the lefthand term is the internal
hom in simplicial sets, and

  3. (3)

the mapping space $[X,Y]$, namely a simplicial set defined by

$[X,Y]_{k}=\mathsf{sPre}(X\otimes\Delta^{k},Y).$

We will also often denote the mapping space by

$\underline{\mathsf{sPre}(\mathsf{Cart})}(X,Y)\coloneqq[X,Y]$

and it is easy to check that these form a two-variable adjunction, namely
there are isomorphisms:

(14) $\underline{\mathsf{sPre}(\mathsf{Cart})}(X\otimes
K,Y)\cong\underline{\mathsf{sSet}}(K,[X,Y])\cong\underline{\mathsf{sPre}(\mathsf{Cart})}(X,Y^{K}),$

natural in $X,Y$ and $K$.

Now let us consider two fully faithful embeddings:

  1. (1)

The Yoneda embedding

$y:\mathsf{Cart}\to\mathsf{sPre}(\mathsf{Cart}),$

and we call objects in the image of this functor representable simplicial
presheaves,

  2. (2)

the functor

${}^{c}(-):\mathsf{Sh}(\mathsf{Cart})\to\mathsf{sPre}(\mathsf{Cart})$

which is induced by precomposition with the inclusion
$i:\mathsf{Set}\to\mathsf{sSet}$, and we call objects in the image of this
functor discrete simplicial presheaves.

The theory of Kan extensions provides us with left and right adjoints to
${}^{c}(-)$:

(15)
${{\mathsf{Pre}(\mathsf{Cart})}}$${\mathsf{sPre}(\mathsf{Cart})}$$\scriptstyle{{}^{c}(-)}$$\scriptstyle{\pi_{0}}$$\scriptstyle{(-)_{0}}$$\scriptstyle{\dashv}$$\scriptstyle{\dashv}$

where $\pi_{0}(X)=\text{coeq}(X_{1}\rightrightarrows X_{0})$ is the presheaf
of connected components of $X$, and $(X)_{0}=X_{0}$ is the presheaf of
$0$-simplices.

######  Remark 4.2.

We will also often not be explicit with the notation of the embeddings $y$ and
${}^{c}(-)$ unless we wish to emphasize the adjunctions.

######  Lemma 4.3.

There is a simplicial Yoneda lemma, namely given a simplicial presheaf $F$ and
a cartesian space $U$, there is a natural isomorphism:

(16) $\underline{\mathsf{sPre}(\mathsf{Cart})}(yU,F)\cong F(U).$

###### Proof.

(17) $\displaystyle\underline{\mathsf{sPre}(\mathsf{Cart})}(yU,F)_{n}$
$\displaystyle=\mathsf{sPre}(\mathsf{Cart})(yU\otimes\Delta^{n},F)$
$\displaystyle\cong\mathsf{sPre}(\mathsf{Cart})(yU,F^{\Delta^{n}})$
$\displaystyle=\mathsf{sPre}(\mathsf{Cart})({}^{c}(yU),F^{\Delta^{n}})$
$\displaystyle\cong\mathsf{Sh}(\mathsf{Cart})(yU,(F^{\Delta^{n}})_{0})$
$\displaystyle\cong\mathsf{Sh}(\mathsf{Cart})(yU,F_{n})$ $\displaystyle\cong
F_{n}(U)$

where the last isomorphism is by the classical Yoneda lemma. ∎

The previous discussion can be summarized by saying that
$\mathsf{sPre}(\mathsf{Cart})$ is enriched, tensored and cotensored over
$\mathsf{sSet}$.

If $X\xrightarrow{f}Y$ is a map of simplicial presheaves such that for each
$U\in\mathsf{Cart}$, the induced map $X(U)\xrightarrow{f_{U}}Y(U)$ is a weak
equivalence of simplicial sets, we call $f$ an objectwise equivalence. If $X$
is a simplicial presheaf such that for each $U\in\mathsf{Cart}$, $X(U)$ is a
Kan complex, we say that $X$ is objectwise fibrant.

######  Lemma 4.4.

Let $X$ be a presheaf, then ${}^{c}X$ is objectwise fibrant.

######  Lemma 4.5.

Let $X\xrightarrow{f}Y$ be a map of presheaves. The induced map
${}^{c}X\xrightarrow{{}^{c}f}{}^{c}Y$ is an objectwise weak equivalence iff
$f$ is an isomorphism.

######  Definition 4.6.

Let $\mathsf{C}$ be a site. A point $p^{*}$ of $\mathsf{C}$ is a functor
$p^{*}:\mathsf{Pre}(\mathsf{C})\to\mathsf{Set}$ that preserves all small
colimits and finite limits. Let $S$ be a set of points of $\mathsf{C}$. We say
that $S$ is a set of enough points if a map of sheaves $f:X\to Y$ on
$\mathsf{C}$ is an isomorphism iff the induced map $p^{*}f:p^{*}X\to p^{*}Y$
is an isomorphism for all $p^{*}\in S$. We say that $\mathsf{C}$ is a site
with enough points if there exists a set $S$ of enough points.

######  Example 4.7.

For each $n\geq 0$, let $p_{n}^{*}$ denote the point on $\mathsf{Cart}$
defined by

$p_{n}^{*}F=\text{colim}_{r\in\mathbb{R}_{\geq 0}}F(B^{n}(1/r))$

where $B^{n}(1/r)$ is the closed ball centered at the origin of radius $1/r$
as a subset of $\mathbb{R}^{n}$.

######  Proposition 4.8 ([Sch13, Proposition 6.3.17]).

The set $\\{p_{n}^{*}\\}_{n\geq 0}$ of Example 4.7 is a set of enough points
for $\mathsf{Cart}$.

Since limits and colimits in $\mathsf{sSet}$ are calculated levelwise, a point
$p_{n}^{*}$ on $\mathsf{Cart}$ extends to a functor
$\mathsf{sPre}(\mathsf{Cart})\to\mathsf{sSet}$. Also, since $p^{*}_{n}$ is a
filtered colimit, and filtered colimits of Kan complexes are Kan complexes, if
$X$ is a simplicial presheaf that is a Kan complex for all $B^{n}(1/r)$, then
$p^{*}_{n}X$ will be a Kan complex as well.

######  Definition 4.9.

Fix a site $\mathsf{C}$ with a set $S$ of enough points. We say that a map
$f:X\to Y$ of simplicial presheaves on $\mathsf{C}$ is a local weak
equivalence if $p^{*}f:p^{*}X\to p^{*}Y$ is a weak equivalence of simplicial
sets for all points $p^{*}\in S$. We say that a map of simplicial presheaves
is a local fibration if $p^{*}f:p^{*}X\to p^{*}Y$ is a Kan fibration. Let $*$
denote the terminal simplicial presheaf that assigns everything to the point
$\Delta^{0}$, then if $X\to*$ is a local fibration, we say that $X$ is locally
fibrant.

######  Theorem 4.10.

There exists a model category structure on the category of simplicial
presheaves over $\mathsf{Cart}$ called the local projective model structure,
which we denote by $\mathsf{s}\widehat{\mathsf{P}}\mathsf{re}$, whose weak
equivalences are the local weak equivalences.

######  Remark 4.11.

To fully construct the local projective model structure would take us too far
away from the main thread of this paper. For more details see [Bla01] and
[Jar15].

######  Lemma 4.12.

If $X\xrightarrow{f}Y$ is an objectwise (fibration) weak equivalence, then $f$
is a local (fibration) weak equivalence.

###### Proof.

This follows from [DHI04], more specifically, if $\mathsf{C}$ is a model
category and $L_{W}\mathsf{C}$ is a left Bousfield localization of it, then if
$f$ is a weak equivalence in $\mathsf{C}$, then it will be a weak equivalence
in $L_{W}\mathsf{C}$, and the local projective model structure is a left
Bousfield localization of the projective model structure whose weak
equivalences are the objectwise weak equivalences. Objectwise fibrations are
local fibrations by the description of local fibrations using local liftings
in [DHI04, Section 3]. ∎

######  Lemma 4.13 ([Bla01, Page 2]).

Consider the set of maps:

$I_{\mathsf{sPre}(\mathsf{Cart})}\coloneqq\left\\{yU\otimes\partial\Delta^{n}\xrightarrow{1_{U}\otimes
i}yU\times\Delta^{n}\right\\},$

where $U\in\mathsf{Cart}$, these are the generating cofibrations, and

$J_{\mathsf{sPre}(\mathsf{Cart})}\coloneqq\left\\{yU\otimes\Lambda^{n}_{k}\xrightarrow{1_{U}\otimes
j}yU\otimes\Delta^{n}\right\\}$

are the generating trivial cofibrations for the local projective model
structure on $\mathsf{sPre}(\mathsf{Cart})$.

