# The Diffeological Čech-de Rham Obstruction

Emilio Minichiello  CUNY Graduate Center, Email address:
<EMAIL_ADDRESS>
###### Abstract.

Using higher topos theory, we explore the obstruction to the Čech-de Rham map
being an isomorphism in each degree for diffeological spaces. In degree 1, we
obtain an exact sequence which interprets Iglesias-Zemmour’s construction
[Igl23] in $\infty$-stack cohomology. We obtain new exact sequences in all
higher degrees. These exact sequences are constructed using homotopy pullback
diagrams that include the $\infty$-stack classifying higher
$\mathbb{R}$-bundle gerbes with connection. We also obtain a conceptual and
succinct proof that the $\infty$-stack cohomology of the irrational torus
$T_{K}$ for $K\subset\mathbb{R}$ a diffeologically discrete subgroup, agrees
with the group cohomology of $K$ with values in $\mathbb{R}$. Finally, for a
Lie group $G$, we prove that the groupoid of diffeological principal
$G$-bundles with connection one obtains via higher topos theory is equivalent
to the groupoid of diffeological principal $G$-bundles with connection defined
in [Wal12].

###### Contents

  1. 1 Introduction
  2. 2 Smooth Sheaves and Diffeological Spaces
  3. 3 Simplicial Presheaves
  4. 4 The Irrational Torus
  5. 5 The Dold-Kan Correspondence
  6. 6 Examples of $\infty$-stacks
  7. 7 The Čech de Rham Obstruction
  8. A Diffeological Principal Bundles with Connection
  9. B Totalization
  10. C Proof of Theorem 7.1

##  1\. Introduction

Classical differential geometry involves the study of finite dimensional
smooth manifolds. As a theory, it has many achievements. One of its most
celebrated is the Čech-de Rham Theorem, more commonly known as the de Rham
Theorem111We call it the Čech-de Rham Theorem because some authors use “the de
Rham Theorem” to refer to the isomorphism between de Rham cohomology and
singular cohomology.. The Čech-de Rham Theorem, proven in 1931 by de Rham
[deR31], states that if $M$ is a finite dimensional smooth manifold, then
there is an isomorphism

(1) $H^{k}_{\text{dR}}(M)\cong\check{H}^{k}(M,\mathbb{R}^{\delta}),$

where $H^{k}_{\text{dR}}(M)$ denotes the de Rham cohomology of $M$, and
$\check{H}^{k}(M,\mathbb{R}^{\delta})$ denotes the Čech cohomology of $M$ with
values in $\mathbb{R}^{\delta}$, the constant sheaf on the discrete group of
real numbers. There are many good textbook accounts of the Čech-de Rham
Theorem, such as [BT+82, Chapter II] and [GQ22, Chapter 9]. The de Rham
cohomology of a finite dimensional smooth manifold $M$ is constructed using
its smooth structure, but the Čech-de Rham Theorem shows that the de Rham
cohomology of $M$ is independent of this smooth structure and depends only on
the topology of $M$.

Diffeology is a modern framework for differential geometry whose main objects
of study are diffeological spaces, encompassing smooth manifolds, orbifolds,
and mapping spaces. The category of diffeological spaces is better behaved
than the category of finite dimensional smooth manifolds, indeed the category
of diffeological spaces is complete, cocomplete and cartesian closed [Igl13].
This makes diffeological spaces attractive to geometers who study spaces that
are not finite dimensional smooth manifolds. However, this generalization
comes at the cost of losing many of the theorems and constructions of
classical differential geometry222Many of these theorems are lost because not
all diffeological spaces have partitions of unity, a crucial ingredient to
many theorems in differential geometry.. Much contemporary work has gone into
extending these constructions and theorems to diffeological spaces. The
textbook [Igl13] by Iglesias-Zemmour has in particular pushed the theory quite
far, defining differential forms, de Rham cohomology, singular cohomology,
fiber bundles, and smooth homotopy groups of diffeological spaces amongst many
other contributions.

In [Igl88], Patrick Iglesias-Zemmour proved that the Čech-de Rham Theorem does
not hold in general for diffeological spaces. Interestingly, this result was
written as a preprint in French in the late 80s and was only recently
published in English as [Igl23]. Furthermore Iglesias-Zemmour obtained an
exact sequence

$0\to
H^{1}_{\text{dR}}(X)\to\check{H}^{1}_{PIZ}(X,\mathbb{R}^{\delta})\to{}^{d}E^{1,0}_{2}(X)\xrightarrow{c_{1}}H^{2}_{\text{dR}}(X)\to\check{H}^{2}_{PIZ}(X,\mathbb{R}^{\delta})$

which is a receptacle for the obstruction to the Čech-de Rham Theorem. The
group ${}^{d}E^{1,0}_{2}(X)$ is the subgroup of the group of isomorphism
classes of diffeological principal $\mathbb{R}$-bundles that admit a
connection. If this group is trivial, as it is for all finite dimensional
smooth manifolds, then $H^{1}_{\text{dR}}(X)\cong\check{H}^{1}_{PIZ}(X)$.
However, the situation for higher degrees is not addressed in [Igl23].
Iglesias-Zemmour writes “We must acknowledge that the geometrical natures of
the higher obstructions of the De Rham theorem still remain uninterpreted. It
would be certainly interesting to pursue this matter further” [Igl23, Page 2].
In this paper, we obtain such an interpretation of the higher obstructions.

In [Min22], we introduced a generalization of Čech cohomology for
diffeological spaces that we call $\infty$-stack cohomology. If $X$ is a
diffeological space and $A$ is a diffeological abelian group, then
$\check{H}^{k}_{\infty}(X,A)$ denotes the $k$th $\infty$-stack cohomology of
$X$ with values in $A$.

Currently, there are four definitions of Čech cohomology for diffeological
spaces in the literature. They are Iglesias-Zemmour’s cohomology from [Igl23],
which we call PIZ cohomology, there is $\infty$-stack cohomology [Min22],
there is Krepski-Watts-Wolbert cohomology [KWW21] and there is Ahmadi’s
cohomology [Ahm23]. In [Min22, Section 5.3], the first three Čech cohomologies
were compared, and some relationships deduced, but it is currently unknown if
any of the above cohomology theories agree in general.

This paper is a sequel to [Min22], where we explored the connection between
diffeological spaces and higher topos theory to study diffeological principal
$G$-bundles. When $G$ is a diffeological group, not necessarily abelian, it is
still possible to define $\infty$-stack cohomology in degree $1$,
$\check{H}^{1}_{\infty}(X,G)$. We proved [Min22, Corollary 6.9] that degree
$1$ $\infty$-stack cohomology is in bijection with isomorphism classes of
diffeological principal $G$-bundles over $X$. In fact, we obtained a much
stronger result by showing that the nerve of the groupoid of diffeological
principal $G$-bundles is weak equivalent to the nerve of the category of
$G$-principal $\infty$-bundles on $X$ [Min22, Theorem 6.8].

In this paper, we study two cases where the tools of higher topos theory help
us better understand diffeological spaces. The first case is studying the
$\infty$-stack cohomology of the irrational torus. The irrational torus was
the first example of a nontrivial diffeological space with trivial underlying
topology, see [Igl20]. In [Igl23], Iglesias-Zemmour proved that if
$K\subset\mathbb{R}$ is a diffeologically discrete subgroup, then the PIZ
cohomology of the irrational torus $T_{K}=\mathbb{R}/K$ is isomorphic to the
group cohomology of $K$ with values in $\mathbb{R}$. However, his proof of
this, [Igl23, Section II], is computational. In Section 4, we prove

######  Theorem 4.4.

There is an isomorphism

$\check{H}^{n}_{\infty}(T_{K},\mathbb{R}^{\delta})\cong
H^{n}_{\text{grp}}(K,\mathbb{R}^{\delta})$

of abelian groups, for every $n\geq 0$, where $\mathbb{R}^{\delta}$ denotes
the discrete group of real numbers, and where
$H^{n}_{\text{grp}}(K,\mathbb{R}^{\delta})$ denotes the group cohomology of
$K$ with coefficients in $\mathbb{R}^{\delta}$.

Theorem 4.4 supports the conjecture that PIZ cohomology and $\infty$-stack
cohomology agree. The proof of Theorem 4.4 is short and conceptual. It uses
the shape functor $\int$, much beloved by higher differential geometers
[BBP22], [Bun22], [Sch13], [Clo23], [Car15], in a crucial way, reducing the
$\infty$-stack cohomology of $T_{K}$ to the singular cohomology of the
classifying space $\textbf{B}K$. This demonstrates the advantage of using
$\infty$-stack cohomology to study diffeological spaces.

The second case, which makes up the bulk of the paper, is to use
$\infty$-stack cohomology, and more generally the framework of higher topos
theory, to study the diffeological Čech-de Rham obstruction. First we obtain a
homotopy pullback diagram of $\infty$-stacks.

######  Theorem 7.1.

For every $k\geq 1$, there exists a commutative diagram of $\infty$-stacks of
the following form

(2)
${{*}}$${{\textbf{B}^{k}\mathbb{R}^{\delta}}}$${{*}}$${{*}}$${{*}}$${{\textbf{B}^{k}_{\nabla}\mathbb{R}}}$${{\Omega^{k+1}_{\text{cl}}}}$${{\Omega^{k+1}}}$${{\textbf{B}^{k}\mathbb{R}}}$${{\textbf{B}^{k}\Omega^{1}_{\text{cl}}}}$${{\Omega^{1\leq\bullet\leq
k+1}}}$${{*}}$${{\textbf{B}^{k+1}\mathbb{R}^{\delta}}}$${{\textbf{B}^{k+1}_{\nabla}\mathbb{R}}}$

furthermore every commutative square in this diagram is a homotopy pullback
square in the Čech model structure on simplicial presheaves over
$\mathsf{Cart}$.

Such diagrams are often used in higher category-theoretic treatments of
differential cohomology, see [Sch13], [ADH21], [Jaz21]. One can think of an
$\infty$-stack as a classifying object for a mathematical structure, such as
diffeological principal $G$-bundles, as in [Min22]. Thus the above diagrams
can be thought of as tight relationships between the corresponding
mathematical structures.

Of particular interest is the $\infty$-stack
$\textbf{B}^{k}_{\nabla}\mathbb{R}$. This is the $\infty$-stack which
classifies diffeological $\mathbb{R}$-bundle $(k-1)$-gerbes with connection.
Cohomology with values in this $\infty$-stack is called the $k$th pure
differential cohomology in [Jaz21]. From Theorem 7.1 we are immediately able
to obtain the following result.

######  Corollary 7.2.

For every diffeological space $X$, there is an exact sequence of vector spaces

(3)
$0\to\check{H}^{k}_{\infty}(X,\mathbb{R}^{\delta})\to\check{H}^{k}_{\infty,\nabla}(X,\mathbb{R})\to\Omega^{k+1}_{\text{cl}}(X)\to\check{H}^{k+1}_{\infty}(X,\mathbb{R}^{\delta}).$

Near the completion of this paper, we learned that an analogous exact sequence
was also obtained in [Jaz21, Page 27] using completely different methods in
the framework of homotopy type theory. The above exact sequence allows us to
compute the pure differential cohomology of the irrational torus.

######  Theorem 7.3.

Let $T_{\alpha}$ denote the irrational torus, then

(4)
$\check{H}^{k}_{\infty,\nabla}(T_{\alpha},R)\cong\begin{cases}\mathbb{R}^{2},&k=1,\\\
\mathbb{R},&k=2,\\\ 0,&k>2.\end{cases}$

While Corollary 7.2 is useful for computations with the irrational torus, it
is desirable to have an exact sequence including de Rham cohomology rather
than closed forms. This is obtained in the following result.

######  Theorem 7.5.

Given a diffeological space $X$ and $k\geq 1$, the sequence of vector spaces

$\check{H}^{k}_{\infty}(X,\mathbb{R}^{\delta})\to\check{H}^{k}_{\text{conn}}(X,\mathbb{R})\to
H^{k+1}_{\text{dR}}(X)\to\check{H}^{k+1}_{\infty}(X,\mathbb{R}^{\delta})$

is exact.

When $k=1$, we obtain an additional piece to this exact sequence.

######  Theorem 7.7.

Given a diffeological space $X$, the sequence of vector spaces

(5) $0\to
H^{1}_{\text{dR}}(X)\to\check{H}^{1}_{\infty}(X,\mathbb{R}^{\delta})\to\check{H}^{1}_{\text{conn}}(X,\mathbb{R})\to
H^{2}_{\text{dR}}(X)\to\check{H}^{2}_{\infty}(X,\mathbb{R}^{\delta})$

is exact.

The above exact sequence is exactly analogous to the exact sequence obtained
by Iglesias-Zemmour in [Igl23].

