Proposition 4.31 also the functor $S_{e}$—in the context of the cohesive
$\infty$-topos ${\mathbf{H}}$ of presheaves of spaces on
${\mathscr{C}\mathrm{art}}$ as follows (see [32] for more background). From
the proof of Proposition 4.31 we see that there are canonical weak
equivalences

$\displaystyle Re\circ Q^{p}(F)$ $\displaystyle\cong
B^{\Delta\mathscr{T}\mathrm{op}}(F,{\mathscr{C}\mathrm{art}},\mathrm{D})$
(4.42) $\displaystyle\simeq
B^{\Delta\mathscr{T}\mathrm{op}}(F,{\mathscr{C}\mathrm{art}},*)$ (4.43)
$\displaystyle\cong\big{|}B^{\mathscr{S}\mathrm{et}_{\Delta}}(F,{\mathscr{C}\mathrm{art}},*)\big{|}\,.$
(4.44)

Using the fact that the topological realisation of a bisimplicial set is
independent of which simplicial direction one realises first (up to canonical
isomorphism), we obtain canonical weak equivalences

$\displaystyle\big{|}B^{\mathscr{S}\mathrm{et}_{\Delta}}(F,{\mathscr{C}\mathrm{art}},*)\big{|}$
$\displaystyle\cong\big{|}B^{\mathscr{S}\mathrm{et}_{\Delta}}(*,{\mathscr{C}\mathrm{art}}^{\mathrm{op}},F)\big{|}$
(4.45)
$\displaystyle\simeq{\mathrm{hocolim}}^{\Delta\mathscr{T}\mathrm{op}}\big{(}|{-}|\circ
F\colon{\mathscr{C}\mathrm{art}}^{\mathrm{op}}\longrightarrow{\Delta\mathscr{T}\mathrm{op}}\big{)}\,.$
(4.46)

Consequently, from Proposition 4.31 we obtain that each of the functors

$Re\circ Q^{p}\simeq|{-}|\circ S_{e}\circ Q^{p}\simeq|{-}|\circ S_{e}$ (4.47)

models the homotopy colimit of the diagram
$|F|\colon{\mathscr{C}\mathrm{art}}^{\mathrm{op}}\to{\Delta\mathscr{T}\mathrm{op}}$,
for any $F\in\mathscr{H}_{\infty}$. Therefore, the functors they present on
the $\infty$-categories underlying $\mathscr{H}_{\infty}^{p}$ and
${\Delta\mathscr{T}\mathrm{op}}$ are equivalent, and they are further
equivalent to the $\infty$-colimit functor. Consequently, on the level of the
underlying $\infty$-categories they each present left-adjoints to the functor
that sends a space to the constant presheaf whose value is that space. This
means that both $Re$ and $S_{e}$ provide explicit presentations for the left-
adjoint $\Pi$ in the three-fold adjunction which implements the cohesive
structure on ${\mathbf{H}}$ (see [32]). This appears to have been known for
$Re$, but the functor $S_{e}$ has not been formally identified as a model for
the cohesion functor $\Pi$ (although this has been indicated in [3]).

##  5 A Whitehead Approximation Theorem

In the last section we established several comparison results for the various
simplicial sets and topological spaces that we can naturally extract from an
object $F\in\mathscr{H}_{\infty}$. It does not seem to be true, however, that
the functors $Re$ and $|{-}|\circ S_{e}$ are generally weakly equivalent. We
were only able to relate these functors on the level of homotopy categories
and their derived functors (Proposition 4.31). The results about diffeological
spaces referred to in Remark 4.39 show that, in general, we cannot expect
anything better for general simplicial presheaves on
${\mathscr{C}\mathrm{art}}$.

In this section, we show that if a simplicial presheaf comes from a smooth
manifold $M$, then the smooth singular complex $S_{e}\underline{M}$ and the
underlying topological space $\mathrm{D}\underline{M}$ are canonically weakly
equivalent. This is a classical result, sometimes referred to as Whitehead’s
Approximation Theorem for manifolds (see [28], for instance). Here, we employ
our results thus far to give a purely homotopy-theoretic proof of this
theorem, which avoids having to approximate continuous maps by smooth ones.

###### Theorem 5.1

The natural transformations

${{\mathscr{M}\mathrm{fd}}}$${{\mathscr{S}\mathrm{et}_{\Delta}}\,.}$$\scriptstyle{\operatorname{Sing}\circ\mathrm{D}}$$\scriptstyle{S_{e}\circ\underline{(-)}}$$\scriptstyle{\varphi}$
${{\mathscr{M}\mathrm{fd}}}$${{\Delta\mathscr{T}\mathrm{op}}\,.}$$\scriptstyle{\mathrm{D}}$$\scriptstyle{|{-}|\circ
S_{e}\circ\underline{(-)}}$$\scriptstyle{\psi}$ (5.2)

introduced in Lemma 4.11 are natural weak equivalences. In particular, the
smooth singular complex of $M$ has the same homotopy type as
$\operatorname{Sing}M$.

The proof of Theorem 5.1 requires a couple of steps, which will occupy the
remainder of this section. To start with, let ${\mathscr{U}}=\\{U_{a}\\}_{a\in
A}$ be a differentiably good open covering of $M$ (see Section 2.1), and let
${\check{C}}{\mathscr{U}}\to\underline{M}$ be the associated Čech covering in
$\mathscr{H}_{\infty}$.

###### Lemma 5.3

The augmentation map ${\check{C}}{\mathscr{U}}\to\underline{M}$ is a weak
equivalence in $\mathscr{H}_{\infty}^{p/i\,\ell}$.

###### Proof.

The morphism ${\check{C}}{\mathscr{U}}_{0}=\coprod_{a\in
A}\underline{U}_{a}\longrightarrow\underline{M}$ is a local epimorphism, or
generalised cover (in the sense of [10, p. 7]) with respect to the
Grothendieck topology of differentiably good open coverings on
${\mathscr{C}\mathrm{art}}$ [17, 6]. Hence, the result follows from [10, Cor.
A.3]. ∎

For a differentiably good open covering ${\mathscr{U}}=\\{U_{a}\\}_{a\in A}$
of $M$, every non-empty finite intersection $\underline{U}_{a_{0}\ldots
a_{n}}$ is representable in $\mathscr{H}_{\infty}$; in particular, each
$\underline{U}_{a_{0}\ldots a_{n}}$, as well as the Čech nerve
${\check{C}}{\mathscr{U}}$, is cofibrant in $\mathscr{H}_{\infty}^{p}$. We let
$sA$ denote the partially ordered set of non-empty finite subsets of $A$,
ordered by inclusion. By abuse of notation, we denote the category associated
to a partially ordered set $P$ also by $P$. For a partially ordered set $P$,
we write $s_{<}P$ for the partially ordered set of totally ordered finite
subsets of $P$. The assignment
$\\{a_{0},\ldots,a_{n}\\}\mapsto\underline{U}_{a_{0}\ldots a_{n}}$ defines a
functor $\underline{U}_{(-)}\colon sA^{\mathrm{op}}\to\mathscr{H}_{\infty}$,
which takes values in cofibrant objects (where $\underline{U}_{a_{0}\ldots
a_{n}}$ is either the presheaf represented by the cartesian space
$U_{a_{0}}\cap\cdots\cap U_{a_{n}}$ if this is non-empty, or it is the initial
presheaf $\emptyset\in\mathscr{H}_{\infty}$). We will show that the Čech nerve
of a differentiably good open covering ${\mathscr{U}}$ of $M$ is equivalent
(more precisely, isomorphic in $\mathrm{h}\mathscr{H}_{\infty}^{p/i}$) to the
homotopy colimit of the functor $\underline{U}_{(-)}\colon
sA^{\mathrm{op}}\to\mathscr{H}_{\infty}^{p/i}$. To that end, we use the
modified two-sided simplicial bar construction
$B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}$, which is introduced in
Appendix C.

###### Proposition 5.4

For any differentiably good open covering ${\mathscr{U}}=\\{U_{a}\\}_{a\in A}$
of a manifold $M$, there is an objectwise weak equivalence in
$\mathscr{H}_{\infty}^{p}$,

${\mathrm{hocolim}}^{\mathscr{H}_{\infty}^{p/i}}(sA^{\mathrm{op}}\xrightarrow{\underline{U}_{(-)}}\mathscr{H}_{\infty}^{p})\simeq
B^{\mathscr{H}_{\infty}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})\overset{\sim}{\longrightarrow}B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})\,.$
(5.5)

###### Proof.

This is a direct application of Proposition C.27 and Corollary C.32 for
objectwise cofibrant diagrams. ∎

###### Example 5.6

A 1-simplex of
$B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})(c)$
consists of a finite subset $\alpha_{0,1}\subset A$, together with a choice of
two non-empty subsets $\alpha_{0},\alpha_{1}\subset\alpha_{0,1}$ and a smooth
map $c\to U_{\alpha_{0,1}}$. Here we have used that $c$ is connected, so that
each map ${\mathscr{Y}}_{c}\to
B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})_{1}$
must factor through exactly one component of the coproduct on the right-hand
side. Note, however, that each of the subsets
$\alpha_{0},\alpha_{1},\alpha_{0,1}\subset A$ may have any finite (non-zero)
number of elements. $\triangleleft$

###### Definition 5.7

We say a covering ${\mathscr{U}}=\\{U_{a}\\}_{a\in A}$ is _closed under finite
intersections_ if it satisfies the following property: for any finite subset
$\alpha=\\{a_{0},\ldots,a_{n}\\}\subset A$ such that $U_{\alpha}\neq\emptyset$
there exists an element $a\in A$ such that $U_{\alpha}=U_{a}$.

###### Lemma 5.8

Given any differentiably good open covering ${\mathscr{U}}=\\{U_{a}\\}_{a\in
A}$ of $M$, there exists a differentiably good open covering
${\mathscr{U}}^{cl}$ of $M$ which is closed under finite intersections, and
such that there is a weak equivalence
${\check{C}}{\mathscr{U}}\overset{\sim}{\longrightarrow}{\check{C}}{\mathscr{U}}^{cl}$
in $\mathscr{H}_{\infty}^{p/i\,\ell}$ over $\underline{M}$.

###### Proof.

Let ${\mathscr{U}}=\\{U_{a}\\}_{a\in A}$ be a differentiably good open
covering of $M$. As before, given a finite subset
$\alpha=\\{a_{0},\ldots,a_{n}\\}\subset A$, we set
$U_{\alpha}\coloneqq\bigcap_{i=0}^{n}U_{a_{i}}$. Let

${\mathscr{U}}^{cl}\coloneqq\\{U_{\alpha}\,|\,\alpha\in
sA,\,U_{\alpha}\neq\emptyset\\}$ (5.9)

be the open covering of $M$ consisting of all non-empty finite intersections
of the patches of the covering ${\mathscr{U}}$. It is indexed over $sA$, and
it is differentiably good if ${\mathscr{U}}$ is so. (Any finite intersection
of elements of ${\mathscr{U}}^{cl}$ can be written as a finite intersection of
elements of ${\mathscr{U}}$.) The canonical inclusion
${\mathscr{U}}\hookrightarrow{\mathscr{U}}^{cl}$ induces a commutative
triangle

${{\check{C}}{\mathscr{U}}}$${{\check{C}}{\mathscr{U}}^{cl}}$${\underline{M}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$
(5.10)

in $\mathscr{H}_{\infty}$. The diagonal arrows are weak equivalences in the
Čech model structures $\mathscr{H}_{\infty}^{p/i\,\ell}$ by Lemma 5.3.
Therefore, it follows that the horizontal morphism is a Čech weak equivalence
as well. ∎

Let $A$ be a set, and let $A^{[\bullet]}$ denote the simplicial set with
$(A^{[\bullet]})_{n}=A^{n+1}$, and whose $i$-th face map forgets the $i$-th
entry of a tuple. In other words, $A^{[\bullet]}$ is the Čech nerve of the
collapse map $A\to*$ in ${\mathscr{S}\mathrm{et}}$. Let
${\operatorname{Ex}}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
be the right adjoint to the simplicial subdivision functor (for more
background, see Appendix C and [7]). There exists a morphism of simplicial
sets,