Taking $\varnothing=\partial\Delta^{0}\to\Delta^{0}$, we see that
$\varnothing\to yU\otimes\Delta^{0}\cong yU$ is a cofibration, thus:

######  Corollary 4.14.

If $U\in\mathsf{Cart}$, then ${}^{c}yU$ is a cofibrant simplicial presheaf.

######  Remark 4.15.

For necessary and sufficient conditions for a simplicial presheaf to be
cofibrant in the local projective model structure see [htt].

We call the fibrant objects of the local projective model structure
$\infty$-sheaves.

######  Theorem 4.16 ([Hol07, Theorem 5.4]).

Let $X:\mathsf{Cart}^{op}\to\mathsf{Gpd}$ be a groupoidal presheaf. Then $X$
is a stack iff the simplicial presheaf $NF$, given by the nerve functor, is an
$\infty$-sheaf.

######  Corollary 4.17.

Let $X:\mathsf{Cart}^{op}\to\mathsf{Set}$ be a presheaf, then ${}^{c}X$ is an
$\infty$-sheaf iff $X$ is a sheaf.

######  Corollary 4.18.

For a diffeological group $G$, the simplicial presheaf

$NBG\simeq N\mathsf{DiffPrinc}_{G}$

is an $\infty$-sheaf.

######  Definition 4.19.

If $X$ and $A$ are simplicial presheaves, then define

(18)
$\mathbb{R}\text{Hom}(X,A)\coloneqq\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,RA)$

where $QX$ is a cofibrant replacement for $X$ and $RA$ is a fibrant
replacement ($\infty$-sheafification) of $A$. We call
$\mathbb{R}\text{Hom}(X,A)$ the derived mapping space of $X$ and $A$.

######  Remark 4.20.

It must be emphasized that the local fibrations are not the fibrations of the
local projective model structure. This leads one to consider the category of
fibrant objects structure on simplicial presheaves, which we turn to in
Section 6.

##  5\. Cofibrant Replacement of Diffeological Spaces

Recall that a diffeological space $X$ is a concrete sheaf on $\mathsf{Cart}$.
By the embedding
$\mathsf{Sh}(\mathsf{Cart})\hookrightarrow\mathsf{sPre}(\mathsf{Cart})$, we
can consider $X$ as a discrete simplicial presheaf on $\mathsf{Cart}$. Thus we
can consider its cofibrant replacement $QX$ in $\mathsf{sPre}(\mathsf{Cart})$
equipped with the local projective model structure.

Since $\mathsf{sPre}(\mathsf{Cart})$ is a combinatorial model category, a
cofibrant replacement functor $Q$ exists. However, we are in the lucky
situation that there is a cofibrant replacement functor $Q$ with a relatively
simple form.

######  Lemma 5.1 ([Dug01, Lemma 2.7]).

Given a diffeological space $X$, thought of as a discrete simplicial presheaf
on $\mathsf{Cart}$, its cofibrant replacement $QX$ is given by the simplicial
presheaf

$QX=\int^{[n]\in\mathsf{\Delta}}\Delta^{n}\times\left(\coprod_{U_{p_{n}}\to\dots\to
U_{p_{0}}\to X}U_{p_{n}}\right)$

Let us examine this coend formula in more detail. In degree $k$, we have

$(QX)_{k}=\coprod_{U_{p_{k}}\to\dots\to U_{p_{0}}\to X}U_{p_{k}}$

where the coproduct is taken over the set $(N\mathsf{Plot}(X))_{k}$, namely
the set of $k$ composable morphisms in the category of plots over $X$. We will
let $N_{k}\coloneqq(N\mathsf{Plot}(X))_{k}$.

The face maps are given as follows:

(19)
$d_{i}(x_{p_{k}},f_{k-1},f_{k-2},\dots,f_{0})=\begin{cases}(f_{k-1}(x_{p_{k}}),f_{k-2},\dots,f_{0})&i=0\\\
(x_{p_{k}},f_{k-1},\dots,f_{k-i-1}f_{k-i},\dots,f_{0})&0<i<k\\\
(x_{p_{k}},f_{k-1},\dots,f_{1})&i=k.\end{cases}$

Degeneracies insert identity maps.

For convenience, we will denote the coproduct over all plots as

$B\coloneqq(QX)_{0}=\coprod_{U_{p}\to X}U_{p}.$

Notice that there is a canonical map

$B\xrightarrow{\pi}X$

given by $\pi(p,x)=p(x)$.

This map induces a map $QX\to{}^{c}X$ of simplicial presheaves and [Dug01,
Lemma 2.7] proves that this is an objectwise weak equivalence, and thus a
local weak equivalence.

Now $B$ is a diffeological space, and we can consider the set of smooth maps
$f:B\to B$ such that the following diagram commutes:

${B}$${B}$${X}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{f}$

Patrick Iglesias-Zemmour calls this the gauge monoid in [Igl20]. It inherits
the subspace diffeology from the functional diffeology on $C^{\infty}(B,B)$.
Notice that $M$ acts on $B$ by $B\times M\xrightarrow{\rho}B$, where
$\rho(b,m)=m(b)$.

Now let us consider the following simplicial diffeological space $B//M$
defined as

$(B//M)_{k}=B\times M^{k}$

with face maps given by

(20)
$d_{i}(b,m_{k-1},\dots,m_{0})=\begin{cases}(m_{k-1}(b),m_{k-2},\dots,m_{0})&i=0\\\
(b,m_{k-1},\dots,m_{k-i-1}m_{k-i},\dots,m_{0})&0<i<k\\\
(b,m_{k-1},\dots,m_{1})&i=k\end{cases}$

We will define a map of simplicial diffeological spaces

$QX\xrightarrow{\phi}B//M$

as follows. Consider the map

$\left(\coprod_{(f_{k-1},\dots,f_{0})\in
N_{k}}U_{p_{k}}\right)\xrightarrow{\phi_{k}}B\times M^{k}$

defined by

$\phi_{k}(b,f_{k-1},\dots,f_{0})=(b,\coprod_{N_{k}}f_{k-1},\dots,\coprod_{N_{k}}f_{0})$

where $\coprod_{N_{k}}f_{i}$ is the coproduct of all maps of plots in the
$i$th position in $N_{k}=(N\mathsf{Plot}(X))_{k}$. In other words, $\phi_{k}$
is constant in $(f_{k-1},\dots,f_{0})$, sending all of them to the complete
coproduct of all the maps of plots.

######  Proposition 5.2.

The map $\phi:QX\to B//M$ is an isomorphism of simplicial diffeological
spaces.

###### Proof.

We leave it to the reader to check that $\phi$ is a map of simplicial
diffeological spaces. We will provide an inverse $\psi:B//M\to QX$ to $\phi$,
defined as follows. In degree $k$, define $\psi_{k}:B\times
M^{k}\to\coprod_{N_{k}}U_{p_{k}}$ as

$\psi_{k}(x_{p_{k}},m_{k-1},\dots,m_{0})=(x_{p_{k}},m_{k-1}|_{U_{p_{k}}},m_{k-2}|_{U_{p_{k-1}}},\dots,m_{0}|_{U_{p_{1}}})$

where $U_{p_{k-i}}$ denotes the image of the map $m_{k-i}|_{U_{p_{k-i+1}}}$.
It is easy to see that this defines a two-sided inverse to $\phi_{k}$. ∎

######  Remark 5.3.

Note that $B//M$ is used in Iglesias-Zemmour’s construction of Cech cohomology
in [Igl20]. Namely his Cech cohomology is constructed by applying the Dold-Kan
correspondence to the cosimplicial abelian group
$(\mathbb{R}^{\delta})^{B\times M^{\bullet}}$, where $\mathbb{R}^{\delta}$ is
the abelian group of real numbers with the discrete diffeology. We will have
more to say about this in section 8.