In Appendix A we turn to the study of connections for diffeological principal
bundles. This theory is still in its infancy, and there are a few references
that give varying definitions of diffeological connections [Igl13, Section
8.32], [Wal12, Section 3], [MW17, Section 4]. The theory of $\infty$-stacks
provides another definition. Let $G$ be a Lie group, and $U$ a cartesian
space. Then let $\Omega^{1}(U,\mathfrak{g})//G$ denote the groupoid whose
objects are differential $1$-forms $\omega\in\Omega^{1}(U,\mathfrak{g})$,
where $\mathfrak{g}$ denotes the Lie algebra of $G$, and where there is a
morphism $g:\omega\to\omega^{\prime}$ if there exists a smooth map $g:U\to G$
such that

$\omega^{\prime}=\text{Ad}_{g}^{-1}(\omega)+g^{*}\text{mc}(G)$

where $\text{mc}(G)$ denotes the Maurer-Cartan form of $G$. Taking the nerve
of this groupoid, and letting $U$ vary defines an $\infty$-stack
$\Omega^{1}(-,\mathfrak{g})//G$, which amongst others has been studied in
[FSS+12], [FH13]. We connect this notion of connection to that given in
[Wal12, Definition 3.2.1] in the following result.

######  Theorem A.3.

Given a diffeological space $X$ and a Lie group $G$, the functor

(6) $\mathsf{Cons}_{\nabla}:\mathsf{Coc}_{\nabla}(X,G)\to\mathsf{Wal}_{G}(X),$

is an equivalence of groupoids, where $\mathsf{Coc}_{\nabla}(X,G)$ is the
groupoid whose objects are maps $QX\to\Omega^{1}(-,\mathfrak{g})//G$, where
$QX$ is a cofibrant replacement of $X$ in the projective model structure on
simplicial presheaves, and $\mathsf{Wal}_{G}(X)$ is the groupoid of
diffeological principal $G$-bundles with connection as defined in [Wal12,
Definition 3.2.1].

To compute $\infty$-stack cohomology, one needs a workable model of the
derived mapping space $\mathbb{R}\text{Hom}(X,A)$, when $X$ is a diffeological
space and $A$ is a presheaf of chain complexes. In Appendix B, we obtain such
a model, which reduces many computations with $\infty$-stacks to manipulations
with double complexes. As a corollary, we obtain a simple and direct proof of
the following well known folklore result.

######  Proposition B.4.

Let $C$ be a cosimplicial chain complex, then

(7) $\text{holim}_{n\in\mathsf{\Delta}}C^{n}\simeq\mathsf{tot}\,C,$

where we are computing the homotopy limit in the category of chain complexes
equipped with the projective model structure, and $\mathsf{tot}\,C$ denotes
the total complex of $C$.

The paper is organized as follows. In Section 2, we introduce diffeological
spaces and place them in the context of sheaf theory. In Section 3, we
introduce simplicial presheaves, show how diffeological spaces embed into
simplicial presheaves, and introduce the shape functor. In Section 4, we prove
that the $\infty$-stack cohomology of the irrational torus $T_{K}$ is
isomorphic to the group cohomology of $K$ with values in $\mathbb{R}$. In
Section 5, we introduce the Dold-Kan correspondence, which is a core tool we
use for the rest of the paper. In Section 6, we introduce the main
$\infty$-stacks that will be used in the paper, and compute various examples
of $\infty$-stack cohomology. In Section 7, we prove the main results of this
paper, Theorem 7.5 and Theorem 7.7. In Appendix A, we prove that our notion of
diffeological principal $G$-bundles with connection using $\infty$-stacks
agrees with Waldorf’s [Wal12]. In Appendix B, we prove a technical result
allowing us to easily compute $\infty$-stack cohomology when the coefficient
$\infty$-stack comes from a presheaf of chain complexes. In Appendix C we
prove Theorem 7.1.

##  2\. Smooth Sheaves and Diffeological Spaces

In this section we briefly describe diffeological spaces and their connection
to sheaves on $\mathsf{Cart}$. See [Min22, Section 2] for more details.

######  Definition 2.1.

Let $M$ be a finite dimensional smooth manifold333We will assume throughout
this paper that manifolds are Hausdorff and paracompact.. We say a collection
of subsets $\mathcal{U}=\\{U_{i}\subseteq M\\}_{i\in I}$ is an open cover if
each $U_{i}$ is an open subset of $M$, and $\bigcup_{i\in I}U_{i}=M$. If $U$
is a finite dimensional smooth manifold diffeomorphic to $\mathbb{R}^{n}$ for
some $n\in\mathbb{N}$, we call $U$ a cartesian space. We call
$\mathcal{U}=\\{U_{i}\subseteq M\\}$ a cartesian open cover of a manifold $M$
if it is an open cover of $M$ and every $U_{i}$ is a cartesian space. We say
that $\mathcal{U}$ is a good open cover if it is a cartesian open cover, and
further every finite non-empty intersection $U_{i_{0}\dots
i_{k}}=U_{i_{0}}\cap\dots\cap U_{i_{k}}$ is a cartesian space.

Let $\mathsf{Man}$ denote the category whose objects are finite dimensional
smooth manifolds and whose morphisms are smooth maps. Let $\mathsf{Cart}$
denote the full subcategory whose objects are cartesian spaces. Given a set
$X$, let $\mathsf{Param}(X)$ denote the set of parametrizations of $X$, namely
the collection of set functions $p:U\to X$, where $U\in\mathsf{Cart}$.

######  Definition 2.2.

A diffeology on a set $X$, consists of a collection $\mathscr{D}$ of
parametrizations $p:U\to X$ satisfying the following three axioms:

  1. (1)

$\mathscr{D}$ contains all points $\mathbb{R}^{0}\to X$,

  2. (2)

If $p:U\to X$ belongs to $\mathscr{D}$, and $f:V\to U$ is a smooth map, then
$pf:V\to X$ belongs to $\mathscr{D}$, and

  3. (3)

If $\\{U_{i}\subseteq U\\}_{i\in I}$ is a good open cover of a cartesian space
$U$, and $p:U\to X$ is a parametrization such that $p|_{U_{i}}:U_{i}\to X$
belongs to $\mathscr{D}$ for every $i\in I$, then $p\in\mathscr{D}$.

A set $X$ equipped with a diffeology $\mathscr{D}$ is called a diffeological
space. Parametrizations that belong to a diffeology are called plots. We say a
set function $f:X\to Y$ between diffeological spaces is smooth if for every
plot $p:U\to X$ in $\mathscr{D}_{X}$, the composition $pf:U\to Y$ belongs to
$\mathscr{D}_{Y}$. We often denote the set of smooth maps from $X$ to $Y$ by
$C^{\infty}(X,Y)$. Let $\mathsf{Diff}$ denote the category of diffeological
spaces.

Every manifold $M$ is canonically a diffeological space by considering the set
of parametrizations $p:U\to M$ that are smooth in the classical sense. This
gives a diffeology on $M$, called the manifold diffeology. One can show
[Igl13, Chapter 4] that the manifold diffeology defines a fully faithful
functor $\mathsf{Man}\hookrightarrow\mathsf{Diff}$.

Diffeology extends many constructions and concepts from classical differential
geometry to diffeological spaces, such as the theory of bundles.

######  Definition 2.3.

We say that a map $\pi:X\to Y$ of diffeological spaces is a subduction if it
is surjective, and for every plot $p:U\to Y$, there exists a good open cover
$\\{U_{i}\subseteq U\\}$, and plots $p_{i}:U_{i}\to X$ making the following
diagram commute

(8)
${{U_{i}}}$${X}$${{U}}$${Y}$$\scriptstyle{\pi}$$\scriptstyle{p_{i}}$$\scriptstyle{p}$

######  Definition 2.4.

A diffeological group is a group $G$ equipped with a diffeology such that the
multiplication map $m:G\times G\to G$, and inverse map $i:G\to G$ are smooth.
A right diffeological group action of a diffeological group $G$ on a
diffeological space $X$ is a smooth map $\rho:X\times G\to X$ such that
$\rho(x,e_{G})=x$, and $\rho(\rho(x,g),h)=\rho(x,gh)$, where $e_{G}$ denotes
the identity element of $G$.

######  Definition 2.5.

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

  1. (1)

the map $\pi:P\to X$ is a subduction, and

  2. (2)

the map $\text{act}:P\times G\to P\times_{X}P$ defined by
$(p,g)\mapsto(p,p\cdot g)$, which we call the action map is a diffeomorphism.

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. A diffeological principal
$G$-bundle $P$ is said to be trivial if there exists an isomorphism
$\varphi:X\times G\to P$, called a trivialization, where
$\text{pr}_{1}:X\times G\to X$ is the product bundle. Let
$\mathsf{DiffPrin}_{G}(X)$ denote the category of diffeological principal
$G$-bundles over a diffeological space $X$.

In [Min22], we proved that diffeological principal bundles can be classified
using cocycles in a way reminiscent of classical differential geometry.
However, rather than using cocycles defined over an open cover, we use
cocycles defined on plots. Let $\mathsf{Plot}(X)$ denote the category whose
objects are plots $p:U\to X$ of $X$ and whose morphisms $f:p\to p^{\prime}$
are smooth maps $f:U\to U^{\prime}$ such that $p^{\prime}f=p$.

######  Definition 2.6.

Given a diffeological space $X$ and a diffeological group $G$, call a
collection $g=\\{g_{f_{0}}\\}$ of smooth maps $g_{f_{0}}:U_{p_{1}}\to G$
indexed by maps of plots $f_{0}:U_{p_{1}}\to U_{p_{0}}$ of $X$ a $G$-cocycle
if for every pair of composable plot maps of $X$

$U_{p_{2}}\xrightarrow{f_{1}}U_{p_{1}}\xrightarrow{f_{0}}U_{p_{0}}$

it follows that

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

We call (9) the diffeological $G$-cocycle condition.

Given two $G$-cocycles, $g,g^{\prime}$, we say a collection
$h=\\{h_{p_{0}}\\}$ of smooth maps $h_{p_{0}}:U_{p_{0}}\to G$ indexed by plots
of $X$ is a morphism of $G$-cocycles $h:g\to g^{\prime}$ if for every map
$f_{0}:U_{p_{1}}\to U_{p_{0}}$ of plots of $X$, it follows that

(10) $g^{\prime}_{f_{0}}\cdot h_{p_{1}}=(h_{p_{0}}\circ f_{0})\cdot
g_{f_{0}}.$

Given a diffeological space $X$ and a $G$-cocycle $g$ on $X$, we can construct
a diffeological principal $G$-bundle $\pi:P\to X$, by taking the quotient

(11) $P=\left(\coprod_{p_{0}\in\mathsf{Plot}(X)}U_{p_{0}}\times
G\right)/{\sim}$

where $\sim$ is the smallest equivalence relation such that
$(x_{p_{1}},k_{1})\sim(x_{p_{0}},k_{0})$ if there exists a map
$f_{0}:U_{p_{1}}\to U_{p_{0}}$ of plots such that $f_{0}(x_{p_{1}})=x_{p_{0}}$
and $k_{0}=g_{f_{0}}(x_{p_{1}})\cdot k_{1}$. We let $\pi=\mathsf{Cons}(g)$,
short for construction. In fact, this construction defines a functor from the
category $\mathsf{Coc}(X,G)$ of $G$-cocycles on $X$ to the category of
diffeological principal $G$-bundles.

######  Theorem 2.7 ([Min22, Theorem 3.15]).

Given a diffeological space $X$ and a diffeological group $G$, the functor

(12) $\mathsf{Cons}:\mathsf{Coc}(X,G)\to\mathsf{DiffPrin}_{G}(X)$

is an equivalence of groupoids.

While extending the classical theory, there are constructions one can do with
diffeological spaces that are not available to smooth manifolds:

  1. (1)

Given a diffeological space $X$, and a subset $A\xhookrightarrow{i}X$ a
subset. Then consider the set of parametrizations $p:U\to A$ such that
$ip:U\to X$ is a plot of $X$. This collection is a diffeology, called the
subspace diffeology on $A$,

  2. (2)

Given a diffeological space $X$ and an equivalence relation $\sim$ on $X$, let
$\pi:X\to X/{\sim}$ denote the resulting quotient function on sets. Consider
the set of parametrizations $p:U\to X/{\sim}$ such that there exists a good
open cover $\\{U_{i}\subseteq U\\}$ and plots $p_{i}:U_{i}\to X$ making the
following diagram commute

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

This forms a diffeology on $X/{\sim}$, called the quotient diffeology,

  3. (3)

Given a pair $X$ and $Y$ of diffeological spaces, the set of parametrizations
$p:U\to X\times Y$ such that the composites $\pi_{1}\circ p$ and $\pi_{2}\circ
p$ are plots of $X$ and $Y$ respectively, forms a diffeology, called the
product diffeology,

  4. (4)

Given diffeological spaces $X$ and $Y$, the set of parametrizations $p:U\to
C^{\infty}(X,Y)$ such that the transposed function $p^{\\#}:U\times X\to Y$ is
a smooth map is a diffeology, called the functional diffeology.

