Hamburger Beiträge zur Mathematik Nr. 857
ZMP–HH/20-13

The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces

Severin Bunk

###### Abstract

Simplicial presheaves on cartesian spaces provide a general notion of smooth
spaces. We define a corresponding smooth version of the singular complex
functor, which maps smooth spaces to simplicial sets. We exhibit this functor
as one of several Quillen equivalences between the Kan-Quillen model category
of simplicial sets and a motivic-style $\mathbb{R}$-localisation of the
(projective or injective) model category of smooth spaces. These Quillen
equivalences and their interrelations are powerful tools: for instance, they
allow us to give a purely homotopy-theoretic proof of a Whitehead
Approximation Theorem for manifolds. Further, we provide a functorial fibrant
replacement in the $\mathbb{R}$-local model category of smooth spaces. This
allows us to compute the homotopy types of mapping spaces in this model
category in terms of smooth singular complexes. We explain the relation of our
fibrant replacement functor to the concordance sheaves introduced recently by
Berwick-Evans, Boavida de Brito, and Pavlov. Finally, we show how the
$\mathbb{R}$-local model category of smooth spaces formalises the homotopy
theory on sheaves used by Galatius, Madsen, Tillmann, and Weiss in their
seminal paper on the homotopy type of the cobordism category.

###### Contents

  1. 1 Introduction and overview
  2. 2 $\mathbb{R}$-local model structures and smooth spaces
    1. 2.1 $\mathbb{R}$-local model structures on simplicial presheaves
    2. 2.2 Evaluation on the point
    3. 2.3 Topological realisation
  3. 3 The smooth singular complex of a simplicial presheaf
    1. 3.1 Extended simplices and the smooth singular complex
    2. 3.2 $S_{e}$ as a left Quillen equivalence
    3. 3.3 $S_{e}$ as a right Quillen equivalence
  4. 4 Comparison of spaces constructed from simplicial presheaves
  5. 5 A Whitehead Approximation Theorem
  6. 6 Local fibrant replacement, concordance, and mapping spaces
  7. A An injective fibrant replacement of simplicial presheaves
  8. B Sheaves on manifolds and sheaves on cartesian spaces
  9. C A modified two-sided bar construction
  10. D Comparing adjunctions and detecting Quillen equivalences

##  1 Introduction and overview

Topological spaces and simplicial sets can be used to construct the same
homotopy theory. This is made rigorous by the fact that the singular complex
and the geometric realisation functors form a Quillen equivalence between the
standard model structure on the category $\mathscr{T}\mathrm{op}$ of
topological spaces and the Kan-Quillen model structure on the category
${\mathscr{S}\mathrm{et}_{\Delta}}$ of simplicial sets. Both of these model
categories formalise what is often called the homotopy theory of spaces, or
$\infty$-groupoids (these are the same according to Grothendieck’s homotopy
hypothesis). The two models differ significantly in their features, though, in
that topological spaces encode homotopies via the notion of continuity, while
simplicial sets are inherently combinatorial. Consequently, each of these two
models for the homotopy theory of spaces has its own merits in different
contexts and applications.

Apart from continuity or combinatorics, another important feature that spaces
can possess and that is relevant in many problems in mathematics is
smoothness. The prime example of a category of smooth spaces is the category
${\mathscr{M}\mathrm{fd}}$ of manifolds and smooth maps, which underlies much
of geometry. There exists a notion of smooth homotopy within the category
${\mathscr{M}\mathrm{fd}}$, and one can find smooth versions of many
topological concepts, such as cohomology, which are invariant under these
smooth homotopies. It would therefore be desirable to have a homotopy theory
on the category of manifolds. However, the category ${\mathscr{M}\mathrm{fd}}$
is poorly behaved in many ways. For instance, it is far from being complete or
cocomplete, thus making it unable to admit a model structure in the sense of
Quillen.

The way to cure this is to weaken—and therefore to generalise—the concept of a
manifold. Here, we take the following approach to smooth spaces, with the main
goal of simultaneously capturing the notions of a manifold and of a (higher)
stack. We start from the category ${\mathscr{C}\mathrm{art}}$ of cartesian
spaces: its objects are all smooth manifolds that are diffeomorphic to
$\mathbb{R}^{n}$ for any $n\in\mathbb{N}_{0}$, and its morphisms are all
smooth maps between these manifolds. We define a smooth space to be a
simplicial presheaf on ${\mathscr{C}\mathrm{art}}$—informally, we understand
the sections of a simplicial presheaf over $c\in{\mathscr{C}\mathrm{art}}$ as
$c$-parameterised families of simplices in a space. We denote the category of
simplicial presheaves on ${\mathscr{C}\mathrm{art}}$ by
$\mathscr{H}_{\infty}$. It contains many geometrically interesting objects
that are not manifolds or even diffeological spaces [22] (for instance the
presheaf of $k$-forms, or the simplicial presheaf of $G$-bundles with
connection, for any Lie group $G$). The category of manifolds, the category of
diffeological spaces, and the category of simplicial sets each include fully
faithfully into $\mathscr{H}_{\infty}$.

The category $\mathscr{H}_{\infty}$ carries two natural model structures,
namely the projective and the injective model structures on functors
${\mathscr{C}\mathrm{art}}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$,
where ${\mathscr{S}\mathrm{et}_{\Delta}}$ carries the Kan-Quillen model
structure. We denote the projective and injective model categories by
$\mathscr{H}_{\infty}^{p}$ and $\mathscr{H}_{\infty}^{i}$, respectively, and
we write $\mathscr{H}_{\infty}^{p/i}$ to refer to either of these model
structures simultaneously. The projective and injective model structures are
canonically Quillen equivalent via the identity functors
$\mathscr{H}_{\infty}^{p}\rightleftarrows\mathscr{H}_{\infty}^{i}$, but they
are not Quillen equivalent to ${\mathscr{S}\mathrm{et}_{\Delta}}$. In that
sense, the model structures $\mathscr{H}_{\infty}^{p/i}$ do not yet define
smooth versions of the homotopy theory of spaces. To achieve that, one needs a
weaker notion of equivalence in $\mathscr{H}_{\infty}$.

There exist (at least) two candidates for such weakened versions of
equivalences in $\mathscr{H}_{\infty}$. First, in [27, 18] a notion of weak
equivalence has been introduced on sheaves on ${\mathscr{M}\mathrm{fd}}$ as
follows: let $\Delta_{e}^{k}\cong\mathbb{R}^{k}$ denote the smooth extended
(affine) $k$-simplex. Extending the usual face and degeneracies of the
topological standard simplices, this gives rise to a cosimplicial cartesian
space
$\Delta_{e}\colon{\mathbbe{\Delta}}\to{\mathscr{C}\mathrm{art}}\subset{\mathscr{M}\mathrm{fd}}$.
Via precomposition, this induces a functor from (pre)sheaves on manifolds to
simplicial sets. In [27, 18], a morphism of sheaves is considered a weak
equivalence of (pre)sheaves whenever it becomes a weak equivalence of
simplicial sets under this functor. We adapt this to our set-up as follows:
for technical reasons, we work with presheaves on cartesian spaces rather than
manifolds, and we work with simplicial (pre)sheaves instead of ordinary
(pre)sheaves. Let
$\delta\colon{\mathbbe{\Delta}}\to{\mathbbe{\Delta}}\times{\mathbbe{\Delta}}$
denote the diagonal functor. We define the _smooth singular complex functor_

$S_{e}\colon\mathscr{H}_{\infty}\xrightarrow{\Delta_{e}^{*}}s{\mathscr{S}\mathrm{et}_{\Delta}}\xrightarrow{\delta^{*}}{\mathscr{S}\mathrm{et}_{\Delta}}\,,$
(1.1)

where the first functor evaluates $F\in\mathscr{H}_{\infty}$ on the extended
simplices to obtain a bisimplicial set, of which the second functor then takes
the diagonal. Let $S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})$ denote the
class of morphisms in $\mathscr{H}_{\infty}$ that are mapped to a weak
equivalence by $S_{e}$. The formal generalisation of the homotopy theory from
[27, 18] is then the localisation

$L_{S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})}\mathscr{H}_{\infty}^{p/i}\,.$
(1.2)

The second notion of weak equivalence in $\mathscr{H}_{\infty}$ is motivated
by motivic homotopy theory (see [35, 26, 12], for instance). Let $I$ denote
the class of all morphisms in $\mathscr{H}_{\infty}$ of the form
$c\times\mathbb{R}\to c$, where $c$ ranges over all objects in
${\mathscr{C}\mathrm{art}}$, and where the morphism is the identity on $c$ and
collapses $\mathbb{R}$ to the point. The localisation

$\mathscr{H}_{\infty}^{p/i\,I}\coloneqq L_{I}\mathscr{H}_{\infty}^{p/i}$ (1.3)

is then a version in smooth geometry of motivic localisation. We call
$\mathscr{H}_{\infty}^{p/i\,I}$ the _$\mathbb{R}$ -local model category_ of
simplicial presheaves on ${\mathscr{C}\mathrm{art}}$, or equivalently of
smooth spaces. This localisation has appeared before in [32, 16] and other
works of these authors. Our first main result is

###### Theorem 1.4

Let $S_{e}\colon\mathscr{H}_{\infty}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ be
as in (1.1).

  1. (1)

The functor
$S_{e}\colon\mathscr{H}_{\infty}^{pI}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ is
a left Quillen equivalence.

  2. (2)

The functor
$S_{e}\colon\mathscr{H}_{\infty}^{iI}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ is
both a left and a right Quillen equivalence.

In particular, the localised model structures $\mathscr{H}_{\infty}^{p/i\,I}$
define homotopy theories of smooth spaces that are equivalent to the usual
homotopy theories of spaces. The key to proving this is to relate the functor
$S_{e}$ to other functors that extract spaces from objects of
$\mathscr{H}_{\infty}$. One of these functors is the left Quillen functor

$Re\colon\mathscr{H}_{\infty}^{p}\to{\Delta\mathscr{T}\mathrm{op}}\,,\qquad
F\longmapsto\int^{c\in{\mathscr{C}\mathrm{art}}^{\mathrm{op}}}|F(c)|\times\mathrm{D}c\,,$
(1.5)

where ${\Delta\mathscr{T}\mathrm{op}}$ is the category of $\Delta$-generated
topological spaces (see Section 2.3 for details). Further,
$|{-}|\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to{\Delta\mathscr{T}\mathrm{op}}$
is the usual geometric realisation functor, and where
$\mathrm{D}c\in{\Delta\mathscr{T}\mathrm{op}}$ is the topological space
underlying $c\in{\mathscr{C}\mathrm{art}}$. We show:

###### Theorem 1.6

Let $Q^{p}\colon\mathscr{H}_{\infty}^{p}\to\mathscr{H}_{\infty}^{p}$ be a
cofibrant replacement functor for the projective model structure. There is a
zig-zag of natural weak equivalences

$|{-}|\circ S_{e}\ \overset{\sim}{\longleftarrow}|{-}|\circ S_{e}\circ Q^{p}\
\overset{\sim}{\longrightarrow}\ Re\circ Q^{p}\,.$ (1.7)

In particular, given any $F\in\mathscr{H}_{\infty}$, we can identify
$|S_{e}(F)|\in{\Delta\mathscr{T}\mathrm{op}}$ with the homotopy colimit of the
diagram
$|F|\colon{\mathscr{C}\mathrm{art}}^{\mathrm{op}}\to{\Delta\mathscr{T}\mathrm{op}}$.

This comparison result has several implications. First, on a very formal
level, it allows us to identify $S_{e}$ as a presentation of the left adjoint
in the cohesive structure on the $\infty$-topos of presheaves of spaces on
${\mathscr{C}\mathrm{art}}$. This has been indicated recently in [3]; here we
prove this formally (see the end of Section 4, and see [5] for an
$\infty$-categorical treatment of this fact).

Further, we can apply Theorem 1.6 to give a purely homotopy-theoretic proof of
the following result, which is sometimes referred to as the Whitehead
Approximation Theorem:

###### Theorem 1.8

Let $M$ be any manifold. The smooth singular complex of $M$—that is, the
simplicial set ${\mathscr{M}\mathrm{fd}}(\Delta_{e},M)$—and the singular
complex $\operatorname{Sing}(M)$ of the topological space underlying $M$ are
weakly equivalent in ${\mathscr{S}\mathrm{et}_{\Delta}}$.

In our proof, at no point do we need to approximate a continuous map by smooth
maps. Instead, the proof relies on a result about Čech nerves of open
coverings from [10], a version of the Nerve Theorem, see e.g. [4, 23, 33, 11]
(concretely, we use [25, Thm. A.3.1], which is sometimes also called Lurie’s
Seifert van Kampen Theorem), and a modified two-sided simplicial bar
construction for simplicial presheaves, which we introduce in Appendix C. This
illustrates that the model categories $\mathscr{H}_{\infty}^{p/i\,I}$ are of
interest beyond their abstract properties: they provide useful tools for doing
smooth homotopy theory.

Next, we construct a fibrant replacement functor ${\mathrm{Cc}}^{p/i}$ for the
localisation $\mathscr{H}_{\infty}^{p/i\,I}$; this construction is motivated
by the _concordance sheaves_ introduced recently in [3]. Thereby, we obtain
explicit access to the mapping spaces in $\mathscr{H}_{\infty}^{p/i\,I}$.
Proving that ${\mathrm{Cc}}^{p/i}$ is indeed a fibrant replacement relies on
the properties of the functor $S_{e}$ that we prove throughout this text, and
in particular on Theorem 1.4. Combining this with Theorems 1.6 and 1.8, we can
compute the mapping spaces in the localisation
$\mathscr{H}_{\infty}^{p/i\,I}$:

###### Theorem 1.9

Let $F,G\in\mathscr{H}_{\infty}$ be any simplicial presheaves on
${\mathscr{C}\mathrm{art}}$. Let $M\in{\mathscr{M}\mathrm{fd}}$ be any
manifold, and define $\underline{M}\in\mathscr{H}_{\infty}$ by setting
$\underline{M}(c)={\mathscr{M}\mathrm{fd}}(c,M)$ for any cartesian space
$c\in{\mathscr{C}\mathrm{art}}$. There are canonical isomorphisms

$\displaystyle\mathrm{Map}_{\mathscr{S}\mathrm{et}_{\Delta}}(S_{e}F,S_{e}G)$
$\displaystyle\cong\mathrm{Map}_{\mathscr{H}_{\infty}^{p/i\,I}}(F,G)\,,$
(1.10) $\displaystyle\mathrm{Map}_{\Delta\mathscr{T}\mathrm{op}}(M,|S_{e}G|)$
$\displaystyle\cong\mathrm{Map}_{\mathscr{H}_{\infty}^{p/i\,I}}(\underline{M},G)$
(1.11)

in $\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$, the homotopy category of
spaces.

Here, the injective case is more easy to treat since every object in
$\mathscr{H}_{\infty}^{iI}$ is cofibrant, so we prove this case first. We then
show the projective case, based on the injective case and on an explicit
Quillen equivalence
$Q^{\prime}:\mathscr{H}_{\infty}^{i}\rightleftarrows\mathscr{H}_{\infty}^{p}:R^{\prime}$
which we construct in Appendix A. (Note that this Quillen equivalence goes in
the opposite direction of the immediate Quillen equivalence
$\mathscr{H}_{\infty}^{p}\rightleftarrows\mathscr{H}_{\infty}^{i}$ induced by
the identity functors.) Further, we relate Theorem 1.9 to [3] explicitly. This
relation relies on results of [3] and on a comparison between homotopy sheaves
on ${\mathscr{C}\mathrm{art}}$ with respect to good open coverings and
homotopy sheaves on ${\mathscr{M}\mathrm{fd}}$ with respect to open coverings:
in Appendix B, we provide an explicit Quillen equivalence between model
categories whose fibrant objects are these two classes of homotopy sheaves.

Finally, we apply Theorem 1.9 to identify the homotopy theory on
$\mathscr{H}_{\infty}$ motivated by the ideas of [27, 18] with the
$\mathbb{R}$-local homotopy theory in the following very strong sense:

###### Theorem 1.12

There is an identity of localisations of $\mathscr{H}_{\infty}^{p/i}$:

$\mathscr{H}_{\infty}^{p/i\,I}=L_{S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})}\mathscr{H}_{\infty}^{p/i}\,.$
(1.13)

This extends and formalises the homotopy theory used in [27, 18] and provides
further interpretation to each of these model structures.

##### Outline.

This paper is organised as follows: We begin in Section 2 by defining the
$\mathbb{R}$-localisations $\mathscr{H}_{\infty}^{p/i\,I}$ of
$\mathscr{H}_{\infty}^{p/i}$. We show that $\mathscr{H}_{\infty}^{pI}$ and
$\mathscr{H}_{\infty}^{iI}$ have the same weak equivalences and that they can
also be seen as further localisations of the local model structures with
respect to differentiably good open coverings. We define the functor
$Re\colon\mathscr{H}_{\infty}^{pI}\to{\Delta\mathscr{T}\mathrm{op}}$ and show
that it is a left Quillen equivalence.

In Section 3, we study the smooth singular complex functor
$S_{e}\colon\mathscr{H}_{\infty}\to{\mathscr{S}\mathrm{et}_{\Delta}}$. We
first show that
$S_{e}\colon\mathscr{H}_{\infty}^{p/i\,I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
is a left Quillen equivalence. Subsequently, we establish
$S_{e}\colon\mathscr{H}_{\infty}^{iI}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ as
a right Quillen equivalence. As an intermediate step, we relate the model
categories $\mathscr{H}_{\infty}^{iI}$ and ${\mathscr{S}\mathrm{et}_{\Delta}}$
to localisations of the model category of complete Segal spaces. This sheds
additional light on the interpretation of the functor $S_{e}$. Section 3
contains the proof of Theorem 1.4.

Section 4 is concerned with the comparison of different ways of extracting
spaces from simplicial presheaves on ${\mathscr{C}\mathrm{art}}$. The key
concept is to extend the homotopy equivalence that embeds the topological
standard simplices into the smooth extended simplices to obtain natural weak
equivalences between functors from $\mathscr{H}_{\infty}$ to
${\mathscr{S}\mathrm{et}_{\Delta}}$ and ${\Delta\mathscr{T}\mathrm{op}}$. Here
we prove Theorem 1.6.

The Whitehead Approximation Theorem, Theorem 1.8, is proven in Section 5. We
start from an open covering of a manifold and manipulate it via the modified
two-sided bar construction from Appendix C until we can apply [25, Thm.
A.3.1].

In Section 6 we construct a fibrant replacement functor for
$\mathscr{H}_{\infty}^{p/i\,I}$ and use it to prove Theorem 1.9. We spell out
the relation of this theorem to [3] and works of Dugger. Then we apply Theorem
6.22 to prove the coincidence of model structures from Theorem 1.12.

Finally, we include four appendices; Appendix A contains the explicit
construction of a fibrant replacement functor for the injective model
structure on $\mathscr{H}_{\infty}$, which features in the proof of Theorem
1.9. Building on this, we provide a Quillen equivalence between model
categories for homotopy sheaves on ${\mathscr{C}\mathrm{art}}$ and homotopy
sheaves on ${\mathscr{M}\mathrm{fd}}$ in Appendix B. In Appendix C, we develop
our modified two-sided bar construction, which we use to prove Theorem 1.8,
and in Appendix D we include some rather standard material on recognising
Quillen equivalences.

##### Notation and conventions

We briefly summarise some notational conventions that will be used throughout
this text.

  * •

We let ${\mathscr{C}\mathrm{at}}$ denote the category of categories, and we
let ${\mathbbe{\Delta}}$ denote the simplex category.

  * •

Given two categories $\mathscr{C},\mathscr{I}$, we write
${\mathscr{C}\mathrm{at}}(\mathscr{I},\mathscr{C})$ or
$\mathscr{C}^{\mathscr{I}}$ for the categories of functors
$\mathscr{I}\to\mathscr{C}$.