$\sigma_{A}\colon A^{[\bullet]}\longrightarrow{\operatorname{Ex}}N(sA)\,,$
(5.11)

which can be described as follows: By adjointness and [7, Lemma 3.1.25], we
have

$\displaystyle\big{(}{\operatorname{Ex}}N(sA)\big{)}_{n}$
$\displaystyle\cong{\mathscr{S}\mathrm{et}_{\Delta}}\big{(}\operatorname{Sd}\Delta^{n},N(sA)\big{)}$
(5.12)
$\displaystyle\cong{\mathscr{S}\mathrm{et}_{\Delta}}\big{(}N(s_{<}[n]),N(sA)\big{)}$
(5.13)
$\displaystyle\cong{\mathscr{P}\mathrm{o}\mathscr{S}\mathrm{et}}\big{(}s_{<}[n],sA\big{)}\,,$
(5.14)

where ${\mathscr{P}\mathrm{o}\mathscr{S}\mathrm{et}}$ is the category of
partially ordered sets, and where in the last step we have used that the nerve
functor $N$ is fully faithful. Given an $n$-simplex $(a_{0},\ldots,a_{n})\in
A^{[n]}=A^{n+1}$, we thus need to describe a map of partially ordered sets
from $s_{<}[n]$ to $sA$. Recall that the elements of $s_{<}[n]$ are the
totally ordered finite subsets of the partially ordered set $[n]$. For any
$0\leq k\leq n$ and any element $\\{i_{0},\ldots,i_{k}\\}\in s_{<}[n]$, we set

$\sigma_{A}(a_{0},\ldots,a_{n})\big{(}\\{i_{0},\ldots,i_{k}\\}\big{)}\coloneqq\\{a_{i_{0}},\ldots,a_{i_{k}}\\}\,.$
(5.15)

Further, there exists a morphism of simplicial sets

$\varrho_{A}\colon{\operatorname{Ex}}\circ
N(sA)\longrightarrow(sA)^{[\bullet]}\,,\qquad\big{(}\alpha\colon s_{<}[n]\to
sA\big{)}\longmapsto\big{(}\alpha(\\{0\\}),\ldots,\alpha(\\{n\\})\big{)}\,.$
(5.16)

The $n$-simplices of $(sA)^{[\bullet]}$ are nothing but $(n{+}1)$-tuples of
finite subsets of $A$. For $(a_{0},\ldots,a_{n})\in A^{[n]}$, we have

$\varrho_{A}\circ\sigma_{A}(a_{0},\ldots,a_{n})=\big{(}\\{a_{0}\\},\ldots,\\{a_{n}\\}\big{)}\quad\in(sA)^{[n]}\,.$
(5.17)

Let ${\mathscr{U}}=\\{U_{a}\\}_{a\in A}$ be an open covering of a manifold
$M$. Consider the morphism

$\psi\colon{\check{C}}{\mathscr{U}}\longrightarrow
B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})$
(5.18)

induced by $\sigma_{A}$; on simplicial level $n$ it reads as

$\psi_{n}\colon\coprod_{a_{0},\ldots,a_{n}\in A}\underline{U}_{a_{0}\ldots
a_{n}}\longrightarrow\coprod_{\alpha\in{\operatorname{Ex}}N(sA)}\underline{U}_{\alpha(\\{0,\ldots,n\\})}\,,$
(5.19)

and it maps the component labelled by $a_{0},\ldots,a_{n}\in A$ to the
component labelled by $\sigma_{A}(a_{0},\ldots,a_{n})$ using the identity on
$\underline{U}_{a_{0}\ldots a_{n}}$. Further, consider the morphism

$\phi\colon
B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})\longrightarrow{\check{C}}{\mathscr{U}}^{cl}$
(5.20)

defined as follows. The morphism $\phi$ maps the component labelled by
$\alpha\in({\operatorname{Ex}}N(sA))_{n}$ to the component labelled by
$\varrho_{A}(\alpha)=(\alpha(\\{0\\}),\ldots,\alpha(\\{n\\}))$ using the
canonical inclusion

$\underline{U}_{\alpha\\{0,\ldots,n\\}}\hookrightarrow\underline{U}_{\bigcup_{i=0}^{n}\alpha(\\{i\\})}\,.$
(5.21)

###### Proposition 5.22

Let ${\mathscr{U}}=\\{U_{a}\\}_{a\in A}$ be a differentiably good open
covering of a manifold $M$. Then, the morphism

$\phi\colon
B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})\longrightarrow{\check{C}}{\mathscr{U}}^{cl}$
(5.23)

induced by $\varrho_{A}$ is a trivial fibration in $\mathscr{H}_{\infty}^{p}$.
Further, the canonical inclusion
${\check{C}}{\mathscr{U}}\hookrightarrow{\check{C}}{\mathscr{U}}^{cl}$ factors
as

${\check{C}}{\mathscr{U}}\overset{\psi}{\longrightarrow}B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})\overset{\phi}{\longrightarrow}{\check{C}}{\mathscr{U}}^{cl}$
(5.24)

in the slice category $(\mathscr{H}_{\infty})_{/\underline{M}}$. In
particular, the morphism $\psi$ is a weak equivalence in
$\mathscr{H}_{\infty}^{p/i\,\ell}$.

###### Proof.

The second part of the claim is straightforward from (5.17); we thus have to
prove that $\phi$ is a projective trivial fibration. To that end, we check
that $\phi$ has the right lifting property with respect to the morphisms
${\mathscr{Y}}_{c}\otimes\partial\Delta^{n}\to{\mathscr{Y}}_{c}\otimes\Delta^{n}$
for $c\in{\mathscr{C}\mathrm{art}}$ and $n\in\mathbb{N}_{0}$. These morphisms
form a set of generating cofibrations for the projective model structure
$\mathscr{H}_{\infty}^{p}$ (see e.g. [2, Proof of Prop. 4.52]). For any
morphism $p\colon Y\to X$ of simplicial presheaves, there is a bijection
between solutions to the lifting problems

${{\mathscr{Y}}_{c}\otimes\partial\Delta^{n}}$${Y}$${{\mathscr{Y}}_{c}\otimes\Delta^{n}}$${X}$$\scriptstyle{p}$
and ${\partial\Delta^{n}}$${Y(c)}$${\Delta^{n}}$${X(c)}$$\scriptstyle{p_{|c}}$
(5.25)

in $\mathscr{H}_{\infty}$ and in ${\mathscr{S}\mathrm{et}_{\Delta}}$,
respectively.

In the case $n=0$, the right-hand side amounts to a commutative diagram

${\emptyset}$${B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})(c)}$${\Delta^{0}}$${{\check{C}}{\mathscr{U}}^{cl}(c)}$$\scriptstyle{\phi_{|c}}$
(5.26)

in ${\mathscr{S}\mathrm{et}_{\Delta}}$. Since $c$ is connected, a vertex in
${\check{C}}{\mathscr{U}}^{cl}(c)$ consists of an element $\alpha\in sA$ and a
smooth map $c\to U_{\alpha}$, which is precisely the same as the data for a
vertex of
$B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})(c)$.
In other words, the map $\phi_{|c}$ is a bijection on vertices, for every
$c\in{\mathscr{C}\mathrm{art}}$.

For $n=1$, we need to consider diagrams in ${\mathscr{S}\mathrm{et}_{\Delta}}$
of the form

${\partial\Delta^{1}}$${B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})(c)}$${\Delta^{1}}$${{\check{C}}{\mathscr{U}}^{cl}(c)}$$\scriptstyle{(\beta,g)}$$\scriptstyle{\phi_{|c}}$$\scriptstyle{(\alpha,f)}$
(5.27)

A 1-simplex in ${\check{C}}{\mathscr{U}}^{cl}(c)$ is a pair
$(\alpha_{0},\alpha_{1})\in(sA)^{2}$ (i.e. a pair of finite subsets of $A$),
together with a smooth map $f\colon c\to U_{\alpha_{0}}\cap
U_{\alpha_{1}}=U_{\alpha_{0}\cup\alpha_{1}}$. The top morphism corresponds to
elements $\beta_{i}\in sA$ and smooth maps $g_{i}\colon c\to U_{\beta_{i}}$
for $i=0,1$. The commutativity of the diagram is precisely the condition that
$\beta_{i}=\alpha_{i}$ and that we have commutative diagrams

${c}$${U_{\alpha_{0}\cup\alpha_{1}}}$${U_{\alpha_{i}}}$$\scriptstyle{f}$$\scriptstyle{g_{i}}$
(5.28)

of smooth maps, for $i=0,1$. Thus, there exists a lift in the diagram,
provided by the 1-simplex

$(\alpha_{0}\hookrightarrow\alpha_{0}\cup\alpha_{1}\hookleftarrow\alpha_{1},f\colon
c\to U_{\alpha_{0}\cup\alpha_{1}})\quad\in
B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})_{1}(c)\,.$
(5.29)

For $n>1$, we need to consider commutative diagrams

${\partial\Delta^{n}}$${B^{\mathscr{H}_{\infty}}_{\operatorname{Ex}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})(c)}$${\Delta^{n}}$${{\check{C}}{\mathscr{U}}^{cl}(c)}$$\scriptstyle{(\beta,g)}$$\scriptstyle{\phi_{|c}}$$\scriptstyle{(\alpha,f)}$
(5.30)

An $n$-simplex in ${\check{C}}{\mathscr{U}}^{cl}(c)$ is an $(n{+}1)$-tuple
$(\alpha_{0},\ldots,\alpha_{n})\in(sA)^{n+1}$, together with a smooth map
$f\colon c\to U_{\bigcup_{i=0}^{n}\alpha_{i}}$. The top morphism is
equivalently the following data: we have $(n{-}1)$-simplices
$\beta_{j}\in({\operatorname{Ex}}N(sA))_{n-1}$ and smooth maps $g_{j}\colon
c\to U_{\beta_{j}(\\{0,\ldots,n-1\\})}$. Via (5.12) we can rephrase
$\beta_{j}$ as a morphism of posets $\beta_{j}\colon s_{<}[n-1]\to sA$. For
notational purposes, we use the canonical identification of the poset $[n-1]$
with the poset $\\{0,\ldots,\hat{j},\ldots,n\\}$, where the hat means that we
are omitting the respective index. Then, we can write the smooth maps $g_{j}$
as

$g_{j}\colon c\longrightarrow
U_{\beta_{j}(\\{0,\ldots,\hat{j},\ldots,n\\})}=U_{\beta_{j}(d_{j}\\{0,\ldots,n\\})}\,.$
(5.31)

Here, $d_{j}$ is the $j$-th face map in the simplicial set $Ns_{<}[n]$. The
fact that these $(n{-}1)$-simplices assemble into a map from
$\partial\Delta^{n}$ amounts to $\\{\beta_{j}\\}_{j=0,\ldots,n}$ forming a map
$\partial\Delta^{n}\to{\operatorname{Ex}}N(sA)$ and to the commutativity of
the diagrams

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i<j\leq n\,.$ (5.32)

This implies that each map $g_{i}$ factors through a unique map $\hat{g}\colon
c\to U_{\bigcup_{j=0}^{n}\beta_{j}[n-1]}\subset U_{\beta_{i}[n-1]}$.

The commutativity of diagram (5.30) means that

$\beta_{j}\\{i\\}=\begin{cases}\alpha_{i}\,,&i<j\,,\\\ \alpha_{i+1}\,,&i\geq
j\end{cases}$ (5.33)

and that the map $f\colon c\to U_{\bigcup_{i=0}^{n}\alpha_{i}}$ factors
through the map

$\hat{g}\colon c\to
U_{\bigcup_{j=0}^{n}\beta_{j}(d_{j}\\{0,\ldots,n\\})}\subset
U_{\bigcup_{i=0}^{n}\alpha_{i}}\,.$ (5.34)

Consequently, there exists an $n$-simplex
$\hat{\beta}\in({\operatorname{Ex}}N(sA))_{n}$, with

$\hat{\beta}(\\{0,\ldots,n\\})=\bigcup_{j=0}^{n}\beta_{j}(d_{j}\\{0,\ldots,n\\})$
(5.35)

and whose boundary is $\beta$. The pair $(\hat{\beta},\hat{g})$ then provides
a lift in diagram (5.30). ∎

###### Corollary 5.36

Let ${\mathscr{U}}=\\{U_{a}\\}_{a\in A}$ be a differentiably good open
covering of a manifold $M$. The canonical morphism

$\underset{sA}{{\mathrm{hocolim}}}^{\mathscr{H}_{\infty}}(\underline{U}_{(-)})=B^{\mathscr{H}_{\infty}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})\longrightarrow\underline{M}$
(5.37)

is a weak equivalence in $\mathscr{H}_{\infty}^{p/i\,\ell}$.