These constructions make the category of diffeological spaces considerably
better than the category of finite dimensional smooth manifolds, as shown in
Corollary 2.10.

Diffeological spaces inherit this nice structure from the category of smooth
sheaves.

######  Definition 2.8.

We briefly recall the relevant definitions for sheaf theory.

  * •

A collection of families $j$ on a category $\mathscr{C}$ consists of a set
$j(U)$ for each $U\in\mathscr{C}$, whose elements $\\{r_{i}:U_{i}\to U\\}\in
j(U)$ are families of morphisms over $U$. We call a collection of families $j$
on $\mathscr{C}$ a coverage if it satisfies the following property: for every
$\\{r_{i}:U_{i}\to U\\}\in j(U)$, and every map $g:V\to U$ in $\mathscr{C}$,
then there exists a family $\\{t_{j}:V_{j}\to V\\}\in j(V)$ such that $gt_{j}$
factors through some $r_{i}$. Namely for every $t_{j}$ there exists some $i$
and some map $s_{j}:V_{j}\to U_{i}$ making the following diagram commute:

(13)
${{V_{j}}}$${{U_{i}}}$${V}$${U}$$\scriptstyle{t_{j}}$$\scriptstyle{s_{j}}$$\scriptstyle{r_{i}}$$\scriptstyle{g}$

The families $\\{r_{i}:U_{i}\to U\\}\in j(U)$ are called covering families
over $U$. If a map $r_{i}:U_{i}\to U$ belongs to a covering family $r\in
j(U)$, then we say that $r_{i}$ is a covering map. If $\mathscr{C}$ is a
category, and $j$ is a coverage on $\mathscr{C}$, then we call the pair
$(\mathscr{C},j)$ a site.

  * •

A presheaf on a category $\mathscr{C}$ is a functor
$F:\mathscr{C}^{op}\to\mathsf{Set}$. A morphism of presheaves is a natural
transformation. An element $x\in F(U)$ for an object $U\in\mathscr{C}$ is
called a section over $U$. If $f:U\to V$ is a map in $\mathscr{C}$, and $x\in
F(V)$, then we sometimes denote $F(f)(x)$ by $x|_{U}$. Let
$\mathsf{Pre}(\mathcal{C})$ denote the category of presheaves on
$\mathcal{C}$.

  * •

If $\\{r_{i}:U_{i}\to U\\}_{i\in I}$ 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 $\mathscr{C}$ 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 $i$.

  * •

Given a family of morphisms $r=\\{r_{i}:U_{i}\to U\\}$ in a category
$\mathscr{C}$, we say that a presheaf $F:\mathscr{C}^{op}\to\mathsf{Set}$ is a
sheaf on $r$ if every matching family $\\{s_{i}\\}$ of $F$ over $r$ has a
unique amalgamation. If $j$ is a coverage on a category $\mathscr{C}$, we call
$F$ a sheaf on $(\mathscr{C},j)$ if it is a sheaf on every covering family of
$j$. Let $\mathsf{Sh}(\mathscr{C})$ denote the full subcategory of
$\mathsf{Pre}(\mathscr{C})$ whose objects are sheaves on $(\mathscr{C},j)$.

One can put a site structure on $\mathsf{Cart}$ using the coverage of good
open covers, see [Min22, Section 4]. We call sheaves on $\mathsf{Cart}$ smooth
sheaves. There are many interesting examples of smooth sheaves. Every
cartesian space defines a representable sheaf $yU$. Every manifold $M$ defines
a sheaf by $U\mapsto C^{\infty}(U,M)$. There are also $\Omega^{n}$ and
$\Omega^{n}_{\text{cl}}$ for every $n\geq 0$, the sheaves of differential
$n$-forms and closed differential $n$-forms respectively. The category
$\mathsf{Sh}(\mathsf{Cart})$ of smooth sheaves is “extremely nice”, being a
Grothendieck topos [MM12].

A sheaf $X$ on $\mathsf{Cart}$ is concrete if $X(U)$ is a subset of the set
functions $U\to X(*)$ where $*$ is the terminal object in $\mathsf{Cart}$. The
representable sheaves $yU$ and the sheaves induced by manifolds $M$ are
concrete, but $\Omega^{n}$ and $\Omega^{n}_{\text{cl}}$ are not.

The full subcategory
$\mathsf{ConSh}(\mathcal{C})\hookrightarrow\mathsf{Sh}(\mathcal{C})$ of
concrete sheaves on a concrete site forms a quasitopos, which while not being
a Grothendieck topos, is still a very “nice” category [BH11, Theorem 52].

######  Theorem 2.9 ([BH11, Prop 24]).

Let $\mathsf{Cart}$ denote the site of cartesian spaces with the coverage of
good open covers. Then there is an equivalence of categories

(14) $\mathsf{Diff}\simeq\mathsf{ConSh}(\mathsf{Cart}),$

where $\mathsf{ConSh}(\mathsf{Cart})$ denotes the category of concrete sheaves
on $\mathsf{Cart}$.

######  Corollary 2.10.

The category $\mathsf{Diff}$ is a quasitopos. This implies that it is a
complete, cocomplete and cartesian closed category.

We refer to Theorem 2.9 as the Baez-Hoffnung Theorem444Strictly speaking, the
Baez-Hoffnung theorem gives an equivalence between the category of what we
call classical diffeological spaces and the category of concrete sheaves on
the site of open subsets of cartesian spaces with open covers, see [Min22,
Appendix A] for a proof that this is an equivalent formulation.. It is the
starting point of the interaction of sheaf theory and diffeology. Many aspects
of the study of diffeological spaces can be restated using sheaf theory, for
example a differential $n$-form $\omega$ on a diffeological space $X$ as
defined in [Igl13, Article 6.28] is equivalently a morphism $X\to\Omega^{n}$
of sheaves.

In [Min22] we took advantage of the Baez-Hoffnung Theorem to embed the
category of diffeological spaces into the category of simplicial presheaves on
$\mathsf{Cart}$. We will delve into this idea in the next section. Once inside
the category of simplicial presheaves, we can then take advantage of many
homotopical tools. This in effect provides a way of obtaining a very powerful
and expressive homotopy theory for diffeological spaces that subsumes the
usual homotopy theory for diffeological spaces as considered in [Igl13,
Chapter 5].

##  3\. Simplicial Presheaves

In this section we detail the model categorical notions we will need for the
remainder of the paper. We assume the reader is comfortable with model
categories and simplicial homotopy theory, and recommend the following
standard sources [Hir09], [Hov07], [GJ12], [GS06] for good references on the
topics. See [Min22, Section 5] for more details.

######  Definition 3.1.

Let $\mathsf{sPre}(\mathsf{Cart})$ denote the category whose objects are
functors $\mathsf{Cart}^{op}\to\mathsf{sSet}$, which we call simplicial
presheaves, and whose morphisms are natural transformations.

Note that $\mathsf{sPre}(\mathsf{Cart})$ is complete and cocomplete, with
limits and colimits computed objectwise. There are two pairs of adjoint
triples that give structure to $\mathsf{sPre}(\mathsf{Cart})$.

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where $(-)_{c}$ is the functor induced by restricting along the unique functor
$\mathsf{Cart}\to*$ and ${}^{c}(-)$ is the functor that sends a presheaf to
the corresponding simplicial presheaf where all the simplicial face and
degeneracy maps are the identity. We often don’t use the notation ${}^{c}(-)$
explicitly, especially for representable presheaves, as it should be clear
from context. The functors $\pi_{0}$ and $(-)_{0}$ are defined objectwise. For
every $U\in\mathsf{Cart}$, and simplicial presheaf $X$ on $\mathsf{Cart}$,
$\pi_{0}X(U)=\pi_{0}(X(U))$, the set of connected components of $X(U)$, and
$(X)_{0}(U)=X(U)_{0}$, the set of vertices of $X(U)$.

######  Remark 3.2.

The above adjoint triples exist for any essentially small category
$\mathcal{C}$ in place of $\mathsf{Cart}$.

The category $\mathsf{sPre}(\mathsf{Cart})$ is tensored, cotensored and
enriched over $\mathsf{sSet}$. Indeed, if $K$ is a simplicial set and $X$ is a
simplicial presheaf, then

  * •

$X\otimes K$ is the simplicial presheaf defined objectwise by

$(X\otimes K)(U)=(X\times K_{c})(U)=X(U)\times K.$

  * •

$X^{K}$ is the simplicial presheaf defined objectwise by

$(X^{K})(U)=X(U)^{K},$

where for simplicial sets $K$ and $L$, $K^{L}$ denotes the simplicial function
complex.

  * •

for any two simplicial presheaves $X$ and $Y$, let
$\underline{\mathsf{sPre}(\mathsf{Cart})}(X,Y)$ denote the simplicial set
defined levelwise by

$\underline{\mathsf{sPre}(\mathsf{Cart})}(X,Y)_{n}=\mathsf{sPre}(\mathsf{Cart})(X\otimes\Delta^{n},Y).$

This structure is compatible in the sense of the following natural
isomorphisms of simplicial sets

(16) $\underline{\mathsf{sPre}(\mathsf{Cart})}(X\otimes
K,Y)\cong\underline{\mathsf{sPre}(\mathsf{Cart})}(X,Y^{K}).$

The category $\mathsf{sPre}(\mathsf{Cart})$ inherits several model structures
from $\mathsf{sSet}$. We say a map $f:X\to Y$ is a projective weak equivalence
if it is an objectwise weak equivalence of simplicial sets, a projective
fibration if it is an objectwise fibration, and a projective cofibration if it
left lifts against all maps that are both projective weak equivalences and
projective fibrations.

######  Theorem 3.3 ([BK72, Page 314], [Lur09, Section A.2.6]).

The projective weak equivalences, fibrations and cofibrations define a proper,
combinatorial, simplicial model category structure on
$\mathsf{sPre}(\mathsf{Cart})$, called the projective model structure on
simplicial presheaves.

Let $\mathbb{H}$ denote the category of simplicial presheaves equipped with
the projective model structure. Note that [Dug01, Corollary 9.4] describes a
sufficient condition on simplicial presheaves to be projective cofibrant, and
it implies that all representable presheaves, denoted $yU$ for a cartesian
space $U$, are projective cofibrant.

Given a cartesian space $U$ and a good cover $\mathcal{U}=\\{U_{i}\subseteq
U\\}$ of $U$, we can form the simplicial presheaf $\check{C}(\mathcal{U})$
defined levelwise by

$\check{C}(\mathcal{U})_{n}=\coprod_{i_{0}\dots i_{n}}y(U_{i_{0}}\cap\dots\cap
U_{i_{n}}).$

We call $\check{C}(\mathcal{U})$ the Čech nerve of $\mathcal{U}$. There is a
canonical map $\pi:\check{C}(\mathcal{U})\to yU$. Let $\check{C}$ denote the
class of morphisms $\pi:\check{C}(\mathcal{U})\to yU$ where $U$ ranges over
the cartesian spaces and $\mathcal{U}$ ranges over the good open covers for
$U$.

######  Theorem 3.4 ([DHI04, Theorem A.6]).

The left Bousfield localization of $\mathbb{H}$ at $\check{C}$ exists. We call
the resulting model structure the Čech model structure on
$\mathsf{sPre}(\mathsf{Cart})$, and denote it by $\check{\mathbb{H}}$. It is
similarly a proper, combinatorial and simplicial model category.555It is
important to note that the projective/objectwise weak equivalences between
simplicial presheaves are still weak equivalences in the Čech model structure.
Furthermore, all Čech weak equivalences between $\infty$-stacks are objectwise
weak equivalences.

The fibrant objects in $\check{\mathbb{H}}$ are called $\infty$-stacks. They
are those projective fibrant simplicial presheaves $X$ such that the canonical
map

(17)
$\underline{\mathsf{sPre}(\mathsf{Cart})}(yU,X)\to\underline{\mathsf{sPre}(\mathsf{Cart})}(\check{C}(\mathcal{U}),X),$

is a weak equivalence of simplicial sets, for every cartesian space $U$ and
good cover $\mathcal{U}$ of $U$. Every sheaf and classical stack of groupoids
on $\mathsf{Cart}$, thought of as simplicial presheaves on $\mathsf{Cart}$, is
an $\infty$-stack. See [Min22, Section 4.1] for more details.

The identity functors define a Quillen adjunction between the projective and
Čech model structure on simplicial presheaves.