  * •

We let
${\mathscr{S}\mathrm{et}_{\Delta}}={\mathscr{C}\mathrm{at}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathscr{S}\mathrm{et}})$
denote the category of simplicial sets. In this article, when viewing
${\mathscr{S}\mathrm{et}_{\Delta}}$ as a model category, we will always use
the cartesian closed Kan-Quillen mode structure on
${\mathscr{S}\mathrm{et}_{\Delta}}$, that is, the model structure for
$\infty$-groupoids.

  * •

We will also be working with the category
$s{\mathscr{S}\mathrm{et}_{\Delta}}={\mathscr{C}\mathrm{at}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathscr{S}\mathrm{et}_{\Delta}})$
of bisimplicial sets. Our convention is always to write a bisimplicial set as
a functor

$X\colon{\mathbbe{\Delta}}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}\,,\quad[n]\mapsto
X_{n}\,,\quad\text{with}\quad X_{n,k}\coloneqq\big{(}X[n]\big{)}[k]\,.$ (1.14)

  * •

If $\mathscr{C}$ is a simplicial category, we write
$\underline{\mathscr{C}}(-,-)\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\longrightarrow{\mathscr{S}\mathrm{et}_{\Delta}}$
for the simplicially enriched hom-functor.

  * •

We let ${\Delta\mathscr{T}\mathrm{op}}$ denote the symmetric monoidal,
simplicial model category of $\Delta$-generated topological spaces.

### Acknowledgements

The author is grateful to Birgit Richter for discussing some of the background
of this paper and for her encouragement to carry out this project. Further,
the author would like to thank Walker Stern for numerous discussions on model
categories and higher structures, Lukas Müller for discussions about smooth
spaces and the string group, as well as Dan Christensen, Enxin Wu, Daniel
Berwick-Evans, Pedro Boavida de Brito, and Dmitri Pavlov for insightful
comments on an earlier version of this paper. The author was partially
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy—EXC 2121 “Quantum
Universe”—390833306.

##  2 $\mathbb{R}$-local model structures and smooth spaces

We start by setting up the model-categorical background used throughout this
paper. Partially following [32] and [14], we consider the category of
simplicial presheaves on cartesian spaces with its canonical projective and
injective model structures. In analogy with ${\mathbb{A}}^{1}$-local homotopy
theory, we localise this category at all the morphisms $c\times\mathbb{R}\to
c$, where $c$ is any cartesian space and where the morphism collapses the real
line. Extending ideas from [14], we establish several Quillen equivalences of
this localised model category with the categories of simplicial sets and
topological spaces.

###  2.1 $\mathbb{R}$-local model structures on simplicial presheaves

In this section we start by setting up the various model structures on
simplicial presheaves that will play a role in this article. Let
${\mathscr{C}\mathrm{art}}$ denote the (small) category of submanifolds of
$\mathbb{R}^{\infty}$ that are diffeomorphic to $\mathbb{R}^{n}$, for any
$n\in\mathbb{N}_{0}$. These manifolds are called _cartesian spaces_. The
morphisms $c\to d$ in ${\mathscr{C}\mathrm{art}}$ are the smooth maps $c\to d$
between these manifolds; in other words, ${\mathscr{C}\mathrm{art}}$ is the
full subcategory of the category ${\mathscr{M}\mathrm{fd}}$ of smooth
manifolds and smooth maps on the cartesian spaces.

Let
$\mathscr{H}_{\infty}\coloneqq{\mathscr{C}\mathrm{at}}({\mathscr{C}\mathrm{art}}^{\mathrm{op}},{\mathscr{S}\mathrm{et}_{\Delta}})$
denote the category of simplicial presheaves on ${\mathscr{C}\mathrm{art}}$.
There is a fully faithful inclusion

$\underline{({-})}\colon{\mathscr{M}\mathrm{fd}}\to\mathscr{H}_{\infty}\,,\qquad\underline{M}(c)={\mathscr{M}\mathrm{fd}}(c,M)\,.$
(2.1)

We view ${\mathscr{S}\mathrm{et}_{\Delta}}$ as endowed with the Kan-Quillen
model structure. The category ${\mathscr{C}\mathrm{art}}$ carries a
Grothendieck pretopology $\tau$ of differentiably good open coverings—see [6,
17] for details. A covering of $c\in{\mathscr{C}\mathrm{art}}$ in this
pretopology is a collection of morphisms $\\{\iota_{i}\colon c_{i}\to
c\\}_{i\in\Lambda}$ in ${\mathscr{C}\mathrm{art}}$ such that each $\iota_{i}$
is an embedding of an open subset, the images of the maps $\iota_{i}$ cover
$c$ (i.e. each $x\in c$ lies in the image of some $\iota_{i}$), and every
finite intersection

$C_{i_{0}\ldots
i_{n}}\coloneqq\bigcap_{a=0}^{n}\iota_{i_{a}}(c_{i_{a}})\subset c$ (2.2)

with $i_{0},\ldots,i_{n}\in\Lambda$ is either empty or a cartesian space. (For
$F\in\mathscr{H}_{\infty}$ we set $F(\emptyset)=*$, in accordance with the
Yoneda Lemma and $\underline{\mathscr{H}}_{\infty}(\emptyset,F)=*$, where here
$\emptyset\in\mathscr{H}_{\infty}$ is the initial object.) We let $\ell$
denote the class of Čech coverings in $\mathscr{H}_{\infty}$ with respect to
the Grothendieck pretopology $\tau$. Given a simplicial model category
${\mathscr{M}}$ and a class $S$ of morphisms in ${\mathscr{M}}$, we denote by
$L_{S}{\mathscr{M}}$ the simplicially enriched left Bousfield localisation of
${\mathscr{M}}$ at the morphisms in $S$ (see [2] for more background).

###### Definition 2.3

We define the following model categories:

  1. (1)

$\mathscr{H}_{\infty}^{p/i}$ is the projective (resp. injective) model
structure on $\mathscr{H}_{\infty}$. We also refer to
$\mathscr{H}_{\infty}^{p/i}$ as the model categories of _smooth spaces_.

  2. (2)

We define the ${\mathscr{S}\mathrm{et}_{\Delta}}$-enriched left Bousfield
localisations

$\mathscr{H}_{\infty}^{p/i\,\ell}\coloneqq
L_{\ell}\mathscr{H}_{\infty}^{p/i}\,.$ (2.4)

This is the projective (resp. injective) model structure for sheaves of
$\infty$-groupoids on ${\mathscr{C}\mathrm{art}}$.

  3. (3)

Let
$I\coloneqq\\{{\mathscr{Y}}_{c}\times{\mathscr{Y}}_{\mathbb{R}}\longrightarrow{\mathscr{Y}}_{c}\\}_{c\in{\mathscr{C}\mathrm{art}}}$
be the set of morphisms obtained by taking the product of the collapse map
$\mathbb{R}\to*$ with all identities
$\\{1_{c}\\}_{c\in{\mathscr{C}\mathrm{art}}}$. We define the projective (resp.
injective) _$\mathbb{R}$ -local model category of smooth spaces_ as the
enriched left Bousfield localisation

$\mathscr{H}_{\infty}^{p/i\,I}\coloneqq L_{I}\mathscr{H}_{\infty}^{p/i}\,.$
(2.5)

  4. (4)

We can further define the model categories

$\mathscr{H}_{\infty}^{p/i\,\ell I}\coloneqq
L_{I}\mathscr{H}_{\infty}^{p/i\,\ell}\,,\qquad\mathscr{H}_{\infty}^{p/i\,I\ell}\coloneqq
L_{\ell}\mathscr{H}_{\infty}^{p/i\,I}\,.$ (2.6)

###### Remark 2.7

We interpret the localisation $\mathscr{H}_{\infty}^{p/i\,I}$ as an
$\mathbb{R}$-localisation of $\mathscr{H}_{\infty}^{p/i}$ akin to motivic
localisation (see, for example, [35, 26, 12]). Though thinking of objects in
$\mathscr{H}_{\infty}$ as smooth spaces, we will mostly refer objects in
$\mathscr{H}_{\infty}$ by the more technically precise term of simplicial
presheaves, and we will sometimes refer to fibrant objects in
$\mathscr{H}_{\infty}^{p/i\,I}$ as _locally constant_ simplicial presheaves.
$\triangleleft$

###### Proposition 2.8

All model structures in Definition 2.3 are simplicial, left proper, tractable,
and symmetric monoidal.

###### Proof.

Except for the claim that the model structures are symmetric monoidal, all
assertions follow from [2, Thm. 4.46]. The model structures for sheaves of
$\infty$-groupoids are symmetric monoidal by [2, Thm. 4.58]. To see that
$\mathscr{H}_{\infty}^{p/i\,I}$ is symmetric monoidal, observe that the
objects ${\mathscr{Y}}_{c}$, $c\in{\mathscr{C}\mathrm{art}}$, form a set of
homotopy generators for $\mathscr{H}_{\infty}^{p/i}$. Let
$F\in\mathscr{H}_{\infty}^{p/i\,I}$ be a local object, and consider the
internal hom object $F^{{\mathscr{Y}}_{d}}$ for any
$d\in{\mathscr{C}\mathrm{art}}$. For any of the morphisms
${\mathscr{Y}}_{c}\times{\mathscr{Y}}_{\mathbb{R}}\to{\mathscr{Y}}_{c}$ in
$I$, the internal hom adjunction yields a commutative diagram of simplicially
enriched hom spaces

${\underline{\mathscr{H}}_{\infty}({\mathscr{Y}}_{c},F^{{\mathscr{Y}}_{d}})}$${\underline{\mathscr{H}}_{\infty}({\mathscr{Y}}_{c}\times{\mathscr{Y}}_{\mathbb{R}},F^{{\mathscr{Y}}_{d}})}$${\underline{\mathscr{H}}_{\infty}({\mathscr{Y}}_{c\times
d},F)}$${\underline{\mathscr{H}}_{\infty}({\mathscr{Y}}_{c\times
d}\times{\mathscr{Y}}_{\mathbb{R}},F)}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$
(2.9)

Here we have used that ${\mathscr{C}\mathrm{art}}$ has finite products. The
bottom horizontal morphism is induced by the morphism $c\times
d\times\mathbb{R}\to c\times d$, which is an element of $I$. Hence, the bottom
morphism is a weak equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$.
Therefore, it follows from [2, Prop. 4.47] that
$\mathscr{H}_{\infty}^{p/i\,I}$ is symmetric monoidal. The exact same proof
shows that $\mathscr{H}_{\infty}^{p/i\,\ell I}$ is symmetric monoidal as well.
The fact that $\mathscr{H}_{\infty}^{p/i\,I\ell}$ is symmetric monoidal will
be seen in Corollary 2.29. ∎

###### Proposition 2.10

There are commutative diagrams of simplicial Quillen adjunctions:

${\mathscr{H}_{\infty}^{p}}$${\mathscr{H}_{\infty}^{i}}$${\mathscr{H}_{\infty}^{p\ell}}$${\mathscr{H}_{\infty}^{i\ell}}$${\mathscr{H}_{\infty}^{p\ell
I}}$${\mathscr{H}_{\infty}^{i\ell I}}$ and
${\mathscr{H}_{\infty}^{p}}$${\mathscr{H}_{\infty}^{i}}$${\mathscr{H}_{\infty}^{pI}}$${\mathscr{H}_{\infty}^{iI}}$${\mathscr{H}_{\infty}^{pI\ell}}$${\mathscr{H}_{\infty}^{iI\ell}}$
(2.11)

where the rightwards and downwards arrows are the left adjoints. All arrows
are identity functors, and all horizontal arrows are Quillen equivalences.

###### Proof.

This follows directly from [6, Prop. 3.13] and the well-known fact that the
identity functor on $\mathscr{H}_{\infty}$ induces a Quillen equivalence
$\mathscr{H}_{\infty}^{p}\rightleftarrows\mathscr{H}_{\infty}^{i}$. ∎

###### Proposition 2.12

Each pair of model categories defined in Definition 2.3(1)–(4) (based on
either the projective or the injective model structure), respectively, has the
same weak equivalences.

###### Proof.

For the model structures in (1) this is clear by definition of the projective
and injective model structures. Regarding the pairs of model categories in
(2), let $Q\colon\mathscr{H}_{\infty}\to\mathscr{H}_{\infty}$ be a cofibrant
replacement functor for the projective model structure
$\mathscr{H}_{\infty}^{p}$. A morphism $f\colon F\to G$ is a weak equivalence
in $\mathscr{H}_{\infty}^{p\ell}$ if and only if $Qf\colon QF\to QG$ is a weak
equivalence in $\mathscr{H}_{\infty}^{p\ell}$. Since the identity functor on
$\mathscr{H}_{\infty}$ induces a Quillen equivalence
$\mathscr{H}_{\infty}^{p\ell}\rightleftarrows\mathscr{H}_{\infty}^{i\ell}$
(Proposition 2.10), and since the left adjoint of a Quillen equivalence
preserves and reflects weak equivalences between cofibrant objects, $Qf$ is a
weak equivalence in $\mathscr{H}_{\infty}^{p\ell}$ if and only if it is a weak
equivalence in $\mathscr{H}_{\infty}^{i\ell}$. Since the canonical natural
transformation $Q\overset{\sim}{\longrightarrow}1$ is an objectwise weak
equivalence, that is equivalent to $f$ being a weak equivalence in
$\mathscr{H}_{\infty}^{p\ell}$ itself. The proofs for (3) and (4) are
analogous. ∎

The reason why we also refer to fibrant objects in
$\mathscr{H}_{\infty}^{p/i\,I}$ as _locally constant_ simplicial presheaves is
the following fact (the second statement is a generalisation of [14, Lemma
3.4.2]):

###### Proposition 2.13

Let $F\in\mathscr{H}_{\infty}$. The following statements hold true:

  1. (1)

The canonical morphism $F\otimes{\mathscr{Y}}_{c}\to F$ is a weak equivalence
in $\mathscr{H}_{\infty}^{p/i\,I}$, for every $c\in{\mathscr{C}\mathrm{art}}$.

  2. (2)

The object $F$ is fibrant in $\mathscr{H}_{\infty}^{p/i\,I}$ if and only if it
is fibrant in $\mathscr{H}_{\infty}^{p/i}$ and the canonical map $F(*)\to
F(c)$ is a weak equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$ for every
$c\in{\mathscr{C}\mathrm{art}}$

###### Proof.

Ad (1): By Proposition 2.10, it suffices to show this for
$\mathscr{H}_{\infty}^{iI}$. There, every object is cofibrant (since every
object in ${\mathscr{S}\mathrm{et}_{\Delta}}$ is cofibrant), so that the
functor
$F\otimes(-)\colon\mathscr{H}_{\infty}^{iI}\to\mathscr{H}_{\infty}^{iI}$ is
left Quillen. Thus, it suffices to show that the morphism
${\mathscr{Y}}_{c}\to*$ is a weak equivalence. Since $c\cong\mathbb{R}^{n}$
for some $n\in\mathbb{N}_{0}$, we can reduce to the case where
$c=\mathbb{R}^{n}$.

We can write the collapse morphism $\mathbb{R}^{n}\to*$ as a composition

$\mathbb{R}^{n}\cong\mathbb{R}^{n-1}\times\mathbb{R}\longrightarrow\mathbb{R}^{n-1}\cong\mathbb{R}^{n-2}\times\mathbb{R}\longrightarrow\cdots\longrightarrow*\,,$
(2.14)

where each arrow is an element of $I$. Thus, the claim follows.

Ad (2): The second condition implies that $F$ is fibrant in
$\mathscr{H}_{\infty}^{p/i\,I}$: since ${\mathscr{C}\mathrm{art}}$ has finite
products, we have a commutative triangle

${F(*)}$${F(c)}$${F(c\times\mathbb{R})}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$
(2.15)

for any $c\in{\mathscr{C}\mathrm{art}}$. The fact that $F$ is $I$-local thus
follows from the two-out-of-three property of weak equivalences in
${\mathscr{S}\mathrm{et}_{\Delta}}$.

It remains to check the other implication. By part (1) we know that for each
$c\in{\mathscr{C}\mathrm{art}}$, the morphism ${\mathscr{Y}}_{c}\to*$ is a
weak equivalence in $\mathscr{H}_{\infty}^{p/i\,I}$. The claim then follows
from the enriched Yoneda Lemma: the top arrow in the commutative diagram

${\underline{\mathscr{H}}_{\infty}(*,F)}$${\underline{\mathscr{H}}_{\infty}({\mathscr{Y}}_{c},F)}$${F(*)}$${F(c)}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$
(2.16)

is a weak equivalence since $\mathscr{H}_{\infty}^{p/i\,I}$ is symmetric
monoidal (Proposition 2.8), so that $\underline{\mathscr{H}}_{\infty}(-,F)$ is
a right Quillen functor. As representables are cofibrant in
$\mathscr{H}_{\infty}^{p/i\,I}$, the functor
$\underline{\mathscr{H}}_{\infty}(-,F)$ thus preserves the weak equivalence
${\mathscr{Y}}_{c}\overset{\sim}{\longrightarrow}*$. ∎

We can show that our definitions of model structures on $\mathscr{H}_{\infty}$
are redundant. This uses the following strong theorem, which goes back to
Joyal.

###### Theorem 2.17

_[ 31, Thm. 15.3.1]_ Let ${\mathscr{M}}$ and ${\mathscr{M}}^{\prime}$ be two
model categories with the same underlying category. Then, ${\mathscr{M}}$ and
${\mathscr{M}}^{\prime}$ coincide as model categories if and only if they have
the same cofibrations and the same fibrant objects.

###### Corollary 2.18

We have to following identities of model categories:

$\mathscr{H}_{\infty}^{p/i\,\ell
I}=\mathscr{H}_{\infty}^{p/i\,I}=\mathscr{H}_{\infty}^{p/i\,I\ell}\,.$ (2.19)

In particular, every Čech-local weak equivalence is an $I$-local weak
equivalence.

###### Proof.

By their construction as left Bousfield localisations, all of the above three
model categories have the same cofibrations. Thus, it suffices to check that
their fibrant objects coincide.

We first show that $\mathscr{H}_{\infty}^{p/i\,\ell
I}=\mathscr{H}_{\infty}^{p/i\,I}$. An object
$F\in\mathscr{H}_{\infty}^{p/i\,\ell I}$ is fibrant if and only if it is
fibrant in $\mathscr{H}_{\infty}^{p/i\,\ell}$ and satisfies that the canonical
map

$F(c)\cong\underline{\mathscr{H}}_{\infty}({\mathscr{Y}}_{c},F)\longrightarrow\underline{\mathscr{H}}_{\infty}({\mathscr{Y}}_{c}\times{\mathscr{Y}}_{\mathbb{R}},F)\cong
F(c\times\mathbb{R})$ (2.20)

is a weak equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$, for every
$c\in{\mathscr{C}\mathrm{art}}$. That is, an object in
$\mathscr{H}_{\infty}^{p/i\,\ell I}$ is fibrant precisely if it is fibrant in
both $\mathscr{H}_{\infty}^{p/i\,\ell}$ and in
$\mathscr{H}_{\infty}^{p/i\,I}$. In particular, this implies that $F$ is
fibrant also in $\mathscr{H}_{\infty}^{p/i\,I}$.