##  6\. Principal Infinity Bundles and Categories of Fibrant Objects

By Remark 4.20, we wish to consider a different homotopical structure on the
category of simplicial presheaves that takes into account the local
fibrations. This homotopical structure is called a category of fibrant objects
structure, and we will use this to describe the main theorem of [NSS14a]
regarding principal $\infty$-bundles.

######  Definition 6.1.

A category of fibrant objects is a category $\mathsf{C}$ with finite products
equipped with two classes of maps $\mathsf{Weak}$ and $\mathsf{Fib}$, called
weak equivalences and fibrations respectively, (we call the intersection
$\mathsf{Fib}\cap\mathsf{Weak}$ trivial fibrations) such that:

  1. (1)

$\mathsf{Weak}$ and $\mathsf{Fib}$ contain all isomorphisms of $\mathsf{C}$,

  2. (2)

weak equivalences satisfy the $2$-of-$3$ property, namely if $f$ and $g$ are
maps with $gf$ defined such that any two of the three $f,g,gf$ are weak
equivalences, then the third is a weak equivalence as well, and $\mathsf{Fib}$
is closed under composition,

  3. (3)

fibrations and trivial fibrations are stable under pullback,

  4. (4)

there exist path objects in $\mathsf{C}$, i.e. for every object $X$, there
exists an object $X^{I}$ such that the diagonal $X\xrightarrow{\Delta}X\times
X$ factors as a weak equivalence followed by a fibration:

(21) ${{X^{I}}}$${X}$${{X\times
X}}$$\scriptstyle{w}$$\scriptstyle{f}$$\scriptstyle{\Delta=(1_{X},1_{X})}$

but this path object does not have to be unique, nor does the factorization.

  5. (5)

every object is fibrant, i.e. the unique map $X\to*$ is a fibration.

######  Remark 6.2.

The full subcategory on the fibrant objects of a model category form a
category of fibrant objects.

######  Theorem 6.3 ([Jar15, Proposition 4.33]).

The category of locally fibrant simplicial presheaves forms a category of
fibrant objects, where the weak equivalences are the local weak equivalences
and the fibrations are the local fibrations.

A simplicial model category has a derived mapping space
$\mathbb{R}\text{Hom}(X,Y)$, and a category of fibrant objects has what is
called the cocycle category

######  Definition 6.4.

Given two objects $X,Y$ in a category of fibrant objects, a cocycle between
them is a span:

${X}$${A}$${Y}$$\scriptstyle{\sim}$

where the left hand map is a trivial fibration. A map of two cocycles is a
diagram:

${A}$${X}$${Y}$${B}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where the vertical map is a weak equivalence. Let $\mathsf{Cocycle}(X,Y)$
denote the category of cocycles between $X$ and $Y$.

######  Proposition 6.5.

Let $X,Y$ be locally fibrant simplicial presheaves, then there exists a weak
equivalence of simplicial sets

$\mathbb{R}\text{Hom}(X,Y)\simeq N\mathsf{Cocycle}(X,Y)$

###### Proof.

Combine [Low15, Lemma 6.4] with [Low15, Theorem 3.12] and [DK80, Corollary
4.7]. ∎

######  Definition 6.6 ([NSS14a, Definition 3.79]).

Let $G$ be a simplicial presheaf which is objectwise a simplicial group acting
on a simplicial presheaf $P$ by $\rho:P\times G\to P$. Then a map
$P\xrightarrow{\pi}X$ is a $G$-Principal $\infty$-bundle333In [NSS14a], what
we call principal $\infty$-bundles are known as weakly principal $G$-bundles.
Also, they only define this for simplicial sheaves, but there are no problems
extending the definition and all of the theorems in that paper to simplicial
presheaves. if:

  1. (1)

$\pi$ is a local fibration,

  2. (2)

The action of $G$ on $P$ is fiberwise, namely $\rho(g,-)$ sends fibers to
fibers, and

  3. (3)

the map

$P\times G\to P\times_{X}P$

given by

$(p,g)\mapsto(p,\rho(p,g))$

is a local weak equivalence.

A map $P\xrightarrow{f}P^{\prime}$ of $G$-principal $\infty$-bundles over $X$
is a map that is $G$-equivariant and commutes with the bundle projections.
Namely it is a map fitting into the following commutative diagram:

${{P\times G}}$${{P^{\prime}\times
G}}$${P}$${{P^{\prime}}}$${X}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{f\times
1_{G}}$

Let $\infty\mathsf{Princ}_{G}(X)$ denote the category of $G$-principal
$\infty$-bundles on $X$.

Let $G$ be a diffeological group, and let $BG$ denote the corresponding stack
on $\mathsf{Cart}$. Abuse notation and let $BG$ also denote $NBG$, the nerve
of $BG$, considered as a simplicial presheaf. Since $BG$ is the nerve of a
group(oid), it will be objectwise fibrant and thus locally fibrant.

######  Theorem 6.7.

Let $X$ be a locally fibrant simplicial presheaf, then there is a weak
equivalence of simplicial sets

$N\mathsf{Cocycle}(X,BG)\simeq N\infty\mathsf{Princ}_{G}(X).$

###### Proof.

This is a special case of [NSS14a, Theorem 3.95]. ∎

######  Corollary 6.8.

For $X$ a diffeological space and $G$ a diffeological group, there is a weak
equivalence of simplicial sets

(22)
$\mathbb{R}\text{Hom}(X,BG)\cong\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,BG)\simeq
N\infty\mathsf{Princ}_{G}(X)$

We wish to show that
$N\mathsf{DiffPrinc}_{G}(X)\simeq\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,BG).$
In the next section we will show that the formulae produced from
$\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,BG)$ exactly match that of a
diffeological principal $G$-bundle. We will follow the same approach as
[FSS11, Section 3.2.1], with a bit more detail.

Recall that $QX$ is defined as a coend
$QX=\int^{[k]}{}^{c}(QX_{k})\otimes\Delta^{k}$, and since colimits are defined
levelwise for simplicial sets, some unravelling gives us

$\displaystyle\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,BG)$
$\displaystyle\cong\int_{[k]}\underline{\mathsf{sPre}(\mathsf{Cart})}({}^{c}QX_{k}\otimes\Delta^{k},BG)$
$\displaystyle\cong\int_{[k]}\underline{\mathsf{sPre}(\mathsf{Cart})}(\coprod_{N_{k}}yU_{p_{k}},BG^{\Delta^{k}})$
$\displaystyle\cong\int_{[k]}\prod_{N_{k}}\underline{\mathsf{sPre}(\mathsf{Cart})}(yU_{p_{k}},BG^{\Delta^{k}})$
$\displaystyle\cong\int_{[k]}\prod_{N_{k}}BG^{\Delta^{k}}(U_{p_{k}})$
$\displaystyle\cong\int_{[k]}\underline{\mathsf{sSet}}\left(\Delta^{k},\prod_{N_{k}}BG(U_{p_{k}})\right)$

This is a well known end444Careful readers might be worried about how we
pulled the coend out of the enriched hom, but this is valid by the dual of
[Rie14, Theorem 7.5.3]. called totalization, which we denote by
$\mathsf{Tot}(BG(QX))$. It is a simplicial set with an explicit description
given as follows. First we note that $BG(QX)$ is a cosimplicial simplicial
set, i.e. for every pair $[m],[n]$, $BG(QX)_{[m],[n]}\coloneqq
BG_{[n]}(QX_{m})=\prod_{N_{m}}C^{\infty}(U_{p_{m}},G^{\times n})$. Then

(23)
$\mathsf{Tot}(BG(QX))_{k}\cong\mathsf{csSet}(\Delta^{\bullet}\times\Delta^{k},BG(QX))$

where $\mathsf{csSet}$ is the category of cosimplicial simplicial sets, and
$\Delta^{\bullet}\times\Delta^{k}$ is the cosimplicial simplicial set defined
by
$\left(\Delta^{\bullet}\times\Delta^{k}\right)_{[m],[n]}\coloneqq\left(\Delta^{m}\times\Delta^{k}\right)_{n}$.
For more details consult [Bou03, Section 2.8].