(18)
${{\mathbb{H}}}$${{\check{\mathbb{H}}}}$$\scriptstyle{1_{\mathsf{sPre}(\mathsf{Cart})}}$$\scriptstyle{1_{\mathsf{sPre}(\mathsf{Cart})}}$$\scriptstyle{\dashv}$

Crucially, the (easy to compute) finite homotopy limits in $\mathbb{H}$ are
preserved as homotopy limits in $\check{\mathbb{H}}$, thanks to the following
result.

######  Proposition 3.5 ([Rez10, Proposition 11.2]).

The left Quillen functor
$1_{\mathsf{sPre}(\mathsf{Cart})}:\mathbb{H}\to\check{\mathbb{H}}$ preserves
finite homotopy limits.

Given simplicial presheaves $X$ and $Y$ on $\mathsf{Cart}$, let $Q$ and $R$
denote cofibrant and fibrant replacement functors for $\check{\mathbb{H}}$
respectively, then let

(19)
$\mathbb{R}\check{\mathbb{H}}(X,Y)=\underline{\mathsf{sPre}(\mathsf{Cart})}(QX,RY).$

We call $\mathbb{R}\check{\mathbb{H}}(X,Y)$ the derived mapping space of $X$
and $Y$. If $X$ is already cofibrant, then we can take
$Q=1_{\mathsf{sPre}(\mathsf{Cart})}$ and if $Y$ is already fibrant, we can
take $R=1_{\mathsf{sPre}(\mathsf{Cart})}$.

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

$\check{H}^{0}_{\infty}(X,A)=\pi_{0}\mathbb{R}\check{\mathbb{H}}(X,A).$

We call this the $0$th $\infty$-stack cohomology of $X$ with values in $A$.

If $A$ is a simplicial presheaf that is objectwise a simplicial group, then we
let

$\check{H}^{1}_{\infty}(X,A)=\pi_{0}\mathbb{R}\check{\mathbb{H}}(X,\overline{W}A),$

where $\overline{W}$ is the delooping functor, see [Min22, Definition 4.29].

If $A$ is a simplicial presheaf such that $\overline{W}^{k}A$, which we call
its $k$-fold delooping, exists for $k\geq 1$, then we say that $A$ is
$k$-deloopable, and we let

$\check{H}^{k}_{\infty}(X,A)=\pi_{0}\mathbb{R}\check{\mathbb{H}}(X,\overline{W}^{k}A).$

We call this the $k$th $\infty$-stack cohomology of $X$ with coefficients in
$A$. If the $k$-fold delooping of a simplicial presheaf $A$ exists and is an
$\infty$-stack, then we denote it by
$\textbf{B}^{k}A\coloneqq\overline{W}^{k}A$. The following result is well
known, see [Min22, Lemma 4.34] for a proof.

######  Lemma 3.6.

If $A$ is a presheaf of simplicial abelian groups, then $A$ is $k$-deloopable
for all $k\geq 1$.

There is a convenient cofibrant replacement functor for $\check{\mathbb{H}}$,
given (in the notation of [Rie14, Section 4.2]) for a simplicial presheaf $X$
by the bar construction $B(X,\mathsf{Cart},y)$, where $y$ denotes the Yoneda
embedding $y:\mathsf{Cart}\hookrightarrow\mathsf{sPre}(\mathsf{Cart})$.

If $X$ is a diffeological space, then by the Baez-Hoffnung Theorem (Theorem
2.9), we can consider it as a sheaf on $\mathsf{Cart}$. Then ${}^{c}X$ is a
simplicial presheaf666We will often not use the notation ${}^{c}X$ for
diffeological spaces in what follows, as it should be apparent from context
what category we are considering $X$ in.. If we apply $Q$ to ${}^{c}X$ then
this formula reduces to the simplicial presheaf given levelwise by

(20) $QX_{n}=\coprod_{(f_{n-1},\dots,f_{0})\in
N(\mathsf{Plot}(X))_{n}}yU_{p_{n}}\otimes\Delta^{n}.$

See [Min22, Section 4.2] for more details.

Using the above cofibrant replacement functor for a diffeological space $X$,
if $A$ is a $\infty$-stack that is also objectwise a simplicial abelian group,
then we can obtain an explicit description of its $k$th $\infty$-stack
cohomology with values in $A$, given by the $k$th cohomology of the cochain
complex obtained by taking the dual Dold-Kan correspondence functor applied to
the cosimplicial abelian group

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the case where $A$ is an abelian diffeological group is given by [Min22,
Corollary 4.38].

######  Example 3.7.

Let $G$ be a diffeological group, and consider the (strict) functor
$\textbf{B}G:\mathsf{Cart}^{op}\to\mathsf{Gpd}$ that sends a cartesian space
$U$ to the groupoid777Here we use the convention discussed in [Min22, Example
5.13] for $\textbf{B}G$.

(22) $[C^{\infty}(U,G)\rightrightarrows*]$

Postcomposing with the nerve functor gives us a simplicial presheaf
$N\textbf{B}G$, which we will often just denote by $\textbf{B}G$. By [Min22,
Theorem 5.17], (referencing [SS21, Lemma 3.3.29] and [Pav22, Proposition
4.13]), $\textbf{B}G$ is an $\infty$-stack.

This $\infty$-stack takes a central role in the theory of diffeological
principal $G$-bundles. For every cartesian space $U$, there is a canonical map
of groupoids

(23) $\textbf{B}G(U)\to\mathsf{DiffPrin}_{G}(U),$

that sends the point to the trivial diffeological principal $G$-bundle, and
sends a map to $G$ to the corresponding automorphism of the trivial bundle.
This map is an equivalence of groupoids.

Furthermore, if $X$ is a diffeological space and $QX$ is its cofibrant
replacement, then $G$-cocycles on $X$ are equivalent to maps of
$\infty$-stacks $QX\to\textbf{B}G$ in the sense of [Min22, Lemma 6.7]. In
other words, for every diffeological space $X$, there is a weak equivalence

(24) $\mathbb{R}\check{\mathbb{H}}(X,\textbf{B}G)\simeq
N\mathsf{DiffPrin}_{G}(X).$

Thus we say that $\textbf{B}G$ classifies diffeological principal $G$-bundles.
This implies that

(25) $\check{H}^{1}_{\infty}(X,G)\cong\pi_{0}\mathsf{DiffPrin}_{G}(X),$

where $\pi_{0}\mathsf{DiffPrin}_{G}(X)$ denotes the set of isomorphism classes
of diffeological principal $G$-bundles on $X$.

Let us now examine the left hand side of (15). If $K$ is a simplicial set,
then $K_{c}$ is the constant simplicial presheaf on $K$, namely $K_{c}(U)=K$
for all $U\in\mathsf{Cart}$. The functors making up (15) are important enough
to warrant renaming. Notice that since $\mathbb{R}^{0}=*$ is the terminal
object in $\mathsf{Cart}$, it is the initial object in $\mathsf{Cart}^{op}$,
thus $\lim_{U\in\mathsf{Cart}^{op}}X(U)\cong X(*)$. For $K\in\mathsf{sSet}$
and $X\in\mathsf{sPre}(\mathsf{Cart})$, we set

(26)
$\text{Disc}(K)=K_{c},\qquad\Gamma(X)=\lim_{U\in\mathsf{Cart}^{op}}X(U)\cong
X(*),\qquad\Pi_{\infty}(X)=\underset{U\in\mathsf{Cart}^{op}}{\text{colim}}X(U).$

It turns out that $\Gamma$ has a further right adjoint,
$\text{CoDisc}:\mathsf{sSet}\to\mathsf{sPre}(\mathsf{Cart})$ defined
objectwise by

$\text{CoDisc}(K)(U)=K^{\Gamma(yU)}.$

We say that $\text{Disc}(K)$ is the discrete simplicial presheaf on $K$,
$\Gamma(X)$ is the global sections of $X$, $\Pi_{\infty}(X)$ is the
fundamental $\infty$-groupoid or shape of $X$, and that $\text{CoDisc}(K)$ is
the codiscrete simplicial presheaf on $K$. In fact, all of these adjunctions
are simplicially enriched adjunctions.

Thus we obtain the following triple of simplicially enriched adjunctions

(27)
${{\mathsf{sPre}(\mathsf{Cart})}}$${{\mathsf{sSet}}}$$\scriptstyle{\Pi_{\infty}}$CoDiscDisc$\scriptstyle{\Gamma}$$\scriptstyle{\dashv}$$\scriptstyle{\dashv}$$\scriptstyle{\dashv}$

######  Proposition 3.8 ([Sch13, Prop 4.1.30 and 4.1.32]).

Each adjunction in (27) is a simplicial Quillen adjunction, where
$\mathsf{sPre}(\mathsf{Cart})$ is given the Čech model structure
$\check{\mathbb{H}}$, and $\mathsf{sSet}$ is given the Kan-Quillen model
structure.

In [Sch13], Schreiber defines the three following endofunctors on the category
of simplicial presheaves on $\mathsf{Cart}$:

(28) $\displaystyle\textstyle{\int}$
$\displaystyle=\text{Disc}\circ\Pi_{\infty}$ $\displaystyle\flat$
$\displaystyle=\text{Disc}\circ\Gamma$ $\displaystyle\sharp$
$\displaystyle=\text{CoDisc}\circ\Gamma$

called shape, flat and sharp respectively.

######  Remark 3.9.

Using the name shape functor for both $\Pi_{\infty}$ and $\int$ is justified
by remembering that $\text{Disc}:\mathsf{sSet}\to\check{\mathbb{H}}$ is fully
faithful.

They give another pair of simplicial Quillen adjunctions

(29)
${{\check{\mathbb{H}}}}$${{\check{\mathbb{H}}}}$$\scriptstyle{\textstyle{\smallint}}$$\scriptstyle{\sharp}$$\scriptstyle{\flat}$$\scriptstyle{\dashv}$$\scriptstyle{\dashv}$

Let us focus further on the shape functor $\Pi_{\infty}$. Let $\Delta^{k}_{a}$
denote the cartesian space defined by

(30)
$\Delta^{k}_{a}=\left\\{(x_{0},\dots,x_{k})\in\mathbb{R}^{k+1}\,:\,\sum_{i=0}^{k}x_{i}=1\right\\}.$

We call these affine simplices.

Let $\text{Sing}_{\infty}:\mathsf{sPre}(\mathsf{Cart})\to\mathsf{sSet}$ be the
functor defined objectwise by

(31)
$\text{Sing}_{\infty}(X)=\text{hocolim}_{\mathsf{\Delta}^{op}}\left(\leavevmode\hbox
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{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right),$

where if we wish to be concrete, we can use the model of the homotopy colimit
given by taking the diagonal of the above bisimplicial set. We call this the
smooth singular complex functor.

######  Lemma 3.10.

The functor $\text{Sing}_{\infty}$ sends objectwise weak equivalences of
simplicial presheaves to weak equivalences.

###### Proof.

This follows from [GJ12, Proposition 1.9] and taking the diagonal to model the
homotopy colimit. ∎

######  Proposition 3.11.

There are natural weak equivalences between the functors

(32) $\text{Sing}_{\infty}\simeq\text{Sing}_{\infty}\circ
Q\simeq\Pi_{\infty}\circ Q$

where $Q$ denotes a cofibrant replacement functor for $\check{\mathbb{H}}$.

###### Proof.

We mirror the proof given in [Bun22, Remark 4.12], and since the left hand
weak equivalence is shown there, we only prove the middle weak equivalence. If
we restrict the smooth singular complex functor along the Yoneda embedding
$\mathsf{Cart}\hookrightarrow\mathsf{sPre}(\mathsf{Cart})$, then we obtain a
functor $\text{Sing}_{\infty}:\mathsf{Cart}\to\mathsf{sSet}$, and there is a
natural weak equivalence of functors
$\text{Sing}_{\infty}\xrightarrow{\sim}*$, where
$*:\mathsf{Cart}\to\mathsf{sSet}$ is the constant functor on a point
$*=\Delta^{0}$, this follows from [Bun22, Proposition 3.11]. Since all
simplicial sets are cofibrant in the Quillen model structure on
$\mathsf{sSet}$, [Rie14, Corollary 5.2.5] implies that this induces a natural
weak equivalence

(33)
$B(X,\mathsf{Cart},\text{Sing}_{\infty})\xrightarrow{\sim}B(X,\mathsf{Cart},*)$

of simplicial presheaves for every $X\in\mathsf{sPre}(\mathsf{Cart})$. But
$B(X,\mathsf{Cart},*)\cong\text{colim}_{\mathsf{Cart}^{op}}B(X,\mathsf{Cart},y)\cong\text{colim}_{\mathsf{Cart}^{op}}QX\cong\Pi_{\infty}QX$.
This follows from the fact that $B(X,\mathsf{Cart},y)\cong QX$, which is just
repackaging the definition of $QX$. Since $\text{Sing}_{\infty}$ is a left
adjoint, we have

$B(X,\mathsf{Cart},\text{Sing}_{\infty})\cong\text{Sing}_{\infty}B(X,\mathsf{Cart},y)\cong\text{Sing}_{\infty}QX.$

This gives the second natural weak equivalence above. ∎

By Proposition 3.11, we will often refer to $\text{Sing}_{\infty}(X)$ as the
shape of $X$ as well. The shape functor has many wonderful properties. While
we will not need all of the following results on the shape functor for this
paper, we provide a concise listing of them here as such results are scattered
throughout the literature.