Conversely, let $F\in\mathscr{H}_{\infty}^{p/i\,I}$ be fibrant. We need to
check that $F$ satisfies descent with respect to the Grothendieck pretopology
$\tau$ of differentiably good open coverings on ${\mathscr{C}\mathrm{art}}$.
To that end, let $c\in{\mathscr{C}\mathrm{art}}$, let
${\mathscr{U}}=\\{c_{i}\to c\\}_{i\in\Lambda}$ be a covering of $c$ in the
site $({\mathscr{C}\mathrm{art}},\tau)$, and consider the commutative diagram

$\displaystyle F(c)\longrightarrow$
$\displaystyle\underset{{\mathbbe{\Delta}}}{\mathrm{holim}}^{\mathscr{S}\mathrm{et}_{\Delta}}\Big{(}\cdots\prod_{i_{0},\ldots,i_{n}}\underline{\mathscr{H}}_{\infty}(QC_{i_{0}\ldots
i_{n}},F)\cdots\Big{)}$ (2.21)
$\displaystyle\simeq\underset{{\mathbbe{\Delta}}}{\mathrm{holim}}^{\mathscr{S}\mathrm{et}_{\Delta}}\Big{(}\cdots\prod_{i_{0},\ldots,i_{n}}F(C_{i_{0}\ldots
i_{n}})\cdots\Big{)}\,,$ (2.22)

where $C_{i_{0}\ldots i_{n}}$ are as in (2.2). (The superscript on
$\mathrm{holim}$ denotes the category in which the homotopy limit is formed.)
Since $F$ is locally constant (i.e. fibrant in
$F\in\mathscr{H}_{\infty}^{p/i\,I}$), by Proposition 2.13 the collapse maps
$c\to*$ induce weak equivalences $F(*)\overset{\sim}{\longrightarrow}F(c)$. We
thus have a commutative diagram

${F(*)}$${\underset{{\mathbbe{\Delta}}}{\mathrm{holim}}^{\mathscr{S}\mathrm{et}_{\Delta}}\Big{(}\cdots\displaystyle{\prod_{i_{0},\ldots,i_{n}\in\Lambda}}F(*)\cdots\Big{)}}$${F(c)}$${\underset{{\mathbbe{\Delta}}}{\mathrm{holim}}^{\mathscr{S}\mathrm{et}_{\Delta}}\Big{(}\cdots\displaystyle{\prod_{i_{0},\ldots,i_{n}\in\Lambda}}F(C_{i_{0}\ldots
i_{n}})\cdots\Big{)}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$ (2.23)

in ${\mathscr{S}\mathrm{et}_{\Delta}}$. We claim that the top morphism in this
diagram is an equivalence: to see this, we first note that, by assumption on
the covering ${\mathscr{U}}$, the collapse morphism
$\operatorname{Sing}(C_{i_{0}\ldots i_{n}})\to*$ is a weak equivalence in
${\mathscr{S}\mathrm{et}_{\Delta}}$ for any $i_{0},\ldots,i_{n}\in\Lambda$
such that $C_{i_{0}\ldots i_{n}}$ is non-empty. For any fibrant
$K\in{\mathscr{S}\mathrm{et}_{\Delta}}$, we thus obtain a weak equivalence

$K\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(*,K)\overset{\sim}{\longrightarrow}\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}\big{(}\operatorname{Sing}(C_{i_{0}\ldots
i_{n}}),K\big{)}\,.$ (2.24)

Since $F(*)\in{\mathscr{S}\mathrm{et}_{\Delta}}$ is fibrant, the product of
these morphisms indexed by $i_{0},\ldots,i_{n}\in\Lambda$ is still a weak
equivalence, so we obtain a commutative diagram

${F(*)}$${\underset{{\mathbbe{\Delta}}}{\mathrm{holim}}^{\mathscr{S}\mathrm{et}_{\Delta}}\Big{(}\cdots\displaystyle{\prod_{i_{0},\ldots,i_{n}\in\Lambda}}F(*)\cdots\Big{)}}$${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}\big{(}\operatorname{Sing}(c),F(*)\big{)}}$${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{hocolim}}}^{\mathscr{S}\mathrm{et}_{\Delta}}\Big{(}\cdots\displaystyle{\coprod_{i_{0},\ldots,i_{n}\in\Lambda}}\operatorname{Sing}(C_{i_{0}\ldots
i_{n}})\cdots\Big{)},F(*)\Big{)}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$
(2.25)

It follows from an application of [24, Thm. A.3.1], or by [11, Thm. 1.1] that
the morphism

$\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{{\mathrm{hocolim}}}^{\mathscr{S}\mathrm{et}_{\Delta}}\Big{(}\cdots\displaystyle{\coprod_{i_{0},\ldots,i_{n}\in\Lambda}}\operatorname{Sing}(C_{i_{0}\ldots
i_{n}})\cdots\Big{)}\longrightarrow\operatorname{Sing}(c)$ (2.26)

is a weak equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$. This is
preserved by the right Quillen functor
$\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(-,F(*))$, and thus the claim
follows. The proof for
$\mathscr{H}_{\infty}^{p/i\,I\ell}=\mathscr{H}_{\infty}^{p/i\,I}$ works
entirely in parallel. ∎

###### Remark 2.27

Corollary 2.18 fails for simplicial (pre)sheaves on manifolds: the proof we
give above relies on the fact that the left-hand vertical morphism in Diagram
(2.25) is a weak equivalence. This is true because every
$c\in{\mathscr{C}\mathrm{art}}$ has an underlying topological space which is
contractible. In contrast, consider a simplicial presheaf on manifolds,
$G\colon{\mathscr{M}\mathrm{fd}}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$,
which is projectively fibrant and $I$-local, i.e. it satisfies that the
canonical morphism $G(M)\to G(M\times\mathbb{R})$ is a weak equivalence for
every $M\in{\mathscr{M}\mathrm{fd}}$. Then, $G$ does not necessarily satisfy
descent with respect to open coverings of manifolds. For instance, consider
the presheaf $[-,\mathbb{S}^{1}]$, sending $M\in{\mathscr{M}\mathrm{fd}}$ to
the set of homotopy classes of continuous (or smooth) maps from $M$ to
$\mathbb{S}^{1}$. This is projectively fibrant and $I$-local. However, let
${\mathscr{U}}=\\{U_{i}\\}_{i\in\Lambda}$ be an open covering of
$M=\mathbb{S}^{1}$ such that each finite intersection $U_{i_{0}\ldots i_{n}}$
is empty or a cartesian space. Then,

$\mathrm{holim}_{{\mathbbe{\Delta}}}^{\mathscr{S}\mathrm{et}_{\Delta}}\prod_{i_{0},\ldots,i_{n}\in\Lambda}[U_{i_{0}\ldots
i_{n}},\mathbb{S}^{1}]\simeq*\,,$ (2.28)

but $[\mathbb{S}^{1},\mathbb{S}^{1}]\not\simeq*$. For more details on the
relation between sheaves on ${\mathscr{C}\mathrm{art}}$ and sheaves on
${\mathscr{M}\mathrm{fd}}$, see Appendix B. $\triangleleft$

###### Corollary 2.29

The model category $\mathscr{H}_{\infty}^{p/i\,I\ell}$ is symmetric monoidal.

###### Proposition 2.30

Let $L_{\mathbb{R}^{\bullet}}\mathscr{H}_{\infty}^{p/i}$ denote the simplicial
left Bousfield localisation of $\mathscr{H}_{\infty}^{p/i}$ at the collapse
morphisms $\\{c\to*\\}_{c\in{\mathscr{C}\mathrm{art}}}$. Further, let
$L_{\mathscr{C}\mathrm{art}}\mathscr{H}_{\infty}^{p/i}$ denote the left
Bousfield localisation of $\mathscr{H}_{\infty}^{p/i}$ at all morphisms in
${\mathscr{C}\mathrm{art}}$. We have the following identities of model
categories:

$\mathscr{H}_{\infty}^{p/i\,I}\overset{(1)}{=}L_{\mathbb{R}^{\bullet}}\mathscr{H}_{\infty}^{p/i}\overset{(2)}{=}L_{\mathscr{C}\mathrm{art}}\mathscr{H}_{\infty}^{p/i}\,.$
(2.31)

###### Proof.

The first identity follows from Proposition 2.13 and Theorem 2.17. The second
identity holds true since for any morphism $c\to d$ in
${\mathscr{C}\mathrm{art}}$ there exists a commutative triangle

${c}$${d}$${*}$ (2.32)

Therefore, the weak equivalences and the cofibrations in
$L_{\mathbb{R}^{\bullet}}\mathscr{H}_{\infty}^{p/i}$ and in
$L_{\mathscr{C}\mathrm{art}}\mathscr{H}_{\infty}^{p/i}$ coincide. ∎

###  2.2 Evaluation on the point

Here we present the first of several ways of extracting a space from an object
$F\in\mathscr{H}_{\infty}$ and show that it provides a Quillen equivalence
between $\mathscr{H}_{\infty}^{p/i\,I}$ and the Kan-Quillen model category
${\mathscr{S}\mathrm{et}_{\Delta}}$.

Consider the adjunction

${\tilde{\mathsf{c}}:{\mathscr{S}\mathrm{et}_{\Delta}}}$${\mathscr{H}_{\infty}^{p/i}:\operatorname{ev}_{*}\,,}$$\scriptstyle{\perp}$
(2.33)

whose left adjoint $\tilde{\mathsf{c}}$ sends a simplicial set $K$ to the
constant simplicial presheaf with value $K$, and whose right adjoint evaluates
a simplicial presheaf at the final object $*\in{\mathscr{C}\mathrm{art}}$.
(Indeed, the adjunction is Quillen for both targets $\mathscr{H}_{\infty}^{p}$
and $\mathscr{H}_{\infty}^{i}$; in the projective case, we readily see that
$\operatorname{ev}_{*}$ is right Quillen, and in the injective case we see
that $\tilde{\mathsf{c}}$ is left Quillen.) Composing this with the
localisation adjunction
$\mathscr{H}_{\infty}^{p/i}\rightleftarrows\mathscr{H}_{\infty}^{p/i\,I}$, we
obtain Quillen adjunctions

${\tilde{\mathsf{c}}:{\mathscr{S}\mathrm{et}_{\Delta}}}$${\mathscr{H}_{\infty}^{p/i\,I}:\operatorname{ev}_{*}\,.}$$\scriptstyle{\perp}$
(2.34)

###### Lemma 2.35

Let $e\colon\tilde{\mathsf{c}}\circ\operatorname{ev}_{*}\to
1_{\mathscr{H}_{\infty}}$ denote the evaluation natural transformation of the
adjunction (2.34). The morphism
$e_{|F}\colon\tilde{\mathsf{c}}\circ\operatorname{ev}_{*}(F)\longrightarrow F$
is an objectwise weak equivalence whenever $F\in\mathscr{H}_{\infty}^{p/i\,I}$
is fibrant.

###### Proof.

For any $F\in\mathscr{H}_{\infty}$, the morphism $e_{|F}$ of simplicial
presheaves is the morphisms $F(c)\to F(*)$ in
${\mathscr{S}\mathrm{et}_{\Delta}}$ induced by the collapse maps $c\to*$. It
readily follows from Proposition 2.13 that $e_{|F}$ is an objectwise weak
equivalence whenever $F$ is fibrant. ∎

###### Lemma 2.36

Let $K\in{\mathscr{S}\mathrm{et}_{\Delta}}$ be any simplicial set. Let
$R^{i}\colon\mathscr{H}_{\infty}^{i}\to\mathscr{H}_{\infty}^{i}$ denote a
fibrant replacement functor in $\mathscr{H}_{\infty}^{i}$ (see Appendix A for
an explicit construction). Then, the simplicial presheaf
$R^{i}\tilde{\mathsf{c}}(K)$ is fibrant in $\mathscr{H}_{\infty}^{iI}$.

###### Proof.

The simplicial presheaf $R^{i}\circ\tilde{\mathsf{c}}(K)$ is fibrant in
$\mathscr{H}_{\infty}^{i}$ by construction. It further comes with an
_objectwise_ weak equivalence
$r^{i}_{\tilde{\mathsf{c}}K}\colon\tilde{\mathsf{c}}K\overset{\sim}{\longrightarrow}R^{i}\tilde{\mathsf{c}}(K)$.
Thus, it follows by the two-out-of-three property of weak equivalences in
${\mathscr{S}\mathrm{et}_{\Delta}}$ that, for any
$c\in{\mathscr{C}\mathrm{art}}$, the canonical map
$(R^{i}\tilde{\mathsf{c}}(K))(*)\longrightarrow(R^{i}\tilde{\mathsf{c}}(K))(c)$
is a weak equivalence. The claim then follows from Proposition 2.30. ∎

We can now prove a version of [14, Thm. 3.4.3] in the context of simplicial
presheaves on cartesian spaces rather than on manifolds. (There, the proof is
outlined for simplicial presheaves on manifolds, where several additional
steps are necessary. Since we work over cartesian spaces, we can employ a
slightly different strategy in our proof that allows us to avoid these
additional steps.)

###### Theorem 2.37

The Quillen adjunction $\tilde{\mathsf{c}}\dashv\operatorname{ev}_{*}$ from
(2.34) is a Quillen equivalence.

###### Proof.

It is evident that $\tilde{\mathsf{c}}$ both preserves and reflects weak
equivalences as a functor
$\tilde{\mathsf{c}}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to\mathscr{H}_{\infty}^{p/i}$.
We claim that it still has that property as a functor
$\tilde{\mathsf{c}}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to\mathscr{H}_{\infty}^{p/i\,I}$.
First, since every weak equivalence in $\mathscr{H}_{\infty}^{p/i}$ is also a
weak equivalence in $\mathscr{H}_{\infty}^{p/i\,I}$, it follows that
$\tilde{\mathsf{c}}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to\mathscr{H}_{\infty}^{p/i\,I}$
still preserves weak equivalences. It thus remains to check that it still
reflects weak equivalences. To see this, let $\psi\colon K\to L$ be an
arbitrary morphism in ${\mathscr{S}\mathrm{et}_{\Delta}}$. Since
$\tilde{\mathsf{c}}$ takes values in cofibrant objects (in both the projective
and the injective situation), it follows that the morphism
$\tilde{\mathsf{c}}\psi\colon\tilde{\mathsf{c}}K\to\tilde{\mathsf{c}}L$ is a
weak equivalence in $\mathscr{H}_{\infty}^{p/i\,I}$ if and only if, for every
fibrant object $G\in\mathscr{H}_{\infty}^{p/i\,I}$, the induced morphism

${\underline{\mathscr{H}}_{\infty}(\tilde{\mathsf{c}}L,G)}$${\underline{\mathscr{H}}_{\infty}(\tilde{\mathsf{c}}K,G)}$$\scriptstyle{(\tilde{\mathsf{c}}\psi)^{*}}$
(2.38)

is a weak equivalence of simplicial sets. By adjointness, this is equivalent
to requiring that the morphism

${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(L,\operatorname{ev}_{*}G)}$${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(K,\operatorname{ev}_{*}G)}$$\scriptstyle{\psi^{*}}$
(2.39)

be a weak equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$ whenever
$G\in\mathscr{H}_{\infty}^{p/i\,I}$ is fibrant. Suppose that this is the case,
i.e. suppose that $\tilde{\mathsf{c}}(\psi)$ is a weak equivalence in
$\mathscr{H}_{\infty}^{p/i\,I}$.

We briefly need to treat the projective and injective cases separately: for a
model category $\mathscr{C}$, let $\mathscr{C}_{f}$ denote the full
subcategory of $\mathscr{C}$ on the fibrant objects. In the projective case,
the functor
$\operatorname{ev}_{*}\colon(\mathscr{H}_{\infty}^{pI})_{f}\to({\mathscr{S}\mathrm{et}_{\Delta}})_{f}$
is surjective; it hits every Kan complex. Therefore, it follows that the map

${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(L,T)}$${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(K,T)}$$\scriptstyle{\psi^{*}}$
(2.40)

is a weak equivalence for every Kan complex
$T\in{\mathscr{S}\mathrm{et}_{\Delta}}$. Hence, $\psi$ is a weak equivalence.
For the injective case, one may now invoke the fact that
$\mathscr{H}_{\infty}^{pI}$ and $\mathscr{H}_{\infty}^{iI}$ have the same weak
equivalences (Proposition 2.12); that already proves the claim in the
injective case as well.

However, for later purposes, we briefly give the following alternative
argument: the construction in Lemma 2.36 shows that every Kan complex is
weakly equivalent to one in the image of the functor
$\operatorname{ev}_{*}\colon(\mathscr{H}_{\infty}^{iI})_{f}\to({\mathscr{S}\mathrm{et}_{\Delta}})_{f}$.
By the two-out-of-three property and the fact that every simplicial set is
cofibrant, it then follows that the map

${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(L,T)}$${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(K,T)}$$\scriptstyle{\psi^{*}}$
(2.41)

is a weak equivalence for _every_ Kan complex $T$, and thus that $\psi$ is a
weak equivalence.

To complete the proof that $\tilde{\mathsf{c}}\dashv\operatorname{ev}_{*}$
provides a Quillen equivalence
${\mathscr{S}\mathrm{et}_{\Delta}}\rightleftarrows\mathscr{H}_{\infty}^{p/i\,I}$,
let $F\in\mathscr{H}_{\infty}^{p/i\,I}$ be fibrant, and consider the
composition

${\tilde{\mathsf{c}}\circ
Q^{\mathscr{S}\mathrm{et}_{\Delta}}\circ\operatorname{ev}_{*}(F)}$${\tilde{\mathsf{c}}\circ\operatorname{ev}_{*}(F)}$${F\,,}$$\scriptstyle{\tilde{\mathsf{c}}(q_{|\operatorname{ev}_{*}F})}$$\scriptstyle{e_{|F}}$
(2.42)

where $Q^{\mathscr{S}\mathrm{et}_{\Delta}}$ is a cofibrant replacement functor
in ${\mathscr{S}\mathrm{et}_{\Delta}}$ with natural weak equivalence $q\colon
Q^{\mathscr{S}\mathrm{et}_{\Delta}}\to 1$, and where the second morphism
$e_{|F}$ is the component at $F$ of the evaluation of the adjunction
$\tilde{\mathsf{c}}\dashv\operatorname{ev}_{*}$. Since every object in
${\mathscr{S}\mathrm{et}_{\Delta}}$ is cofibrant and since
$\tilde{\mathsf{c}}$ is left Quillen, we have that
$\tilde{\mathsf{c}}(q_{|\operatorname{ev}_{*}F})$ is a weak equivalence.
Further, the morphism $e_{|F}$ is an equivalence by Lemma 2.35. The claim now
follows by [21, Cor. 1.3.6]. ∎

###### Example 2.43

Let $G$ be a Lie group with Lie algebra ${\mathfrak{g}}$. Consider the object
${\mathscr{B}\mathrm{un}}_{G,0}^{\nabla}\in\mathscr{H}_{\infty}^{p}$, whose
value on $c\in{\mathscr{C}\mathrm{art}}$ is the nerve of the following
groupoid (in particular, ${\mathscr{B}\mathrm{un}}_{G,0}^{\nabla}$ is fibrant
in $\mathscr{H}_{\infty}^{p}$ by construction): its objects are
${\mathfrak{g}}$-valued 1-forms $A\in\Omega^{1}(c,{\mathfrak{g}})$ such that
$\mathrm{d}A+\frac{1}{2}[A,A]=0$, and its morphisms $A\to A^{\prime}$ are
smooth maps $g\colon c\to G$ such that $A^{\prime}={\mathrm{Ad}}(g^{-1})\circ
A+g^{*}\mu_{G}$, where $\mu_{G}$ is the Maurer-Cartan form on $G$. In other
words, $A$ is a flat $G$-connection on a trivial principal $G$-bundle on $c$,
and $g$ is equivalently a morphism of flat principal $G$-bundles on $c$. In
particular, any such morphism $g$ is actually a _constant_ map $g\colon c\to
G$. Observe that ${\mathscr{B}\mathrm{un}}_{G,0}^{\nabla}(*)$ is the nerve of
the groupoid with one object and the group underlying $G$ as its morphisms. It
hence follows that the functor
${\mathscr{B}\mathrm{un}}_{G,0}^{\nabla}(*)\longrightarrow{\mathscr{B}\mathrm{un}}_{G,0}^{\nabla}(c)$
is fully faithful (on the underlying groupoids), for any
$c\in{\mathscr{C}\mathrm{art}}$. Since any flat $G$-bundle on $c$ is
isomorphic to the trivial flat $G$-bundle (because $c\cong\mathbb{R}^{n}$ for
some $n\in\mathbb{N}_{0}$), the functor
${\mathscr{B}\mathrm{un}}_{G,0}^{\nabla}(*)\longrightarrow{\mathscr{B}\mathrm{un}}_{G,0}^{\nabla}(c)$
is also essentially surjective. Since the nerve of an equivalence of groupoids
is an equivalence of Kan complexes, it follows that
${\mathscr{B}\mathrm{un}}_{G,0}^{\nabla}$ is a fibrant object in
$\mathscr{H}_{\infty}^{pI}$. $\triangleleft$

###  2.3 Topological realisation

In this subsection we further build on and extend ideas from [14] to
investigate a second way of obtaining a space from a simplicial presheaf on
${\mathscr{C}\mathrm{art}}$. This time, we send a simplicial presheaf to a
certain coend valued in topological spaces.