A $k$-simplex of $\mathsf{Tot}(BG(QX))$ thus contains a huge amount of
information, but in this case, since $BG$ comes from a groupoid, most of this
information will be redundant. Let us describe what a vertex of this
simplicial set is. It is a map of cosimplicial simplicial sets
$\Delta^{\bullet}\times\Delta^{0}\cong\Delta^{\bullet}\to BG(QX)$. This means
it is a commutative diagram of the form555Where we exclude the codegeneracy
maps from the notation for clarity.:

${{\Delta^{0}}}$${{\Delta^{1}}}$${{\Delta^{2}}}$${\dots}$${{\prod_{\mathsf{Plot}(X)}BG(U_{p_{0}})}}$${{\prod_{N_{1}}BG(U_{p_{1}})}}$${{\prod_{N_{2}}BG(U_{p_{2}})}}$${\dots}$$\scriptstyle{g^{1}}$$\scriptstyle{g^{2}}$$\scriptstyle{g^{0}}$

Let us unravel what this means. Firstly, $g^{0}$ contributes no information,
as $BG(U_{p_{0}})_{0}=C^{\infty}(U_{p_{0}},*)$. However, $g^{1}$ is the data
of maps $g^{1}(f_{0})\coloneqq g_{f_{0}}:U_{p_{1}}\to G$ for every map of
plots $f_{0}:U_{p_{1}}\to U_{p_{0}}$. Now $g^{2}$ is the data of a map
$g^{2}(f_{1},f_{0})\coloneqq g_{f_{1},f_{0}}:U_{p_{2}}\to G\times G$ for every
pair of composable maps of plots
$U_{p_{2}}\xrightarrow{f_{1}}U_{p_{1}}\xrightarrow{f_{0}}U_{p_{0}}$. Let
$g^{2}(f_{1},f_{0})=(h,k)$. The data of the cosimplicial maps commuting
insists that

$(d^{0}g^{1})(f_{1},f_{0})=g^{1}(f_{0})\circ
f_{1}=(g^{2}d^{0})(f_{1},f_{0})=d_{0}(g^{2}(f_{1},f_{0}))=d_{0}(h,k)=k$
$(d^{1}g^{1})(f_{1},f_{0})=g^{1}(f_{0}f_{1})=(g^{2}d^{1})(f_{1},f_{0})=d_{1}(g^{2}(f_{1},f_{0}))=d_{1}(h,k)=kh$
$(d^{2}g^{1})(f_{1},f_{0})=g^{1}(f_{1})=(g^{2}d^{2})(f_{1},f_{0})=d_{2}(g^{2}(f_{1},f_{0}))=d_{2}(h,k)=h$

In other words

(24) $g_{f_{0}f_{1}}=(g_{f_{0}}\circ f_{1})\cdot g_{f_{1}}.$

We can visualize this as a triangle:

${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{g_{f_{0}f_{1}}}$$\scriptstyle{g_{f_{1}}}$$\scriptstyle{g_{f_{0}}\circ
f_{1}}$$\scriptstyle{g^{2}(f_{1},f_{0})}$

which is filled in if the condition (24) holds. The codegeneracy maps will
guarantee that if $f$ is an identity map of plots $f=1_{U_{p}}$, then
$g_{f}=e_{G}$ is the constant map at the unit of $G$.

Now here’s an important point: $g^{3}$ will provide no further data. We will
explain why using the notion of being coskeletal.

######  Definition 6.9.

A simplicial set $X$ is $k$-coskeletal if for every boundary
$\partial\Delta^{n}\to X$, there exists a unique $n$-simplex $\Delta^{n}\to X$
making the following diagram commute:

${{\partial\Delta^{n}}}$${X}$${{\Delta^{n}}}$

for all $n>k$.

It is well known that if $X=N(\mathsf{C})$ is the nerve of a category
$\mathsf{C}$, then $X$ will be $2$-coskeletal. Intuitively this is because if
we have the boundary of a triangle:

${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$

in $N(\mathsf{C})$, then this corresponds to three maps in $\mathsf{C}$ with
$s(f)=s(h),t(f)=s(g)$ and $t(h)=t(g)$. This boundary is filled iff $h=gf$.
Thus not every $2$-boundary is filled, however, there cannot be two distinct
$2$-simplexes with the same boundary. Filling $2$-boundaries is unique, but
may not always exist. A $3$-boundary would then be a collection of $6$
morphisms that are appropriately composable, and a filler exists iff they are
that composition, i.e. of the form $\\{f,g,h,gf,hg,hgf\\}$. But a $3$-boundary
requires four $2$-simplices, which will cause any three maps along the
boundary of the $2$-simplex to be composed. This implies that for every
$3$-boundary, there will always exist a filler, and it will be unique. This
extends upwards, causing every $n$-boundary to have a unique filler for $n>2$.

In our case $BG(QX)$ is a cosimplicial simplicial set such that $BG(QX_{i})$
is $2$-coskeletal for every $i$. Now as we’ve seen, the $3$-simplex $g^{3}\in
BG(QX_{3})$ is required to satisfy that $\partial
g^{3}=(d^{0}g^{1},d^{1}g^{1},d^{2}g^{1},d^{3}g^{1})$. But that means we’ve
just specified a $3$-boundary in a $2$-coskeletal simplicial set. Thus there
exists a unique filler $g^{3}$. This of course continues, so that a vertex
$g\in\mathsf{Tot}(BG(QX))$ determines and is completely determined by $g^{1}$
and $g^{2}$.

Let us repeat this analysis for a $1$-simplex in $\mathsf{Tot}(BG(QX))$. This
is the data of a commutative diagram:

${{\Delta^{0}\times\Delta^{1}}}$${{\Delta^{1}\times\Delta^{1}}}$${{\Delta^{2}\times\Delta^{1}}}$${\dots}$${{\prod_{N_{0}}BG(U_{p_{0}})}}$${{\prod_{N_{1}}BG(U_{p_{1}})}}$${{\prod_{N_{2}}BG(U_{p_{2}})}}$${\dots}$$\scriptstyle{h^{0}}$$\scriptstyle{h^{1}}$$\scriptstyle{h^{2}}$

Now unravelling this diagram, skipping some similar details, such a
$1$-simplex consists of the following data. If $g$ and $g^{\prime}$ are
$0$-simplices in $\mathsf{Tot}(BG(QX))$ consisting of collections of maps
$\\{g_{f}\\}$ and $\\{g^{\prime}_{f}\\}$, then a $1$-simplex is a collection
of maps $\\{h_{p_{0}}:U_{p_{0}}\to G\\}$ indexed by plots $p_{0}:U_{p_{0}}\to
X$ such that if $f_{0}:U_{p_{1}}\to U_{p_{0}}$ is a map of plots, then

(25) $g^{\prime}_{f}\cdot h_{p_{1}}=h_{p_{0}}\cdot g_{f}.$

By the same reasoning as before, the rest of the diagram provides no further
conditions on this data, as the maps $\Delta^{k}\times\Delta^{1}\to
BG(QX_{k})$ will consist of $(k+1)$-simplices, and $BG(QX_{k})$ is
$2$-coskeletal, so that $h$ depends only on $h^{0}$ and $h^{1}$.

A $2$-simplex in $\mathsf{Tot}(BG(QX))$ will similarly be completely
determined by its boundary $h,h^{\prime},h^{\prime\prime}$.

Similar reasoning also proves that there are no additional conditions coming
from higher $k$-simplices of $\mathsf{Tot}(BG(QX))$. In other words,
$\mathsf{Tot}(BG(QX))$ is $2$-coskeletal.

######  Lemma 6.10 ([nLa22, Proposition 2.1]).

If $X$ is a $k$-coskeletal Kan complex, then $\pi_{n}(X,x)=0$ for all $n>k-1$
and $x\in X_{0}$.