######  Remark 3.12.

Since $\int$ is just applying the shape functor and then treating the
resulting simplicial set as a constant simplicial presheaf, along with the
fact that $\int\simeq\text{Disc}\circ\text{Sing}_{\infty}$, we will blur the
distinction between $\Pi_{\infty},\text{Sing}_{\infty}$ and $\int$. We will
use $\int$ when we wish to be ambiguous about which particular model of the
shape functor we wish to use.

######  Theorem 3.13 ([Bun21a, Theorem 4.15]).

Let $M$ be a finite dimensional smooth manifold, thought of as a simplicial
presheaf on $\mathsf{Cart}$, then

(34) $\int M\simeq\text{Sing}(M^{top})$

where $M^{top}$ is the underlying topological space of $M$ and
$\text{Sing}:\mathsf{Top}\to\mathsf{sSet}$ is the classical singular complex.
In other words, the shape of $M$ is its underlying homotopy type.

######  Remark 3.14.

It should be noted that Theorem 3.13 is really a consequence of a classical
result known as the nerve theorem [Bor48].

######  Proposition 3.15 ([Pav22a, Example 14.1]).

Let $\Omega^{n}_{\text{cl}}$ denote the sheaf on $\mathsf{Cart}$ of closed
differential $n$-forms. Its shape is

(35) $\int\Omega^{n}_{\text{cl}}\simeq\textbf{B}^{n}\mathbb{R}^{\delta}.$

######  Lemma 3.16.

Let $K$ be a simplicial set, then the counit

(36) $\varepsilon_{K}:(\Pi_{\infty}\circ\text{Disc})(K)\to K.$

is an isomorphism. In other words, for discrete simplicial presheaves $K_{c}$,
we have

$\int K_{c}\cong K.$

###### Proof.

This follows from the fact that Disc is fully faithful, and the unit of an
adjunction where the right adjoint is fully faithful is an isomorphism. ∎

######  Proposition 3.17 ([Car15, Theorem 3.4]).

Let $M$ denote a simplicial manifold. Let $\underline{M}$ denote the
simplicial presheaf on $\mathsf{Cart}$ defined degreewise by

$\underline{M}(U)_{k}=C^{\infty}(U,M_{k}).$

Then the shape of $\underline{M}$ is

(37) $\int\underline{M}\simeq||M||$

where $||M||$ denotes the homotopy type of the “fat” geometric realization of
$M$, see [Car15, Section 3.2].

######  Remark 3.18.

The above result is especially interesting when $M$ is the nerve of a Lie
groupoid, as this says that the shape of a Lie groupoid (thought of as a
simplicial presheaf) is weak equivalent to the homotopy type of the Lie
groupoid’s classifying space [Car15, Section 2.2].

######  Remark 3.19.

We should also mention that the shape functor has been used to great effect in
what is now called the Smooth Oka Principle. See the following references
[BBP22], [SS21], [Clo23], [Pav22, Section 10].

Let $\text{Sing}_{D}:\mathsf{Diff}\to\mathsf{sSet}$ be the functor defined
levelwise by

(38) $\text{Sing}_{D}(X)_{n}=\mathsf{Diff}(\Delta^{n}_{a},X).$

We call this the diffeological singular complex. Note that by using the
diagonal as a model for the homotopy colimit, for a diffeological space $X$
(actually any sheaf of sets on $\mathsf{Cart}$), we have

(39) $\text{Sing}_{D}(X)\cong\text{Sing}_{\infty}(X).$

######  Proposition 3.20.

Let $X$ be a diffeological space, then

(40) $\Pi_{\infty}(QX)\simeq\text{Sing}_{D}(X)\simeq N\mathsf{Plot}(X)$

where $N\mathsf{Plot}(X)$ is the nerve of the category of plots of $X$.

###### Proof.

We will only prove the second weak equivalence, as the first holds by the
previous discussion. It is shown in [Bun22, Proposition 3.6] that
$\text{Sing}_{\infty}$ has a left and right adjoint (though only the right
adjoint forms a Quillen adjunction), therefore we have

(41) $\displaystyle\Pi_{\infty}(QX)$
$\displaystyle=\Pi_{\infty}\left(\int^{n\in\mathsf{\Delta}}\coprod_{N\mathsf{Plot}(X)_{n}}yU_{p_{n}}\otimes\Delta^{n}\right)$
$\displaystyle\cong\int^{n\in\mathsf{\Delta}}\coprod_{N\mathsf{Plot}(X)_{n}}\Pi_{\infty}(yU_{p_{n}})\times(\Pi_{\infty}\circ\text{Disc})(\Delta^{n})$
$\displaystyle\simeq\int^{n\in\mathsf{\Delta}}\coprod_{N\mathsf{Plot}(X)_{n}}*\times\Delta^{n}$
$\displaystyle\cong N\mathsf{Plot}(X).$

where the weak equivalence is given by Theorem 3.13 and Lemma 3.16. ∎

##  4\. The Irrational Torus

In this section, we will show that if $K\subset\mathbb{R}$ is a
diffeologically discrete subgroup of the real numbers, then the infinity stack
cohomology of the irrational torus $T_{K}=\mathbb{R}^{n}/K$ (for any $n\geq
1$) with values in $\mathbb{R}^{\delta}$, is isomorphic to the group
cohomology of $K$ with values in $\mathbb{R}^{\delta}$. This was first proved
by Iglesias-Zemmour [Igl23, Page 15] with his own version of diffeological
Čech cohomology, which we will refer to as PIZ cohomology. In [Min22] we found
maps between $\infty$-stack cohomology and PIZ cohomology, and showed that one
of these maps is a retract, but it is still an open question as to whether
these two cohomologies are isomorphic.

The motivation for this section is two-fold. One is to support the conjecture
that $\infty$-stack cohomology is isomorphic to PIZ cohomology. We do this by
showing that they agree on one of the most important class of examples of
diffeological spaces, the irrational tori. The second motivation is to show
the power of $\infty$-topos theory and in particular $\infty$-stack cohomology
through the use of the shape operation. The proof of Theorem 4.4 is completely
different than that in [Igl23], and it is conceptually more straightforward.

######  Definition 4.1.

Suppose that $K\subset\mathbb{R}^{n}$ is a subgroup, and furthermore, when $K$
is given the subset diffeology of $\mathbb{R}$, it coincides with the discrete
diffeology, so that every plot is constant. We call its quotient
$T_{K}=\mathbb{R}^{n}/K$ the $n$-dimensional $K$-irrational torus888The word
irrational comes from the example where $n=1$, and
$K=\mathbb{Z}+\alpha\mathbb{Z}$ with $\alpha$ an irrational number. This is
the most studied example of an irrational torus in diffeology. Interestingly,
$\mathbb{Z}+\alpha\mathbb{Z}\subset\mathbb{R}$ with the subset topology is
dense in $\mathbb{R}$, hence not discrete, however the subset diffeology is
discrete..

We define the quotient map $\pi:\mathbb{R}^{n}\to T_{K}$ of diffeological
spaces. This map is a diffeological principal $K$-bundle [Igl13, Article
8.15], where $K$ is given the discrete diffeology. By the discussion in
Example 3.7, the bundle $\pi$ is classified by a map of $\infty$-stacks
$g_{K}:T_{K}\to N\mathsf{DiffPrin}_{K}$. Furthermore, by [Min22, Corollary
6.12], we obtain the following cube, where the front face and back face are
homotopy pullback squares and the maps going from the back face to the front
face are all objectwise weak equivalences

(42)
${{\widetilde{\mathbb{R}^{n}}}}$${{\mathbf{E}K}}$${{\mathbb{R}^{n}}}$${{*}}$${{QT_{K}}}$${{\textbf{B}K}}$${{T_{K}}}$${{N\mathsf{DiffPrin}_{K}}}$$\scriptstyle{\pi}$$\scriptstyle{{g_{K}}}$

Now if we apply the shape functor to this cube, the back face remains a
homotopy pullback by the following result.

######  Proposition 4.2 ([SS21, Proposition 3.3.8]).

Let $K$ be a simplicial set, and let $f:X\to\text{Disc}(K)$ and
$g:Y\to\text{Disc}(K)$ be maps of simplicial presheaves on $\mathsf{Cart}$.
Then there is a Čech weak equivalence

(43) $\left(\smallint X\right)\times^{h}_{\text{Disc}(K)}\left(\smallint
Y\right)\simeq\smallint\left(X\times^{h}_{\text{Disc}(K)}Y\right)$

between the homotopy pullbacks of the maps $\int f$ and $\int g$ and the shape
of the homotopy pullback of the maps $f$ and $g$.

######  Remark 4.3.

For a model-category theoretic proof of Proposition 4.2, use the argument of
[Sch13, Theorem 4.1.34], and mirror the argument of [Sch13, Proposition
4.1.35] model categorically.

Therefore the front face must also be a homotopy pullback square. Now by Lemma
3.10, Lemma 3.16, and Theorem 3.13 it follows that

(44)
$\smallint\widetilde{\mathbb{R}^{n}}\simeq\smallint\mathbb{R}^{n}\simeq*,\qquad\smallint
QT_{K}\simeq\smallint
T_{K},\qquad\smallint\mathbf{E}K\simeq\smallint*\cong*,\qquad\textbf{B}K\cong\smallint\textbf{B}K\simeq\smallint\mathsf{DiffPrin}_{K}.$

From this we obtain the main result of this section.

######  Theorem 4.4.

There is an isomorphism

(45) $\check{H}^{n}_{\infty}(T_{K},\mathbb{R}^{\delta})\cong
H^{n}_{\text{grp}}(K,\mathbb{R}^{\delta})$

of abelian groups, for every $n\geq 0$, where $\mathbb{R}^{\delta}$ denotes
the discrete group of real numbers, and where
$H^{n}_{\text{grp}}(K,\mathbb{R}^{\delta})$ denotes the group cohomology of
$K$ with coefficients in $\mathbb{R}^{\delta}$.

###### Proof.

First we note that since $\mathbb{R}^{\delta}$ is discrete,
$\mathbb{R}^{\delta}\cong\flat\mathbb{R}^{\delta}$. Similarly,
$\textbf{B}^{n}\mathbb{R}^{\delta}\cong\flat\textbf{B}^{n}\mathbb{R}^{\delta}$
for all $n\geq 0$. Thus

(46)
$\mathbb{R}\check{\mathbb{H}}(T_{K},\textbf{B}^{n}\mathbb{R}^{\delta})\cong\mathbb{R}\check{\mathbb{H}}(T_{K},\flat\textbf{B}^{n}\mathbb{R}^{\delta})\cong\mathbb{R}\check{\mathbb{H}}(\smallint
T_{K},\textbf{B}^{n}\mathbb{R}^{\delta}).$

Now from Proposition 3.11, there is a weak equivalence
$\text{Sing}_{D}(T_{K})\to\int T_{K}$ of simplicial sets, and by [CW14,
Proposition 4.30], $\text{Sing}_{D}(T_{K})$ is a Kan complex. Now $\int
N\mathsf{DiffPrin}_{K}$ is a Kan complex since it is a simplicial abelian
group. Furthermore since $T_{K}$ is diffeologically connected [Igl23, Section
II], $\text{Sing}_{D}(T_{K})$ is connected. Thus the composite map
$\text{Sing}_{D}(T_{K})\to\int N\mathsf{DiffPrin}_{K}$ is a map of connected
Kan complexes whose homotopy fiber is contractible. Therefore by the long
exact sequence of homotopy groups [Cis19, Theorem 3.8.12], it is a weak
equivalence, which implies that $\int g_{K}$ is a weak equivalence. Therefore
we have

(47) $\mathbb{R}\check{\mathbb{H}}(\smallint
T_{K},\textbf{B}^{n}\mathbb{R}^{\delta})\simeq\mathbb{R}\check{\mathbb{H}}(\smallint
N\mathsf{DiffPrin}_{K},\textbf{B}^{n}\mathbb{R}^{\delta})\simeq\mathbb{R}\check{\mathbb{H}}(\textbf{B}K,\textbf{B}^{n}\mathbb{R}^{\delta}).$

Which implies that

(48)
$\check{H}^{n}_{\infty}(T_{K},\mathbb{R}^{\delta})=\pi_{0}\mathbb{R}\check{\mathbb{H}}(\smallint
T_{K},\textbf{B}^{n}\mathbb{R}^{\delta})\cong\pi_{0}\mathbb{R}\check{\mathbb{H}}(\textbf{B}K,\textbf{B}^{n}\mathbb{R}^{\delta})=\check{H}^{n}_{\infty}(\textbf{B}K,\mathbb{R}^{\delta}).$

However since $\textbf{B}K$ and $\textbf{B}^{n}\mathbb{R}^{\delta}$ are
discrete, Disc is fully faithful and $\textbf{B}^{n}\mathbb{R}^{\delta}$ is a
Kan complex, we have

$\mathbb{R}\check{\mathbb{H}}(\textbf{B}K,\textbf{B}^{n}\mathbb{R}^{\delta})\simeq\underline{\mathsf{sSet}}(\textbf{B}K,\textbf{B}^{n}\mathbb{R}^{\delta}),$

where
$\underline{\mathsf{sSet}}(\textbf{B}K,\textbf{B}^{n}\mathbb{R}^{\delta})$ is
the usual simplicial set function complex. It is then well known (see [Wei95,
Example 8.2.3] for instance) that

(49)
$\pi_{0}\underline{\mathsf{sSet}}(\textbf{B}K,\textbf{B}^{n}\mathbb{R}^{\delta})\cong
H^{n}_{\text{grp}}(K,\mathbb{R}^{\delta}).$

This proves the theorem. ∎

##  5\. The Dold-Kan Correspondence

In this section, we discuss the Dold-Kan correspondence, which is central to
Section 7.