More precisely, we let ${\Delta\mathscr{T}\mathrm{op}}$ denote the category of
$\Delta$-generated topological spaces (see [13, 36] for background). We will
be working with ${\Delta\mathscr{T}\mathrm{op}}$ as our choice of category of
topological spaces throughout; however, most of the theory in this paper also
works with the category of Kelley spaces (see, for instance, [21]), except for
where we work explicitly with diffeological spaces (Lemma 2.61, Remark 4.39).

We provide some very compact background on $\Delta$-generated topological
spaces. A topological space $X$ is $\Delta$-generated precisely if its
topology coincides with the final topology induced by all continuous maps
$|\Delta^{n}|\to X$, for all $n\in\mathbb{N}_{0}$. (Here, $|\Delta^{n}|$ is
the standard topological $n$-simplex.) In particular, the category of
$\Delta$-generated topological spaces and continuous maps, denoted
${\Delta\mathscr{T}\mathrm{op}}$, is symmetric monoidal and cartesian closed
[36]. Note, however, that the product in ${\Delta\mathscr{T}\mathrm{op}}$ is
not the usual product of topological spaces—one has to pass to the
$\Delta$-generated topology after taking the usual product of topological
spaces.

The category ${\Delta\mathscr{T}\mathrm{op}}$ carries a cofibrantly generated
model structure, having the same generating cofibrations and generating
trivial cofibrations as the standard model structure on topological spaces
[13]. The geometric realisation functor
$|{-}|\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to{\Delta\mathscr{T}\mathrm{op}}$
takes values in $\Delta$-generated spaces, since
${\Delta\mathscr{T}\mathrm{op}}$ is closed under colimits of topological
spaces and contains $|\Delta^{n}|$ for each $n\in\mathbb{N}_{0}$. By
construction of the model structure on ${\Delta\mathscr{T}\mathrm{op}}$, the
induced adjunction
$|{-}|:{\mathscr{S}\mathrm{et}_{\Delta}}\rightleftarrows{\Delta\mathscr{T}\mathrm{op}}:\operatorname{Sing}$
is a Quillen adjunction ($|{-}|$ sends generating (trivial) cofibrations to
(trivial) cofibrations). Further, by the same proof as in [21, Lemma 3.18], it
follows that $|{-}|$ preserves finite products. Then, the proof of [21, Prop.
4.2.11] applies as well, showing that ${\Delta\mathscr{T}\mathrm{op}}$ is a
symmetric monoidal model category. It also follows that
${\Delta\mathscr{T}\mathrm{op}}$ is a simplicial model category and that
$|{-}|$ is a monoidal left Quillen functor. Finally, the Quillen adjunction
$|{-}|\dashv\operatorname{Sing}$ is even a Quillen equivalence, since the
inclusion of ${\Delta\mathscr{T}\mathrm{op}}$ into Kelley spaces (or all
topological spaces) is a Quillen equivalence [13]; the claim then follows from
the two-out-of-three property of Quillen equivalences.

In this section, we provide a left Quillen equivalence
$\mathscr{H}_{\infty}^{pI}\to{\Delta\mathscr{T}\mathrm{op}}$. The main ideas
for this section stem from [14]; there the full proof is technically rather
involved. Again, we circumvent these problems here by working over cartesian
spaces rather than over the category of manifolds.

Let ${\mathscr{D}\mathrm{fg}}$ denote the category of diffeological spaces as
defined in [6]—this is the full subcategory of
${\mathscr{C}\mathrm{at}}({\mathscr{C}\mathrm{art}}^{\mathrm{op}},{\mathscr{S}\mathrm{et}})$
on the concrete sheaves with respect to the Grothendieck pretopology $\tau$
(see the beginning of Section 2.1). Concretely, a diffeological space can be
defined as a pair $(X,\mathrm{Plot}_{X})$, where
$X\in{\mathscr{S}\mathrm{et}}$, and where $\mathrm{Plot}_{X}$ assigns to every
cartesian space $c\in{\mathscr{C}\mathrm{art}}$ a subset
$\mathrm{Plot}_{X}(c)\subset{\mathscr{S}\mathrm{et}}(c,X)$ of the maps from
the underlying set of $c$ to $X$. These maps are called _plots of $X$_ and
have to satisfy that

  1. (1)

$\mathrm{Plot}_{X}(*)=X$ (every constant map is a plot),

  2. (2)

for every $f\in{\mathscr{C}\mathrm{art}}(c,d)$ and every
$g\in\mathrm{Plot}_{X}(d)$, we have that $g\circ f\in\mathrm{Plot}_{X}(c)$
(i.e. $\mathrm{Plot}_{X}$ is a presheaf on ${\mathscr{C}\mathrm{art}}$), and

  3. (3)

the presheaf $\mathrm{Plot}_{X}$ is a sheaf with respect to $\tau$.

We will often identify a diffeological space $(X,\mathrm{Plot}_{X})$ with the
sheaf it defines (see [6] for more background), and we will denote this simply
by $X$.

###### Example 2.44

For any manifold $M\in{\mathscr{M}\mathrm{fd}}$, the presheaf $\underline{M}$,
given by $c\mapsto{\mathscr{M}\mathrm{fd}}(c,M)$, is a diffeological space.
$\triangleleft$

###### Definition 2.45

Let
$\mathrm{D}\colon{\mathscr{D}\mathrm{fg}}\to{\Delta\mathscr{T}\mathrm{op}}$ be
the functor defined as follows: for $X\in{\mathscr{D}\mathrm{fg}}$, we let
$\mathrm{D}X$ be the underlying set of the diffeological space
$X\in{\mathscr{D}\mathrm{fg}}$, endowed with the final topology defined by its
plots $c\to X$, where $c$ ranges over all cartesian spaces. A morphism
$f\in{\mathscr{D}\mathrm{fg}}(X,Y)$ is sent to the map it defines on the sets
underlying $X$ and $Y$. We call $\mathrm{D}$ the _diffeological topology_
functor and $\mathrm{D}(X)$ the _underlying topological space of $X$_.

The $\Delta$-generated topological spaces are in fact precisely those
topological spaces that arise as the underlying topological spaces of
diffeological spaces [34, 8].

###### Proposition 2.46

_[ 9, 34]_ There exists an adjunction

${\mathrm{D}:{\mathscr{D}\mathrm{fg}}}$${{\Delta\mathscr{T}\mathrm{op}}:\mathrm{C}\,,}$$\scriptstyle{\perp}$
(2.47)

where (under the embedding of ${\mathscr{D}\mathrm{fg}}$ into presheaves on
${\mathscr{C}\mathrm{art}}$) we have
$\mathrm{C}(T)(c)={\Delta\mathscr{T}\mathrm{op}}(c,T)$ for any topological
space $T$ and any cartesian space $c$.

The following proposition consists of results that can already be found in
[8]; we only include the proofs here for completeness.

###### Proposition 2.48

Let $(-)\times(-)$ denote the product in ${\Delta\mathscr{T}\mathrm{op}}$, and
let $(-)\times^{t}(-)$ denote the usual product of topological spaces. The
functor $\mathrm{D}$ has the following properties:

  1. (1)

For any manifold $M$, the space $\mathrm{D}(\underline{M})$ coincides with the
underlying topological space of $M$.

  2. (2)

For any manifolds $M,N\in{\mathscr{M}\mathrm{fd}}$, the canonical maps
$\mathrm{D}(M\times
N)\to\mathrm{D}M\times\mathrm{D}N\to\mathrm{D}M\times^{t}\mathrm{D}N$ are
homeomorphisms.

  3. (3)

$\mathrm{D}\colon{\mathscr{D}\mathrm{fg}}\to{\Delta\mathscr{T}\mathrm{op}}$
preserves finite products.

###### Proof.

Part (1) is [9, E.g. 3.7]: it is clear that any subset $U\subset M$ which is
open in the manifold topology is also open in the diffeological topology.
Conversely, if $U$ is open in $\mathrm{D}(\underline{M})$, then its
intersection with all images of charts of $M$ must be open. As these images
form a basis for the manifold topology, $U$ is open in the manifold topology.

Part (2) follows readily from Part (1) together with the fact that $M\times N$
is again a manifold.

Part (3) is merely [8, Lemma 4.1] and the remarks following that lemma. For
completeness, we fill in the details omitted there. In [8] it is proven that
the natural map $\mathrm{D}(X\times Y)\to\mathrm{D}X\times\mathrm{D}Y$ is a
homeomorphism whenever $\mathrm{D}X$ is locally compact Hausdorff. Since
$\mathrm{D}\underline{c}$ is locally compact Hausdorff for any
$c\in{\mathscr{C}\mathrm{art}}$, and since $\mathrm{D}$ preserves colimits, we
have the following canonical isomorphisms in ${\Delta\mathscr{T}\mathrm{op}}$:
let $X,Y\in{\mathscr{D}\mathrm{fg}}$ be arbitrary. Using that
${\mathscr{D}\mathrm{fg}}$ and ${\Delta\mathscr{T}\mathrm{op}}$ are cartesian
closed, we compute

$\displaystyle\mathrm{D}(X\times Y)$
$\displaystyle\cong\mathrm{D}\big{(}(\underset{{\mathscr{C}\mathrm{art}}/X}{{\mathrm{colim}}}^{\mathscr{D}\mathrm{fg}}\underline{c})\times
Y\big{)}$ (2.49)
$\displaystyle\cong\underset{{\mathscr{C}\mathrm{art}}/X}{{\mathrm{colim}}}^{\Delta\mathscr{T}\mathrm{op}}\mathrm{D}(\underline{c}\times
Y)$ (2.50)
$\displaystyle\cong(\underset{{\mathscr{C}\mathrm{art}}/X}{{\mathrm{colim}}}^{\Delta\mathscr{T}\mathrm{op}}\mathrm{D}\underline{c})\times
Y$ (2.51) $\displaystyle\cong\mathrm{D}X\times\mathrm{D}Y\,.$ (2.52)

In the third isomorphism we have used the above-mentioned result [8, Lemma
4.1]. ∎

We point out that we only use manifolds without boundary or corners here. For
manifolds with boundary, part (1) of Proposition 2.48 fails—see, for instance,
[9, Cor. 4.47] and [28, Warning 2.22]. Since each cartesian space
$c\in{\mathscr{C}\mathrm{art}}$ is diffeomorphic to $\mathbb{R}^{n}$ for some
$n\in\mathbb{N}_{0}$, and since $\mathbb{R}^{n}$ is (isomorphic to) a CW
complex for any $n\in\mathbb{N}_{0}$, it follows that
$\mathrm{D}\underline{c}$ is cofibrant in ${\Delta\mathscr{T}\mathrm{op}}$ for
every $c\in{\mathscr{C}\mathrm{art}}$. We have the following version of [16,
Prop. 2.3]:

###### Theorem 2.53

There exists a Quillen adjunction $Re\dashv S$, sitting inside a weakly
commutative diagram

${{\mathscr{C}\mathrm{art}}}$${{\Delta\mathscr{T}\mathrm{op}}}$${\mathscr{H}_{\infty}^{p}}$$\scriptstyle{{\mathscr{Y}}}$$\scriptstyle{\mathrm{D}}$$\scriptstyle{S}$$\scriptstyle{Re}$
(2.54)

Further, there is a canonical natural isomorphism
$Re\circ{\mathscr{Y}}\cong\mathrm{D}$.

###### Proof.

The functor $Re$ is defined as the (enriched) left Kan extension of
$\mathrm{D}$ along ${\mathscr{Y}}$ in digram (2.54). Explicitly, we can write

$\displaystyle Re(F)$
$\displaystyle=\int^{c\in{\mathscr{C}\mathrm{art}}}F(c)\otimes\mathrm{D}\underline{c}$
(2.55)
$\displaystyle=\int^{c\in{\mathscr{C}\mathrm{art}}}|F(c)|\times\mathrm{D}\underline{c}\,,$
(2.56) $\displaystyle S(T)(c)$
$\displaystyle={\Delta\mathscr{T}\mathrm{op}}(|\Delta^{\bullet}|\times\mathrm{D}\underline{c},T)$
(2.57) $\displaystyle\cong\operatorname{Sing}(T^{\mathrm{D}\underline{c}})\,.$
(2.58)

Since $\mathrm{D}c$ is cofibrant in ${\Delta\mathscr{T}\mathrm{op}}$ and
$\operatorname{Sing}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to{\Delta\mathscr{T}\mathrm{op}}$
is right Quillen, it follows that $S$ maps fibrations (resp. trivial
fibrations) in ${\Delta\mathscr{T}\mathrm{op}}$ to objectwise fibrations
(resp. trivial fibrations) in $\mathscr{H}_{\infty}$. Thus, $S$ is right
Quillen.

The second claim follow from the canonical isomorphisms

$\displaystyle
Re({\mathscr{Y}}_{d})=\int^{c\in{\mathscr{C}\mathrm{art}}}{\mathscr{Y}}_{d}(c)\otimes\mathrm{D}\underline{c}\cong\int^{c\in{\mathscr{C}\mathrm{art}}}{\mathscr{C}\mathrm{art}}(c,d)\times\mathrm{D}\underline{c}\cong\mathrm{D}\underline{d}\,.$
(2.59)

The statement now follows from Proposition 2.48(1). ∎

###### Lemma 2.60

The adjunction $Re\dashv S$ has the following properties:

  1. (1)

It is a simplicial adjunction.

  2. (2)

$S$ is monoidal.

###### Proof.

Part (1) holds true since geometric realisation preserves finite products of
simplicial sets and since the functor
$K\otimes(-)\colon{\Delta\mathscr{T}\mathrm{op}}\to{\Delta\mathscr{T}\mathrm{op}}$
is a left adjoint, for any $K\in{\mathscr{S}\mathrm{et}_{\Delta}}$. Part (2)
holds true since $S$ is right adjoint and ${\Delta\mathscr{T}\mathrm{op}}$ is
cartesian monoidal. ∎

###### Lemma 2.61

Consider the fully faithful inclusion
$\iota\colon{\mathscr{D}\mathrm{fg}}\hookrightarrow{\mathscr{C}\mathrm{at}}({\mathscr{C}\mathrm{art}}^{\mathrm{op}},{\mathscr{S}\mathrm{et}})\hookrightarrow\mathscr{H}_{\infty}$.
The diagram

${{\mathscr{D}\mathrm{fg}}}$${\mathscr{H}_{\infty}}$${{\Delta\mathscr{T}\mathrm{op}}}$$\scriptstyle{\iota}$$\scriptstyle{\mathrm{D}}$$\scriptstyle{Re}$
(2.62)

commutes up to natural isomorphism. In particular, for any manifold
$M\in{\mathscr{M}\mathrm{fd}}$, $Re\underline{M}$ is homeomorphic to the
underlying topological space of $M$.

###### Proof.

For $X\in{\mathscr{D}\mathrm{fg}}$, we have canonical natural isomorphisms

$Re\circ\iota(X)\cong
Re\int^{c\in{\mathscr{C}\mathrm{art}}}\iota(X)(c)\otimes{\mathscr{Y}}_{c}\cong\int^{c\in{\mathscr{C}\mathrm{art}}}\iota(X)(c)\otimes\mathrm{D}\underline{c}$
(2.63)

and

$\mathrm{D}X\cong\mathrm{D}\int^{c\in{\mathscr{C}\mathrm{art}}}\iota(X)(c)\otimes\underline{c}\cong\int^{c\in{\mathscr{C}\mathrm{art}}}\iota(X)(c)\otimes\mathrm{D}\underline{c}\,.$
(2.64)

Combining this with Proposition 2.48 completes the proof. ∎

###### Proposition 2.65

The pair $Re\dashv S$ induces a Quillen adjunction

${Re:\mathscr{H}_{\infty}^{pI}}$${{\Delta\mathscr{T}\mathrm{op}}:S\,.}$$\scriptstyle{\perp}$
(2.66)

###### Proof.

By [19, Prop. 3.1.6, Prop. 3.3.18], it suffices to show that $S$ preserves
fibrant objects as a functor
${\Delta\mathscr{T}\mathrm{op}}\to\mathscr{H}_{\infty}^{pI}$. By Proposition
2.30 we are thus left to check that, for every
$T\in{\Delta\mathscr{T}\mathrm{op}}$ and any $c\in{\mathscr{C}\mathrm{art}}$,
the canonical morphism $S(T)(*)\to S(T)(c)$ is a weak equivalence in
${\mathscr{S}\mathrm{et}_{\Delta}}$. However, recalling the canonical
isomorphism $S(T)(c)\cong\operatorname{Sing}(T^{\mathrm{D}\underline{c}})$,
this follows readily from the fact that both $*$ and $c$ are cofibrant in
${\Delta\mathscr{T}\mathrm{op}}$, that $T$ is fibrant, and that
$\operatorname{Sing}$ is a right Quillen functor. Consequently, the functor
$c\mapsto S(T)(c)$ maps the weak equivalence $c\to*$ in
${\Delta\mathscr{T}\mathrm{op}}$ to a weak equivalence in
${\mathscr{S}\mathrm{et}_{\Delta}}$. ∎

###### Proposition 2.67

The functor $Re$ from diagram (2.54) has the following properties:

  1. (1)

$Re$ sends the morphism $F\times{\mathscr{Y}}_{\mathbb{R}}\to F$ to a weak
equivalence in ${\Delta\mathscr{T}\mathrm{op}}$, for every cofibrant
$F\in\mathscr{H}_{\infty}^{p}$.

  2. (2)

$Re$ sends every Čech nerve ${\check{C}}{\mathscr{U}}\to{\mathscr{Y}}_{c}$ to
a weak equivalence in ${\Delta\mathscr{T}\mathrm{op}}$, for every
differentiably good open covering ${\mathscr{U}}=\\{c_{a}\to c\\}_{a\in A}$ in
${\mathscr{C}\mathrm{art}}$.

###### Proof.

Ad (1): The morphism is a weak equivalence in $\mathscr{H}_{\infty}^{pI}$
between cofibrant objects by Proposition 2.13. Therefore, the claim follows
from Proposition 2.65.

Ad (2): Let ${\check{C}}{\mathscr{U}}\to{\mathscr{Y}}_{c}$ denote Čech nerve
of the covering ${\mathscr{U}}$. We view this as a morphism from a simplicial
presheaf ${\check{C}}{\mathscr{U}}$ to a simplicially constant presheaf
${\mathscr{Y}}_{c}$. Since ${\mathscr{U}}$ is a differentiably good open
covering, ${\check{C}}{\mathscr{U}}$ is levelwise a coproduct of representable
presheaves on ${\mathscr{C}\mathrm{art}}$; hence, ${\check{C}}{\mathscr{U}}$
is cofibrant in $\mathscr{H}_{\infty}^{p}$. By construction of the Čech model
structure $\mathscr{H}_{\infty}^{p\ell}$, we have that the morphism
${\check{C}}{\mathscr{U}}\to{\mathscr{Y}}_{c}$ is a weak equivalence in
$\mathscr{H}_{\infty}^{p\ell}$. By Corollary 2.18, this is also a weak
equivalence in $\mathscr{H}_{\infty}^{pI}$. The result now follows from
Proposition 2.65 and since both ${\check{C}}{\mathscr{U}}$ and
${\mathscr{Y}}_{c}$ are cofibrant. ∎

We now prove an important property of the model categories
$\mathscr{H}_{\infty}^{p/i\,I}$ that allows us to detect $I$-local weak
equivalences. Dugger calls this property _rigidity_ in [14].