Thanks to the lemma, we know that if $\mathsf{Tot}(BG(QX))$ is a Kan complex,
then it has no homotopy groups except for $\pi_{0}$ and $\pi_{1}$. We will use
this observation to show that
$N\mathsf{DiffPrinc}_{G}(X)\simeq\mathsf{Tot}(BG(QX))$ in the next section.
But first we must show that $\mathsf{Tot}(BG(QX))$ is indeed a Kan complex.
However this follows from the isomorphism

$\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,BG)\cong\mathsf{Tot}(BG(QX)),$

because the local projective model structure on $\mathsf{sPre}(\mathsf{Cart})$
is a simplicial model category, and
$\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,BG)$ is a Kan complex because
$QX$ is locally projective cofibrant and by Lemma 3.8 and Theorem 4.16 $BG$ is
locally projective fibrant, i.e. an $\infty$-sheaf.

##  7\. Principal Infinity Bundles and Diffeological Principal Bundles

In this section, we will show how to produce a diffeological principal
$G$-bundle from the data of a vertex of $\mathsf{Tot}(BG(QX))$ from the last
section, and a map of such bundles from a $1$-simplex.

Consider the diffeological groupoid

$G\times G\underset{m}{\overset{\text{pr}_{1}}{\rightrightarrows}}G$

where the source map is the first projection and the target map is the
multiplication map. This defines a groupoidal presheaf

$EG\coloneqq\left[U\mapsto\left(C^{\infty}(U,G\times G)\rightrightarrows
C^{\infty}(U,G)\right)\right].$

There is a canonical map of simplicial presheaves $EG\to BG$ induced by the
corresponding map of diffeological groupoids:

${{G\times
G}}$${G}$${G}$${{*}}$$\scriptstyle{\text{pr}_{1}}$$\scriptstyle{m}$$\scriptstyle{s}$

where $s$ is the map $(g_{1},g_{2})\mapsto g_{2}g_{1}^{-1}$. It is worth
noting that $(NEG)_{k}\cong G^{\times k+1}$ and $(NBG)_{k}\cong G^{\times k}$
are diffeological spaces for each $k\geq 0$, and thus we can also think of
$BG$ and $EG$ as simplicial diffeological spaces. Further, $EG$ is a
simplicial object in the category of diffeological groups, equivalently a
group object in simplicial diffeological spaces.

Now recall if $M$ is a finite dimensional smooth manifold, and $G$ is a Lie
group, there is an infinite dimensional Frechet manifold called $BG$, and
smooth homotopy classes of maps $M\to BG$ are in bijection with isomorphism
classes of smooth principal $G$-bundles on $M$. There is a particularly
pleasant way to think about this in terms of stacks. Here we merely reproduce
the exposition in [Sch13, Section 1.2.6.1]. We know that $\underline{M}$ and
$BG$ are stacks on $\mathsf{Cart}$, and a map $\underline{M}\to BG$ is
equivalent to a map of Lie groupoids $\check{C}(\mathcal{U})\to BG$ , where
$\mathcal{U}$ is a differentiably good cover. Such a map is equivalently a
cocycle $\\{g_{ij}:U_{ij}\to G\\}$. Now to produce the desired principal
$G$-bundle one considers the $2$-pullback

${{\widetilde{P}}}$${EG}$${\check{C}(\mathcal{U})}$${BG}$$\scriptstyle{g}$$\scriptstyle{\lrcorner}$

in the category of Lie groupoids. Then $P\coloneqq\pi_{0}(\widetilde{P})$
turns out to be a smooth finite dimensional manifold, and there is a canonical
map $P\xrightarrow{\pi}X$ which is exactly the principal $G$-bundle one would
construct using the bundle construction Proposition 3.6. We will use this same
idea, generalized to simplicial presheaves and diffeological spaces.

Given a diffeological group $G$ and a diffeological space $X$, suppose there
is a map $g:QX\to BG$. Since $BG$ is the nerve of a diffeological groupoid,
the only data of this map is in degrees $0$ and $1$. Unravelling the
definitions, we see that a such a map $g$ is determined by a collection of
maps $\\{g_{f_{0}}:U_{p_{1}}\to G\\}$ for every map of plots

${{U_{p_{1}}}}$${{U_{p_{0}}}}$${X}$$\scriptstyle{f_{0}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{0}}$

such that if we have a composition of such maps:

${{U_{p_{1}}}}$${{U_{p_{2}}}}$${{U_{p_{0}}}}$${X}$$\scriptstyle{f_{0}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{0}}$$\scriptstyle{f_{1}}$$\scriptstyle{p_{2}}$

then

(26) $g_{f_{0}f_{1}}(x)=(g_{f_{0}}\circ f_{1})(x)\cdot g_{f_{1}}(x),\text{ and
if }f_{0}=1_{U_{p_{0}}},\text{ then }g_{1_{U_{p_{0}}}}=e_{G}$

where $e_{G}:U_{p_{0}}\to G$ is the constant map at the unit of $G$. This is
the analogue of the cocycle condition for diffeological spaces, where instead
of indexing cocycles by open covers we index them by maps in the plot
category, and is precisely the condition (24). This means that a vertex of
$\mathsf{Tot}(BG(QX))$ is precisely a map $QX\to BG$.

With such a map $g$, we can consider the pullback:

${{\widetilde{P}}}$${EG}$${QX}$${BG}$$\scriptstyle{g}$$\scriptstyle{\lrcorner}$

in the category of simplicial presheaves (equivalently in the category of
simplicial diffeological spaces), so that $\widetilde{P}$ is the simplicial
diffeological space such that

$\widetilde{P}_{1}=\coprod_{U_{p_{1}}\xrightarrow{f_{0}}U_{p_{0}}}U_{p_{1}}\times_{G}(G\times
G),\qquad\widetilde{P}_{0}=\coprod U_{p_{0}}\times G.$

We can visualize the elements of $\widetilde{P}_{0}$ as points labeled by
$(p,u,h)$, where $p:U_{p}\to X$ is a plot, $u\in U_{p}$, and $h\in G$, and a
map (i.e. element of $\widetilde{P}_{1}$) from $(p,u,h)$ to $(q,v,k)$ is a map
$U_{p}\xrightarrow{f}V_{q}$ of plots, such that $f(u)=v$ and $k=g_{f}(u)h$.
Now consider the relation $(p,u,h)\sim(q,v,k)$ where two points are equivalent
if there exists a morphism between them. This is not a symmetric relation, so
consider the smallest equivalence relation containing this relation, namely
its symmetric closure. We can quotient $\widetilde{P}_{0}$ by this equivalence
relation, obtaining a diffeological space $P$. It is easy to see that

$\pi_{0}(\widetilde{P})\cong P.$

There is a canonical map

(27) $P\xrightarrow{\pi}X,\qquad[p,u,h]\mapsto p(u)$

and this is well defined because if $[p,u,h]=[q,v,k]$, then there exists a map
$f$ such that $qf=p$, and $f(u)=v$. Then
$\pi([q,v,k])=q(v)=q(f(u))=qf(u)=p(u)$.

######  Theorem 7.1.

Given a diffeological group $G$, diffeological space $X$ and map
$QX\xrightarrow{g}BG$, the map $P\xrightarrow{\pi}X$ defined in (27) is a
diffeological principal $G$-bundle.