######  Remark 5.1.

For the remainder of this paper, by a vector space, we mean a real vector
space, not necessarily of finite dimension. By a chain complex we mean a non-
negatively graded chain complex of vector spaces. Let $\mathsf{Ch}$ denote the
category of chain complexes.

######  Definition 5.2.

Let $\mathsf{Vect}$ denote the category whose objects are vector spaces and
whose morphisms are linear maps. Let
$\mathsf{sVect}\coloneqq\mathsf{Vect}^{\mathsf{\Delta}^{op}}$ denote the
category of simplicial vector spaces.

######  Proposition 5.3 ([GS06, Proposition 4.2 and Theorem 4.13], [Jar03,
Section 1]).

The category $\mathsf{sVect}$ admits a proper, combinatorial, simplicial model
category structure where a morphism $f:X\to Y$ is a

  1. (1)

weak equivalence if it is a weak homotopy equivalence of the underlying
simplicial sets,

  2. (2)

fibration if it is a Kan fibration of the underlying simplicial sets,

  3. (3)

cofibration if it is degreewise a monomorphsim.

We call this the Kan-Quillen model structure999Note that every object in
$\mathsf{sVect}$ is fibrant and cofibrant. on $\mathsf{sVect}$.

######  Proposition 5.4 ([GS06, Theorem 1.5], [Jar03, Section 1]).

The category $\mathsf{Ch}$ admits a proper, combinatorial, simplicial model
category structure where a morphism $f:C\to D$ is a

  1. (1)

weak equivalence if it is a quasi-isomorphism of chain complexes, and

  2. (2)

fibration if $f_{k}:C_{k}\to D_{k}$ is surjective in degrees $k\geq 1$.

  3. (3)

cofibration if it is degreewise a monomorphism101010For the projective model
structure on chain complexes of $R$-modules for a general commutative ring
$R$, we require these to be degreewise monomorphisms with projective cokernel.
Since all vector spaces are projective as $\mathbb{R}$-modules, this condition
is always satisfied..

We call this the projective model structure111111Note that every object in
$\mathsf{Ch}$ is fibrant and cofibrant. on chain complexes.

There is an adjoint pair of functors,

(50)
${{\mathsf{Ch}}}$${{\mathsf{sVect}}}$$\scriptstyle{\mathsf{DK}}$$\scriptstyle{N}$$\scriptstyle{\dashv}$

which by the Dold-Kan correspondence [GS06, Theorem 4.1] form an adjoint
equivalence121212This result actually holds for chain complexes and simplicial
objects taking values in any idempotent complete, additive category, see
[Lur17, Theorem 1.2.3.7]..

######  Lemma 5.5 ([SS03, Section 4.1]).

The adjunction $N\dashv\mathsf{DK}$ is a Quillen adjunction ([Hir09,
Definition 8.5.2]) between the model category structures on $\mathsf{Ch}$ and
$\mathsf{sVect}$. Furthermore, the functors form a Quillen equivalence
([Hir09, Definition 8.5.20]). In fact, since $N$ and $\mathsf{DK}$ form an
adjoint equivalence, it follows that $\mathsf{DK}\dashv N$ is also an adjoint
equivalence. Furthermore $N$ and $\mathsf{DK}$ are both left and right Quillen
functors.

If we consider the category of simplicial sets $\mathsf{sSet}$ with its usual
Kan-Quillen model structure [GS06, Theorem 1.22], there is a simplicial
Quillen adjunction

(51)
${{\mathsf{sSet}}}$${{\mathsf{sVect}}}$$\scriptstyle{\mathbb{R}[-]}$$\scriptstyle{U}$$\scriptstyle{\dashv}$

where $U$ denotes the forgetful functor, and $\mathbb{R}[-]$ denotes the
functor that sends a simplicial set $X$ to the free simplicial vector space
$\mathbb{R}X$, defined degreewise by $\mathbb{R}X_{n}=\mathbb{R}(X_{n})$,
where $\mathbb{R}(X_{n})$ is the free vector space on the set $X_{n}$. Thus we
obtain a Quillen adjunction

(52)
${{\mathsf{sSet}}}$${{\mathsf{Ch}}}$$\scriptstyle{N\mathbb{R}[-]}$$\scriptstyle{U\mathsf{DK}}$$\scriptstyle{\dashv}$

which is furthermore a simplicial Quillen adjunction131313We will often omit
the functor $U$ in our notation..

Note that the Dold-Kan correspondence also provides a simplicial enrichment of
$\mathsf{Ch}$. Indeed, suppose $C$ and $D$ are chain complexes, then let
$\underline{\mathsf{Ch}}(C,D)$ denote the simplicial vector space defined
degreewise by

(53)
$\underline{\mathsf{Ch}}(C,D)_{k}=\mathsf{Ch}(N\mathbb{R}\Delta^{k}\otimes
C,D).$

This makes the Dold-Kan correspondence an enriched adjoint equivalence. This
is also the simplicial enrichment mentioned in Proposition 5.4.

This also supplies $\mathsf{Ch}$ with tensoring and cotensoring over
$\mathsf{sSet}$. Namely if $K$ is a simplicial set and $C$ is a chain complex
then $C\otimes K$ is the chain complex $C\otimes N\mathbb{R}K$, and
$C^{K}=N\mathbb{R}\underline{\mathsf{Ch}}(N\mathbb{R}K,C)$.

Now the category of chain complexes $\mathsf{Ch}$ is also enriched over
itself. Indeed, if $C$ and $D$ are chain complexes, then let
$\underline{\text{Map}_{\mathsf{Ch}}}(C,D)$ denote the chain complex defined
as follows. First let us define the unbounded chain complex
$\underline{\text{Map}_{\mathsf{Ch}}}^{\mathbb{Z}}(C,D)$ defined in degree
$k\in\mathbb{Z}$ by

(54) $\underline{\text{Map}_{\mathsf{Ch}}}^{\mathbb{Z}}(C,D)_{k}=\prod_{i\geq
0}\mathsf{Vect}(C_{i},D_{i+k}),$

with
$d:\underline{\text{Map}_{\mathsf{Ch}}}^{\mathbb{Z}}(C,D)_{k}\to\underline{\text{Map}_{\mathsf{Ch}}}^{\mathbb{Z}}(C,D)_{k-1}$
defined for a map $f$ by

$df=d_{D}f-(-1)^{k}fd_{C}.$

We call an element of degree $k$ in
$\underline{\text{Map}_{\mathsf{Ch}}}^{\mathbb{Z}}(C,D)$ a degree $k$ map from
$C$ to $D$.

######  Definition 5.6.

If $C$ is an unbounded ($\mathbb{Z}$-graded) chain complex, then let
$\tau_{\geq 0}C$ denote the chain complex defined degreewise by $(\tau_{\geq
0}C)_{k}=C_{k}$ for $k>0$, and $(\tau_{\geq 0}C)_{0}=\mathbb{Z}_{0}C$, the set
of $0$-cycles of $C$, i.e. those $x\in C_{0}$ such that $dx=0$, the
differential on $\tau_{\geq 0}C$ is induced by the differential on $C$. We
call $\tau_{\geq 0}C$ the smart truncation of $C$.

Now given chain complexes $C$ and $D$, let

(55) $\underline{\text{Map}_{\mathsf{Ch}}}(C,D)=\tau_{\geq
0}\underline{\text{Map}_{\mathsf{Ch}}}^{\mathbb{Z}}(C,D),$

denote the smart truncation applied to
$\underline{\text{Map}_{\mathsf{Ch}}}^{\mathbb{Z}}(C,D)$. This means that
$\underline{\text{Map}_{\mathsf{Ch}}}(C,D)_{k}=\underline{\text{Map}_{\mathsf{Ch}}}^{\mathbb{Z}}(C,D)$
for $k>0$, and
$\underline{\text{Map}_{\mathsf{Ch}}}(C,D)_{0}\cong\mathsf{Ch}(C,D)$. We refer
to $\underline{\text{Map}_{\mathsf{Ch}}}(C,D)$ as the mapping chain complex
between $C$ and $D$.

######  Lemma 5.7 ([Opa21, Example 4.3.2]).

Let $C$ and $D$ be chain complexes, then we have an isomorphism of simplicial
vector spaces

(56)
$\mathsf{DK}\,\underline{\text{Map}_{\mathsf{Ch}}}(C,D)\cong\underline{\mathsf{Ch}}(C,D).$

Further, this provides an isomorphism

(57) $\underline{\text{Map}_{\mathsf{Ch}}}(C,D)\cong
N\underline{\mathsf{Ch}}(C,D).$

An explicit description for the path space of a chain complex $C$,
equivalently the cotensoring $C^{\Delta^{1}}$, is given in Appendix C.

######  Definition 5.8.

Let $\mathcal{C}$ be a small category. Then let $\mathsf{ChPre}(\mathcal{C})$
denote the category whose objects are functors
$\mathcal{C}^{op}\to\mathsf{Ch}$, and whose morphisms are natural
transformations. We call such functors presheaves of chain complexes.

######  Proposition 5.9 ([Hir09, Section 11.6]).

The category $\mathsf{ChPre}(\mathcal{C})$ admits a proper, combinatorial,
simplicial model category structure where a morphism $f:C\to D$ is a

  1. (1)

weak equivalence if it objectwise a weak equivalence in the projective model
structure on chain complexes, and

  2. (2)

fibration if it is objectwise a fibration in the projective model structure on
chain complexes.

We refer to this as the (global) projective model structure on presheaves of
chain complexes.

Thus we obtain a similar simplicial Quillen pair

(58)
${{\mathsf{sPre}(\mathcal{C})}}$${{\mathsf{ChPre}(\mathcal{C})}}$$\scriptstyle{N\mathbb{R}[-]}$$\scriptstyle{U\mathsf{DK}}$$\scriptstyle{\dashv}$

where $\mathsf{sPre}(\mathcal{C})$ is equipped with the projective model
structure. In Appendix C, we will use (58), along with Proposition 3.5, to
compute homotopy pullbacks in $\check{\mathbb{H}}$.

##  6\. Examples of $\infty$-stacks

In this section we detail the $\infty$-stacks involved in this paper, and
examine the $\infty$-stack cohomology of a diffeological space with
coefficients in some of these example $\infty$-stacks.

######  Example 6.1.

Given a finite dimensional smooth manifold $M$, the functor $U\mapsto
C^{\infty}(U,M)$ defines a sheaf on $\mathsf{Cart}$, and therefore an
$\infty$-stack. The same goes for diffeological spaces. Given a diffeological
space $X$, let $X^{\delta}$ denote the diffeological space with the same
underlying set, but equipped with the discrete diffeology. As sheaves we have
$X^{\delta}\cong\text{Disc}(X(*))=\flat X$.

######  Example 6.2.

The presheaf of differential $k$-forms $\Omega^{k}$ and the presheaf of closed
differential $k$-forms $\Omega^{k}_{\text{cl}}$ are sheaves of vector spaces
on $\mathsf{Cart}$ for every $k\geq 0$. Thus they are $\infty$-stacks. The de
Rham differential defines a map of $\infty$-stacks
$d:\Omega^{k}\to\Omega^{k+1}$ for all $k\geq 0$.