###### Proposition 2.68

_[ 14, Lemma 3.4.4]_ If $F,G\in\mathscr{H}_{\infty}^{p/i\,I}$ are fibrant,
then a morphism $\psi\colon F\to G$ is an $I$-local weak equivalence if and
only if the morphism $\psi_{|*}\colon F(*)\to G(*)$ is a weak equivalence in
${\mathscr{S}\mathrm{et}_{\Delta}}$.

###### Proof of Proposition 2.68.

Since $\psi$ is a morphism between local objects in a left Bousfield
localisation of $\mathscr{H}_{\infty}^{p/i}$, it is an equivalence in
$\mathscr{H}_{\infty}^{p/i\,I}$ if and only if it is a weak equivalence in
$\mathscr{H}_{\infty}^{p/i}$. That is, $\psi$ is an $I$-local weak equivalence
if and only if it is an objectwise weak equivalence.

Therefore, it immediately follows that if $\psi$ is an $I$-local weak
equivalence, then $\psi_{|*}$ is a weak equivalence of simplicial sets.

Conversely, suppose that $\psi_{|*}$ is a weak equivalence in
${\mathscr{S}\mathrm{et}_{\Delta}}$. For any $c\in{\mathscr{C}\mathrm{art}}$,
we have the following commutative square in
${\mathscr{S}\mathrm{et}_{\Delta}}$:

${F(*)=\big{(}\tilde{\mathsf{c}}\operatorname{ev}_{*}(F)\big{)}}$${F(c)}$${G(*)=\big{(}\tilde{\mathsf{c}}\operatorname{ev}_{*}(G)\big{)}}$${G(c)}$$\scriptstyle{e_{|F}}$$\scriptstyle{\psi_{|*}}$$\scriptstyle{\psi_{|c}}$$\scriptstyle{e_{|G}}$
(2.69)

As both $F$ and $G$ are $I$-local objects, it follows from Lemma 2.35 that the
horizontal morphisms are weak equivalences in
${\mathscr{S}\mathrm{et}_{\Delta}}$. Together with the assumption that
$\psi_{|*}$ is a weak equivalence on simplicial sets, it now follows that
$\psi$ is, in fact, an objectwise weak equivalence. ∎

###### Theorem 2.70

There is a commutative diagram of simplicial Quillen equivalences

${\mathscr{H}_{\infty}^{pI}}$${{\mathscr{S}\mathrm{et}_{\Delta}}}$${{\Delta\mathscr{T}\mathrm{op}}}$$\scriptstyle{Re}$$\scriptstyle{\operatorname{ev}_{*}}$$\scriptstyle{|{-}|}$$\scriptstyle{\tilde{\mathsf{c}}}$$\scriptstyle{\operatorname{Sing}}$$\scriptstyle{S}$
(2.71)

where $\tilde{\mathsf{c}}$, $Re$, and $|{-}|$ are the left adjoints.

###### Proof.

It is well-established that the pair $|{-}|\dashv\operatorname{Sing}$ is a
simplicial Quillen equivalence (see e.g. [21]). We have also seen in Theorem
2.37 that the adjunction $\tilde{\mathsf{c}}\dashv\operatorname{ev}_{*}$ is a
simplicial Quillen equivalence. The commutativity of (2.71) follows from the
definitions (2.55) of the functors $Re$ and $S$, which use $|{-}|$ and
$\operatorname{Sing}$, respectively. The fact that $Re\dashv S$ is a Quillen
equivalence then follows from the two-out-of-three property of Quillen
equivalences and Corollary D.12. ∎

###### Remark 2.72

A slightly different version of Theorem 2.70 has been found previously in [16,
14], working over ${\mathscr{M}\mathrm{fd}}$ instead of
${\mathscr{C}\mathrm{art}}$. We found that ${\mathscr{C}\mathrm{art}}$ has
several technical advantages (in particular, we do not need to additionally
consider stalk-wise weak equivalences) and provides a sufficiently large
category of parameter spaces to describe geometric and topological structures,
as Theorem 2.70 shows (see also [6] for a treatment of geometric structures).
$\triangleleft$

##  3 The smooth singular complex of a simplicial presheaf

In this section we introduce the _smooth singular complex_ , sometimes also
called the _concordance space_ , of a simplicial presheaf on
${\mathscr{C}\mathrm{art}}$. We investigate its homotopical properties—for
instance, it sends smooth homotopies to simplicial homotopies—and we establish
it both as a left Quillen equivalence
$\mathscr{H}_{\infty}^{p/i\,I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ and as a
right Quillen equivalence
$\mathscr{H}_{\infty}^{iI}\to{\mathscr{S}\mathrm{et}_{\Delta}}$.

###  3.1 Extended simplices and the smooth singular complex

In a fashion similar to motivic homotopy theory (see e.g. [26, 35, 12]), we
consider the extended affine simplices in order to build our smooth singular
complex functor. However, we purely rely on the smooth manifold structure of
the affine cartesian simplices rather than on their function algebras.

###### Definition 3.1

The _extended $n$-simplex_ is the cartesian space

$\Delta_{e}^{n}\coloneqq\big{\\{}(t^{0},\ldots,t^{n})\in\mathbb{R}^{n+1}\,\big{|}\,\sum_{i=0}^{n}t^{i}=1\big{\\}}\subset\mathbb{R}^{n+1}\,.$
(3.2)

Face and degeneracy maps are defined as the affine linear extensions of the
face and degeneracy maps of the standard simplices $|\Delta^{n}|$. The
extended simplices thus define a functor
$\Delta_{e}\colon{\mathbbe{\Delta}}\to{\mathscr{C}\mathrm{art}}$.

By construction, the topological standard simplex

$|\Delta^{n}|=\\{t\in\mathbb{R}^{n+1}\,|\,\sum_{i=0}^{n}t^{i}=1,\,0\leq
t^{i}\leq 1\ \forall i=0,\ldots,n\\}$ (3.3)

is a subset of the extended simplex $\Delta_{e}^{n}$, for any
$n\in\mathbb{N}_{0}$. This inclusion
$|\Delta^{n}|\hookrightarrow\Delta_{e}^{n}$ is compatible with the face and
degeneracy maps. Recalling the functor
$\mathrm{D}\colon{\mathscr{D}\mathrm{fg}}\to{\Delta\mathscr{T}\mathrm{op}}$
from Definition 2.45, we see that there is a morphism

$\iota\colon|\Delta|\to\mathrm{D}\Delta_{e}$ (3.4)

of functors ${\mathbbe{\Delta}}\to{\Delta\mathscr{T}\mathrm{op}}$. In
particular, the diagram

${{|\Delta^{n}|}}$${\mathrm{D}\Delta_{e}^{n}}$${{|\Delta^{k}|}}$${\mathrm{D}\Delta_{e}^{k}}$$\scriptstyle{\iota^{n}}$$\scriptstyle{|\Delta|(\sigma)}$$\scriptstyle{\mathrm{D}\Delta_{e}(\sigma)}$$\scriptstyle{\iota^{k}}$
(3.5)

in ${\Delta\mathscr{T}\mathrm{op}}$ commutes for every morphism
$\sigma\in{\mathbbe{\Delta}}([n],[k])$.

The extended simplices functor $\Delta_{e}$ induces a Quillen adjunction

${\mathscr{H}_{\infty}^{p}}$${\mathscr{H}_{\infty}^{i}}$${({\mathscr{S}\mathrm{et}_{\Delta}}^{{\mathbbe{\Delta}}^{\mathrm{op}}})^{i}=({\mathscr{S}\mathrm{et}_{\Delta}}^{{\mathbbe{\Delta}}^{\mathrm{op}}})^{Reedy}\,.}$$\scriptstyle{1}$$\scriptstyle{\perp}$$\scriptstyle{\Delta_{e}^{*}}$$\scriptstyle{\perp}$$\scriptstyle{1}$$\scriptstyle{(\Delta_{e})_{!}}$
(3.6)

Here we have made use of [19, Thm. 15.8.7], which implies that the injective
model structure on bisimplicial sets agrees with the Reedy model structure.

###### Theorem 3.7

_[ 31, Thm. 5.2.3]_ Let ${\mathscr{M}}$ be a simplicial model category. Then,
the realisation functor

$|{-}|_{\mathscr{M}}\colon{\mathscr{M}}^{{\mathbbe{\Delta}}^{\mathrm{op}}}\to{\mathscr{M}}\,,\qquad
X_{\bullet}\longmapsto\int^{[n]\in{\mathbbe{\Delta}}^{\mathrm{op}}}\Delta^{n}\otimes
X_{n}$ (3.8)

is a left Quillen functor with respect to the Reedy model category structure
on ${\mathscr{M}}^{{\mathbbe{\Delta}}^{\mathrm{op}}}$.

###### Proposition 3.9

Let
$\delta\colon{\mathbbe{\Delta}}\to{\mathbbe{\Delta}}\times{\mathbbe{\Delta}}$
be the diagonal functor. There exists a canonical isomorphism

$|{-}|_{\mathscr{S}\mathrm{et}_{\Delta}}\cong\delta^{*}\,,$ (3.10)

of functors
$s{\mathscr{S}\mathrm{et}_{\Delta}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$,
where $\delta^{*}(X)_{n}=X_{n,n}$ is the pullback along the diagonal functor.

###### Proof.

This is a straightforward application of the Yoneda Lemma in the (co)end
calculus. ∎

###### Corollary 3.11

The diagonal functor is a left Quillen functor

$\delta^{*}\colon\big{(}{\mathscr{S}\mathrm{et}_{\Delta}}^{({\mathbbe{\Delta}}^{\mathrm{op}})}\big{)}^{i}\longrightarrow{\mathscr{S}\mathrm{et}_{\Delta}}\,.$
(3.12)

In particular, it is homotopical, i.e. it preserves all weak equivalences.

Consequently, we can define a left Quillen functor as the composition

$S_{e}\coloneqq\delta^{*}\circ\Delta_{e}^{*}\colon\mathscr{H}_{\infty}^{p/i}\longrightarrow{\mathscr{S}\mathrm{et}_{\Delta}}\,.$
(3.13)

Consider a complete and cocomplete category $\mathscr{E}$, two categories
$\mathscr{C},\mathscr{D}$, and a functor $F\colon\mathscr{C}\to\mathscr{D}$.
Recall that, in this situation, the functor
$F^{*}\colon{\mathscr{C}\mathrm{at}}(\mathscr{D},\mathscr{E})\longrightarrow{\mathscr{C}\mathrm{at}}(\mathscr{C},\mathscr{E})$
has a left adjoint $F_{!}$ and a right adjoint $F_{*}$, which are given by the
left and the right Kan extension along $F$. By the construction of $S_{e}$ as
a composition of pullback functors which act on categories of simplicial
presheaves, we infer:

###### Proposition 3.14

The functor $S_{e}=\delta^{*}\circ\Delta_{e}^{*}$ has both adjoints. We thus
obtain a triple of adjunctions $L_{e}\dashv S_{e}\dashv R_{e}$, where $L_{e}$
and $R_{e}$ are given by the compositions

$L_{e}=\Delta_{e!}\circ\delta_{!}\,,\qquad\text{and}\qquad
R_{e}=\Delta_{e*}\circ\delta_{*}\,.$ (3.15)

The adjunction $S_{e}\dashv R_{e}$ is a simplicial Quillen adjunction.

###### Definition 3.16

We call the functor
$S_{e}\colon\mathscr{H}_{\infty}^{p/i}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
the _smooth singular complex functor_. For $F\in\mathscr{H}_{\infty}$, the
simplicial set $S_{e}F$ is called the _smooth singular complex of $F$_.

###  3.2 $S_{e}$ as a left Quillen equivalence

We further investigate the homotopical properties of the smooth singular
complex functor $S_{e}$. So far, we know that the adjunction
$S_{e}:\mathscr{H}_{\infty}^{p/i}\rightleftarrows{\mathscr{S}\mathrm{et}_{\Delta}}:R_{e}$
is Quillen. Our goal here is to show that this Quillen adjunction descends to
the localisation $\mathscr{H}_{\infty}^{p/i\,I}$ and that there it even forms
a Quillen equivalence.

###### Definition 3.17

Let $F,G\in\mathscr{H}_{\infty}$ be two simplicial presheaves on
${\mathscr{C}\mathrm{art}}$, and let $f_{0},f_{1}\colon F\to G$ be a pair of
morphisms. A _smooth homotopy from $f_{0}$ to $f_{1}$_ is a commutative
diagram

${F\times\Delta^{\\{0\\}}}$${F\times\underline{\mathbb{R}}}$${G}$${F\times\Delta^{\\{1\\}}}$$\scriptstyle{f_{0}}$$\scriptstyle{h}$$\scriptstyle{f_{1}}$
(3.18)

in $\mathscr{H}_{\infty}$, where the vertical inclusions are induced by the
maps $*\to\mathbb{R}$, given by $*\mapsto 0$ and $*\mapsto 1$.

###### Lemma 3.19

The functor
$S_{e}\colon\mathscr{H}_{\infty}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ maps
smoothly homotopic morphisms to simplicially homotopic morphisms.

###### Proof.

The projection $(t^{0},t^{1})\mapsto t^{0}$ yields a diffeomorphism
$\psi\colon\Delta_{e}^{1}\to\mathbb{R}$ of cartesian spaces. Observe that
there is a morphism of simplicial sets

$\nu\colon\Delta^{1}\longrightarrow
S_{e}\underline{\mathbb{R}}={\mathscr{C}\mathrm{art}}(\Delta_{e}^{\bullet},\mathbb{R})\,,$
(3.20)

defined by sending the generating non-degenerate 1-simplex of $\Delta^{1}$ to
the 1-simplex $\psi$. Hence, using the fact that $S_{e}$ preserves products,
we apply $S_{e}$ to diagram (3.18) and augment it using $\nu$ to obtain a
commutative diagram

${S_{e}F\times\Delta^{\\{0\\}}}$${S_{e}F\times\Delta^{1}}$${S_{e}F\times
S_{e}\mathbb{R}}$${S_{e}G}$${S_{e}F\times\Delta^{\\{1\\}}}$$\scriptstyle{S_{e}f_{0}}$$\scriptstyle{1\times\nu}$$\scriptstyle{S_{e}h}$$\scriptstyle{S_{e}f_{1}}$
(3.21)

This establishes a simplicial homotopy $S_{e}h\circ(1_{X}\times\nu)$ from
$S_{e}f_{0}$ to $S_{e}f_{1}$. ∎

Lemma 3.19 can be seen as a generalisation of [9, Lemma 4.10] away from
diffeological spaces to simplicial presheaves. Indeed, the composition

${{\mathscr{D}\mathrm{fg}}}$${\mathscr{H}_{\infty}}$${{\mathscr{S}\mathrm{et}_{\Delta}}}$$\scriptstyle{\iota}$$\scriptstyle{S_{e}}$
(3.22)

is precisely the _smooth singular functor_ from [9].

###### Proposition 3.23

For any $c\in{\mathscr{C}\mathrm{art}}$, the functor $S_{e}$ sends the
collapse morphism $\mathsf{c}\colon{\mathscr{Y}}_{c}\to*$ to a weak
equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$.

###### Proof.

Let $c\in{\mathscr{C}\mathrm{art}}$, and let $x\in c$ be any point. The
inclusion $x\colon*\to c$ induces a smooth homotopy equivalence
$*\rightleftarrows c$. The functor $S_{e}$ maps this to a simplicial homotopy
equivalence according to Lemma 3.19. ∎

###### Corollary 3.24

The functor $S_{e}$ induces Quillen adjunctions

${S_{e}:\mathscr{H}_{\infty}^{p/i\,I}}$${{\mathscr{S}\mathrm{et}_{\Delta}}:R_{e}\,.}$$\scriptstyle{\perp}$
(3.25)

###### Proof.

Each morphism
${\mathscr{Y}}_{c}\times{\mathscr{Y}}_{\mathbb{R}}\to{\mathscr{Y}}_{c}$ in $I$
is a morphism between cofibrant objects in $\mathscr{H}_{\infty}^{p/i}$.
Therefore, by [19, Prop. 3.3.18] it suffices to show that $S_{e}$ sends each
morphism in $I$ to a weak equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$.
One way to see that this holds true is by observing that $S_{e}$ preserves
finite products (it is a monoidal functor). Therefore, we have that

$\displaystyle
S_{e}({\mathscr{Y}}_{c}\times{\mathscr{Y}}_{\mathbb{R}}\to{\mathscr{Y}}_{c})$
$\displaystyle=S_{e}\big{(}{\mathscr{Y}}_{c}\xrightarrow{1_{{\mathscr{Y}}_{c}}}{\mathscr{Y}}_{c}\big{)}\times
S_{e}\big{(}{\mathscr{Y}}_{\mathbb{R}}\xrightarrow{\mathsf{c}}*\big{)}$ (3.26)
$\displaystyle=S_{e}(1_{{\mathscr{Y}}_{c}})\otimes
S_{e}\big{(}{\mathscr{Y}}_{\mathbb{R}}\xrightarrow{\mathsf{c}}*\big{)}\,,$
(3.27)

where $\mathsf{c}\colon{\mathscr{Y}}_{\mathbb{R}}\to*$ is the collapse
morphism. By Proposition 3.23, the morphism $S_{e}(\mathsf{c})$ is a weak
equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$, and since
${\mathscr{S}\mathrm{et}_{\Delta}}$ is a monoidal model category in which
every object is cofibrant, it follows that
$S_{e}(1_{{\mathscr{Y}}_{c}})\otimes S_{e}(\mathsf{c})$ is a weak equivalence
as well. ∎

###### Proposition 3.28

The functors
$S_{e}\colon\mathscr{H}_{\infty}^{p/i\,I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
are homotopical.

###### Proof.

$S_{e}\colon\mathscr{H}_{\infty}^{iI}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ is
homotopical because it is left Quillen and every object in
$\mathscr{H}_{\infty}^{iI}$ is cofibrant. The corresponding statement for the
projective model structure now follows from Proposition 2.12. ∎

Note, in particular, that by Proposition 2.18 the functor $S_{e}$ also sends
weak equivalences in the Čech local model structures
$\mathscr{H}_{\infty}^{p/i\,\ell}$ to weak equivalences in
${\mathscr{S}\mathrm{et}_{\Delta}}$.

###### Theorem 3.29

The Quillen adjunctions

${S_{e}:\mathscr{H}_{\infty}^{p/i\,I}}$${{\mathscr{S}\mathrm{et}_{\Delta}}:R_{e}\,}$$\scriptstyle{\perp}$
(3.30)

are Quillen equivalences.

###### Proof.

We have Quillen adjunctions

${{\mathscr{S}\mathrm{et}_{\Delta}}}$${\mathscr{H}_{\infty}^{p/i\,I}}$${{\mathscr{S}\mathrm{et}_{\Delta}}}$$\scriptstyle{\perp}$$\scriptstyle{\tilde{\mathsf{c}}}$$\scriptstyle{\perp}$$\scriptstyle{S_{e}}$$\scriptstyle{\operatorname{ev}_{*}}$$\scriptstyle{R_{e}}$
(3.31)

and we know from Theorem 2.37 that the Quillen adjunction
$\tilde{\mathsf{c}}\dashv\operatorname{ev}_{*}$ is even a Quillen equivalence.
We readily see that $S_{e}\circ\tilde{\mathsf{c}}$ is the identity functor on
${\mathscr{S}\mathrm{et}_{\Delta}}$. Therefore, $S_{e}\dashv R_{e}$ is a
Quillen equivalence by Corollary D.12. ∎

###### Corollary 3.32

The functor $S_{e}$ both preserves and reflects weak equivalences in
$\mathscr{H}_{\infty}^{p/i\,I}$.