###### Proof.

This is a combination of Lemma 5.3, Proposition 5.4, and Proposition 5.22. ∎

###### Proof of Theorem 5.1.

Choose a differentiably good open covering ${\mathscr{U}}=\\{U_{a}\\}_{a\in
A}$ of $M$ and denote by $\pi_{\mathscr{U}}\colon
B^{\mathscr{H}_{\infty}}(*,sA^{\mathrm{op}},\underline{U}_{(-)})\longrightarrow\underline{M}$
the canonical Čech weak equivalence from Corollary 5.36. By Propositions 2.10
and 2.12, this is a weak equivalence in $\mathscr{H}_{\infty}^{iI}$. Since
each of the functors
$|{-}|\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to{\Delta\mathscr{T}\mathrm{op}}$,
$S_{e}\colon\mathscr{H}_{\infty}\to{\mathscr{S}\mathrm{et}_{\Delta}}$, and
$Re\colon\mathscr{H}_{\infty}\to{\Delta\mathscr{T}\mathrm{op}}$ is left
adjoint and simplicial, each of them preserves two-sided bar constructions.
Therefore, applying $|{-}|\circ S_{e}$ to the morphism $\pi_{\mathscr{U}}$, we
obtain a morphism

$|S_{e}\pi_{\mathscr{U}}|\colon
B^{\Delta\mathscr{T}\mathrm{op}}\big{(}*,sA^{\mathrm{op}},|S_{e}\underline{U}_{(-)}|\big{)}\longrightarrow|S_{e}\underline{M}|\,,$
(5.38)

which is a weak equivalence since both $|{-}|$ and $S_{e}$ are homotopical. On
the other hand, applying the functor $Re$ to $\pi_{\mathscr{U}}$ yields a
morphism

$Re(\pi_{\mathscr{U}})\colon
B^{\Delta\mathscr{T}\mathrm{op}}\big{(}*,sA^{\mathrm{op}},Re\underline{U}_{(-)}\big{)}\longrightarrow
Re\underline{M}\,.$ (5.39)

Observing that the functor $\underline{U}_{(-)}\colon
sA^{\mathrm{op}}\to\mathscr{H}_{\infty}$ factors through the category of
manifolds, we can use Proposition 2.48 to replace $Re$ by the functor
$\mathrm{D}$ from Definition 2.45; we can thus equivalently (up to canonical
isomorphism) write $Re(\pi_{\mathscr{U}})$ as a morphism

$Re(\pi_{\mathscr{U}})\colon
B^{\Delta\mathscr{T}\mathrm{op}}\big{(}*,sA^{\mathrm{op}},\mathrm{D}\underline{U}_{(-)}\big{)}\longrightarrow\mathrm{D}\underline{M}\,.$
(5.40)

By Lemma 4.11, we obtain a diagram

${B^{\Delta\mathscr{T}\mathrm{op}}\big{(}*,sA^{\mathrm{op}},|S_{e}\underline{U}_{(-)}|\big{)}}$${B^{\Delta\mathscr{T}\mathrm{op}}\big{(}*,sA^{\mathrm{op}},\mathrm{D}\underline{U}_{(-)}\big{)}}$${{|S_{e}\underline{M}|}}$${\mathrm{D}\underline{M}}$$\scriptstyle{B^{\Delta\mathscr{T}\mathrm{op}}(*,sA^{\mathrm{op}},\psi)}$$\scriptstyle{|S_{e}\pi_{\mathscr{U}}|}$$\scriptstyle{\sim}$$\scriptstyle{Re(\pi_{\mathscr{U}})}$$\scriptstyle{\psi_{M}}$
(5.41)

This diagram commutes by the naturality of $\psi$.

Both functors $|S_{e}\underline{U}_{(-)}|$ and $\mathrm{D}\underline{U}_{(-)}$
take values in cofibrant topological spaces. This has two implications: first,
by Lemma 4.17 and by the homotopical properties of the bar construction [31,
Sec. 5] the top morphism in diagram (5.41) is a weak equivalence. Second, both
bar constructions model the homotopy colimits in
${\Delta\mathscr{T}\mathrm{op}}$ of the diagrams
$|S_{e}\underline{U}_{(-)}|\colon
sA^{\mathrm{op}}\to{\Delta\mathscr{T}\mathrm{op}}$ and $\mathrm{D}\colon
sA^{\mathrm{op}}\to{\Delta\mathscr{T}\mathrm{op}}$, respectively.

We now aim to show that the right-hand vertical map is a weak equivalence. To
that end, we consider the diagram $\mathrm{D}\underline{U}_{(-)}\colon
sA^{\mathrm{op}}\to{\Delta\mathscr{T}\mathrm{op}}$. For a topological space
$T$, let $\mathcal{O}(T)$ denote its partially ordered set of open subsets.
Then, we can write $\mathrm{D}\underline{U}_{(-)}$ as a diagram
$\mathrm{D}\underline{U}_{(-)}\colon
sA^{\mathrm{op}}\to\mathcal{O}(\mathrm{D}M)$. For
$x\in\mathrm{D}\underline{M}$, let $(sA^{\mathrm{op}})_{x}\subset
sA^{\mathrm{op}}$ denote the full subcategory of $sA^{\mathrm{op}}$ on those
objects $\alpha\in sA$ satisfying that $x\in U_{\alpha}\subset M$. Since
${\mathscr{U}}$ is an open covering of $M$, the category
$(sA^{\mathrm{op}})_{x}$ is non-empty and filtered, for each $x\in M$. We can
therefore apply Lurie’s Seifert-van Kampen Theorem [25, Thm. A.3.1] (which, in
this particular incarnation is often called the Nerve Theorem): the map

${\mathrm{hocolim}}^{\mathscr{S}\mathrm{et}_{\Delta}}\big{(}sA^{\mathrm{op}}\xrightarrow{\mathrm{D}\underline{U}_{(-)}}{\Delta\mathscr{T}\mathrm{op}}\xrightarrow{\operatorname{Sing}}{\mathscr{S}\mathrm{et}_{\Delta}}\big{)}\longrightarrow\operatorname{Sing}\mathrm{D}M$
(5.42)

is a weak equivalence of simplicial sets. Applying the realisation functor and
using that the evaluation morphism
$|{-}|\circ\operatorname{Sing}\overset{\sim}{\longrightarrow}1_{\Delta\mathscr{T}\mathrm{op}}$
is a natural weak equivalence (every object in
${\Delta\mathscr{T}\mathrm{op}}$ is fibrant, and every object in
${\mathscr{S}\mathrm{et}_{\Delta}}$ is cofibrant), we obtain a commutative
diagram

${{\mathrm{hocolim}}^{\Delta\mathscr{T}\mathrm{op}}\big{(}sA^{\mathrm{op}}\xrightarrow{\mathrm{D}\underline{U}_{(-)}}{\Delta\mathscr{T}\mathrm{op}}\xrightarrow{|{-}|\circ\operatorname{Sing}}{\Delta\mathscr{T}\mathrm{op}}\big{)}}$${{|\operatorname{Sing}\mathrm{D}M|}}$${{\mathrm{hocolim}}^{\Delta\mathscr{T}\mathrm{op}}\big{(}sA^{\mathrm{op}}\xrightarrow{\mathrm{D}\underline{U}_{(-)}}{\Delta\mathscr{T}\mathrm{op}}\big{)}}$${\mathrm{D}M}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$
(5.43)

It follows that the morphism
$Re(\pi_{\mathscr{U}})\colon{\mathrm{hocolim}}^{\Delta\mathscr{T}\mathrm{op}}(sA^{\mathrm{op}}\xrightarrow{\mathrm{D}\underline{U}_{(-)}}{\Delta\mathscr{T}\mathrm{op}})\longrightarrow\mathrm{D}M$
is a weak equivalence of topological spaces. Thus, in the commutative diagram
(5.41), the vertical morphisms and the top morphism are weak equivalences of
topological spaces. It follows that the bottom morphism $\psi_{M}$ is a weak
equivalence as well. ∎

##  6 Local fibrant replacement, concordance, and mapping spaces

In this section, we present a fibrant replacement functor in the model
structures $\mathscr{H}_{\infty}^{p/i\,I}$. Its construction is motivated by
the _concordance sheaf_ construction from [3]. This functor allows us to
compute mapping spaces in $\mathscr{H}_{\infty}^{p/i\,I}$ in Theorem 6.22. We
start by presenting the fibrant replacement functor:

###### Lemma 6.1

Suppose that $F_{0},F_{1}\in\mathscr{H}_{\infty}$ and that $h\colon
F_{0}\times\underline{\mathbb{R}}\to F_{1}$ is a smooth homotopy between
morphisms $f,g\colon F_{0}\to F_{1}$. Then, for any
$G\in\mathscr{H}_{\infty}$, there is a smooth homotopy $\tilde{h}\colon
G^{F_{1}}\times\underline{\mathbb{R}}\to G^{F_{0}}$ from $G^{f}$ to
$G^{g}\colon G^{F_{1}}\to G^{F_{0}}$.

###### Proof.

Applying the exponential functor $G^{(-)}$ to the morphism $h$, we obtain a
morphism

$G^{h}\in\mathscr{H}_{\infty}(G^{F_{1}},G^{F_{0}\times\underline{\mathbb{R}}})\cong\mathscr{H}_{\infty}(G^{F_{1}},(G^{F_{0}}){}^{\underline{\mathbb{R}}})\,.$
(6.2)

Using the internal-hom adjunction of $\mathscr{H}_{\infty}$ then yields a
morphism
$\tilde{h}=(G^{h})^{\dashv}\in\mathscr{H}_{\infty}(G^{F_{1}}\times\underline{\mathbb{R}},G^{F_{0}})$
as desired. ∎

We consider the following construction: let
$\underline{\Delta}_{e}^{k}\in\mathscr{H}_{\infty}$ denote the simplicial
presheaf represented by the extended affine simplex
$\Delta_{e}^{k}\in{\mathscr{C}\mathrm{art}}$. This provides a functor

$\underline{\Delta}_{e}\colon{\mathbbe{\Delta}}\to\mathscr{H}_{\infty}\,,\qquad[k]\longmapsto\underline{\Delta}_{e}^{k}={\mathscr{Y}}_{\Delta_{e}^{k}}\,.$
(6.3)

Given an object $F\in\mathscr{H}_{\infty}$, we can compose this functor by the
functor $F^{(-)}\colon\mathscr{H}_{\infty}\to\mathscr{H}_{\infty}$, obtaining
an object
$F^{\underline{\Delta}_{e}}\in{\mathscr{C}\mathrm{at}}({\mathbbe{\Delta}}^{\mathrm{op}},\mathscr{H}_{\infty})$.
Equivalently, we can view this as a bisimplicial presheaf

$F^{\underline{\Delta}_{e}}\colon{\mathscr{C}\mathrm{art}}^{\mathrm{op}}\longrightarrow
s{\mathscr{S}\mathrm{et}_{\Delta}}\,,\qquad c\longmapsto F(\Delta_{e}\times
c)\,,$ (6.4)

which we can now compose by the diagonal functor $\delta^{*}\colon
s{\mathscr{S}\mathrm{et}_{\Delta}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ to
obtain a new simplicial presheaf on ${\mathscr{C}\mathrm{art}}$. Putting
everything together, this defines a functor

$\delta^{*}\circ(-)^{\underline{\Delta}_{e}}\colon\mathscr{H}_{\infty}\to\mathscr{H}_{\infty}\,,\qquad
F\longmapsto\delta^{*}\circ F^{\underline{\Delta}_{e}}\,.$ (6.5)

The collapse morphisms $\Delta_{e}^{k}\to*$ induce a natural transformation
$\underline{\Delta}_{e}\to*$ of functors
${\mathbbe{\Delta}}\to\mathscr{H}_{\infty}$ (this even consists of $I$-local
equivalences by Proposition 2.13). From this we obtain a natural
transformation

$\gamma\colon
1_{\mathscr{H}_{\infty}}\longrightarrow\delta^{*}\circ(-)^{\underline{\Delta}_{e}}\,.$
(6.6)

Now, let $R^{p/i}\colon\mathscr{H}_{\infty}\to\mathscr{H}_{\infty}$ be a
fibrant replacement functor for the projective (resp. injective) model
structure, with natural objectwise weak equivalence $r^{p/i}\colon
1_{\mathscr{H}_{\infty}}\overset{\sim}{\longrightarrow}R^{p/i}$. (Observe, in
particular, that a fibrant replacement functor $R^{p}$ for the projective
model structure can be obtained by postcomposition with a fibrant replacement
functor in ${\mathscr{S}\mathrm{et}_{\Delta}}$, i.e. we can use
$R^{p}(F)=R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ F$ for
$F\in\mathscr{H}_{\infty}$. An explicit model for an injective fibrant
replacement functor is given in Appendix A.) We define functors

${\mathrm{Cc}}^{p/i}\colon\mathscr{H}_{\infty}\to\mathscr{H}_{\infty}\,,\qquad
F\longmapsto R^{p/i}\big{(}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\big{)}\,,$ (6.7)

for the projective and for the injective model structure, respectively.
Further, we define the natural transformation

${\mathrm{cc}}^{p/i}_{|F}\colon F\xrightarrow{\gamma_{|F}}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\xrightarrow[\sim]{r^{p/i}}{\mathrm{Cc}}^{p/i}F\,.$
(6.8)

###### Proposition 6.9

The functors ${\mathrm{Cc}}^{p/i}$, together with the natural morphisms
${\mathrm{cc}}^{p/i}$ provide a functorial fibrant replacement in
$\mathscr{H}_{\infty}^{p/i\,I}$.