There is a canonical map

(59) $\text{mc}(\mathbb{R}):\mathbb{R}\to\Omega^{1}$

of $\infty$-stacks, defined by the Yoneda Lemma as follows. Notice that
$\mathbb{R}\in\mathsf{Cart}$, so by the Yoneda lemma, a map
$\omega:y\mathbb{R}\to\Omega^{1}$ is equivalent to an element
$\omega\in\Omega^{1}(\mathbb{R})$. There is a canonical element of the set of
$1$-forms on $\mathbb{R}$, called the Maurer-Cartan form of $\mathbb{R}$. For
a general Lie group $G$, we let $\text{mc}(G)$ denote its Maurer-Cartan form.
If we label the coordinate of $\mathbb{R}$ by $t$, then the Maurer-Cartan form
is simply given by $\text{mc}(\mathbb{R})=dt$. Thus for a cartesian space $U$,
the function $\text{mc}(\mathbb{R})(U):\mathbb{R}(U)\to\Omega^{1}(U)$ acts by
taking a smooth map $f:U\to\mathbb{R}$ and pulling back the Maurer-Cartan form
$f^{*}\text{mc}(\mathbb{R})\in\Omega^{1}(U)$. Note that this is the same thing
as $df$. In other words as maps of $\infty$-stacks, we have
$\text{mc}(\mathbb{R})=d$.

######  Example 6.3.

Let $G$ be a diffeological group. As discussed in Example 3.7 the presheaf of
groupoids $\textbf{B}G$ given by

$U\mapsto[C^{\infty}(U,G)\rightrightarrows*]$

is a stack, and is objectwise weak equivalent to the stack of diffeological
principal $G$-bundles. We abuse notation as in [Min22, Example 5.13] and also
let $\textbf{B}G$ denote the corresponding simplicial presheaf, which is an
$\infty$-stack. Given a diffeological space $X$, a $G$-cocycle on $X$ as in
Definition 2.6 is precisely a map $QX\to\textbf{B}G$ of simplicial presheaves.
Theorem 2.7 shows that the resulting groupoid of $G$-cocycles on $X$ is
equivalent to the groupoid of diffeological principal $G$-bundles on $X$. Thus
$\check{H}^{1}_{\infty}(X,G)\coloneqq\check{H}^{0}_{\infty}(X,\textbf{B}G)$ is
the set of isomorphism classes of $G$-cocycles, which is isomorphic to the set
of isomorphism classes of diffeological principal $G$-bundles.

######  Example 6.4.

If $G$ is a Lie group with Lie algebra $\mathfrak{g}$, then let
$\Omega^{1}(-,\mathfrak{g})//G$ denote the presheaf of groupoids

$U\mapsto[\Omega^{1}(U,\mathfrak{g})\times
C^{\infty}(U,G)\underset{s}{\overset{t}{\rightrightarrows}}\Omega^{1}(U,\mathfrak{g})]$

where $t(\omega,g)=\omega$ and
$s(\omega,g)=\text{Ad}_{g}^{-1}(\omega)+g^{*}\text{mc}(G)$. The nerve of this
presheaf of groupoids is an $\infty$-stack [FSS+12, Proposition
3.2.5]141414Notice that the definition above is precisely the opposite of the
corresponding $\infty$-stack considered in [FSS+12, Section 3]. This is
because of the convention we use in [Min22, Example 5.13]. However this makes
no difference on the theory as we show in Appendix A.. This is the
$\infty$-stack that classifies principal $G$-bundles with connection. We will
often abuse notation and write $\Omega^{1}(-,\mathfrak{g})//G$ to refer to the
presheaf of groupoids and the simplicial presheaf obtained by taking the nerve
construction. Note that a map $QX\to\Omega^{1}(-,\mathfrak{g})//G$ is
equivalent to the data of a $G$-cocycle $g_{f_{0}}:U_{p_{1}}\to G$ and a
collection $\\{A_{p_{0}}\\}_{p_{0}\in\mathsf{Plot}(X)}$ of $1$-forms
$A_{p_{0}}\in\Omega^{1}(U_{p_{0}},\mathfrak{g})$ such that for every map
$f_{0}:U_{p_{1}}\to U_{p_{0}}$ of plots we have

(60)
$A_{p_{1}}=\text{Ad}_{g_{f_{0}}}^{-1}(f_{0}^{*}A_{p_{0}})+g_{f_{0}}^{*}\text{mc}(G).$

Let us call this collection of data $(g,A)=(\\{g_{f_{0}}\\},\\{A_{p_{0}}\\})$
a $G$-cocycle with connection. We show that this definition of connection is
equivalent to the one given in [Wal12, Definition 3.2.1] in Appendix A.

######  Remark 6.5.

The following examples of simplicial presheaves can be checked to be
$\infty$-stacks by using [Pav22, Corollary 6.2]. One simply needs to notice
that the examples that follow are presheaves of bounded chain complexes, and
can thus be thought of equivalently as presheaves of cochain complexes, and
that the homotopy descent condition for presheaves of cochain complexes is
equivalent to the condition of Dold-Kan applied to the presheaves of chain
complexes to be $\infty$-stacks.

######  Example 6.6.

Given a sheaf $A$ of abelian groups on $\mathsf{Cart}$, with $k\geq 1$, the
simplicial presheaf $\textbf{B}^{k}A$ is obtained by taking Dold-Kan of the
presheaf of chain complexes $[A\to 0\to\dots\to 0]$. When $A$ is an abelian
diffeological group, then $\textbf{B}A$ is the $\infty$-stack that classifies
diffeological principal $A$-bundles, as shown in [Min22].

In this paper we will consider the example $\textbf{B}^{k}\mathbb{R}$. Given a
diffeological space $X$, a map $QX\to\textbf{B}^{k}\mathbb{R}$ consists of a
$g\in\mathbb{R}(QX_{k})$ such that $\delta g=0$, see Appendix B. We call these
$\mathbb{R}$-bundle $(k-1)$-gerbes. Thus a diffeological principal
$\mathbb{R}$-bundle is precisely a $\mathbb{R}$-bundle $0$-gerbe.

There is a vast literature on bundle gerbes in differential geometry such as
[Mur96], [Bun21], [Ste04]. Typically bundle gerbes are defined as geometric
objects, and then shown to define cohomology classes through cocycles such as
above. However, giving descriptions of bundle $k$-gerbes as geometric objects
becomes difficult and tedious as $k$ grows. Their description as cocycles is
much more economical, and is all we need for this paper. There should be no
difficulty in translating between the geometric description of diffeological
bundle $1$-gerbes, such as in [Wal12] and the cocycle description we give
here, but we leave this to future work.

######  Example 6.7.

For $k\geq 1$, let $\textbf{B}^{k}_{\nabla}\mathbb{R}$151515Note that the
$\textbf{B}^{k}$ in $\textbf{B}^{k}_{\nabla}\mathbb{R}$ is just notation, it
is not actually the delooping of anything. denote the simplicial presheaf
obtained by applying Dold-Kan to the following presheaf of chain
complexes161616Note that the way this chain complex is written, $\Omega^{k}$
is in degree $0$.

(61)
$[\mathbb{R}\xrightarrow{d}\Omega^{1}\xrightarrow{d}\Omega^{2}\to\dots\to\Omega^{k}].$

This complex is often referred to as the Deligne complex when $\mathbb{R}$ is
replaced with $U(1)$ and $d:\mathbb{R}\to\Omega^{1}$ is replaced with
$d\log:U(1)\to\Omega^{1}$. For $k\geq 2$, $\textbf{B}^{k}_{\nabla}\mathbb{R}$
classifies $\mathbb{R}$-bundle $(k-1)$-gerbes with connection, and
$\infty$-stack cohomology with values in $\textbf{B}^{k}_{\nabla}\mathbb{R}$
is called pure differential cohomology171717We also recommend [Jaz21, Section
3.2] for a discussion of how pure differential cohomology fits into the
hexagon diagram of differential cohomology. in [Jaz21, Definition 3.2.10].
Thus we call $\textbf{B}^{k}_{\nabla}\mathbb{R}$ the pure $k$-Deligne complex.

Given a diffeological space $X$, we will denote $\infty$-stack cohomology with
values in the pure $k$-Deligne complex by

$\check{H}^{k}_{\infty,\nabla}(X,\mathbb{R})\coloneqq\check{H}^{0}_{\infty}(X,\textbf{B}^{k}_{\nabla}\mathbb{R}).$

Let us also note that when $k=1$, we have an objectwise weak equivalence of
$\infty$-stacks

$\textbf{B}_{\nabla}\mathbb{R}\simeq(\Omega^{1}(-)//\mathbb{R})^{op}.$

To see this, note that both of the simplicial presheaves are identical in
simplicial degrees $0$ and $1$. This is because $\mathbb{R}$ is abelian, so
$t(\omega,g)=\text{Ad}_{g}^{-1}(\omega)+g^{*}\text{mc}(\mathbb{R})=\omega+dg$
in $(\Omega^{1}(-)//\mathbb{R})^{op}$, which is precisely the face map
$d_{0}:\textbf{B}_{\nabla}\mathbb{R}_{1}\to\textbf{B}_{\nabla}\mathbb{R}_{0}$.
Since $(\Omega^{1}(-)//\mathbb{R})^{op}$ is the nerve of a presheaf of
groupoids, it is $2$-coskeletal, and therefore its $k$-homotopy groups are
trivial for $k\geq 2$. The objectwise homotopy groups of
$\textbf{B}_{\nabla}\mathbb{R}$ are given by the objectwise homology of the
chain complex by the Dold-Kan correspondence, and thus are also trivial for
$k\geq 2$, thus they are objectwise weak equivalent. The distinction between
$\Omega^{1}(-)//\mathbb{R}$ and $(\Omega^{1}(-)//\mathbb{R})^{op}$ is because
of [Min22, Example 5.13], so technically $\textbf{B}_{\nabla}\mathbb{R}$
classifies diffeological principal $\mathbb{R}^{op}$-bundles with opposite
connection, but this distinction is immaterial to the theory and we sweep it
under the rug, and we say that the above classifies diffeological principal
$\mathbb{R}$-bundles with connection.

If $X$ is a diffeological space, then as we will see in Example 6.13, a
$\mathbb{R}$-bundle $(k-1)$-gerbe with connection on $X$ is given by the data

(62)
$(\omega^{k},\omega^{k-1},\,\dots\,,\omega^{1},g)\in\Omega^{k}(QX_{0})\oplus\Omega^{k-1}(QX_{1})\oplus\dots\oplus\Omega^{1}(QX_{k-1})\oplus\mathbb{R}(QX_{k}),$

such $D(\omega^{k},\dots,g)=0$ in the double complex $\Omega^{i}(QX_{j})$, see
Appendix B. We will let $[\omega^{k},\,\dots\,,g]$ denote the isomorphism
class it represents in $\check{H}^{k}_{\infty,\nabla}(X,\mathbb{R})$.

######  Example 6.8.

For $k\geq 1$, consider the $\infty$-stack
$\textbf{B}^{k}\mathbb{R}^{\delta}$. This $\infty$-stack classifies
diffeological principal $\mathbb{R}^{\delta}$-bundles. However, note that
there is a map of presheaves of chain complexes

(63) $[\mathbb{R}^{\delta}\to 0\to\dots\to
0]\to[\mathbb{R}\xrightarrow{d}\Omega^{1}\xrightarrow{d}\cdots\xrightarrow{d}\Omega^{k}_{\text{cl}}]$

given by the inclusion $\mathbb{R}^{\delta}\hookrightarrow\mathbb{R}$.
Furthermore, this map is an objectwise quasi-isomorphism, by the Poincare
lemma. Thus we will take the right hand side of (63) to be the model of
$\textbf{B}^{k}\mathbb{R}^{\delta}$ we will use for the rest of this paper.
From this presentation, it is easy to see that
$\textbf{B}^{k}\mathbb{R}^{\delta}$ is equivalently the $\infty$-stack that
classifies diffeological principal $\mathbb{R}$-bundle $(k-1)$-gerbes with
flat connection.

######  Example 6.9.

For $k\geq 1$, consider the $\infty$-stack
$\textbf{B}^{k}\Omega^{1}_{\text{cl}}$. There is a map of presheaves of chain
complexes

(64) $[\Omega^{1}_{\text{cl}}\to 0\to\dots\to
0]\to[\Omega^{1}\xrightarrow{d}\Omega^{2}\xrightarrow{d}\dots\xrightarrow{d}\Omega^{k}_{\text{cl}}]$

and this map is an objectwise quasi-isomorphism again by the Poincare lemma.
We take the right hand side to be the model we will use for
$\textbf{B}^{k}\Omega^{1}_{\text{cl}}$ for the rest of the paper.