###### Proof.

In the injective case, this follows from Theorem 3.29 since every object in
$\mathscr{H}_{\infty}^{iI}$ is cofibrant and $S_{e}$ is a left Quillen
equivalence (see e.g. [21, Prop. 1.3.16]). The projective case then follows
from Proposition 2.12. ∎

###### Corollary 3.33

Any smooth homotopy equivalence in $\mathscr{H}_{\infty}$ is a weak
equivalence in $\mathscr{H}_{\infty}^{p/i\,I}$.

###### Proof.

By Proposition 3.19, $S_{e}$ sends smooth homotopy equivalences to simplicial
homotopy equivalences, which are, in particular, weak equivalences in
${\mathscr{S}\mathrm{et}_{\Delta}}$. Thus, the claim follows from Corollary
3.32. ∎

###### Remark 3.34

Let $W_{\mathscr{S}\mathrm{et}_{\Delta}}$ denote the class of weak
equivalences in ${\mathscr{S}\mathrm{et}_{\Delta}}$, and let
$S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})$ denote the class of
morphisms in $\mathscr{H}_{\infty}$ whose image under $S_{e}$ is in
$W_{\mathscr{S}\mathrm{et}_{\Delta}}$. Corollary 3.32 lets us suspect that
there is an equivalence of model categories

$\mathscr{H}_{\infty}^{p/i\,I}\simeq
L_{S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})}\mathscr{H}_{\infty}^{p/i}\,.$
(3.35)

Using properties of local weak equivalences in Bousfield localisation should
allow us to prove that conjecture here already, but instead we give a very
direct proof later in Theorem 6.54. $\triangleleft$

###  3.3 $S_{e}$ as a right Quillen equivalence

The goal of this subsection is to establish the smooth singular functor as a
right Quillen functor
$S_{e}\colon\mathscr{H}_{\infty}^{iI}\to{\mathscr{S}\mathrm{et}_{\Delta}}$.
Apart from having convenient technical implications on the functor
$S_{e}\colon\mathscr{H}_{\infty}^{iI}\to{\mathscr{S}\mathrm{et}_{\Delta}}$,
the appearance of several intermediate model structures of bisimplicial sets
sheds additional light on the way that the functor $S_{e}$ works. We already
know from Proposition 3.14 that $S_{e}=\delta^{*}\circ\Delta_{e}^{*}$ has a
left adjoint $L_{e}=\Delta_{e!}\circ\delta_{!}$. We will show that both its
constituting functors $\Delta_{e!}$ and $\delta_{!}$ are left Quillen
functors.

####  3.3.1 Model structures for $\infty$-groupoids on the category of
bisimplicial sets

We start by analysing the functor $\delta_{!}$ in more detail. Let

${\iota_{n}\colon{\mathrm{Sp}}_{n}\coloneqq\Delta^{1}\underset{\Delta^{0}}{\sqcup}\cdots\underset{\Delta^{0}}{\sqcup}\Delta^{1}}$${\Delta^{n}}$
(3.36)

denote the _spine-inclusion_ of the $n$-simplex $\Delta^{n}$, for $n\geq 1$.
(Note that for $n=1$ the morphism $\iota_{1}$ is an isomorphism.) Let
${\mathrm{Sp}}\coloneqq\\{\iota_{n}\colon{\mathrm{Sp}}_{n}\hookrightarrow\Delta^{n}\,|\,n\geq
1\\}$ denote the set of all spine inclusions.

We write
$s{\mathscr{S}\mathrm{et}_{\Delta}}={\mathscr{C}\mathrm{at}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathscr{S}\mathrm{et}_{\Delta}})$
for the category of bisimplicial sets. There exists a bifunctor

$\boxtimes\colon{\mathscr{S}\mathrm{et}_{\Delta}}\times{\mathscr{S}\mathrm{et}_{\Delta}}\longrightarrow
s{\mathscr{S}\mathrm{et}_{\Delta}}\,,\qquad(K\boxtimes L)_{m,n}\coloneqq
K_{m}\times L_{n}.$ (3.37)

We view a bisimplicial set $X$ as a simplicial diagram $m\mapsto
X_{m,\bullet}$ in ${\mathscr{S}\mathrm{et}_{\Delta}}$. Let
$J\in{\mathscr{S}\mathrm{et}_{\Delta}}$ denote the nerve of the groupoid with
two objects and a unique isomorphism between them. The following definitions
are taken from [30, 1, 20].

###### Definition 3.38

We define the following model structures on the category
$s{\mathscr{S}\mathrm{et}_{\Delta}}$ of bisimplicial sets:

  1. (1)

We view
$s{\mathscr{S}\mathrm{et}_{\Delta}}={\mathscr{C}\mathrm{at}}({\mathbbe{\Delta}}^{\mathrm{op}},{\mathscr{S}\mathrm{et}_{\Delta}})$
as endowed with the injective model structure. Recall that this coincides with
the Reedy model structure [19].

  2. (2)

We let ${\mathscr{SS}\mathrm{p}}\coloneqq
L_{\mathrm{Sp}}s{\mathscr{S}\mathrm{et}_{\Delta}}$ be the left Bousfield
localisation of $s{\mathscr{S}\mathrm{et}_{\Delta}}$ at the spine inclusions.
This is the _model category for Segal spaces_.

  3. (3)

The _model category for complete Segal spaces_ is the localisation
$\textsc{css}\coloneqq L_{J\boxtimes\Delta^{0}}{\mathscr{SS}\mathrm{p}}$.

Let
$L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$
denote the left Bousfield localisation of the injective model category of
bisimplicial sets at all collapse morphisms
$\\{\Delta^{n}\boxtimes\Delta^{0}\to\Delta^{0}\boxtimes\Delta^{0}\,|\,n\in\mathbb{N}_{0}\\}$.
Let
$L_{{\mathbbe{\Delta}}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$
denote the left Bousfield localisation of $s{\mathscr{S}\mathrm{et}_{\Delta}}$
at all morphisms
$\\{\Delta^{n}\boxtimes\Delta^{0}\to\Delta^{m}\boxtimes\Delta^{0}\,|\,n,m\in\mathbb{N}_{0}\\}$.
(Compare these localisations to those in Proposition 2.30.) We will mostly be
using the model category
$L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$.
However, for conceptual clarity and for an interpretation as model categories
for $\infty$-groupoids, we include the following proposition.

###### Proposition 3.39

The following left Bousfield localisations yield identical model categories:

$L_{{\mathbbe{\Delta}}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}\overset{(1)}{=}L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}\overset{(2)}{=}L_{\Delta^{1}\boxtimes\Delta^{0}}{\mathscr{SS}\mathrm{p}}\overset{(3)}{=}L_{\Delta^{1}\boxtimes\Delta^{0}}\textsc{css}\,.$
(3.40)

###### Proof.

By Theorem 2.17 it suffices to check that all four model categories have the
same cofibrations and fibrant objects. For cofibrations, this is trivial since
each of the model categories is a left Bousfield localisation of
$s{\mathscr{S}\mathrm{et}_{\Delta}}$. It thus remains to check that the
fibrant objects of the three model categories coincide.

Identity (1) is a direct consequence of the two-out-of-three property of weak
equivalences.

For identity (2), let $X\in
L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$ be
fibrant. That is, $X$ is injective fibrant in
$s{\mathscr{S}\mathrm{et}_{\Delta}}$ and the canonical map $X_{0}\to X_{n}$ is
a weak equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$ for any
$n\in\mathbb{N}_{0}$. We have to show that $X$ satisfies the Segal condition,
i.e. that for every $n\geq 2$ the spine inclusion
${\mathrm{Sp}}_{n}\hookrightarrow\Delta^{n}$ induces a weak equivalence

$X_{n}\longrightarrow
X_{1}\underset{X_{0}}{\times}\cdots\underset{X_{0}}{\times}X_{1}\,.$ (3.41)

(As pointed out in [30], the strict pullback is a homotopy pullback here
because $X$ is Reedy fibrant.) Consider the commutative diagram

${X_{n}}$${X_{1}\underset{X_{0}}{\times}\cdots\underset{X_{0}}{\times}X_{1}}$${X_{0}}$${X_{0}\underset{X_{0}}{\times}\cdots\underset{X_{0}}{\times}X_{0}}$$\scriptstyle{s}$$\scriptstyle{(1_{X_{0}},\ldots,1_{X_{0}})}$$\scriptstyle{\cong}$$\scriptstyle{s_{0}\times\cdots\times
s_{0}}$ (3.42)

Since $X$ is Reedy fibrant, the pullbacks on the right-hand side are homotopy
pullbacks. Therefore, both vertical maps in (3.42) are weak equivalence. It
follows by the commutativity of the diagram that $X$ satisfies the Segal
condition. Then, $X$ is fibrant in
$L_{\Delta^{1}\boxtimes\Delta^{0}}{\mathscr{SS}\mathrm{p}}$ since, by
assumption, the morphism $X_{0}\to X_{1}$ is a weak equivalence.

Conversely, if $X$ is fibrant in
$L_{\Delta^{1}\boxtimes\Delta^{0}}{\mathscr{SS}\mathrm{p}}$, then the top
horizontal morphism in diagram (3.42) is a weak equivalence because $X$
satisfies the Segal condition, and the right-hand vertical morphism is a weak
equivalence because $X$ is injective fibrant and $X$ is local with respect to
$\Delta^{1}\boxtimes\Delta^{0}\to\Delta^{0}\boxtimes\Delta^{0}$. It thus
follows by the commutativity of the diagram that also the left vertical
morphism is an equivalence, for any $n\geq 2$, so that $X$ is fibrant in
$L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$.

For identity (3), recall that in any Segal space $X$ there is a notion of when
a morphism $f\in X_{1}$ is invertible (or a ‘homotopy equivalence’ in the
language of [30]). One defines the _space $X_{weq}$ of homotopy equivalences_
in $X$ to be the union of those connected components of $X_{1}$ that contain
invertible morphisms (by [30, Lemma 5.8], if $X\in{\mathscr{SS}\mathrm{p}}$ is
fibrant, then a connected component of $X_{1}$ contains a homotopy equivalence
if and only if it consists purely of homotopy equivalences). For any Segal
space, the degeneracy morphism $s_{0}\colon X_{0}\to X_{1}$ factors as

${X_{weq}}$${X_{0}}$${X_{1}}$$\scriptstyle{\iota_{X}}$$\scriptstyle{\widehat{s_{0}}}$$\scriptstyle{s_{0}}$
(3.43)

Let $X\in\textsc{css}$ be fibrant. In other words, $X$ is a fibrant object in
${\mathscr{SS}\mathrm{p}}$ and the morphism $\widehat{s_{0}}\colon X_{0}\to
X_{weq}$ is a weak equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$. Then,
$X$ is local in $L_{\Delta^{1}\boxtimes\Delta^{0}}\textsc{css}$ precisely if,
additionally, the morphism $s_{0}\colon X_{0}\to X_{1}$ is a weak equivalence
of simplicial sets. Since every fibrant object in css is also fibrant in
${\mathscr{SS}\mathrm{p}}$, this implies that every fibrant object in
$L_{\Delta^{1}\boxtimes\Delta^{0}}\textsc{css}$ is fibrant in
$L_{\Delta^{1}\boxtimes\Delta^{0}}{\mathscr{SS}\mathrm{p}}$.

Conversely, let $Y\in{\mathscr{SS}\mathrm{p}}$ be fibrant. Then, $Y$ is local
in $L_{\Delta^{1}\boxtimes\Delta^{0}}{\mathscr{SS}\mathrm{p}}$ precisely if
the morphism $s_{0}\colon Y_{0}\to Y_{1}$ is a weak equivalence. We need to
show that if this criterion is satisfied, then $Y$ satisfies that
$\widehat{s_{0}}\colon Y_{0}\to Y_{weq}$ is a weak equivalence in
${\mathscr{S}\mathrm{et}_{\Delta}}$. However, since $\iota_{Y}\colon
Y_{weq}\to Y$ is the inclusion of a union of connected components of $Y_{1}$,
diagram (3.43) and the fact that $s_{0}$ is a weak equivalence imply that
$\iota_{Y}$ hits every connected component of $Y_{1}$. Therefore, $\iota_{Y}$
is a weak equivalence; it follows from the two-out-of-three property that
$\widehat{s_{0}}\colon Y_{0}\to Y_{weq}$ is a weak equivalence as well. ∎

###### Remark 3.44

Let $X\in{\mathscr{SS}\mathrm{p}}$ be fibrant. Since $\iota_{X}\colon
X_{weq}\to X_{1}$ is the inclusion of a union of connected components of
$X_{1}$, it follows that $\iota_{X}$ is a weak equivalence precisely if it is
an isomorphism, i.e. precisely if $X_{weq}=X_{1}$. In other words, a fibrant
object in both $L_{\Delta^{1}\boxtimes\Delta^{0}}{\mathscr{SS}\mathrm{p}}$ and
$L_{\Delta^{1}\boxtimes\Delta^{0}}\textsc{css}$ is a complete Segal space with
all 1-morphisms invertible. In that sense, the fibrant objects in these model
categories are $\infty$-groupoids. The model category
$L_{{\mathbbe{\Delta}}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$
can be seen as the model category of essentially constant simplicial diagrams
of spaces, in analogy to how
$L_{\mathscr{C}\mathrm{art}}\mathscr{H}_{\infty}^{p/i}$ describes locally
constant simplicial presheaves. $\triangleleft$

###### Remark 3.45

The model structures in Proposition 3.39 agree with the _diagonal model
structure_ on bisimplicial sets: these model categories have the same
underlying categories, the same cofibrations, an, by [29, Thm. 2.1], they also
have the same fibrant objects. Thus, the claimed equality follows from Theorem
2.17. $\triangleleft$

###### Proposition 3.46

The diagonal $\delta^{*}\colon
s{\mathscr{S}\mathrm{et}_{\Delta}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
induces a Quillen adjunction

${\delta_{!}:{\mathscr{S}\mathrm{et}_{\Delta}}}$${L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}:\delta^{*}\,.}$$\scriptstyle{\perp}$
(3.47)

###### Proof.

This is a direct consequence of Remark 3.45 and [29, Thm. 2.4]. ∎

From now on, we will understand the adjunction $\delta_{!}\dashv\delta^{*}$ as
the above Quillen adjunction. There is another Quillen adjunction that relates
$L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$ to
the model category of simplicial sets, in analogy with Theorem 2.37.

###### Proposition 3.48

Consider the adjoint pair
$\mathsf{c}_{\mathbbe{\Delta}}:{\mathscr{S}\mathrm{et}_{\Delta}}\rightleftarrows
s{\mathscr{S}\mathrm{et}_{\Delta}}:\operatorname{ev}_{[0]}$, where
$\mathsf{c}_{\mathbbe{\Delta}}=\Delta^{0}\boxtimes(-)$, and where
$\operatorname{ev}_{[0]}(X)=X_{0,\bullet}$. This satisfies:

  1. (1)

$\mathsf{c}_{\mathbbe{\Delta}}\dashv\operatorname{ev}_{[0]}$ is a Quillen
adjunction ${\mathscr{S}\mathrm{et}_{\Delta}}\rightleftarrows
s{\mathscr{S}\mathrm{et}_{\Delta}}$.

  2. (2)

Composing the Quillen adjunction from (1) with the localisation adjunction
$s{\mathscr{S}\mathrm{et}_{\Delta}}\rightleftarrows
L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$
yields a Quillen equivalence

${\mathsf{c}_{\mathbbe{\Delta}}:{\mathscr{S}\mathrm{et}_{\Delta}}}$${L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}:\operatorname{ev}_{[0]}\,.}$$\scriptstyle{\perp}$
(3.49)

###### Proof.

It is straightforward to see that
$\mathsf{c}_{\mathbbe{\Delta}}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to
s{\mathscr{S}\mathrm{et}_{\Delta}}$ preserves cofibrations and further
preserves as well as reflects weak equivalences. This proves claim (1).

To see part (2), we first show that
$\mathsf{c}_{\mathbbe{\Delta}}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to
L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$
still reflects weak equivalences. The logic of the proof is entirely parallel
to the proof of Theorem 2.37 for the injective case: consider any morphism
$\psi\colon K\to L$ of simplicial sets and suppose that
$\mathsf{c}_{\mathbbe{\Delta}}\psi\colon\mathsf{c}_{\mathbbe{\Delta}}K\to\mathsf{c}_{\mathbbe{\Delta}}L$
is a weak equivalence in
$L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$.
This is the case if and only if, for any fibrant $X\in
L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$,
the induced map

${\underline{s{\mathscr{S}\mathrm{et}_{\Delta}}}(\mathsf{c}_{\mathbbe{\Delta}}L,X)}$${\underline{s{\mathscr{S}\mathrm{et}_{\Delta}}}(\mathsf{c}_{\mathbbe{\Delta}}K,X)}$$\scriptstyle{(\mathsf{c}_{\mathbbe{\Delta}}\psi)^{*}}$
(3.50)

is an equivalence in ${\mathscr{S}\mathrm{et}_{\Delta}}$. By adjointness, that
is equivalent to the map

${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(L,\operatorname{ev}_{[0]}X)}$${\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(K,\operatorname{ev}_{[0]}X)}$$\scriptstyle{\psi^{*}}$
(3.51)

being a weak equivalence of simplicial sets for every
$X\in(L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}})_{f}$.
Using a fibrant replacement in the injective model category
$s{\mathscr{S}\mathrm{et}_{\Delta}}$, we see in analogy with Lemma 2.36 and
the proof of Theorem 2.37 that every Kan complex is weakly equivalent to an
object in the image of the (Kan-complex-valued) functor
$\operatorname{ev}_{[0]}\colon(L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}})_{f}\longrightarrow({\mathscr{S}\mathrm{et}_{\Delta}})_{f}$.
Thus, it follows that $\psi^{*}$ is a weak equivalence.

Now consider a fibrant object $X\in
L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}$ and
the composition

${\mathsf{c}_{\mathbbe{\Delta}}\circ
Q^{\mathscr{S}\mathrm{et}_{\Delta}}\circ\operatorname{ev}_{[0]}(X)}$${\mathsf{c}_{\mathbbe{\Delta}}\circ\operatorname{ev}_{[0]}(X)}$${X\,,}$$\scriptstyle{\mathsf{c}_{\mathbbe{\Delta}}(q_{|\operatorname{ev}_{[0]}X})}$$\scriptstyle{e_{|X}}$
(3.52)

where $e\colon\mathsf{c}_{\mathbbe{\Delta}}\circ\operatorname{ev}_{[0]}\to
1_{s{\mathscr{S}\mathrm{et}_{\Delta}}}$ is the evaluation natural
transformation of the adjunction
$\mathsf{c}_{\mathbbe{\Delta}}\dashv\operatorname{ev}_{[0]}$. Further,
$q\colon
Q^{\mathscr{S}\mathrm{et}_{\Delta}}\overset{\sim}{\longrightarrow}1_{\mathscr{S}\mathrm{et}_{\Delta}}$
is a cofibrant replacement functor in ${\mathscr{S}\mathrm{et}_{\Delta}}$
(which we can take to be the identity, since
$({\mathscr{S}\mathrm{et}_{\Delta}})_{c}={\mathscr{S}\mathrm{et}_{\Delta}}$).
The first morphism is a weak equivalence since $\mathsf{c}_{\mathbbe{\Delta}}$
preserves weak equivalences, and the second morphism is a weak equivalence as
a consequence of the fibrancy of $X$; compare to the proof of Theorem 2.37.
This proves the claim by [21, Cor. 1.3.16]. ∎

####  3.3.2 The functors $\Delta_{e!}$ and $L_{e}$

Next, we show that $\Delta_{e!}\colon
s{\mathscr{S}\mathrm{et}_{\Delta}}\to\mathscr{H}_{\infty}^{iI}$ is left
Quillen, and that the Quillen adjunction $\Delta_{e!}\dashv\Delta_{e*}$
descends to the localisation
$L_{\Delta^{1}\boxtimes\Delta^{0}}{\mathscr{SS}\mathrm{p}}$ of
$s{\mathscr{S}\mathrm{et}_{\Delta}}$. The functor $\Delta_{e!}$ acts as

$\Delta_{e!}(X)=\int^{n}X_{n,\bullet}\otimes\Delta_{e}^{n}\,.$ (3.53)

In particular, for bisimplicial sets in the image of $(-)\boxtimes(-)$ we find

$\displaystyle\Delta_{e!}(K\boxtimes L)$ $\displaystyle=\int^{n}K_{n}\otimes
L\otimes\Delta_{e}^{n}$ (3.54) $\displaystyle\cong
L\otimes\int^{n}K_{n}\otimes\Delta_{e}^{n}$ (3.55) $\displaystyle\cong
L\otimes\Delta_{e!}(K\boxtimes\Delta^{0})\,.$ (3.56)

It follows that

$\Delta_{e!}(\Delta^{0}\boxtimes L)\cong\tilde{\mathsf{c}}L$ (3.57)

for any $L\in{\mathscr{S}\mathrm{et}_{\Delta}}$, where
$\tilde{\mathsf{c}}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to\mathscr{H}_{\infty}$
is the constant-presheaf functor.