###### Proof.

First, we show that ${\mathrm{Cc}}^{p/i}F$ is indeed fibrant in the $I$-local
model structure $\mathscr{H}_{\infty}^{p/i\,I}$, for every
$F\in\mathscr{H}_{\infty}$. By construction, ${\mathrm{Cc}}^{p/i}F$ is a
fibrant object in $\mathscr{H}_{\infty}^{p/i}$. It thus remains to show that
it is $\mathbb{R}$-local. Given any $c\in{\mathscr{C}\mathrm{art}}$, we have

$\big{(}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\big{)}(c)=\delta^{*}\big{(}F(c\times\Delta_{e})\big{)}=\delta^{*}\big{(}F^{{\mathscr{Y}}_{c}}(\Delta_{e})\big{)}=S_{e}(F^{{\mathscr{Y}}_{c}})\,.$
(6.10)

By Lemma 6.1, the smooth homotopy equivalence ${\mathscr{Y}}_{c}\to*$ induces
a smooth homotopy equivalence $F\to F^{{\mathscr{Y}}_{c}}$, which is a weak
equivalence in $\mathscr{H}_{\infty}^{p/i\,I}$ by Corollary 3.33. Since
$r^{p/i}$ is a natural weak equivalence of functors valued in
$\mathscr{H}_{\infty}^{p/i}$, its component

$\delta^{*}\circ
F^{\underline{\Delta}_{e}}\xrightarrow[\sim]{r^{p/i}}R^{p/i}\big{(}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\big{)}$ (6.11)

is an objectwise weak equivalence. Consequently, for every
$c\in{\mathscr{C}\mathrm{art}}$, we have a commutative square

${\big{(}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\big{)}(c)}$${R^{p/i}\big{(}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\big{)}(c)}$${\big{(}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\big{)}(*)}$${R^{p/i}\big{(}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\big{)}(*)}$$\scriptstyle{r^{p/i}}$$\scriptstyle{\sim}$$\scriptstyle{r^{p/i}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$
(6.12)

whose vertical morphisms are induced from the collapse morphism $c\to*$. It
follows that ${\mathrm{Cc}}^{p/i}F$ is $\mathbb{R}$-local and hence a fibrant
object in $\mathscr{H}_{\infty}^{p/i\,I}$.

Finally, we need to show that the morphism $\gamma_{|F}\colon
F\longrightarrow\delta^{*}\circ F^{\underline{\Delta}_{e}}$, induced by the
collapse $\Delta_{e}\to*$, is a weak equivalence in
$\mathscr{H}_{\infty}^{p/i\,I}$. Since the functor
$S_{e}\colon\mathscr{H}_{\infty}^{p/i\,I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
preserves as well as reflects weak equivalences, that is equivalent to showing
that the induced morphism

$S_{e}(\gamma_{|F})\colon S_{e}F\longrightarrow S_{e}\big{(}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\big{)}$ (6.13)

is a weak equivalence of simplicial sets. More explicitly,
$S_{e}(\gamma_{|F})$ is the morphism

$\tilde{\delta}^{*}\big{(}F(\tilde{\Delta}_{e})\big{)}\longrightarrow\tilde{\delta}^{*}\big{(}\delta^{*}F(\tilde{\Delta}_{e}\times\Delta_{e})\big{)}$
(6.14)

induced by collapsing the extended simplices $\Delta_{e}$ without the tilde.
Note that we have only added the tilde in order to keep track of which
diagonal functor refers to which copy of $\Delta_{e}$. We have canonical
isomorphisms

$S_{e}\big{(}\delta^{*}\circ
F^{\underline{\Delta}_{e}}\big{)}\cong\tilde{\delta}^{*}\big{(}\delta^{*}F(\tilde{\Delta}_{e}\times\Delta_{e})\big{)}\cong\delta^{*}\big{(}(\tilde{\delta}^{*}\circ
F^{\tilde{\Delta}_{e}})(\Delta_{e})\big{)}$ (6.15)

and we know from the first part of this proof that the morphism

$\tilde{\delta}^{*}\big{(}F(\tilde{\Delta}_{e})\big{)}=(\tilde{\delta}^{*}\circ
F^{\tilde{\Delta}_{e}})(*)\longrightarrow(\tilde{\delta}^{*}\circ
F^{\tilde{\Delta}_{e}})(\Delta_{e}^{k})$ (6.16)

is a weak equivalence, for every $k\in\mathbb{N}_{0}$. This induces a
(levelwise) weak equivalence of bisimplicial sets

$\Delta^{0}\boxtimes\tilde{\delta}^{*}\big{(}F(\tilde{\Delta}_{e})\big{)}\overset{\sim}{\longrightarrow}(\tilde{\delta}^{*}\circ
F^{\tilde{\Delta}_{e}})(\Delta_{e})\,,$ (6.17)

which under $\delta^{*}$ maps to the morphism $S_{e}(\gamma_{|F})$. Since the
diagonal functor $\delta^{*}\colon
s{\mathscr{S}\mathrm{et}_{\Delta}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ is
homotopical, we obtain that $S_{e}(\gamma_{|F})$, and thus also $\gamma_{|F}$,
is indeed a weak equivalence. ∎

###### Corollary 6.18

Mapping spaces in $\mathscr{H}_{\infty}^{p/i\,I}$ can be computed (up to
isomorphism in $\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$) as the
simplicially enriched hom spaces

$\displaystyle\mathrm{Map}_{\mathscr{H}_{\infty}^{iI}}(F,G)$
$\displaystyle\simeq\underline{\mathscr{H}}_{\infty}\big{(}F,R^{i}(\delta^{*}\circ
G^{\underline{\Delta}_{e}})\big{)}\,,$ (6.19)
$\displaystyle\mathrm{Map}_{\mathscr{H}_{\infty}^{pI}}(F,G)$
$\displaystyle\simeq\underline{\mathscr{H}}_{\infty}\big{(}Q^{p}F,R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ\delta^{*}\circ
G^{\underline{\Delta}_{e}}\big{)}\,,$ (6.20)

where $Q^{p}$ is a cofibrant replacement functor for the projective model
structure $\mathscr{H}_{\infty}^{p}$.

###### Definition 6.21

Given an object $F\in\mathscr{H}_{\infty}$, we call ${\mathrm{Cc}}^{p/i}F$ its
_(projective/injective) derived concordance sheaf_. For
$G\in\mathscr{H}_{\infty}$ we refer to the spaces in (6.19) as the spaces of
_derived concordances of morphisms_ from $F$ to $G$.

We can apply these insights to describe the mapping spaces in the model
categories $\mathscr{H}_{\infty}^{p/i\,I}$. In particular, given any
$G\in\mathscr{H}_{\infty}$, part (3) of the following theorem shows that the
derived sections of the derived concordance sheaf of $G$ (in the sense of
Definition 6.21) on manifolds are represented by maps from the space
underlying $M$ to the smooth singular complex of $G$.

###### Theorem 6.22

There are natural isomorphisms in
$\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$ as follows:

  1. (1)

For $F,G\in\mathscr{H}_{\infty}$, we have

$\mathrm{Map}_{\mathscr{S}\mathrm{et}_{\Delta}}(S_{e}F,S_{e}G)\cong\mathrm{Map}_{\mathscr{H}_{\infty}^{p/i\,I}}(F,G)\,.$
(6.23)

  2. (2)

For $F,G\in\mathscr{H}_{\infty}$, we have

$\mathrm{Map}_{\Delta\mathscr{T}\mathrm{op}}(|S_{e}F|,|S_{e}G|)\cong\mathrm{Map}_{\mathscr{H}_{\infty}^{p/i\,I}}(F,G)\,.$
(6.24)

  3. (3)

For any manifold $M$ and $G\in\mathscr{H}_{\infty}$, we have

$\mathrm{Map}_{\Delta\mathscr{T}\mathrm{op}}(M,|S_{e}G|)\cong\mathrm{Map}_{\mathscr{H}_{\infty}^{p/i\,I}}(\underline{M},G)\,.$
(6.25)

###### Proof.

For claim (1), we consider the injective case first. The projective case then
follows immediately from Corollary A.24. Using that every object in
$\mathscr{H}_{\infty}^{iI}$ is cofibrant, we have the following isomorphisms
in $\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$:

$\displaystyle\mathrm{Map}_{\mathscr{S}\mathrm{et}_{\Delta}}(S_{e}F,S_{e}G)$
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(S_{e}F,R^{\mathscr{S}\mathrm{et}_{\Delta}}S_{e}G)$
(6.26)
$\displaystyle\cong\underline{\mathscr{H}}_{\infty}\big{(}F,R_{e}R^{\mathscr{S}\mathrm{et}_{\Delta}}S_{e}(G)\big{)}$
(6.27)
$\displaystyle\cong\underline{\mathscr{H}}_{\infty}\big{(}F,R_{e}R^{\mathscr{S}\mathrm{et}_{\Delta}}S_{e}{\mathrm{Cc}}^{i}(G)\big{)}$
(6.28)
$\displaystyle\cong\underline{\mathscr{H}}_{\infty}\big{(}F,{\mathrm{Cc}}^{i}(G)\big{)}$
(6.29) $\displaystyle\cong\mathrm{Map}_{\mathscr{H}_{\infty}^{iI}}(F,G)\,.$
(6.30)

The first isomorphism is merely the fact that
${\mathscr{S}\mathrm{et}_{\Delta}}$ is a simplicial model category in which
every object is cofibrant. In the second isomorphism, we use the fact that
$S_{e}\dashv R_{e}$ is a simplicial adjunction. To see the third isomorphism,
we use the weak equivalence ${\mathrm{cc}}^{i}_{|G}\colon
G\overset{\sim}{\longrightarrow}{\mathrm{Cc}}^{i}(G)$ from Proposition 6.9.
Since both $S_{e}$ and $R^{\mathscr{S}\mathrm{et}_{\Delta}}$ are homotopical
functors, and since $R_{e}$ preserves fibrant objects, the morphism
$R_{e}R^{\mathscr{S}\mathrm{et}_{\Delta}}S_{e}({\mathrm{cc}}^{i}_{|G})$ is a
weak equivalence between fibrant objects in $\mathscr{H}_{\infty}^{iI}$, which
is thus preserved by $\underline{\mathscr{H}}_{\infty}(F,-)$. The fourth
isomorphism stems from the fact that $S_{e}\dashv R_{e}$ is a Quillen
equivalence, so that the canonical natural transformation
$1_{\mathscr{H}_{\infty}}\longrightarrow
R_{e}R^{\mathscr{S}\mathrm{et}_{\Delta}}S_{e}$ is a weak equivalence in
$\mathscr{H}_{\infty}^{iI}$ on every cofibrant object—that is, it is a weak
equivalence on every object, since all objects in $\mathscr{H}_{\infty}^{iI}$
are cofibrant. Further, its component at the object ${\mathrm{Cc}}^{i}(G)$ is
a weak equivalence between fibrant objects in $\mathscr{H}_{\infty}^{iI}$,
which is again preserved by $\underline{\mathscr{H}}_{\infty}(F,-)$. The final
isomorphism directly follows from the insight that ${\mathrm{Cc}}^{i}$ is a
fibrant replacement functor for $\mathscr{H}_{\infty}^{iI}$ (Proposition 6.9).