######  Example 6.10.

For $k\geq 1$, let $\Omega^{1\leq\bullet\leq k}$ denote the simplicial
presheaf obtained by applying Dold-Kan to the following presheaf of chain
complexes

(65) $[\Omega^{1}\xrightarrow{d}\Omega^{2}\to\dots\to\Omega^{k}].$

If $X$ is a diffeological space and $A$ is a $k$-deloopable $\infty$-stack,
recall that its $\infty$-stack cohomology is given by

$\check{H}^{k}_{\infty}(X,A)=\mathbb{R}\check{\mathbb{H}}(X,\textbf{B}^{k}A).$

Let us compute an example of $\infty$-stack cohomology for a diffeological
space $X$ with values in the $\infty$-stack $\textbf{B}\Omega^{1}$, as it will
be emblematic of how we compute $\infty$-stack cohomology for all of the
relevant examples presented in this section. In Appendix B we go into detail
on how to compute such examples.

######  Example 6.11.

Let $A^{\prime}=\textbf{B}\Omega^{1}=[\Omega^{1}\to 0]$ and
$A=\mathsf{DK}\,A^{\prime}$. Let us compute $H^{0}(X,A)$ for a diffeological
space $X$ using Proposition B.3. Consider the double complex $A^{\prime}(QX)$,

(66)
${{\Omega^{1}(QX_{0})}}$${{\Omega^{1}(QX_{1})}}$${{\Omega^{1}(QX_{2})}}$${\dots}$${0}$${0}$${0}$${\dots}$$\scriptstyle{\delta}$$\scriptstyle{-\delta}$

so that $\mathsf{tot}\,A^{\prime}(QX)$ is the chain complex

(67)
$\mathsf{tot}\,A^{\prime}(QX)=[\Omega^{1}(QX_{0})\xrightarrow{D}\ker(-\delta:\Omega^{1}(QX_{1})\to\Omega^{1}(QX_{2}))]$

Thus a $0$-cycle in $\mathsf{tot}\,A^{\prime}(QX)$ consists of a collection
$\\{\omega_{f_{0}}\in\Omega^{1}(U_{p_{1}})\\}_{f_{0}:U_{p_{1}}\to U_{p_{0}}}$
of $1$-forms for every map of plots of $X$ such that $-\delta\omega=0$, which
is equivalent to the condition that for every pair of composable plot maps
$U_{p_{2}}\xrightarrow{f_{1}}U_{p_{1}}\xrightarrow{f_{0}}U_{p_{0}}$ we have

$(\delta\omega)_{(f_{1},f_{0})}=f_{1}^{*}\omega_{f_{0}}-\omega_{f_{0}f_{1}}+\omega_{f_{1}}=0.$

Two such $0$-cocycles are cohomologous $\omega\sim\omega^{\prime}$ if there
exists a collection
$\\{\lambda_{p_{0}}\in\Omega^{1}(U_{p_{0}})\\}_{p_{0}\in\mathsf{Plot}(X)}$ of
$1$-forms for every plot of $X$ such that

$(D\lambda)_{f_{0}}=(\delta\lambda)_{f_{0}}=\omega^{\prime}_{f_{0}}-\omega_{f_{0}}$

for every map $f_{0}:U_{p_{1}}\to U_{p_{0}}$ of plots of $X$. Thus
$\check{H}^{0}_{\infty}(X,\textbf{B}\Omega^{1})=\check{H}^{1}_{\infty}(X,\Omega^{1})$
is precisely analogous to the term $H^{1,1}_{\delta}$ in [Igl23, Section 4.4].

######  Example 6.12.

Let us compute
$\check{H}^{1}_{\infty,\nabla}(X,\mathbb{R})\coloneqq\check{H}^{0}_{\infty}(X,\textbf{B}_{\nabla}\mathbb{R})$
for a diffeological space $X$. Consider the double complex

(68)
${{\mathbb{R}(QX_{0})}}$${{\mathbb{R}(QX_{1})}}$${{\mathbb{R}(QX_{2})}}$${\dots}$${{\Omega^{1}(QX_{0})}}$${{\Omega^{1}(QX_{1})}}$${{\Omega^{1}(QX_{2})}}$${\dots}$$\scriptstyle{d}$$\scriptstyle{\delta}$$\scriptstyle{-\delta}$$\scriptstyle{-\delta}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{\delta}$

where $\mathbb{R}(QX_{i})=C^{\infty}(QX_{i},\mathbb{R})$. A $0$-cocycle is the
data of a map $g:QX_{1}\to\mathbb{R}$ and a $1$-form $A\in\Omega^{1}(QX_{0})$
such that $-\delta g=0$ and $-\delta A=dg$. The condition $-\delta g=\delta
g=0$ is equivalent to the condition
$g_{f_{0}f_{1}}=f_{1}^{*}g_{f_{0}}+g_{f_{1}}$ for every pair of composable
plot maps $U_{p_{2}}\xrightarrow{f_{1}}U_{p_{1}}\xrightarrow{f_{0}}U_{p_{0}}$,
which is precisely the condition for $g$ to be a cocycle defining a
diffeological principal $\mathbb{R}$-bundle on $X$, see [Min22, Section 5].
The condition $\delta A=dg$ is equivalent to the condition that for every map
of plots $f_{0}:U_{p_{1}}\to U_{p_{0}}$ we have
$A_{p_{1}}-f_{0}^{*}A_{p_{0}}=dg_{f_{0}}$, which is precisely the equation for
a connection on a diffeological principal $\mathbb{R}$-bundle, see Appendix A.
Given two $0$-cocycles $(A,g)$ and $(A^{\prime},g^{\prime})$, a $1$-coboundary
consists of an element $h\in\mathbb{R}(QX_{0})$ such that $\delta
h=g^{\prime}-g$ and $dh=A^{\prime}-A$. This is precisely the definition of a
morphism of $G$-cocycles with connection, see Definition A.1. Thus an element
of $\check{H}^{1}_{\infty,\nabla}(X,\mathbb{R})$ is an isomorphism class of a
diffeological principal $\mathbb{R}$-bundle on $X$ with connection.

######  Example 6.13.

Let us compute $\check{H}^{2}_{\infty,\nabla}(X,\mathbb{R})$ for a
diffeological space $X$. Consider the double complex

(69)
${{\mathbb{R}(QX_{0})}}$${{\mathbb{R}(QX_{1})}}$${{\mathbb{R}(QX_{2})}}$${\dots}$${{\Omega^{1}(QX_{0})}}$${{\Omega^{1}(QX_{1})}}$${{\Omega^{1}(QX_{2})}}$${\dots}$${{\Omega^{2}(QX_{0})}}$${{\Omega^{2}(QX_{1})}}$${{\Omega^{2}(QX_{2})}}$${\dots}$$\scriptstyle{d}$$\scriptstyle{-\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{-\delta}$$\scriptstyle{d}$$\scriptstyle{-\delta}$$\scriptstyle{d}$$\scriptstyle{\delta}$$\scriptstyle{d}$

Then a $0$-cycle in $\mathsf{tot}\,\textbf{B}^{2}_{\nabla}\mathbb{R}(QX)$ is
an element
$(\omega,A,g)\in\Omega^{2}(QX_{0})\oplus\Omega^{1}(QX_{1})\oplus\mathbb{R}(QX_{2})$
such that $D(\omega,A,g)=0$. This is equivalent to the equations
$-\delta\omega=dA$, $-\delta A=dg$ and $-\delta g=0$. The equation $-\delta
g=\delta g=0$ is equivalent to the condition that for every triple of
composable maps of plots $(f_{2},f_{1},f_{0})$ of $X$, we have

(70) $(\delta
g)_{(f_{2},f_{1},f_{0})}=f_{2}^{*}g_{(f_{1},f_{0})}-g_{(f_{1}f_{2},f_{0})}+g_{(f_{2},f_{0}f_{1})}-g_{(f_{2},f_{1})}=0.$

This is the diffeological analogue of the cocycle data of a
$\mathbb{R}$-bundle gerbe on $X$. The other two equations $-\delta\omega=dA$
and $-\delta A=dg$ are the diffeological analogue of the cocycle data of a
connection on a $\mathbb{R}$-bundle gerbe. Thus
$\check{H}^{2}_{\infty,\nabla}(X,\mathbb{R})=\check{H}^{0}_{\infty}(X,\textbf{B}^{2}_{\nabla}\mathbb{R})$
can be taken as the definition of the abelian group of isomorphism classes of
diffeological $\mathbb{R}$-bundle $1$-gerbes with connection on $X$. The story
for $k\geq 2$ is exactly analogous, and thus we take our definition of the
abelian group of isomorphism classes of diffeological $\mathbb{R}$-bundle
$(k-1)$-gerbes with connection to be
$\check{H}^{k}_{\infty,\nabla}(X,\mathbb{R})=\check{H}^{0}_{\infty}(X,\textbf{B}^{k}_{\nabla}\mathbb{R})$.

##  7\. The Čech de Rham Obstruction

In this section, we obtain a diffeological Čech-de Rham obstruction exact
sequence in every degree from a homotopy pullback diagram of $\infty$-stacks.
In degree $1$, our exact sequence is analogous to [Igl23].

Let $X$ be a diffeological space. In [Igl23], Iglesias-Zemmour constructs the
following exact sequence of vector spaces

(71) $0\to
H^{1}_{dR}(X)\to\check{H}^{1}_{PIZ}(X,\mathbb{R}^{\delta})\to{}^{d}E^{1,0}_{2}(X)\xrightarrow{c_{1}}H^{2}_{dR}(X)\to\check{H}^{2}_{PIZ}(X,\mathbb{R}^{\delta})$

using the five term exact sequence coming from a diffeological version of the
Čech-de Rham bicomplex spectral sequence. The vector space
${}^{d}E^{1,0}_{2}(X)$ is the subspace of isomorphism classes of diffeological
principal $\mathbb{R}$-bundles on $X$ that admit a connection, and the vector
spaces $\check{H}^{k}_{\text{PIZ}}(X,\mathbb{R}^{\delta})$ are Iglesias-
Zemmour’s version of diffeological Čech cohomology, which we refer to as PIZ
cohomology. The relationship between PIZ cohomology and $\infty$-stack
cohomology is only partially understood, see [Min22, Section 5].

The exact sequence (71) demonstrates the obstruction to the Čech-de Rham
Theorem holding for diffeological spaces. For finite dimensional smooth
manifolds, all principal $\mathbb{R}$-bundles are trivial, as they have
contractible fiber, and thus the obstruction vanishes. However, there are
diffeological spaces (the irrational torus for example) that have nontrivial
principal $\mathbb{R}$-bundles that admit connections [Igl13, Article 8.39].

We construct and geometrically interpret the obstruction to the Čech-de Rham
isomorphism in each degree $k\geq 1$ via $\infty$-stacks.

######  Theorem 7.1.

For every $k\geq 1$, there exists a commutative diagram of $\infty$-stacks of
the following form

(72)
${{*}}$${{\textbf{B}^{k}\mathbb{R}^{\delta}}}$${{*}}$${{*}}$${{*}}$${{\textbf{B}^{k}_{\nabla}\mathbb{R}}}$${{\Omega^{k+1}_{\text{cl}}}}$${{\Omega^{k+1}}}$${{\textbf{B}^{k}\mathbb{R}}}$${{\textbf{B}^{k}\Omega^{1}_{\text{cl}}}}$${{\Omega^{1\leq\bullet\leq
k+1}}}$${{*}}$${{\textbf{B}^{k+1}\mathbb{R}^{\delta}}}$${{\textbf{B}^{k+1}_{\nabla}\mathbb{R}}}$

furthermore every commutative square in this diagram is a homotopy pullback
square in $\check{\mathbb{H}}$.

We prove Theorem 7.1 in Appendix C.

######  Corollary 7.2.

For every diffeological space $X$, there is an exact sequence of vector
spaces181818Near the completion of this paper, we learned that an analogous
exact sequence was also obtained in [Jaz21, Page 27] using completely
different methods in the framework of homotopy type theory.

(73)
$0\to\check{H}^{k}_{\infty}(X,\mathbb{R}^{\delta})\to\check{H}^{k}_{\infty,\nabla}(X,\mathbb{R})\to\Omega^{k+1}_{\text{cl}}(X)\to\check{H}^{k+1}_{\infty}(X,\mathbb{R}^{\delta}).$

###### Proof.

This follows from Theorem 7.1 and Lemma C.6. ∎

Let us explore the consequences of Corollary 7.2 in the case where $X$ is the
irrational torus. Let $K=\mathbb{Z}+\alpha\mathbb{Z}$ be the subgroup of
$\mathbb{R}$ consisting of those $x\in\mathbb{R}$ of the form $n+\alpha m$
where $n,m$ are integers and $\alpha$ is an irrational number. Let
$T_{\alpha}=\mathbb{R}/\mathbb{Z}+\alpha\mathbb{Z}$. We can fully compute the
de Rham and $\infty$-stack cohomology of $T_{\alpha}$. Every differential form
on $T_{\alpha}$ is closed [Igl13, Exercise 119] so