###### Lemma 3.58

For any $n\in\mathbb{N}_{0}$, the morphism
$\Delta_{e!}(\partial\Delta^{n}\boxtimes\Delta^{0}\longrightarrow\Delta^{n}\boxtimes\Delta^{0})$
is a cofibration in $\mathscr{H}_{\infty}^{i}$.

###### Proof.

For $n=0,1$ this is straightforward. Consider the presentation of
$\partial\Delta^{n}$ as a coequaliser,

$\partial\Delta^{n}\cong{\mathrm{coeq}}\bigg{(}\coprod_{0\leq i<j\leq
n}\Delta^{n-2}\rightrightarrows\coprod_{0\leq k\leq n}\Delta^{n-1}\bigg{)}\,.$
(3.59)

Since $\Delta_{e!}$ preserves colimits, we obtain

$\partial\Delta_{e}^{n}\coloneqq\Delta_{e!}(\partial\Delta^{n}\boxtimes\Delta^{0})\cong{\mathrm{coeq}}\bigg{(}\coprod_{0\leq
i<j\leq n}\Delta_{e}^{n-2}\rightrightarrows\coprod_{0\leq k\leq
n}\Delta_{e}^{n-1}\bigg{)}\,.$ (3.60)

The colimit is taken in $\mathscr{H}_{\infty}$ (not in
${\mathscr{D}\mathrm{fg}}$, even though all $\Delta_{e}^{k}$ are diffeological
spaces), and so we have

$\partial\Delta_{e}^{n}(c)\cong{\mathrm{coeq}}\bigg{(}\coprod_{0\leq i<j\leq
n}\Delta_{e}^{n-2}(c)\rightrightarrows\coprod_{0\leq k\leq
n}\Delta_{e}^{n-1}(c)\bigg{)}$ (3.61)

for any $c\in{\mathscr{C}\mathrm{art}}$. In particular, any section of
$\partial\Delta_{e}^{n}$ over $c\in{\mathscr{C}\mathrm{art}}$ comes from some
section $f\in\Delta_{e}^{n-1}(c)$ of a face of $\Delta_{e}^{n}$. Two such
sections $f,g\in\Delta_{e}^{n-1}(c)$ are identified precisely if they factor
through the copy of $\Delta_{e}^{n-2}$ that joins the respective faces of
$\Delta_{e}^{n}$ and if, further, $f$ and $g$ agree as maps
$c\to\Delta_{e}^{n-2}$.

Let $f,g\in\partial\Delta_{e}^{n}(c)$ be any two elements, and assume that
$\iota_{e}^{n}\circ f=\iota_{e}^{n}\circ g$, where
$\iota_{e}^{n}\colon\partial\Delta_{e}^{n}\longrightarrow\Delta_{e}^{n}$ is
the canonical morphism. Observe that $\iota_{e}^{n}$ is injective as a map on
the underlying sets
$\partial\Delta_{e}^{n}(*)\hookrightarrow\Delta_{e}^{n}(*)$. Since every
section $f\colon{\mathscr{Y}}_{c}\to\partial\Delta_{e}^{n}$ is, in particular,
a map ${\mathscr{Y}}_{c}(*)\to\partial\Delta_{e}^{n}(*)$ of the underlying
sets, and analogously a section ${\mathscr{Y}}_{c}\to\Delta_{e}^{n}$ is, in
particular, a map ${\mathscr{Y}}_{c}(*)\to\Delta_{e}^{n}(*)$ of underlying
sets, it follows that
$\iota_{e}^{n}\colon\partial\Delta_{e}^{n}\to\Delta_{e}^{n}$ is an objectwise
monomorphism. ∎

###### Remark 3.62

We point out that $\partial\Delta_{e}^{n}$, as defined in (3.60), is not a
diffeological space for $n\geq 2$. For instance, consider a differentiably
good open covering $c=c_{0}\cup c_{1}$ of a cartesian space $c$. We denote the
intersection $c_{0}\cap c_{1}$ by $c_{01}\in{\mathscr{C}\mathrm{art}}$. Let
$f_{i}\colon c_{i}\to\Delta_{e}^{n-1}$ be smooth maps, for $i=0,1$, to
adjacent faces of $\Delta_{e}^{n}$, such that $f_{i|c_{01}}\colon
c_{01}\to\Delta_{e}^{n-2}$ factors through the $n{-}2$-simplex which joins the
two faces. These data to not lift to a section
$f\in\partial\Delta_{e}^{n}(c)$, since such an $f$ must factor through one of
the faces $\partial\Delta_{e}^{n-1}$. That is, $\partial\Delta_{e}^{n}$ does
not satisfy the sheaf condition. $\triangleleft$

Let $\mathscr{C},\mathscr{D},\mathscr{E}$ be categories. Recall the notion of
an _adjunction of two variables_ $\mathscr{C}\times\mathscr{D}\to\mathscr{E}$
(see, for example, [21, Def. 4.1.12]). We will denote an adjunction of two
variables only by its tensor functor
$\otimes\colon\mathscr{C}\times\mathscr{D}\to\mathscr{E}$. If $\mathscr{E}$
has pushouts, then there is an induced _pushout product_ , or _box product_ on
morphisms: given morphisms $f\colon A\to B$ in $\mathscr{C}$ and $g\colon X\to
Y$ in $\mathscr{D}$, their pushout product (relative to $\otimes$) is the
induced morphism in $\mathscr{E}$ given by

${A\otimes Y\underset{A\otimes B}{\sqcup}B\otimes X}$${B\otimes
Y\,.}$$\scriptstyle{f\square g}$ (3.63)

We recall following definitions:

###### Definition 3.64

_[ 21, Def. 4.2.1]_ Let $\mathscr{C},\mathscr{D},\mathscr{E}$ be model
categories, and let $\otimes\colon\mathscr{C}\times\mathscr{D}\to\mathscr{E}$
be an adjunction of two variables. Then, $\otimes$ is a _Quillen adjunction of
two variables_ if the induced pushout product $f,g\mapsto f\square g$
satisfies the _pushout-product axiom_ :

  1. (1)

if both $f$ and $g$ are cofibrations, then so is $f\square g$, and

  2. (2)

if, _in addition_ , $f$ or $g$ is a weak equivalence, then so is $f\square g$.

###### Definition 3.65

_[ 21, Def. 4.2.6]_ A _(symmetric) monoidal model category_ is a closed
(symmetric) monoidal category $(\mathscr{C},\otimes)$ together with a model
structure on the underlying category $\mathscr{C}$ such that:

  1. (1)

the closed monoidal structure is a Quillen adjunction of two variables
$\otimes\colon\mathscr{C}\times\mathscr{C}\to\mathscr{C}$.

  2. (2)

Let $u\in\mathscr{C}$ be the monoidal unit, and let $q_{u}\colon
Q^{\mathscr{C}}u\overset{\sim}{\longrightarrow}u$ be a cofibrant replacement.
Then, tensoring with any cofibrant object from the left or the right sends
$q_{u}$ to a weak equivalence.

###### Example 3.66

The model category $s{\mathscr{S}\mathrm{et}_{\Delta}}$ with the injective
model structure is symmetric monoidal. Similarly, each of the model categories
$\mathscr{H}_{\infty}^{p/i}$, $\mathscr{H}_{\infty}^{p/i\,\ell}$, and
$\mathscr{H}_{\infty}^{p/i\,I}$ is symmetric monoidal by Proposition 2.8. In
each of the monoidal model structures we encounter here, the monoidal unit is
already cofibrant, so that the second axiom of Definition 3.65 is trivially
satisfied. $\triangleleft$

The injective model structure on ${\mathscr{S}\mathrm{et}_{\Delta}}$ is
cofibrantly generated (see e.g. [30]), with generating cofibrations

$\mathcal{I}=\big{\\{}(\partial\Delta^{n}\boxtimes\Delta^{0}\hookrightarrow\Delta^{n}\boxtimes\Delta^{0})\square(\Delta^{0}\boxtimes\partial\Delta^{m}\hookrightarrow\Delta^{0}\boxtimes\Delta^{m})\,\big{|}\,n,m\in\mathbb{N}_{0}\big{\\}}$
(3.67)

and generating trivial cofibrations

$\mathcal{J}=\big{\\{}(\partial\Delta^{n}\boxtimes\Delta^{0}\hookrightarrow\Delta^{n}\boxtimes\Delta^{0})\square(\Delta^{0}\boxtimes\Lambda^{m}_{k}\hookrightarrow\Delta^{0}\boxtimes\Delta^{m})\,\big{|}\,m,n\in\mathbb{N}_{0},\,0\leq
k\leq m\big{\\}}\,.$ (3.68)

###### Proposition 3.69

There is a Quillen adjunction

${\Delta_{e!}:s{\mathscr{S}\mathrm{et}_{\Delta}}}$${\mathscr{H}_{\infty}^{i}:\Delta_{e}^{*}\,.}$$\scriptstyle{\perp}$
(3.70)

###### Proof.

We have already seen in Lemma 3.58 that $\Delta_{e!}$ sends the morphism
$\partial\Delta^{n}\boxtimes\Delta^{0}\hookrightarrow\Delta^{n}\boxtimes\Delta^{0}$
to the injective cofibration
$\partial\Delta_{e}^{n}\hookrightarrow\Delta_{e}^{n}$ in
$\mathscr{H}_{\infty}^{i}$. Further, it follows from Equations (3.54) and
(3.57) that

$\Delta_{e!}(\Delta^{0}\boxtimes\partial\Delta^{m}\hookrightarrow\Delta^{0}\boxtimes\Delta^{m})=\big{(}\tilde{\mathsf{c}}\partial\Delta^{m}\longrightarrow\tilde{\mathsf{c}}\Delta^{m}\big{)}\,,$
(3.71)

which is an injective cofibration, and that

$\Delta_{e!}(\Delta^{0}\boxtimes\Lambda^{m}_{k}\hookrightarrow\Delta^{0}\boxtimes\Delta^{m})=\big{(}\tilde{\mathsf{c}}\Lambda^{m}_{k}\longrightarrow\tilde{\mathsf{c}}\Delta^{m}\big{)}\,,$
(3.72)

which is an injective trivial cofibration. Since $\mathscr{H}_{\infty}^{i}$ is
a symmetric monoidal model category, and since $\Delta_{e!}$ preserves
pushouts, it now follows that $\Delta_{e!}$ sends the generating (trivial)
cofibrations of ${\mathscr{S}\mathrm{et}_{\Delta}}$ to (trivial) cofibrations
in $\mathscr{H}_{\infty}^{i}$. Thus, $\Delta_{e!}$ is a left Quillen functor
by [21, Lemma 2.1.20]. ∎

###### Corollary 3.73

There is a Quillen adjunction

${\Delta_{e!}:s{\mathscr{S}\mathrm{et}_{\Delta}}}$${\mathscr{H}_{\infty}^{iI}:\Delta_{e}^{*}\,.}$$\scriptstyle{\perp}$
(3.74)

###### Proposition 3.75

The Quillen adjunction $\Delta_{e!}\dashv\Delta_{e}^{*}$ descends to a Quillen
adjunction

${\Delta_{e!}:L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}}$${\mathscr{H}_{\infty}^{iI}:\Delta_{e}^{*}\,.}$$\scriptstyle{\perp}$
(3.76)

###### Proof.

This is a direct consequence of the fact that
$\Delta_{e!}(\Delta^{n}\boxtimes\Delta^{0})\cong\Delta_{e}^{n}$. The collapse
morphism $\Delta_{e}^{n}\to*$ is a weak equivalence in
$\mathscr{H}_{\infty}^{iI}$ by Proposition 2.13. ∎

###### Corollary 3.77

The adjunction
$L_{e}:{\mathscr{S}\mathrm{et}_{\Delta}}\rightleftarrows\mathscr{H}_{\infty}^{iI}:S_{e}$
is a Quillen adjunction.

###### Proof.

This is a direct consequence of Proposition 3.46 and Proposition 3.75. ∎

###### Theorem 3.78

There is a commutative triangle of Quillen equivalences

${L_{\Delta^{\bullet}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}}$${{\mathscr{S}\mathrm{et}_{\Delta}}}$${\mathscr{H}_{\infty}^{iI}}$$\scriptstyle{\Delta_{e!}}$$\scriptstyle{\operatorname{ev}_{[0]}}$$\scriptstyle{\tilde{\mathsf{c}}}$$\scriptstyle{\mathsf{c}_{\mathbbe{\Delta}}}$$\scriptstyle{\operatorname{ev}_{*}}$$\scriptstyle{\Delta_{e}^{*}}$
(3.79)

where $\mathsf{c}_{\mathbbe{\Delta}}$, $\Delta_{e!}$, and $\tilde{\mathsf{c}}$
are the left adjoints.

###### Proof.

We know from Theorem 2.37 that the bottom adjunction is a Quillen equivalence,
and we know from Proposition 3.48 that the left diagonal adjunction is a
Quillen equivalence. Further, it is evident that the diagram of right adjoints
commutes strictly. Thus, the claim follows from the two-out-of-three property
of Quillen equivalences. ∎

###### Lemma 3.80

Consider the functors
$\operatorname{ev}_{*},S_{e}\colon\mathscr{H}_{\infty}\to{\mathscr{S}\mathrm{et}_{\Delta}}$.

  1. (1)

There is a canonical natural transformation
$\gamma\colon\operatorname{ev}_{*}\to S_{e}$.

  2. (2)

The restriction of $\gamma$ to a morphism between functors
$(\mathscr{H}_{\infty}^{p/i\,I})_{f}\longrightarrow({\mathscr{S}\mathrm{et}_{\Delta}})_{f}$
is a natural weak equivalence.

###### Proof.

Consider first the functors
$\Delta_{e}^{*},\mathsf{c}_{\mathbbe{\Delta}}\circ\operatorname{ev}_{[0]}\colon\mathscr{H}_{\infty}^{p/i\,I}\longrightarrow
s{\mathscr{S}\mathrm{et}_{\Delta}}$. For any $F\in\mathscr{H}_{\infty}$, the
collapse map $\Delta_{e}^{n}\to*$ induces a morphism
$\widehat{\gamma}_{|F,n}\colon F(*)\to F(\Delta_{e}^{n})$ of simplicial sets.
Since $*\in{\mathscr{C}\mathrm{art}}$ is final, this induces a natural
transformation
$\widehat{\gamma}\colon\mathsf{c}_{\mathbbe{\Delta}}\circ\operatorname{ev}_{*}\longrightarrow\Delta_{e}^{*}$.
Applying the diagonal functor $\delta^{*}$ to this natural transformation, we
obtain a natural transformation

$\gamma\colon\operatorname{ev}_{*}=\delta^{*}\circ\mathsf{c}_{\mathbbe{\Delta}}\circ\operatorname{ev}_{*}\longrightarrow\delta^{*}\circ\Delta_{e}^{*}=S_{e}\,.$
(3.81)

This shows part (1). Part (2) then follows from the fact that, whenever
$F\in\mathscr{H}_{\infty}^{p/i\,I}$ is fibrant, the morphism $F(*)\to
F(\Delta_{e}^{n})$ is a weak equivalence for every $n\in\mathbb{N}_{0}$.
Therefore, if $F$ is fibrant, then
$\widehat{\gamma}_{|F}\colon\mathsf{c}_{\mathbbe{\Delta}}(F(*))\longrightarrow\Delta_{e}^{*}F$
is an objectwise weak equivalence in $s{\mathscr{S}\mathrm{et}_{\Delta}}$. The
claim now follows from the fact that the diagonal functor $\delta^{*}$ is
homotopical. ∎

###### Theorem 3.82

The Quillen adjunction

${L_{e}:{\mathscr{S}\mathrm{et}_{\Delta}}}$${\mathscr{H}_{\infty}^{iI}:S_{e}\,.}$$\scriptstyle{\perp}$
(3.83)

is a Quillen equivalence.

###### Proof.

We will show that the total right derived functor
$\mathbb{R}S_{e}\colon\mathrm{h}\mathscr{H}_{\infty}^{iI}\longrightarrow\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$
is an equivalence of categories. If
$R^{iI}\colon\mathscr{H}_{\infty}^{iI}\to\mathscr{H}_{\infty}^{iI}$ is a
fibrant replacement functor in $\mathscr{H}_{\infty}^{iI}$, we can write
$\mathbb{R}S_{e}$ as the composition [21, Def. 1.3.6]

${\mathrm{h}\mathscr{H}_{\infty}^{iI}}$${\mathrm{h}(\mathscr{H}_{\infty}^{iI})_{f}}$${{\mathscr{S}\mathrm{et}_{\Delta}}\,.}$$\scriptstyle{\mathrm{h}R^{iI}}$$\scriptstyle{\mathrm{h}S_{e}}$
(3.84)

Consider the natural transformation $\gamma\colon\operatorname{ev}_{*}\to
S_{e}$ from Lemma 3.80. Since its component on each fibrant object
$F\in\mathscr{H}_{\infty}^{iI}$ is a weak equivalence, $\gamma$ induces a
natural isomorphism

$\mathbb{R}\gamma=\mathrm{h}R^{iI}(\mathrm{h}\gamma)\colon\mathbb{R}\operatorname{ev}_{*}\overset{\cong}{\longrightarrow}\mathbb{R}S_{e}$
(3.85)

of total right derived functors. Since $\operatorname{ev}_{*}$ is a right
Quillen equivalence by Theorem 2.37, its total right derived functor
$\mathbb{R}\operatorname{ev}_{*}$ is an equivalence of categories. Therefore,
it now follows that also $\mathbb{R}S_{e}$ is an equivalence of categories.
Thus, it follows from Proposition D.3 that $L_{e}\dashv S_{e}$ is a Quillen
equivalence. ∎

###### Corollary 3.86

The Quillen adjunction

${\delta_{!}:{\mathscr{S}\mathrm{et}_{\Delta}}}$${L_{\Delta^{1}\boxtimes\Delta^{0}}s{\mathscr{S}\mathrm{et}_{\Delta}}:\delta^{*}\,.}$$\scriptstyle{\perp}$
(3.87)

is a Quillen equivalence.

###### Proof.

This follows from the fact that $L_{e}=\Delta_{e!}\circ\delta_{!}$, together
with Theorem 3.78 and the two-out-of-three property for Quillen equivalences.
∎

Corollary 3.86 becomes particularly interesting in light of Proposition 3.39:
it establishes a Quillen equivalence between the Kan-Quillen model structure
on simplicial sets and each of the model structures in Proposition 3.39. In
other words, Corollary 3.86 shows that each of the model categories from
Proposition 3.39 is a model category for $\infty$-groupoids.

##  4 Comparison of spaces constructed from simplicial presheaves

In Sections 2 and 3 we have seen several ways of extracting a space from a
simplicial presheaf on ${\mathscr{C}\mathrm{art}}$. The main goal of this
section is to establish comparisons between the resulting spaces. In
particular, these comparisons are useful tools in applications of the
$\mathbb{R}$-local homotopy theory of simplicial presheaves, such as in
Section 5.