Claim (2) then follows from the fact that every object in
${\Delta\mathscr{T}\mathrm{op}}$ is fibrant and that every object of
${\mathscr{S}\mathrm{et}_{\Delta}}$ is cofibrant: if
$K,L\in{\mathscr{S}\mathrm{et}_{\Delta}}$, then there are canonical
isomorphisms in $\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$

$\displaystyle\mathrm{Map}_{\Delta\mathscr{T}\mathrm{op}}\big{(}|K|,|L|\big{)}$
$\displaystyle\cong\underline{{\Delta\mathscr{T}\mathrm{op}}}\big{(}|K|,|L|\big{)}$
(6.31)
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}\big{(}K,\operatorname{Sing}|L|\big{)}$
(6.32)
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}\big{(}K,\operatorname{Sing}|R^{\mathscr{S}\mathrm{et}_{\Delta}}(L)|\big{)}$
(6.33)
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}\big{(}K,\operatorname{Sing}|R^{\mathscr{S}\mathrm{et}_{\Delta}}(L)|\big{)}$
(6.34)
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}\big{(}K,R^{\mathscr{S}\mathrm{et}_{\Delta}}(L)\big{)}$
(6.35)
$\displaystyle\cong\mathrm{Map}_{\mathscr{S}\mathrm{et}_{\Delta}}(K,L)\,.$
(6.36)

Claim (3) now follows from combining part (2) with Theorem 5.1. ∎

###### Remark 6.37

Part (3) of Theorem 6.22 is related to recent results by Berwick-Evans,
Boavida de Brito, and Pavlov from [3]: in that paper, the authors work with
the category $\widetilde{\mathscr{H}}_{\infty}$ of simplicial presheaves on
${\mathscr{M}\mathrm{fd}}$. Given
$\widetilde{G}\in\widetilde{\mathscr{H}}_{\infty}$, they consider the
concordance sheaf

${\mathrm{B}}\widetilde{G}=\delta^{*}\widetilde{G}^{\underline{\Delta}_{e}}\,,$
(6.38)

where the exponential is taken in $\widetilde{\mathscr{H}}_{\infty}$. The
inclusion functor
$\iota\colon{\mathscr{C}\mathrm{art}}\hookrightarrow{\mathscr{M}\mathrm{fd}}$
induces a homotopy right Kan extension
$\operatorname{hoRan}_{\iota}^{\prime}\colon\mathscr{H}_{\infty}\to\widetilde{\mathscr{H}}_{\infty}$
(see Appendix B for details). Explicitly, working with projective model
structures, we have

$\operatorname{hoRan}_{\iota}(F)(M)\cong\underline{\mathscr{H}}_{\infty}(Q^{\prime}M,F)$
(6.39)

for any $M\in{\mathscr{M}\mathrm{fd}}$, where
$Q^{\prime}\colon\mathscr{H}_{\infty}^{p}\to\mathscr{H}_{\infty}^{p}$ is the
cofibrant replacement functor introduced in Appendix A. The functor
$\operatorname{hoRan}_{\iota}^{\prime}\colon\mathscr{H}_{\infty}^{p\ell}\to\widetilde{\mathscr{H}}_{\infty}^{p\ell}$
is a right Quillen functor, where on the target side we consider the Čech
localisation with respect to open coverings—this is a consequence of [6, Prop.
3.16, Thm. 3.18]. It provides a concrete way of comparing sheaves on
${\mathscr{C}\mathrm{art}}$ to sheaves on ${\mathscr{M}\mathrm{fd}}$; the
functor

$\operatorname{hoRan}_{\iota}^{\prime}\colon\mathscr{H}_{\infty}^{p\ell}\to\widetilde{\mathscr{H}}_{\infty}^{p\ell}$

is a right Quillen equivalence by Theorem B.26.

We can now compare the derived concordance sheaves from Definition 6.21 to the
concordance sheaf construction (6.38) from [3]: let $F\in\mathscr{H}_{\infty}$
be any simplicial presheaf on ${\mathscr{C}\mathrm{art}}$. On the one hand, we
have

$\operatorname{hoRan}_{\iota}^{\prime}({\mathrm{Cc}}^{p}F)(M)=\underline{\mathscr{H}}_{\infty}(Q^{\prime}M,{\mathrm{Cc}}^{p}F)\,,$
(6.40)

and on the other hand, we have

$\big{(}{\mathrm{B}}\operatorname{hoRan}_{\iota}^{\prime}(F)\big{)}(M)=\delta^{*}\big{(}\underline{\mathscr{H}}_{\infty}(Q^{\prime}(M\times\Delta_{e}),F)\big{)}\,.$
(6.41)

Given that $\operatorname{hoRan}_{\iota}^{\prime}(F)$ is a sheaf on
${\mathscr{M}\mathrm{fd}}$, it now follows from [3, Thm. 1.1, Thm. 1.2] that
${\mathrm{B}}\operatorname{hoRan}_{\iota}^{\prime}(F)$ is a sheaf on
${\mathscr{M}\mathrm{fd}}$ as well. Then, the results in Appendix B imply that
$\operatorname{hoRan}_{\iota}^{\prime}({\mathrm{Cc}}^{p}F)$ and
${\mathrm{B}}\operatorname{hoRan}_{\iota}^{\prime}(F)$ are isomorphic in
$\mathrm{h}\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ if and only if their
images under the restriction functor
$\iota^{*}\colon\widetilde{\mathscr{H}}_{\infty}^{p\ell}\to\mathscr{H}_{\infty}^{p\ell}$
are isomorphic in $\mathrm{h}\mathscr{H}_{\infty}^{p\ell}$.

If $M=c$ is a cartesian space, we readily obtain a natural weak equivalence

$\operatorname{hoRan}_{\iota}^{\prime}({\mathrm{Cc}}^{p}F)(c)=\underline{\mathscr{H}}_{\infty}(Q^{\prime}c,{\mathrm{Cc}}^{p}F)\overset{\sim}{\longleftarrow}{\mathrm{Cc}}^{p}(F)(c)\,,$
(6.42)

and we further obtain natural weak equivalences

$\displaystyle\big{(}{\mathrm{B}}\operatorname{hoRan}_{\iota}^{\prime}(F)\big{)}(M)$
$\displaystyle=\delta^{*}\big{(}\underline{\mathscr{H}}_{\infty}(Q^{\prime}(c\times\Delta_{e}),F)\big{)}$
(6.43)
$\displaystyle\overset{\sim}{\longleftarrow}\delta^{*}\big{(}\underline{\mathscr{H}}_{\infty}(c\times\Delta_{e},F)\big{)}$
(6.44) $\displaystyle\cong\delta^{*}F(c\times\Delta_{e})$ (6.45)
$\displaystyle\overset{\sim}{\longrightarrow}{\mathrm{Cc}}^{p}(F)(c)\,.$
(6.46)

The first weak equivalence uses that $\Delta_{e}^{k}$ is a cartesian space,
for each $k\in\mathbb{N}_{0}$, so that $c\times\Delta_{e}^{k}$ is
representable in $\mathscr{H}_{\infty}$, and the last morphism arises from
postcomposing with a fibrant replacement functor in
${\mathscr{S}\mathrm{et}_{\Delta}}$. This establishes a natural zig-zag of
weak equivalences between
$\operatorname{hoRan}_{\iota}^{\prime}({\mathrm{Cc}}^{p}F)$ and
${\mathrm{B}}\operatorname{hoRan}_{\iota}^{\prime}(F)$. Note that the results
[3, Thm. 1.1, Thm. 1.2] enter crucially in the construction of this zig-zag
because we use the fact that
${\mathrm{B}}\operatorname{hoRan}_{\iota}^{\prime}(F)$ is a sheaf on
${\mathscr{M}\mathrm{fd}}$. $\triangleleft$

###### Remark 6.47

We outline an alternative proof of Theorem 6.22, based on Proposition 6.9 and
Theorem 2.37 (which we recall goes back to [14]). Let
$F,G\in\mathscr{H}_{\infty}$, and let
$Q\colon\mathscr{H}_{\infty}^{p}\to\mathscr{H}_{\infty}^{p}$ denote a
cofibrant replacement functor. In the homotopy category
$\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$ of spaces, we have natural
isomorphisms

$\displaystyle\mathrm{Map}_{\mathscr{H}_{\infty}^{pI}}(F,G)$
$\displaystyle\cong\underline{\mathscr{H}}_{\infty}(QF,{\mathrm{Cc}}^{p}G)$
(6.48)
$\displaystyle\cong\underline{\mathscr{H}}_{\infty}(QF,\tilde{\mathsf{c}}R^{\mathscr{S}\mathrm{et}_{\Delta}}S_{e}G)$
(6.49)
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(\underset{{\mathscr{C}\mathrm{art}}^{\mathrm{op}}}{{\mathrm{colim}}}\,QF,R^{\mathscr{S}\mathrm{et}_{\Delta}}S_{e}G)$
(6.50)
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(\underset{{\mathscr{C}\mathrm{art}}^{\mathrm{op}}}{{\mathrm{hocolim}}}\,F,R^{\mathscr{S}\mathrm{et}_{\Delta}}S_{e}G)$
(6.51)
$\displaystyle\cong\underline{{\mathscr{S}\mathrm{et}_{\Delta}}}(S_{e}F,R^{\mathscr{S}\mathrm{et}_{\Delta}}S_{e}G)$
(6.52)
$\displaystyle\cong\mathrm{Map}_{\mathscr{S}\mathrm{et}_{\Delta}}(S_{e}F,S_{e}G)\,.$
(6.53)

In the second isomorphism, we have used that ${\mathrm{Cc}}^{p}G$ is locally
constant and that
${\mathrm{Cc}}^{p}G(*)=R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ S_{e}G$. The
third isomorphism arises from the Quillen adjunction
${\mathrm{colim}}:\mathscr{H}_{\infty}^{p}\rightleftarrows{\mathscr{S}\mathrm{et}_{\Delta}}:\tilde{\mathsf{c}}$,
and the fourth isomorphism stems from fact that ${\mathrm{colim}}\circ Q$
models the homotopy colimit. Finally, the fifth isomorphism arises from our
observation at the end of Section 4 that $S_{e}F$ is a model for the homotopy
colimit of the diagram
$F\colon{\mathscr{C}\mathrm{art}}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$.
Parts (2) and (3) of Theorem 6.22 then follow as in our proof above.
$\triangleleft$

We can now give a direct proof of the relation between model categories from
Remark 3.34; that is, we identify the homotopy theory induced on
$\mathscr{H}_{\infty}$ by the smooth singular complex functor $S_{e}$:

###### Theorem 6.54

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
$W_{\mathscr{S}\mathrm{et}_{\Delta}}$. There is an identity of model
categories

$\mathscr{H}_{\infty}^{p/i\,I}=L_{S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})}\mathscr{H}_{\infty}^{p/i}\,.$
(6.55)

###### Proof.

We set

${\mathscr{M}}^{p/i}\coloneqq
L_{S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})}\mathscr{H}_{\infty}^{p/i}\,.$
(6.56)

The model categories $\mathscr{H}_{\infty}^{p/i\,I}$ and ${\mathscr{M}}^{p/i}$
have the same cofibrations, since they are left Bousfield localisations of the
same model category. (Here we use either the projective or the injective model
structure on _both_ sides.) By Theorem 2.17 it now suffices to show that they
have the same fibrant objects. Corollary 3.32 implies that $I\subset
S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})$; thus, any fibrant object in
${\mathscr{M}}^{p/i}$ is also fibrant in $\mathscr{H}_{\infty}^{p/i\,I}$.

To see that any fibrant object of $\mathscr{H}_{\infty}^{p/i\,I}$ is also
fibrant in ${\mathscr{M}}^{p/i}$, consider a fibrant object
$G\in\mathscr{H}_{\infty}^{p/i\,I}$ and a morphism $f\colon F_{0}\to F_{1}$ in
$S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})$. By Theorem 6.22, we have a
commutative diagram

${\mathrm{Map}_{\mathscr{H}_{\infty}^{p/i\,I}}(F_{1},G)}$${\mathrm{Map}_{\mathscr{H}_{\infty}^{p/i\,I}}(F_{0},G)}$${\mathrm{Map}_{\mathscr{S}\mathrm{et}_{\Delta}}(S_{e}F_{1},S_{e}G)}$${\mathrm{Map}_{\mathscr{S}\mathrm{et}_{\Delta}}(S_{e}F_{0},S_{e}G)}$$\scriptstyle{f^{*}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{(S_{e}f)^{*}}$
(6.57)

in $\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$. By assumption on $f$, the
morphism $S_{e}f$ is a weak equivalence in
${\mathscr{S}\mathrm{et}_{\Delta}}$. Hence, it induces a weak equivalence on
mapping spaces; that is, the bottom morphism in the diagram is an isomorphism
in $\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$. From that, it follows that
also the top morphism is an isomorphism in
$\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$, which implies that $G$ is
$S_{e}^{-1}(W_{\mathscr{S}\mathrm{et}_{\Delta}})$-local, and therefore fibrant
in ${\mathscr{M}}^{p/i}$. ∎

##  Appendix A An injective fibrant replacement of simplicial presheaves

From the definition of the projective and the injective model structure on
$\mathscr{H}_{\infty}$, it follows directly that there is a Quillen
equivalence

${\mathscr{H}_{\infty}^{p}}$${\mathscr{H}_{\infty}^{i}\,.}$$\scriptstyle{\perp}$
(A.1)

Both of the functors in this adjunction are the identity on
$\mathscr{H}_{\infty}$. Here, we will construct a Quillen equivalence in the
opposite direction, i.e.