###### Proof.

Let us show that if $p:U\to X$ is a plot, then $p^{*}P$ is $G$-equivariantly
diffeomorphic to $U\times G$. Note that

$p^{*}P=\\{(u,[q,v,k])\in U\times P\,:\,p(u)=q(v)\\}$

So given a point $(u,[q,v,k])\in p^{*}P$, we have a diagram

${{*}}$${V}$${U}$${X}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{u}$$\scriptstyle{v}$$\scriptstyle{x}$

Now let us define a map $p^{*}P\xrightarrow{\varphi}U\times G$ as follows.
Note that $[q,v,k]=[x,*,g^{-1}_{v}(*)k]$, as $v:*\to V$ is a map of plots
$x\xrightarrow{v}q$. Similarly
$[p,u,g_{u}(*)g_{v}^{-1}(*)k]=[x,*,g_{v}^{-1}(*)k]$. For brevity, if $u:*\to
U$ is a constant map of plots, then denote $g_{u}(*)$ by $g_{u}$. So define
$\varphi(u,[q,v,k])=(u,g_{u}g_{v}^{-1}k)$. Let us show that this is well
defined. If $[r,w,l]=[q,v,k]$, then there exists a diagram:

${{*}}$${W}$${V}$${X}$$\scriptstyle{q}$$\scriptstyle{r}$$\scriptstyle{v}$$\scriptstyle{w}$$\scriptstyle{f}$

and thus $l=g_{f}(v)k$. Notice that $g_{w}=g_{fv}=(g_{f}\circ v)g_{v}$ by the
cocycle condition (26). Thus $g_{w}=g_{f}(v)g_{v}$, and therefore
$g_{v}^{-1}=g_{w}^{-1}g_{f}(v)$. Thus

$\varphi(u,[r,w,l])=(u,g_{u}g_{w}^{-1}l)=(u,g_{u}g_{w}^{-1}g_{f}(v)k)=(u,g_{u}g_{v}^{-1}k)=\varphi(u,[q,v,k]).$

So $\varphi$ is well defined. It is also $G$-equivariant, as
$\varphi(u,[q,v,k]\cdot
g)=\varphi(u,[q,v,kg])=(u,g_{u}g_{v}^{-1}kg)=(u,g_{u}g_{v}^{-1}k)\cdot g$. We
define an inverse $\psi:U\times G\to p^{*}P$ by $(u,g)\mapsto(u,[p,u,g])$.
This is clearly $G$-equivariant, and it is easy to see that
$\varphi\psi=1_{U\times G}$ and $\psi\varphi=1_{p^{*}P}$.

Now let us show that $G$ acts on the fibers of $\pi$ freely and transitively.
However this is imediate, as a fiber of $\pi$ is in particular a pullback:

${{\pi^{-1}(x)}}$${P}$${{*}}$${X}$$\scriptstyle{x}$$\scriptstyle{\pi}$$\scriptstyle{\lrcorner}$

and every constant map $*\to X$ is a plot, and we’ve already shown that for
any plot this pullback is trivial, and thus $\pi^{-1}(x)\cong*\times G\cong
G$, which acts freely and transitively on itself by right multiplication.

Thus by Lemma 2.24, $P\xrightarrow{\pi}X$ is a diffeological principal
$G$-bundle. ∎

Now given a diffeological principal $G$-bundle $P\xrightarrow{\pi}X$, suppose
that we have a map of plots $U\xrightarrow{f}V$ over $X$. Then pulling back
$P$ along these maps gives a map $U\times G\xrightarrow{f^{*}}V\times G$ over
$X$. This map is given by $(u,h)\mapsto(f(u),g_{f}(u)h)$. If we have a
composition of maps of plots over $X$:
$U\xrightarrow{f_{1}}V\xrightarrow{f_{0}}W$, then we get
$(u,h)\mapsto(f_{1}(u),g_{f_{1}}(u)h)\mapsto(f_{0}f_{1}(u),g_{f_{0}}(f_{1}(u))g_{f_{1}}(u)h)$
and this has to be equal to $(u,h)\mapsto(f_{0}f_{1}(u),g_{f_{0}f_{1}}(u)h)$.
Thus we recover the cocycle condition (26) and thus a map $g:QX\to BG$.

######  Proposition 7.2.

Suppose we have two maps $g,g^{\prime}:QX\to BG$, equivalently $0$-simplices
of $\prod_{N_{0}}BG(U_{p_{0}})$, then a $1$-simplex
$h\in\left(\prod_{N_{0}}BG(U_{p_{0}})\right)_{1}$ between them satisfying
condition (25) is equivalent to a $G$-equivariant map of the resulting
diffeological principal $G$-bundles.

###### Proof.

Suppose $P$ and $P^{\prime}$ are the diffeological principal $G$-bundles
represented by the maps $g,g^{\prime}$ respectively. Then we define a map
$h:P\to P^{\prime}$ as follows. For $[p,u,k]\in P$, let
$h([p,u,k])=[p,u,h_{p}(u)k]$. If $[q,v,l]=[p,u,k]$, then $l=g_{f}(u)k$, and
thus
$h([q,v,l])=[q,v,h_{q}(v)l]=[q,v,h_{q}(v)g_{f}(u)k]=[q,v,g^{\prime}_{f}(v)h_{p}(u)k]=[p,u,h_{p}(u)k]$
by (25), so the map is well defined and is clearly smooth and $G$-equivariant.

If one has a map $h:P\to P^{\prime}$ of diffeological principal $G$-bundles,
then one can consider the map pulled back along plots. This will reproduce the
collection $\\{h_{p}\\}$ along with condition (25) as desired. ∎

What we have shown is that we can produce diffeological principal $G$-bundles
using local data, where in this case we interpret local to mean ”plotwise”. We
can thus construct a map of simplicial sets

$\varphi:\mathsf{Tot}(BG(QX))\to N\mathsf{DiffPrinc}_{G}(X)$

as follows. Since the codomain is the nerve of a groupoid we need only define
this map in degrees $0$ and $1$. Thus we send a vertex in
$\mathsf{Tot}(BG(QX))$, which is a collection $\\{g_{f}\\}$ to the
diffeological principal $G$-bundle it constructs, and similarly send a
$1$-simplex, which is a collection $\\{h_{p}\\}$ to the map of diffeological
principal $G$-bundles it determines. Since $\mathsf{Tot}(BG(QX))$ is
$2$-coskeletal by the discussion in Section 6,
$\pi_{n}(\mathsf{Tot}(BG(QX))=0$ for $n\neq 0,1$, and thus it is easy to show
that this map is a weak equivalence of simplicial sets. We need only to show
that it is an isomorphism of $\pi_{0}$ and $\pi_{1}$.

Now $\varphi$ inducing an isomorphism on $\pi_{0}$ is equivalent to saying
that $\varphi$ is a bijection between isomorphism classes of cocycles and
isomorphisms classes of diffeological principal $G$-bundles. This has already
been established above.

If $g=\\{g_{f}\\}$ is a cocycle, then we can consider
$\pi_{1}(\mathsf{Tot}(BG(QX)),g)$. This is precisely all of the maps of
cocycles $\\{h_{p}\\}$ from $g$ to $g$. It is easy to check that this is
precisely the set of automorphisms of the principal bundle $P$ constructed
from $g$, which is exactly the homotopy group
$\pi_{1}(\mathsf{DiffPrinc}_{G}(X),P)$, which is often called the gauge group
of the principal bundle.

This proves:

######  Theorem 7.3.

For any diffeological group $G$ and diffeological space $X$ the above map
$\varphi$ is a weak equivalence of Kan complexes:

(28) $N\mathsf{DiffPrinc}_{G}(X)\simeq\mathsf{Tot}(BG(QX))$

######  Theorem 7.4.

There is a weak equivalence of simplicial sets

(29) $N\mathsf{DiffPrinc}_{G}(X)\simeq N\infty\mathsf{Princ}_{G}(X)$

###### Proof.

Combine Theorem 7.3 and Corollary 6.8. ∎

##  8\. Comparison to Čech Cohomology

In this section we discuss different definitions of Čech cohomology of a
diffeological space in the literature, that of [KWW21] and [Igl20], and offer
the perspective of higher topos theory.

######  Remark 8.1.

In what follows we will always consider chain complexes and cochain complexes
to be non-negatively graded.

In [Igl20], Čech cohomology of a diffeological space $X$ is defined666Modulo
some details, Iglesias-Zemmour defines diffeological spaces with open subsets
of cartesian spaces and uses a generating family of open balls, but they are
clearly equivalent constructions. as follows. First one considers $B//M$, the
simplicial diffeological space defined in section 5. Let $\mathbb{R}^{\delta}$
denote the real numbers equipped with the discrete diffeology. Then
$(R^{\delta})^{B\times M^{\times k}}\coloneqq C^{\infty}(B\times M^{\times
k},\mathbb{R}^{\delta})$ is precisely the set of locally constant real valued
maps $(B//M)_{k}\to\mathbb{R}$. The set
$C^{\infty}((B//M)_{k},\mathbb{R}^{\delta})$ is then an abelian group by
pointwise addition. Thus we obtain a cosimplicial abelian group

${{(\mathbb{R}^{\delta})^{B}}}$${{(\mathbb{R}^{\delta})^{B\times
M}}}$${{(\mathbb{R}^{\delta})^{B\times M\times M}}}$${\dots}$

and from this one can obtain a cochain complex as follows.