The right adjoint of $S_{e}=\delta^{*}\circ\Delta_{e}^{*}$ is given as
$R_{e}=\Delta_{e*}\circ\delta_{*}$. We start by making this functor more
explicit: consider a simplicial set $K\in{\mathscr{S}\mathrm{et}_{\Delta}}$
and a cartesian space $c\in{\mathscr{C}\mathrm{art}}$. Since the adjunction
$S_{e}\dashv R_{e}$ is simplicial, there are natural isomorphisms

$\displaystyle(R_{e}K)(c)$
$\displaystyle\cong\underline{\mathscr{H}}_{\infty}({\mathscr{Y}}_{c},R_{e}K)$
(4.1)
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(S_{e}{\mathscr{Y}}_{c},K)$
(4.2) $\displaystyle\cong K^{S_{e}({\mathscr{Y}}_{c})}\,.$ (4.3)

The following lemma is then immediate:

###### Lemma 4.4

There exist canonical natural isomorphisms

$S_{e}\circ\tilde{\mathsf{c}}\cong
1_{\mathscr{S}\mathrm{et}_{\Delta}}\,,\qquad\text{and}\qquad\operatorname{ev}_{*}\circ
R_{e}\cong 1_{\mathscr{S}\mathrm{et}_{\Delta}}\,.$ (4.5)

It follows that there exists a natural isomorphism $\operatorname{ev}_{*}\circ
R_{e}\circ\operatorname{Sing}\cong\operatorname{Sing}$.

###### Lemma 4.6

There is an isomorphism

$\operatorname{Sing}(T)^{K}\cong\operatorname{Sing}\big{(}T^{|K|}\big{)}\,,$
(4.7)

natural in both $T\in{\Delta\mathscr{T}\mathrm{op}}$ and in
$K\in{\mathscr{S}\mathrm{et}_{\Delta}}$.

###### Proof.

Since the adjunction $|{-}|\dashv\operatorname{Sing}$ is simplicial (because
$|{-}|$ preserves finite products), we have binatural isomorphisms

$\displaystyle\operatorname{Sing}(T)^{K}$
$\displaystyle=\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}\big{(}K,\operatorname{Sing}(T)\big{)}$
(4.8) $\displaystyle\cong\underline{{\Delta\mathscr{T}\mathrm{op}}}(|K|,T)$
(4.9) $\displaystyle=\operatorname{Sing}(T^{|K|})\,.$ (4.10)

Here we have used that ${\Delta\mathscr{T}\mathrm{op}}$ is cartesian closed
and simplicially enriched. ∎

###### Lemma 4.11

For every manifold $M\in{\mathscr{M}\mathrm{fd}}$ there exist morphisms

$\varphi_{M}\colon
S_{e}\underline{M}\longrightarrow\operatorname{Sing}(\mathrm{D}M)\qquad\text{and}\qquad\psi_{M}\colon|{-}|\circ
S_{e}(\underline{M})\longrightarrow\mathrm{D}M$ (4.12)

of simplicial sets and of topological spaces, respectively. These assemble
into natural transformations

$\varphi\colon
S_{e}\longrightarrow\operatorname{Sing}\circ\mathrm{D}\qquad\text{and}\qquad\psi\colon|{-}|\circ
S_{e}\longrightarrow\mathrm{D}$ (4.13)

of functors ${\mathscr{M}\mathrm{fd}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ and
${\mathscr{M}\mathrm{fd}}\to{\Delta\mathscr{T}\mathrm{op}}$, respectively.

###### Proof.

Let $n\in\mathbb{N}_{0}$ and consider the set
$\underline{M}(\Delta_{e}^{n})={\mathscr{M}\mathrm{fd}}(\Delta_{e}^{n},M)$; it
is the set of all smooth maps $\Delta_{e}^{n}\to M$ of manifolds. Recall the
morphism
$\iota^{\bullet}\colon|\Delta^{\bullet}|\longrightarrow\mathrm{D}\Delta_{e}^{\bullet}$
of cosimplicial topological spaces from (3.5). If $f\colon\Delta_{e}^{n}\to M$
is any smooth map, then the composition
$f\circ\iota^{n}\colon|\Delta^{n}|\to\mathrm{D}M$ is continuous. Here we have
used that $\mathrm{D}M$ coincides with the underlying topological space of the
manifold $M$ (see Proposition 2.48). This provides a map

$\varphi_{M|n}\colon\underline{M}(\Delta_{e}^{n})={\mathscr{M}\mathrm{fd}}(\Delta_{e}^{n},M)\overset{\mathrm{D}}{\longrightarrow}{\Delta\mathscr{T}\mathrm{op}}(\mathrm{D}\Delta_{e}^{n},\mathrm{D}M)\overset{(\iota_{n})^{*}}{\longrightarrow}{\Delta\mathscr{T}\mathrm{op}}(|\Delta^{n}|,\mathrm{D}M)\,.$
(4.14)

Since $\iota^{\bullet}$ is a morphism of cosimplicial topological spaces, and
since the maps $\varphi_{M|n}$ are defined by precomposition by $\iota^{n}$,
it readily follows that $\varphi_{M|n}$ is natural in both
$M\in{\mathscr{M}\mathrm{fd}}$ and $n\in{\mathbbe{\Delta}}$. Thus, we obtain
the desired morphism of simplicial sets

$\varphi_{M}\colon
S_{e}\underline{M}\longrightarrow\operatorname{Sing}(\mathrm{D}M)\,.$ (4.15)

The composition

$\psi_{M}\colon|S_{e}\underline{M}|\xrightarrow{|\varphi_{M}|}|\operatorname{Sing}(\mathrm{D}M)|\xrightarrow[\sim]{e_{M}}\mathrm{D}M$
(4.16)

then defines the morphism $\psi_{M}$, where
$e\colon|{-}|\circ\operatorname{Sing}\overset{\sim}{\longrightarrow}1_{\Delta\mathscr{T}\mathrm{op}}$
is the evaluation morphism of the adjunction $|{-}|\dashv\operatorname{Sing}$.
∎

###### Lemma 4.17

The restrictions of $\varphi$ and $\psi$ to
${\mathscr{C}\mathrm{art}}\subset{\mathscr{M}\mathrm{fd}}$ are natural weak
equivalences of functors
${\mathscr{C}\mathrm{art}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ and
${\mathscr{C}\mathrm{art}}\to{\Delta\mathscr{T}\mathrm{op}}$, respectively.

###### Proof.

This follows readily from the observation that, for any cartesian space
$c\in{\mathscr{C}\mathrm{art}}$, both $|S_{e}{\mathscr{Y}}_{c}|$ and
$\mathrm{D}({\mathscr{Y}}_{c})$ are weakly equivalent to
$*\in{\Delta\mathscr{T}\mathrm{op}}$. Hence, by the two-out-of-three property
of weak equivalences any morphism
$|S_{e}{\mathscr{Y}}_{c}|\longrightarrow\mathrm{D}({\mathscr{Y}}_{c})$ is a
weak equivalence. ∎

###### Proposition 4.18

There exists a natural weak equivalence

${{\Delta\mathscr{T}\mathrm{op}}}$${\mathscr{H}_{\infty}^{p/i\,I}\,.}$$\scriptstyle{R_{e}\circ\operatorname{Sing}}$$\scriptstyle{S}$$\scriptstyle{\eta}$$\scriptstyle{\sim}$
(4.19)

###### Proof.

We consider the projective case; the injective case then follows since
$\mathscr{H}_{\infty}^{pI}$ and $\mathscr{H}_{\infty}^{iI}$ have the same weak
equivalences (Proposition 2.12). Given a topological space
$T\in{\Delta\mathscr{T}\mathrm{op}}$, by Equation (4.1) and Lemma 4.6 we have

$R_{e}\circ\operatorname{Sing}(T)(c)\cong\operatorname{Sing}\big{(}T^{|S_{e}{\mathscr{Y}}_{c}|}\big{)}\qquad\text{and}\qquad
S(T)(c)\cong\operatorname{Sing}\big{(}T^{\mathrm{D}c}\big{)}\,.$ (4.20)

The natural morphisms $\psi$ from Lemma 4.11 induce a morphism

$\eta_{|T}\coloneqq\operatorname{Sing}\big{(}T^{\psi}\big{)}\colon
S(T)\longrightarrow R_{e}\circ\operatorname{Sing}(T)$ (4.21)

in $\mathscr{H}_{\infty}$, which is natural in
$T\in{\Delta\mathscr{T}\mathrm{op}}$.

In order to see that the morphism $\operatorname{Sing}(T^{\psi})$ is a weak
equivalence in $\mathscr{H}_{\infty}^{pI}$, we first observe that both $S(T)$
and $R_{e}\circ\operatorname{Sing}(T)$ are fibrant in
$\mathscr{H}_{\infty}^{pI}$. We have that

$\operatorname{ev}_{*}\big{(}S(T)\big{)}=\operatorname{Sing}(T)\qquad\text{and}\qquad\operatorname{ev}_{*}\big{(}R_{e}\circ\operatorname{Sing}(T)\big{)}=\operatorname{Sing}\big{(}T^{|S_{e}(*)|}\big{)}=\operatorname{Sing}(T)\,.$
(4.22)

Further, the morphism $\psi_{|*}\colon|S_{e}(*)|=*\to*=\mathrm{D}(*)$ is the
identity, so that also the morphism

$\Big{(}\operatorname{Sing}\big{(}T^{\psi}\big{)}\Big{)}_{|*}\colon
S(T)(*)=\operatorname{Sing}(T)\longrightarrow\operatorname{Sing}(T)=R_{e}\circ\operatorname{Sing}(T)(*)$
(4.23)

is the identity. The fact that $\eta$ is a natural weak equivalence now
follows from the fact that the right Quillen equivalence
$\operatorname{ev}_{*}\colon\mathscr{H}_{\infty}^{pI}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
reflects weak equivalences between fibrant objects (which was also the content
of Proposition 2.68). ∎

###### Corollary 4.24

There is a natural weak equivalence

${{\Delta\mathscr{T}\mathrm{op}}}$${{\mathscr{S}\mathrm{et}_{\Delta}}\,.}$$\scriptstyle{\operatorname{Sing}}$$\scriptstyle{S_{e}\circ
S}$$\scriptstyle{\eta^{\prime}}$$\scriptstyle{\sim}$ (4.25)

###### Proof.

Since the functor
$S_{e}\colon\mathscr{H}_{\infty}^{p/i\,I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
is homotopical, we obtain a natural weak equivalence

$S_{e}\eta\colon S_{e}\circ S\overset{\sim}{\longrightarrow}S_{e}\circ
R_{e}\circ\operatorname{Sing}\,.$ (4.26)

Let $e\colon S_{e}\circ R_{e}\longrightarrow
1_{\mathscr{S}\mathrm{et}_{\Delta}}$ denote the evaluation morphism of the
adjunction $S_{e}\dashv R_{e}$. The fact that every object in
${\mathscr{S}\mathrm{et}_{\Delta}}$ is cofibrant, together with the fact that
$S_{e}\dashv R_{e}$ is a Quillen equivalence imply that the morphism
$e_{|K}\colon S_{e}\circ R_{e}(K)\longrightarrow K$ is a weak equivalence in
${\mathscr{S}\mathrm{et}_{\Delta}}$ for every fibrant simplicial set $K$.
Since
$\operatorname{Sing}\colon{\Delta\mathscr{T}\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
takes values in fibrant simplicial sets, it follows that the composition

$\eta^{\prime}\colon S_{e}\circ S\xrightarrow[\sim]{S_{e}\eta}S_{e}\circ
R_{e}\circ\operatorname{Sing}\xrightarrow[\sim]{e_{\operatorname{Sing}}}\operatorname{Sing}$
(4.27)

is a natural weak equivalence. ∎

###### Corollary 4.28

Let $Q^{p}\colon\mathscr{H}_{\infty}^{pI}\to\mathscr{H}_{\infty}^{pI}$ be a
cofibrant replacement functor, with associated natural weak equivalence
$q^{p}\colon Q^{p}\to 1_{\mathscr{H}_{\infty}}$. There is a zig-zag of natural
weak equivalences

$S_{e}\ \xleftarrow[\sim]{S_{e}q^{p}}\ S_{e}\circ Q^{p}\
\xrightarrow[\sim]{\eta^{\prime\prime}}\ \operatorname{Sing}\circ Re\circ
Q^{p}$ (4.29)

of functors
$\mathscr{H}_{\infty}^{pI}\longrightarrow{\Delta\mathscr{T}\mathrm{op}}$.

###### Proof.

The left-facing natural transformation is a weak equivalence since $S_{e}$ is
homotopical. The right-facing morphism is the composition

$\eta^{\prime\prime}\colon S_{e}\circ
Q^{p}\xrightarrow{S_{e}co_{Q^{p}}}S_{e}\circ S\circ Re\circ
Q^{p}\xrightarrow[\sim]{(\eta^{\prime})_{ReQ^{p}}}\operatorname{Sing}\circ
Re\circ Q^{p}\,,$ (4.30)

where $co\colon 1_{\Delta\mathscr{T}\mathrm{op}}\longrightarrow S\circ Re$ is
the coevaluation morphism of the adjunction $Re\dashv S$. Since this
adjunction is a Quillen equivalence and since every object in
${\Delta\mathscr{T}\mathrm{op}}$ is fibrant, it follows that $co_{F}\colon
F\to S\circ Re(F)$ is a weak equivalence for every cofibrant object
$F\in\mathscr{H}_{\infty}^{pI}$. ∎

###### Proposition 4.31

There exists a zig-zag of natural weak equivalences

$|{-}|\circ S_{e}\ \xleftarrow[\sim]{|S_{e}(q^{p})|}\ |{-}|\circ S_{e}\circ
Q^{p}\ \xrightarrow[\sim]{\eta^{\prime\prime\prime}}\ Re\circ Q^{p}$ (4.32)

of functors
$\mathscr{H}_{\infty}^{pI}\longrightarrow{\Delta\mathscr{T}\mathrm{op}}$. In
particular, there exists a natural isomorphism of total left derived functors

${\mathrm{h}\mathscr{H}_{\infty}^{pI}}$${\mathrm{h}{\Delta\mathscr{T}\mathrm{op}}\,.}$$\scriptstyle{\mathbb{L}Re}$$\scriptstyle{\mathrm{h}|{-}|\circ\mathrm{h}S_{e}}$$\scriptstyle{\eta^{\prime\prime\prime}}$$\scriptstyle{\cong}$
(4.33)

Observe that $S_{e}$ and $|{-}|$ are already homotopical, so that we do not
need to precompose them by a cofibrant replacement in order to obtain their
total left derived functors.

###### Proof.

We readily obtain a zig-zag as in (4.33) by applying the functor $|{-}|$ to
the zig-zag (4.29) and then postcomposing by the evaluation transformation
$e\colon|{-}|\circ\operatorname{Sing}\overset{\sim}{\longrightarrow}1_{\Delta\mathscr{T}\mathrm{op}}$.
However, there is an alternative way of obtaining a zig-zag as in (4.33)
directly and explicitly, which we think is worth showing: let $Q^{p}$ be
Dugger’s cofibrant replacement functor for $\mathscr{H}_{\infty}^{p}$ [16].
Explicitly, it sends a simplicial presheaf $F$ to the two-sided bar
construction

$Q^{p}F=B^{\mathscr{H}_{\infty}}(F,{\mathscr{C}\mathrm{art}},{\mathscr{Y}})\,,$
(4.34)

in the notation of [31]. (The superscript indicates in which simplicial
category we are forming the bar construction.) Using that $Re$ is simplicial
and commutes with colimits, and that there is a natural isomorphism
$Re\circ{\mathscr{Y}}_{c}\cong\mathrm{D}c$ for any cartesian space
$c\in{\mathscr{C}\mathrm{art}}$ (cf. Lemma 2.61), we obtain a canonical
isomorphism

$Re\circ Q^{p}(F)\cong
B^{\Delta\mathscr{T}\mathrm{op}}(F,{\mathscr{C}\mathrm{art}},\mathrm{D})\,.$
(4.35)

Now we use that the morphism $\psi$ from Lemma 4.11 induces a natural weak
equivalence $\psi\colon|{-}|\circ
S_{e}\overset{\sim}{\longrightarrow}\mathrm{D}$ of functors
${\mathscr{C}\mathrm{art}}\to{\Delta\mathscr{T}\mathrm{op}}$ (cf. Lemma 4.17).
Since each of the functors
$F\colon{\mathscr{C}\mathrm{art}}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
and $\mathrm{D},|{-}|\circ
S_{e}\colon{\mathscr{C}\mathrm{art}}\to{\Delta\mathscr{T}\mathrm{op}}$ are
objectwise cofibrant, [31, Cor. 5.2.5] implies that $\psi$ induces a weak
equivalence

$B^{\Delta\mathscr{T}\mathrm{op}}(-,{\mathscr{C}\mathrm{art}},\psi)\colon
B^{\Delta\mathscr{T}\mathrm{op}}(-,{\mathscr{C}\mathrm{art}},|{-}|\circ
S_{e})\overset{\sim}{\longrightarrow}B^{\Delta\mathscr{T}\mathrm{op}}(-,{\mathscr{C}\mathrm{art}},\mathrm{D})=Re\circ
Q^{p}$ (4.36)

of functors $\mathscr{H}_{\infty}\to{\Delta\mathscr{T}\mathrm{op}}$.

On the other hand, since both $|{-}|$ and $S_{e}$ are left adjoints, we have a
natural isomorphism

$\displaystyle
B^{\Delta\mathscr{T}\mathrm{op}}(F,{\mathscr{C}\mathrm{art}},|{-}|\circ
S_{e})$ $\displaystyle\cong|{-}|\circ
S_{e}\big{(}B^{\mathscr{H}_{\infty}}(F,{\mathscr{C}\mathrm{art}},{\mathscr{Y}})\big{)}$
(4.37) $\displaystyle=|{-}|\circ S_{e}\circ Q^{p}(F)\,.$ (4.38)

Now, the morphism $q^{p}\colon
Q^{p}\overset{\sim}{\longrightarrow}1_{\mathscr{H}_{\infty}}$, together with
the fact that both $|{-}|$ and $S_{e}$ preserve weak equivalences, yield the
claim. ∎

###### Remark 4.39

Recall the embedding
$\iota\colon{\mathscr{D}\mathrm{fg}}\hookrightarrow\mathscr{H}_{\infty}$ of
diffeological spaces into simplicial presheaves. By Lemma 2.61, the
composition $Re\circ\iota$ agrees with the functor
$\mathrm{D}\colon{\mathscr{D}\mathrm{fg}}\to{\Delta\mathscr{T}\mathrm{op}}$
that sends a diffeological space to its underlying topological space, whose
topology is induced by its plots. It is interesting to ask whether the
homotopy type of $\mathrm{D}X$ agrees with that of the smooth singular complex
$S_{e}\iota(X)$ of $X$, for any diffeological space
$X\in{\mathscr{D}\mathrm{fg}}$. So far, however, we only see that the homotopy
type of $S_{e}\iota X$ agrees with the cobar construction

$\displaystyle S_{e}\iota(X)$ $\displaystyle\simeq
B^{\Delta\mathscr{T}\mathrm{op}}(X,{\mathscr{C}\mathrm{art}},\mathrm{D})$
(4.40)
$\displaystyle=\int^{n}|\Delta^{n}|\times\Big{(}\coprod_{c_{0},\ldots,c_{n}\in{\mathscr{C}\mathrm{art}}}\mathrm{D}c_{0}\times{\mathscr{C}\mathrm{art}}(c_{0},c_{1})\times{\mathscr{C}\mathrm{art}}(c_{n-1},c_{n})\times
X(c_{n})\Big{)}$ (4.41)

rather than with the underlying topological space $\mathrm{D}X$ of $X$. This
is in accordance with—and maybe provides some further insight to—results from
[8, 28] that the smooth singular complex of a diffeological space $X$ is not
in general equivalent to the smooth singular complex of $\mathrm{D}X$.
$\triangleleft$

To conclude this section, we can interpret the functor $Re$—and because of