${Q^{\prime}:\mathscr{H}_{\infty}^{i}}$${\mathscr{H}_{\infty}^{p}:R^{\prime}\,.}$$\scriptstyle{\perp}$
(A.2)

We start by defining the functor $Q^{\prime}$. Its construction is not
specific to simplicial presheaves on ${\mathscr{C}\mathrm{art}}$, but it works
for simplicial presheaves over any small category. Thus, let $\mathscr{C}$ be
a small category, let ${\mathscr{K}}_{\infty}$ denote the category of
simplicial presheaves on $\mathscr{C}$, and let
${\mathscr{Y}}\colon\mathscr{C}\to{\mathscr{K}}_{\infty}$ denote the Yoneda
embedding. We denote the projective and the injective model structures on
${\mathscr{K}}_{\infty}$ by ${\mathscr{K}}_{\infty}^{p}$ and by
${\mathscr{K}}_{\infty}^{i}$, respectively. The conventional two-sided
simplicial bar construction provides a functor [31]

$B_{\bullet}\big{(}(-),\mathscr{C},{\mathscr{Y}}\big{)}\colon{\mathscr{K}}_{\infty}\longrightarrow({\mathscr{K}}_{\infty})^{{\mathbbe{\Delta}}^{\mathrm{op}}}\,,\qquad
F\longmapsto B_{\bullet}(F,\mathscr{C},{\mathscr{Y}})\,.$ (A.3)

###### Lemma A.4

The functor $B_{\bullet}((-),\mathscr{C},{\mathscr{Y}})$ sends injective
cofibrations in ${\mathscr{K}}_{\infty}$ to injective cofibrations in
$({\mathscr{K}}_{\infty}^{p})^{{\mathbbe{\Delta}}^{\mathrm{op}}}$. That is, if
$f\colon F\to G$ is an objectwise cofibration of simplicial presheaves on
$\mathscr{C}$, then $B_{n}(f,\mathscr{C},{\mathscr{Y}})$ is a projective
cofibration of simplicial presheaves, for each $n\in\mathbb{N}_{0}$.

###### Proof.

We have that

$B_{n}(F,\mathscr{C},{\mathscr{Y}})=\coprod_{c_{0},\ldots,c_{n}\in\mathscr{C}}F(c_{n})\otimes\mathscr{C}(c_{n-1},c_{n})\otimes\cdots\otimes\mathscr{C}(c_{0},c_{1})\otimes{\mathscr{Y}}_{c_{0}}\,.$
(A.5)

Observing that ${\mathscr{Y}}_{c}\in{\mathscr{K}}_{\infty}^{p}$ is cofibrant
for every $c\in\mathscr{C}$ and recalling that ${\mathscr{K}}_{\infty}^{p}$ is
a simplicial model category, we see that, in each part of the coproduct, the
functor

$(-)\otimes\mathscr{C}(c_{n-1},c_{n})\otimes\cdots\otimes\mathscr{C}(c_{0},c_{1})\otimes{\mathscr{Y}}_{c_{0}}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to{\mathscr{K}}_{\infty}$
(A.6)

preserves cofibrations. Since the objects under the coproduct are each
cofibrant, these cofibrations induce a cofibration between the coproducts. ∎

Let $\mathscr{I}$ be a small category, ${\mathscr{V}}$ a symmetric monoidal
category, and ${\mathscr{M}}$ a model category category enriched, tensored and
cotensored over ${\mathscr{V}}$ (i.e. a model ${\mathscr{V}}$-category in the
terminology of [2]). Following the notation in [31], we let

$(-)\underset{\mathscr{I}}{\otimes}(-)\colon{\mathscr{V}}^{\mathscr{I}^{\mathrm{op}}}\times{\mathscr{M}}^{\mathscr{I}}\longrightarrow{\mathscr{M}}\qquad\text{and}\qquad\\{-,-\\}^{\mathscr{I}}\colon{\mathscr{V}}^{\mathscr{I}}\times{\mathscr{M}}^{\mathscr{I}}\longrightarrow{\mathscr{M}}$
(A.7)

denote the functor tensor product and the functor hom, respectively. Let
${\mathbbe{\Delta}}_{/(-)}\colon{\mathbbe{\Delta}}\to{\mathscr{C}\mathrm{at}}$
denote the functor that sends $[k]\in{\mathbbe{\Delta}}$ to the slice category
${\mathbbe{\Delta}}_{/[k]}$, and let $N$ denote the nerve functor. We now
define the functor

$Q^{\prime}\colon{\mathscr{K}}_{\infty}\longrightarrow{\mathscr{K}}_{\infty}\,,\qquad
Q^{\prime}\coloneqq
N({\mathbbe{\Delta}}_{/(-)})^{\mathrm{op}}\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{\otimes}B_{\bullet}\big{(}(-),\mathscr{C},{\mathscr{Y}}\big{)}\,.$
(A.8)

###### Lemma A.9

The functor
$Q^{\prime}\colon{\mathscr{K}}_{\infty}^{i}\to{\mathscr{K}}_{\infty}^{p}$
preserves cofibrations.

###### Proof.

We view the functor tensor product in the definition of $Q^{\prime}$ as

$(-)\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{\otimes}(-)\colon({\mathscr{S}\mathrm{et}_{\Delta}})^{\mathbbe{\Delta}}\times({\mathscr{K}}_{\infty}^{p})^{{\mathbbe{\Delta}}^{\mathrm{op}}}\longrightarrow{\mathscr{K}}_{\infty}^{p}\,.$
(A.10)

Further, the functor tensor product is a left Quillen bifunctor when endowing
the two source categories with any of the pairs of model structures
(projective, injective), (Reedy, Reedy), or (injective, projective) [31, Thms.
11.5.9, 14.3.1]. Further, the functor
$N({\mathbbe{\Delta}}_{/(-)})^{\mathrm{op}}\colon{\mathbbe{\Delta}}\longrightarrow{\mathscr{S}\mathrm{et}_{\Delta}}$
is projectively cofibrant [19, Prop. 14.8.8]. (Note that it is then also Reedy
cofibrant.) Consequently, the functor

$N({\mathbbe{\Delta}}_{/(-)})^{\mathrm{op}}\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{\otimes}(-)\colon({\mathscr{K}}_{\infty}^{p})^{{\mathbbe{\Delta}}^{\mathrm{op}}}\longrightarrow{\mathscr{K}}_{\infty}^{p}$
(A.11)

is left Quillen with respect to the injective model structure on
$({\mathscr{K}}_{\infty}^{p})^{{\mathbbe{\Delta}}^{\mathrm{op}}}$ (and hence
also with respect to the Reedy model structure). We can write $Q^{\prime}$ as
the composition

${{\mathscr{K}}^{i}_{\infty}}$${\big{(}({\mathscr{K}}_{\infty}^{p})^{{\mathbbe{\Delta}}^{\mathrm{op}}}\big{)}_{inj}}$${{\mathscr{K}}_{\infty}^{p}\,.}$$\scriptstyle{B_{\bullet}((-),\mathscr{C},{\mathscr{Y}})}$$\scriptstyle{N({\mathbbe{\Delta}}_{/(-)})^{\mathrm{op}}\otimes_{{\mathbbe{\Delta}}^{\mathrm{op}}}(-)}$
(A.12)

The model category of simplicial diagrams in the middle carries the injective
model structure. We have shown in Lemma A.4 that the first functor preserves
cofibrations, and it follows from our arguments above that also the second
functor preserves cofibrations. ∎

Next, we are going to employ the Bousfield-Kan map to show that $Q^{\prime}$
can also be seen as a cofibrant replacement functor on
${\mathscr{K}}_{\infty}^{p}$. The Bousfield-Kan map is a morphism

$bk\colon
N({\mathbbe{\Delta}}_{/(-)})^{\mathrm{op}}\longrightarrow\Delta^{\bullet}$
(A.13)

of cosimplicial simplicial sets. We will use the following two statements:

###### Proposition A.14

_[ 19, Prop. 18.7.2]_ The Bousfield-Kan map is a Reedy weak equivalence
between Reedy cofibrant cosimplicial simplicial sets.

###### Proposition A.15

For any $F\in{\mathscr{K}}_{\infty}$, the simplicial object in
${\mathscr{K}}_{\infty}$ given as $B_{\bullet}(F,\mathscr{C},{\mathscr{Y}})$
is Reedy cofibrant.

###### Proof.

This follows directly from [31, Rmk. 5.2.2], applied to the functor
${\mathscr{Y}}\colon\mathscr{C}\to{\mathscr{K}}_{\infty}^{p}$. ∎

We now consider the induced natural transformation

$bk\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{\otimes}B_{\bullet}\big{(}(-),\mathscr{C},{\mathscr{Y}}\big{)}\colon
Q^{\prime}\longrightarrow\Delta^{\bullet}\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{\otimes}B_{\bullet}\big{(}(-),\mathscr{C},{\mathscr{Y}}\big{)}=B\big{(}(-),\mathscr{C},{\mathscr{Y}}\big{)}\,.$
(A.16)

By Proposition A.15, the functor

$(-)\underset{{\mathbbe{\Delta}}^{\mathrm{op}}}{\otimes}B_{\bullet}\big{(}F,\mathscr{C},{\mathscr{Y}}\big{)}\colon\big{(}({\mathscr{S}\mathrm{et}_{\Delta}})^{\mathbbe{\Delta}}\big{)}_{Reedy}\longrightarrow{\mathscr{K}}_{\infty}^{p}$
(A.17)

is a left Quillen functor for every $F\in{\mathscr{K}}$. It then follows from
Proposition A.14 and the arguments in the proof of Proposition A.9 that the
natural transformation (A.16) is a natural weak equivalence of functors
${\mathscr{K}}_{\infty}\to{\mathscr{K}}_{\infty}^{p}$. Finally, we use that
the functor $B((-),\mathscr{C},{\mathscr{Y}})$ agrees with Dugger’s cofibrant
replacement functor $Q^{p}$ for ${\mathscr{K}}_{\infty}^{p}$ from [16]. In
particular, it comes with a natural weak equivalence $q^{p}\colon Q^{p}\to
1_{{\mathscr{K}}_{\infty}}$. Composing $q^{p}$ with the morphism (A.16), we
obtain a natural weak equivalence $q^{\prime}\colon Q^{\prime}\to
1_{{\mathscr{K}}_{\infty}}$. Putting everything together, we have proven

###### Proposition A.18

The functor $Q^{\prime}$ from (A.8), together with the natural weak
equivalence $q^{\prime}$ provide a cofibrant replacement functor for
${\mathscr{K}}_{\infty}^{p}$. In particular, $Q^{\prime}$ preserves objectwise
weak equivalences.

Finally, we observe that $Q^{\prime}$ has a right adjoint, which is explicitly
given by

$R^{\prime}\colon{\mathscr{K}}_{\infty}\longrightarrow{\mathscr{K}}_{\infty}\,,\qquad
R^{\prime}(G)=\big{\\{}N({\mathbbe{\Delta}}_{/(-)})^{\mathrm{op}},C^{\bullet}(G,\mathscr{C}^{\mathrm{op}},{\mathscr{Y}}^{(-)})\big{\\}}^{\mathbbe{\Delta}}\,,$
(A.19)

where
${\mathscr{Y}}^{(-)}\colon\mathscr{C}^{\mathrm{op}}\to{\mathscr{C}\mathrm{at}}(\mathscr{C},{\mathscr{S}\mathrm{et}_{\Delta}})$
denotes the co-Yoneda embedding of $\mathscr{C}$.

###### Theorem A.20

The functors $Q^{\prime}$ and $R^{\prime}$ satisfy the following properties:

  1. (1)

The adjunction $Q^{\prime}\dashv R^{\prime}$ is a Quillen equivalence

${Q^{\prime}:\mathscr{H}_{\infty}^{i}}$${\mathscr{H}_{\infty}^{p}:R^{\prime}\,.}$$\scriptstyle{\perp}$
(A.21)

  2. (2)

There is a natural transformation $r^{\prime}\colon
1_{{\mathscr{K}}_{\infty}}\to R^{\prime}$ such that $r^{\prime}_{|G}\colon
G\to R^{\prime}G$ is a weak equivalence in ${\mathscr{K}}_{\infty}^{i}$ for
every projectively fibrant $G\in{\mathscr{K}}_{\infty}$.