If $A$ is a cosimplicial abelian group, then we can define a cochain complex
$C^{\text{co}}A$ called the alternating coface cochain complex by

$(C^{\text{co}}A)^{k}\coloneqq A^{k},\qquad
d:(C^{\text{co}}A)^{k}\to(C^{\text{co}}A)^{k+1},\qquad
d=\sum_{i=0}^{k}(-1)^{i}d^{i}.$

This makes $C^{co}$ a functor $\mathsf{cAb}\to\mathsf{CoCh}$, where
$\mathsf{cAb}$ denotes the category of cosimplicial abelian groups and
$\mathsf{CoCh}$ is the category of cochain complexes. For some applications
this functor is very useful, but for us we want a ”smaller” model of this
functor. Let if $A$ is a cosimplicial abelian group, then let $N^{co}A$ be the
cochain complex which in degree $n+1$ is given by $A_{n+1}/D(A_{n+1})$ where
$D(A_{n+1})$ denotes the degenerate $n+1$-simplices with differential induced
by the quotient. We call this the normalized cochain complex, and it is
naturally quasi-isomorphic to $C^{co}$. In other words for every cosimplicial
abelian group $A$, there is a quasi-isomorphism $C^{co}A\to N^{co}A$, natural
in $A$.

One important reason why we prefer $N^{co}$ over $C^{co}$ is that $N^{co}$ has
a right adjoint
$N^{\text{co}}:\mathsf{cAb}\rightleftarrows\mathsf{CoCh}:\Gamma^{\text{co}}$,
which is actually an adjoint equivalence, called the dual Dold-Kan
correspondence [Stacks, Tag 019H]. The classical Dold-Kan correspondence
$N:\mathsf{sAb}\rightleftarrows\mathsf{Ch}:\Gamma$ associates to a chain
complex a simplicial abelian group and vice versa, and also forms an adjoint
equivalence, and has a similar natural quasi-isomorphism $C\simeq N$. See
[GJ12, Section III.2] for more information and a definition of $\Gamma$.

The Iglesias-Zemmour Čech cohomology of $X$ is then defined as the cohomology
of this cochain complex: $\check{H}^{k}_{PIZ}(X,\mathbb{R}^{\delta})\coloneqq
H^{k}(N^{\text{co}}\left[(\mathbb{R}^{\delta})^{B//M}\right])\cong
H^{k}(C^{co}\left[(\mathbb{R}^{\delta})^{B//M}\right])$. We will show that we
can produce these cohomology groups using the local projective model structure
on simplicial presheaves.

Given an abelian group $A$ and a non-negative integer $n$, there exists a
topological space 777Only defined up to weak homotopy equivalence $K(A,n)$,
called the $n$th Eilenberg-Maclane space, which has trivial homotopy groups in
all degrees except for $n$, which has $\pi_{n}(K(A,n))=A$. One can construct
the underlying homotopy type of this space using the Dold-Kan correspondence.
Namely consider the chain complex $A[k]$, defined by

$\left(A[k]\right)_{n}=\begin{cases}A&\text{if }n=k\\\ 0&\text{if }n\neq
k\end{cases},\qquad d=0$

Then $\Gamma(A[k])$ is a Kan complex, equipped with basepoint $*$ such that

$\pi_{n}(\Gamma(A[k]),*)=\begin{cases}A&\text{if }n=k\\\ 0&\text{if }n\neq
k\end{cases}$

so the geometric realization of $\Gamma(A[k])$ is $K(A,k)$.

######  Remark 8.2.

For future reference, if $C$ is a chain complex, then let $C[k]$ denote the
chain complex such that $C[k]_{n}=C_{n-k}$, so that we identify an abelian
group A with the chain complex $A[0]$, and then $(A[0])[k]=A[k]$.

If $X$ is a diffeological space, we can take its cofibrant replacement
$QX\cong B//M$. Higher topos theory tells us that if $A$ is locally projective
fibrant, i.e. an $\infty$-sheaf, then the cohomology of $X$ with coefficients
in $A$ is defined as

$H^{0}(X,A)\coloneqq\pi_{0}\mathbb{R}\text{Hom}(X,A)\cong\pi_{0}\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,A).$

For the rest of this section we will refer to this as the $\infty$-sheaf
cohomology of $X$ with values in $A$.

When $A$ is a presheaf of simplicial groups, that is also locally projective
fibrant as a simplicial presheaf, i.e. an $\infty$-sheaf, then one defines

$H^{1}(X,A)\coloneqq\pi_{0}\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,\overline{W}A)$

where $\overline{W}:\mathsf{sGrp}\to\mathsf{sSet}$ is the delooping functor,
defined as follows.

######  Definition 8.3.

Given a simplicial group $G$, let $\overline{W}G$ denote the simplicial set
with

$\displaystyle\overline{W}G_{0}$ $\displaystyle=*$
$\displaystyle\overline{W}G_{n}$ $\displaystyle=G_{n-1}\times
G_{n-2}\times\dots\times G_{0}$

with face and degeneracy maps given by

$\displaystyle d_{i}(g_{n-1},\dots,g_{0})$
$\displaystyle=\begin{cases}(g_{n-2},\dots,g_{0})&\text{ if }i=0\\\
(d_{i-1}(g_{n-1}),\dots,d_{1}(g_{n-i+1}),g_{n-i-1}\cdot
d_{0}(g_{n-i}),g_{n-i-2},\dots,g_{0}))&\text{ if }1\leq i\leq n\end{cases}$
$\displaystyle s_{i}(g_{n-1},\dots,g_{0})$
$\displaystyle=\begin{cases}(1,g_{n-1},\dots,g_{0})&\text{ if }i=0\\\
(s_{i-1}(g_{n-1}),\dots,s_{0}(g_{n-i}),1,g_{n-i-1},\dots,g_{0})&\text{ if
}1\leq i\leq n\end{cases}$

If $\overline{W}A$ happens to be a simplicial group, then we can iterate this
construction once more. Namely if $\overline{W}^{k-1}A$ is a simplicial group,
then $\overline{W}^{k}A$ exists, and we define higher $\infty$-sheaf
cohomology groups as

$H^{k}(X,A)\coloneqq\pi_{0}\mathbb{R}\text{Hom}(X,\overline{W}^{k}A)\cong\pi_{0}\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,\overline{W}^{k}A).$

######  Lemma 8.4.

If $A$ is a simplicial abelian group, then $\overline{W}A$ will be a
simplicial abelian group.

###### Proof.

This follows from the alternate descriptions of $\overline{W}A$ given in
[Jar97, Section 4.5]. ∎

Thus if $A$ is a simplicial abelian group, $\overline{W}^{k}A$ exists for all
$k$.

######  Lemma 8.5 ([Jar97, Section 4.6]).

Let $A$ be a simplicial abelian group. Then there is an isomorphism of chain
complexes

$N\overline{W}A\cong(NA)[1]$

where $(NA)[1]$ is the chain complex $NA$ shifted up by $1$, i.e.
$(NA[1])_{k}=(NA)_{k-1}$.