  3. (3)

Let $R^{\mathscr{S}\mathrm{et}_{\Delta}}$ be a fibrant replacement functor for
simplicial sets. Then, $G\mapsto
R^{\prime}(R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ G)$ is a fibrant
replacement functor on ${\mathscr{K}}_{\infty}^{i}$.

###### Proof.

Ad (1): Proposition A.18, together with the observation that $Q^{\prime}$ is a
left adjoint, readily implies that $Q^{\prime}$ is a left Quillen functor. The
fact that this is a Quillen equivalence follows from the existence of the
natural weak equivalence $q^{\prime}\colon
Q^{\prime}\overset{\sim}{\longrightarrow}1_{{\mathscr{K}}_{\infty}}$.
Formally, this implies that the composition of $Q^{\prime}$ by the left
Quillen equivalence
$1_{{\mathscr{K}}_{\infty}}\colon{\mathscr{K}}_{\infty}^{p}\to{\mathscr{K}}_{\infty}^{i}$
is weakly equivalent to the identity functor on ${\mathscr{K}}_{\infty}$.
Thus, the statement follows from the two-out-of-three property of Quillen
equivalences [21] and Corollary D.12.

Ad (2): Let

$\tau_{F,G}\colon{\mathscr{K}}_{\infty}(Q^{\prime}F,G)\longrightarrow{\mathscr{K}}_{\infty}(F,R^{\prime}G)$
(A.22)

be the natural isomorphism that establishes the adjunction $Q^{\prime}\dashv
R^{\prime}$. We define $r^{\prime}\colon 1_{{\mathscr{K}}_{\infty}}\to
R^{\prime}$ to be the image under $\tau$ of the natural transformation
$q^{\prime}$.

Let $G\in{\mathscr{K}}_{\infty}$ be a projectively fibrant object. Since every
object in ${\mathscr{K}}_{\infty}^{i}$ is cofibrant and since
$Q^{\prime}\dashv R^{\prime}$ is a Quillen equivalence, a morphism
$\varphi\colon Q^{\prime}F\to G$ is a weak equivalence (in
${\mathscr{K}}_{\infty}^{p}$) if and only if $\tau_{F,G}(\varphi)\colon F\to
R^{\prime}G$ is a weak equivalence (in ${\mathscr{K}}_{\infty}^{i}$).

Now consider the weak equivalence $q^{\prime}_{|G}\colon Q^{\prime}G\to G$,
for $G\in{\mathscr{K}}_{\infty}^{p}$ fibrant. This is a weak equivalence of
the form considered above; thus, the morphism $r^{\prime}_{|G}\colon G\to
R^{\prime}G$ is a weak equivalence whenever $G$ is projectively fibrant.

Ad (3): Let $R^{\mathscr{S}\mathrm{et}_{\Delta}}$ be a fibrant replacement
functor in ${\mathscr{S}\mathrm{et}_{\Delta}}$, with associated natural weak
equivalence $r^{\mathscr{S}\mathrm{et}_{\Delta}}\colon
1_{\mathscr{S}\mathrm{et}_{\Delta}}\overset{\sim}{\longrightarrow}R^{\mathscr{S}\mathrm{et}_{\Delta}}$.
Let $G\in{\mathscr{K}}_{\infty}$ be arbitrary and consider the composition

${G}$${R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ
G}$${R^{\prime}(R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ
G)\,.}$$\scriptstyle{r^{\mathscr{S}\mathrm{et}_{\Delta}}\circ
1_{G}}$$\scriptstyle{r^{\prime}{|R^{\mathscr{S}\mathrm{et}_{\Delta}}G}}$
(A.23)

The first morphism is an objectwise weak equivalence by definition. Since
$R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ G$ is projectively fibrant, the
second morphism is a weak equivalence as well by part (2). ∎

###### Corollary A.24

Let $F,G$ be any two objects in ${\mathscr{K}}_{\infty}$. There is a canonical
isomorphism between (the homotopy types of) the mapping spaces of the
projective and the injective model structures

$\mathrm{Map}_{{\mathscr{K}}_{\infty}^{p}}(F,G)\cong\mathrm{Map}_{{\mathscr{K}}_{\infty}^{i}}(F,G)$
(A.25)

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

###### Proof.

Since both ${\mathscr{K}}_{\infty}^{p}$ and ${\mathscr{K}}_{\infty}^{i}$ are
simplicial model categories, we have the following isomorphisms in
$\mathrm{h}{\mathscr{S}\mathrm{et}_{\Delta}}$:

$\displaystyle\mathrm{Map}_{{\mathscr{K}}_{\infty}^{p}}(F,G)$
$\displaystyle\cong\underline{{\mathscr{K}}}_{\infty}(Q^{\prime}F,R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ
G)$ (A.26)
$\displaystyle\cong\underline{{\mathscr{K}}}_{\infty}(F,R^{\prime}R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ
G)$ (A.27)
$\displaystyle\cong\mathrm{Map}_{{\mathscr{K}}_{\infty}^{i}}(F,G)\,.$ (A.28)

The first isomorphism arises from the fact that ${\mathscr{K}}_{\infty}^{p}$
is simplicial and that $Q^{\prime}$ is a cofibrant replacement functor in
${\mathscr{K}}_{\infty}^{p}$ (Proposition A.18). The second isomorphism is the
adjointness $Q^{\prime}\dashv R^{\prime}$, and the third isomorphism stems
from the facts that every object in ${\mathscr{K}}_{\infty}^{i}$ is cofibrant
and that the functor $G\mapsto
R^{\prime}(R^{\mathscr{S}\mathrm{et}_{\Delta}}\circ G)$ is a fibrant
replacement functor in ${\mathscr{K}}_{\infty}^{i}$ (Theorem A.20). ∎

##  Appendix B Sheaves on manifolds and sheaves on cartesian spaces

This appendix is devoted to the comparison of two Čech localisations: on the
one hand, we consider the localisation $\mathscr{H}_{\infty}^{p\ell}$ of the
projective model structure $\mathscr{H}_{\infty}^{p}$ of simplicial presheaves
on cartesian spaces at the differentiably good open coverings (see Section
2.1). On the other hand, we have the Čech localisation
$\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ of the projective model structure
$\widetilde{\mathscr{H}}_{\infty}^{p}$ of simplicial presheaves on manifolds
at the open coverings. (The same arguments also apply to the localisation of
$\widetilde{\mathscr{H}}_{\infty}^{p}$ at the surjective submersions.)

We will compare these localisations by means of the functors

$\displaystyle\operatorname{hoRan}_{\iota}\colon\mathscr{H}_{\infty}\longrightarrow\widetilde{\mathscr{H}}_{\infty}\,,\qquad
F\mapsto\underline{\mathscr{H}}_{\infty}\big{(}Q(-),F\big{)}\,,$ (B.1)
$\displaystyle\operatorname{hoRan}^{\prime}_{\iota}\colon\mathscr{H}_{\infty}\longrightarrow\widetilde{\mathscr{H}}_{\infty}\,,\qquad
F\mapsto\underline{\mathscr{H}}_{\infty}\big{(}Q^{\prime}(-),F\big{)}\,,$
(B.2)

where
$\iota\colon{\mathscr{C}\mathrm{art}}\hookrightarrow{\mathscr{M}\mathrm{fd}}$
is the canonical inclusion, and where
$Q\colon\mathscr{H}_{\infty}^{p}\to\mathscr{H}_{\infty}^{p}$ is Dugger’s
cofibrant replacement functor for the projective model structure on simplicial
presheaves (see Appendix A). Further, $Q^{\prime}\dashv R^{\prime}$ is the
Quillen equivalence from Theorem A.20. The natural transformation (A.16)
induces a natural transformation
$\eta\colon\operatorname{hoRan}_{\iota}\longrightarrow\operatorname{hoRan}_{\iota}^{\prime}$
whose component $\eta_{|F}$ on every fibrant object
$F\in\mathscr{H}_{\infty}^{p}$ is a projective weak equivalence.

The functor
$\operatorname{hoRan}_{\iota}\colon\mathscr{H}_{\infty}^{p}\to\widetilde{\mathscr{H}}_{\infty}^{p}$
is a right Quillen functor with left adjoint $Q\circ\iota^{*}$, and this
Quillen adjunction descends to a Quillen adjunction on Čech localisations,

${Q\circ\iota^{*}:\widetilde{\mathscr{H}}_{\infty}^{p\ell}}$${\mathscr{H}_{\infty}^{p\ell}:\operatorname{hoRan}_{\iota}\,,}$$\scriptstyle{\perp}$
(B.3)

which follows from [6, Thm. 3.18]. Analogously,
$\operatorname{hoRan}_{\iota}^{\prime}\colon\mathscr{H}_{\infty}^{p}\to\widetilde{\mathscr{H}}_{\infty}^{p}$
is a right Quillen functor (since $Q^{\prime}$ is homotopical and valued in
projectively cofibrant simplicial presheaves). Further, the natural
transformation
$\eta\colon\operatorname{hoRan}_{\iota}\longrightarrow\operatorname{hoRan}_{\iota}^{\prime}$
establishes that $\operatorname{hoRan}_{\iota}^{\prime}$ maps local objects in
$\mathscr{H}_{\infty}^{p\ell}$ to local objects in
$\widetilde{\mathscr{H}}_{\infty}^{p\ell}$, and hence that we also have a
Quillen adjunction

${Q^{\prime}\circ\iota^{*}:\widetilde{\mathscr{H}}_{\infty}^{p\ell}}$${\mathscr{H}_{\infty}^{p\ell}:\operatorname{hoRan}_{\iota}^{\prime}\,.}$$\scriptstyle{\perp}$
(B.4)

Note that we can also write the right adjoint as

$\operatorname{hoRan}_{\iota}^{\prime}\cong\iota_{*}\circ R^{\prime}\,,$ (B.5)

with $R^{\prime}\colon\mathscr{H}_{\infty}\to\mathscr{H}_{\infty}$ defined as
in (A.19), and where $\iota_{*}$ is the right Kan extension along $\iota$.

###### Lemma B.6

The derived counit of the Quillen adjunction
$Q^{\prime}\circ\iota^{*}\dashv\operatorname{hoRan}_{\iota}^{\prime}$ is a
projective weak equivalence on every fibrant $F\in\mathscr{H}_{\infty}^{p}$.

###### Proof.

Let
$\widetilde{Q}\colon\widetilde{\mathscr{H}}_{\infty}^{p}\to\widetilde{\mathscr{H}}_{\infty}^{p}$
be a cofibrant replacement functor with associated natural weak equivalence
$\widetilde{q}\colon\widetilde{Q}\overset{\sim}{\longrightarrow}1_{\widetilde{\mathscr{H}}_{\infty}}$.
Consider a fibrant object $F\in\mathscr{H}_{\infty}^{p}$ and the diagram

${(Q^{\prime}\circ\iota^{*})\circ\widetilde{Q}\circ(\iota_{*}\circ
R^{\prime})(F)}$${(Q^{\prime}\circ\iota^{*})\circ(\iota_{*}\circ
R^{\prime})(F)}$${(Q^{\prime}\circ
R^{\prime})(F)}$${F}$$\scriptstyle{(Q^{\prime}\circ\iota^{*})\widetilde{q}_{|\iota_{*}\circ
R^{\prime}(F)}}$$\scriptstyle{\cong}$ (B.7)

The left-hand vertical morphism is a weak equivalence in
$\mathscr{H}_{\infty}^{p}$ because $\widetilde{q}_{|\iota_{*}\circ
R^{\prime}(F)}$ is a projective weak equivalence and
$\iota^{*}\colon\widetilde{\mathscr{H}}_{\infty}^{p}\to\mathscr{H}_{\infty}^{p}$
is homotopical, as is $Q^{\prime}$. The left-hand bottom morphism is the
counit of the adjunction $\iota^{*}\dashv\iota_{*}$, which is an isomorphism
by the Yoneda Lemma. The right-hand bottom morphism is a weak equivalence by
Theorem A.20 and the fact that every object in $\mathscr{H}_{\infty}^{i}$ is
cofibrant. ∎

It follows that the total derived functor
$\mathbb{R}\operatorname{hoRan}_{\iota}^{\prime}\colon\mathrm{h}\mathscr{H}_{\infty}^{p}\to\mathrm{h}\widetilde{\mathscr{H}}_{\infty}^{p}$
is fully faithful, and hence that also
$\mathbb{R}\operatorname{hoRan}_{\iota}^{\prime}\colon\mathrm{h}\mathscr{H}_{\infty}^{p\ell}\to\mathrm{h}\widetilde{\mathscr{H}}_{\infty}^{p\ell}$
is fully faithful.