######  Lemma 8.6.

If $A$ is an abelian group, thought of as a discrete simplicial abelian group
${}^{c}A$, then $\overline{W}^{k}A$ exists for every $k\geq 0$, and there
exists an isomorphism

$\overline{W}^{k}{}^{c}A\cong\Gamma(A[k])$

###### Proof.

We proceed by induction. For the base case, we have

$N\overline{W}{}^{c}A\cong(N{}^{c}A)([1]).$

But $N{}^{c}A\cong A[0]$ as is easily checked, and $(A[0])[1]=A[1]$, so
$N\overline{W}{}^{c}A\cong A[1]$, thus

$\Gamma N\overline{W}{}^{c}A\cong\overline{W}{}^{c}A\cong\Gamma A[1].$

Now suppose $\overline{W}^{k-1}{}^{c}A\cong\Gamma A[k-1]$. Then by Lemma 8.5

$N\overline{W}\left(\overline{W}^{k-1}{}^{c}A\right)\cong\left(N\Gamma
A[k-1]\right)[1]$

but $N\Gamma A[k-1]\cong A[k-1]$ since $N$ and $\Gamma$ form an adjoint
equivalence, thus:

$N\overline{W}\left(\overline{W}^{k-1}{}^{c}A\right)\cong
N\overline{W}^{k}{}^{c}A\cong A[k-1]([1])=A[k]$

taking the adjoint gives

$\overline{W}^{k}{}^{c}A\cong\Gamma A[k].$

∎

Setting $A=\mathbb{R}^{\delta}$, we obtain for any diffeological space $X$ its
$k$-th $\infty$-sheaf cohomology with coefficients in $\mathbb{R}^{\delta}$ is
defined as

$H^{k}(X,\mathbb{R}^{\delta})\coloneqq\pi_{0}\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,\Gamma(\mathbb{R}^{\delta}[k]))\cong\pi_{0}\mathsf{Tot}(\Gamma(\mathbb{R}^{\delta}[k])(QX)).$

Now $\pi_{0}\mathsf{Tot}(X)$ simplifies greatly if $X$ is a cosimplicial
simplicial abelian group, rather than a general cosimplicial simplicial set.

######  Proposition 8.7 ([Jar19, Lemma 19]).

If $A$ is a cosimplicial abelian group, then

(30) $\pi_{0}\mathsf{Tot}(\Gamma A[k])\cong H^{k}(N^{\text{co}}A)$

where $\Gamma A[k]$ denotes the cosimplicial simplicial abelian group obtained
by considering the abelian group $A^{i}$ as a simplicial abelian group
$\Gamma(A^{i}[k])$.

Now setting $A=\mathbb{R}^{\delta}(QX)$ as the cosimplicial abelian group
gives us:

######  Corollary 8.8.

For a diffeological space $X$, the $\infty$-sheaf cohomology groups with
values in $\mathbb{R}^{\delta}$ and Iglesias-Zemmour’s Čech cohomology groups
coincide:

(31) $\check{H}^{k}_{PIZ}(X,\mathbb{R}^{\delta})\cong
H^{k}(X,\mathbb{R}^{\delta})$

######  Remark 8.9.

It is trivial to extend Iglesias-Zemmour’s construction for arbitrary abelian
groups, and we have shown that this will also agree with $\infty$-sheaf
cohomology.

####  8.0.1. Comparing Diffeological Čech cohomologies

In the literature there are two definitions [Igl20] and [KWW21], of
diffeological Čech cohomology that we wish to compare. We admit outright that
we are unable to show using our framework that these cohomologies coincide,
although we document some interesting relationships between their
constructions.

The Iglesias-Zemmour Čech cochain complex888Which we shall now refer to as the
PIZ complex. is defined as

$\check{C}_{PIZ}(X,\mathbb{R}^{\delta})=N^{\text{co}}\left[\left(\mathbb{R}^{\delta}\right)^{B//M}\right].$

We’ve proven that the cohomology of the PIZ complex is in fact given by
$\infty$-sheaf cohomology. Now we shall discuss the Čech cohomology of [KWW21]
with coefficients in $\mathbb{R}^{\delta}$.

Given a diffeological space $X$, we obtain its nebula
$B=\coprod_{p\in\mathsf{Plot}(X)}U_{p}$, which is equipped with a map
$B\xrightarrow{\pi}X$ defined by $\pi(p,u)=p(u)$. We can then form the
simplicial diffeological space

${{\check{C}(B\to
X)}}$${B}$${{B\times_{X}B}}$${{B\times_{X}B\times_{X}B}}$${\dots}$$\scriptstyle{\coloneqq}$

where $B\times_{X}B$ is the pullback in the category of diffeological spaces.
Homming this simplicial diffeological space into $\mathbb{R}^{\delta}$ gives
us a cosimplicial abelian group $(\mathbb{R}^{\delta})^{\check{C}(B\to X)}$
and thus a cochain complex:

$\check{C}_{KWW}(X,\mathbb{R}^{\delta})\coloneqq
N^{\text{co}}\left[\left(\mathbb{R}^{\delta}\right)^{\check{C}(B\to
X)}\right].$

It might seem then that we can apply a similar analysis to
$\check{C}_{KWW}(X,\mathbb{R}^{\delta})$, but unfortunately that is not the
case, as $\check{C}(B\to X)$ is not a cofibrant simplicial presheaf in the
local projective model structure in general. A necessary condition for a
simplicial presheaf to be cofibrant is to be degreewise a coproduct of
retracts of representables, i.e. must be degreewise a coproduct of retracts of
cartesian spaces. However the pullback $B\times_{X}B$ will not satisfy this
condition in general.

######  Example 8.10.

Let us find an example for which $\check{C}(B\to X)$ will not be cofibrant.
Consider the circle $S^{1}$ with two plots $p,q:\mathbb{R}\to S^{1}$ defined
as follows. Let $p$ first map $\mathbb{R}$ diffeomorphically to
$(-\varepsilon,\pi+\varepsilon)$ and then map $t\mapsto e^{it}$. Similarly for
$q$ first map $\mathbb{R}$ diffeomorphically to
$(\pi-\varepsilon,2\pi+\varepsilon)$ and then map $t\mapsto e^{it}$. Then
$B\times_{X}B$ will include a term
$\mathbb{R}_{p}\times_{S^{1}}\mathbb{R}_{q}$999This follows from the category
of diffeological spaces being a quasitopos, so that we can pull out the
coproducts from the fiber product. which will clearly be homotopy equivalent
to $S^{0}$. This cannot be a retract of a cartesian space by the classical
Brouwer fixed point argument, that the identity map on the reduced homology of
$S^{0}$ cannot factor through the zero map.

However, it does happen that we can say something about $\check{C}(B\to X)$.

######  Proposition 8.11.

The natural map $\check{C}(B\to X)\to{}^{c}X$ of simplicial presheaves is a
local weak equivalence.

###### Proof.

We will prove this using [DHI04, Corollary A.3]. First one shows that $B\to X$
is a generalized cover or local epimorphism of presheaves of sets. However
both $B$ and $X$ are sheaves on $\mathsf{Cart}$, and the map $B\to X$ is
clearly objectwise a surjection of sets, and therefore an epimorphism of
sheaves. By [MM12, Corollary III.7.5] an epimorphism of sheaves is
equivalently a local epimorphism of presheaves. Thus [DHI04, Corollary A.3]
proves that $\check{C}(B\to X)\to{}^{c}X$ is a weak equivalence in the Čech
model structure on simplicial presheaves. Since the local projective model
structure is a Left Bousfield localization of the Čech model structure, every
Čech weak equivalence is a local weak equivalence, proving the claim. ∎

Applying the cofibrant replacement functor gives us a commutative diagram

${{Q\check{C}(B\to X)}}$${{\check{C}(B\to X)}}$${{QX\cong
B//M}}$${X}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

and by the $2$-of-$3$ property, this implies that $Q\check{C}(B\to X)\simeq
QX$.

Unfortunately this is not enough to prove an equivalence of the two cohomology
theories. To do so, one would need to prove that the cohomology of
$\check{C}_{KWW}(X,\mathbb{R}^{\delta})$ agrees with that of
$N^{\text{co}}\left[\left(\mathbb{R}^{\delta}\right)^{Q\check{C}(B\to
X)}\right]$, which we have not been able to show.

We do conjecture that these cohomology theories agree:

######  Conjecture 8.12.

For a diffeological space $X$, the Iglesias-Zemmour Čech cohomology, Krepski-
Wolbert-Watts Čech cohomology and $\infty$-sheaf cohomology of $X$ with values
in $\mathbb{R}^{\delta}$ agree:

(32)
$H^{k}(X,\mathbb{R}^{\delta})\cong\check{H}^{k}_{PIZ}(X,\mathbb{R}^{\delta})\cong\check{H}^{k}_{KWW}(X,\mathbb{R}^{\delta})$

for $k\geq 0$.

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