Next, we would like to show that
$\mathbb{R}\operatorname{hoRan}_{\iota}^{\prime}$ is also essentially
surjective. We start by recalling that, for any open covering
${\mathscr{U}}=\\{U_{a}\\}_{a\in A}$ of a manifold $M$, the induced Čech nerve
${\check{C}}{\mathscr{U}}\to\underline{M}$ is a weak equivalence in both
$\mathscr{H}_{\infty}^{p\ell}$ and $\widetilde{\mathscr{H}}_{\infty}^{p\ell}$
(this follows directly from [10, Cor. A.3]). If ${\mathscr{U}}$ is even a
differentiably good open covering of $M$ (i.e. each finite intersection of
patches is either empty or a cartesian space), then ${\check{C}}{\mathscr{U}}$
is levelwise a coproduct of representables in $\mathscr{H}_{\infty}$, and
hence it is cofibrant in $\mathscr{H}_{\infty}^{p\ell}$.

Suppose that $G\in\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ is fibrant. Given
a manifold $M$ with a differentiably good open covering ${\mathscr{U}}$, we
have weak equivalences

$G(M)\cong\underline{\widetilde{\mathscr{H}}}_{\infty}(\underline{M},G)\overset{\sim}{\longrightarrow}\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{U}},G)\cong\underline{\mathscr{H}}_{\infty}({\check{C}}{\mathscr{U}},\iota^{*}G)\,.$
(B.8)

In this sense, the value of a sheaf $G$ on manifolds is determined by its
values on cartesian spaces. However, the above weak equivalence is by no means
functorial in $M$, so to make this statement precise, we need to improve on
(B.8).

For $M\in{\mathscr{M}\mathrm{fd}}$, let ${\mathrm{Cov}}(M)$ denote the
following category: an object of ${\mathrm{Cov}}(M)$ is an open covering
${\mathscr{U}}=\\{U_{a}\\}_{a\in A}$ of $M$, and given another open covering
${\mathscr{V}}$ of $M$ there is a unique morphism
${\mathscr{V}}\to{\mathscr{U}}$ precisely if ${\mathscr{V}}$ refines
${\mathscr{U}}$. Let ${\mathrm{GCov}}(M)$ denote the full subcategory of
${\mathrm{Cov}}(M)$ on the differentiably good open coverings. Note that
${\mathrm{Cov}}(M)$ is cofiltered since every pair
$({\mathscr{U}},{\mathscr{V}})$ of open coverings has a common refinement
${\mathscr{U}}\times_{M}{\mathscr{V}}$. Moreover, any open covering of $M$ has
a differentiably good refinement [17, App. A], which implies that
${\mathrm{GCov}}(M)$ is cofiltered and that the canonical inclusion
$j_{M}\colon{\mathrm{GCov}}(M)\hookrightarrow{\mathrm{Cov}}(M)$ is final,
since for any ${\mathscr{U}}\in{\mathrm{Cov}}(M)$ the slice category
$j/{\mathscr{U}}$ is cofiltered as well.

Given $G\in\widetilde{\mathscr{H}}_{\infty}$, we consider the diagrams in
simplicial sets

${\mathrm{Cov}}(M)^{\mathrm{op}}\longrightarrow{\mathscr{S}\mathrm{et}_{\Delta}}\,,\qquad{\mathscr{U}}\longmapsto\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{U}},G)\quad\text{and}\quad{\mathscr{U}}\longmapsto
G(M)\,.$ (B.9)

Since ${\mathrm{Cov}}(M)^{\mathrm{op}}$ is filtered and
${\mathscr{S}\mathrm{et}_{\Delta}}$ is combinatorial, the ordinary colimits of
these diagrams model their homotopy colimits [15, Prop. 7.3], so that we
obtain a morphism

$G(M)\longrightarrow\underset{{\mathscr{U}}\in{\mathrm{Cov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{U}},G)\Big{)}\simeq\underset{{\mathscr{U}}\in{\mathrm{Cov}}(M)^{\mathrm{op}}}{{\mathrm{hocolim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{U}},G)\Big{)}\,.$
(B.10)

(The right-hand side of (B.10) is Grothendieck’s Plus Construction, applied to
$M$ and $G$.) If $G\in\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ is fibrant,
then the morphism in (B.10) is a weak equivalence, for each manifold
$M\in{\mathscr{M}\mathrm{fd}}$. Since
$j_{M}^{\mathrm{op}}\colon{\mathrm{GCov}}(M)^{\mathrm{op}}\hookrightarrow{\mathrm{Cov}}(M)^{\mathrm{op}}$
is cofinal, we further obtain a canonical isomorphism

$\underset{{\mathscr{U}}\in{\mathrm{Cov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{U}},G)\Big{)}\cong\underset{{\mathscr{U}}\in{\mathrm{GCov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{U}},G)\Big{)}\,.$
(B.11)

If ${\mathscr{U}}\in{\mathrm{GCov}}(M)$, then ${\check{C}}{\mathscr{U}}$ is
levelwise a coproduct of presheaves represented by cartesian spaces. Hence,
there is a further canonical isomorphism

$\underset{{\mathscr{U}}\in{\mathrm{GCov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{U}},G)\Big{)}\cong\underset{{\mathscr{U}}\in{\mathrm{GCov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\mathscr{H}}_{\infty}({\check{C}}{\mathscr{U}},\iota^{*}G)\Big{)}\,,$
(B.12)

where
$\iota\colon{\mathscr{C}\mathrm{art}}\hookrightarrow{\mathscr{M}\mathrm{fd}}$
is the canonical inclusion functor. Thus, for any
$G\in\widetilde{\mathscr{H}}_{\infty}$ and any $M\in{\mathscr{M}\mathrm{fd}}$
we obtain a morphism

$\gamma_{G,M}\colon
G(M)\longrightarrow\underset{{\mathscr{U}}\in{\mathrm{GCov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\mathscr{H}}_{\infty}({\check{C}}{\mathscr{U}},\iota^{*}G)\Big{)}$
(B.13)

which is natural in $G$ and which is a weak equivalence whenever
$G\in\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ is fibrant.

Next we show that $\gamma_{G,M}$ is also natural in $M$. Let $f\colon M\to N$
be a smooth map. This induces a functor
$f^{*}\colon{\mathrm{Cov}}(N)\to{\mathrm{Cov}}(M)$, acting via
$f^{*}(\\{U_{a}\\}_{a\in A})=\\{f^{-1}(U_{a})\\}_{a\in A}$. This establishes
the assignment $M\mapsto{\mathrm{Cov}}(M)$ as a (strict) functor
${\mathrm{Cov}}\colon{\mathscr{M}\mathrm{fd}}^{\mathrm{op}}\to{\mathscr{C}\mathrm{at}}$.
Given an open covering ${\mathscr{U}}$ of $N$, the map $f$ induces a canonical
morphism
$\widehat{f}\colon{\check{C}}(f^{*}{\mathscr{U}})\to{\check{C}}{\mathscr{U}}$
in $\widetilde{\mathscr{H}}_{\infty}$. This induces a natural transformation

${{{\mathrm{Cov}}(N)^{\mathrm{op}}}}$${{{\mathrm{Cov}}(M)^{\mathrm{op}}}}$${{\mathscr{S}\mathrm{et}_{\Delta}}}$$\scriptstyle{f^{*}}$$\scriptstyle{\widetilde{\mathscr{H}}_{\infty}({\check{C}}(-),G)}$$\scriptstyle{\widetilde{\mathscr{H}}_{\infty}({\check{C}}(-),G)}$$\scriptstyle{\varphi}$
(B.14)

Further, recall that for any composition of functors
$\mathscr{I}\overset{F}{\longrightarrow}\mathscr{J}\overset{D}{\longrightarrow}\mathscr{C}$
there is a canonical morphism ${\mathrm{colim}}(D\circ
F)\to{\mathrm{colim}}(D)$. Thus, we have a canonical morphism

$\underset{{\mathscr{U}}\in{\mathrm{Cov}}(N)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}\big{(}{\check{C}}(f^{*}{\mathscr{U}}),G\big{)}\Big{)}\longrightarrow\underset{{\mathscr{U}}\in{\mathrm{Cov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{U}},G)\Big{)}\,.$
(B.15)

Combining this morphism with the natural transformation $\varphi$ yields a
morphism

$\underset{{\mathscr{U}}\in{\mathrm{Cov}}(N)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}\big{(}{\check{C}}{\mathscr{U}},G\big{)}\Big{)}\longrightarrow\underset{{\mathscr{V}}\in{\mathrm{Cov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{V}},G)\Big{)}$
(B.16)

in ${\mathscr{S}\mathrm{et}_{\Delta}}$ which is compatible with composition of
smooth maps of manifolds. Finally, we use the isomorphisms (B.12) to define
the top morphism in the diagram

${\underset{{\mathscr{U}}\in{\mathrm{GCov}}(N)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\mathscr{H}}_{\infty}\big{(}{\check{C}}{\mathscr{U}},\iota^{*}G\big{)}\Big{)}}$${\underset{{\mathscr{V}}\in{\mathrm{GCov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\mathscr{H}}_{\infty}({\check{C}}{\mathscr{V}},\iota^{*}G)\Big{)}}$${\underset{{\mathscr{U}}\in{\mathrm{Cov}}(N)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}\big{(}{\check{C}}{\mathscr{U}},G\big{)}\Big{)}}$${\underset{{\mathscr{V}}\in{\mathrm{Cov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\widetilde{\mathscr{H}}}_{\infty}({\check{C}}{\mathscr{V}},G)\Big{)}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$
(B.17)

It follows that the assignment

$M\longmapsto\underset{{\mathscr{V}}\in{\mathrm{GCov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\mathscr{H}}_{\infty}({\check{C}}{\mathscr{V}},\iota^{*}G)\Big{)}$
(B.18)

defines a functor
${\mathscr{M}\mathrm{fd}}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$.
That is, we obtain a functor

$\operatorname{Pl}\colon\widetilde{\mathscr{H}}_{\infty}\to\widetilde{\mathscr{H}}_{\infty}\,,\qquad\operatorname{Pl}G(M)=\underset{{\mathscr{V}}\in{\mathrm{GCov}}(M)^{\mathrm{op}}}{{\mathrm{colim}}^{\mathscr{S}\mathrm{et}_{\Delta}}}\Big{(}\underline{\mathscr{H}}_{\infty}({\check{C}}{\mathscr{V}},\iota^{*}G)\Big{)}\,,$
(B.19)

coming with a natural transformation $\gamma\colon
1_{\widetilde{\mathscr{H}}_{\infty}}\to\operatorname{Pl}$ which is an
objectwise weak equivalence on every fibrant
$G\in\widetilde{\mathscr{H}}_{\infty}^{p\ell}$. In particular, we have shown:

###### Proposition B.20

Let $G,G^{\prime}\in\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ be fibrant. A
morphism $g\colon G\to G^{\prime}$ is a weak equivalence in
$\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ if and only if
$\iota^{*}g\colon\iota^{*}G\to\iota^{*}G^{\prime}$ is a weak equivalence in
$\mathscr{H}_{\infty}^{p\ell}$.

Note that for $G,G^{\prime}\in\widetilde{\mathscr{H}}_{\infty}^{p\ell}$
fibrant, a morphism $g\colon G\to G^{\prime}$ is a weak equivalence in
$\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ precisely if it is an objectwise
equivalence (since both $G$ and $G^{\prime}$ are local objects), and
analogously $\iota^{*}g\colon\iota^{*}G\to\iota^{*}G^{\prime}$ is a weak
equivalence if and only if it is an objectwise equivalence (since
$\iota^{*}\colon\widetilde{\mathscr{H}}_{\infty}^{p\ell}\to\mathscr{H}_{\infty}^{p\ell}$
preserves fibrant objects).

###### Lemma B.21

The functor
$\mathbb{R}\operatorname{hoRan}_{\iota}^{\prime}\colon\mathrm{h}\mathscr{H}_{\infty}^{p\ell}\longrightarrow\mathrm{h}\widetilde{\mathscr{H}}_{\infty}^{p\ell}$
is essentially surjective.

###### Proof.

Let $G\in\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ be fibrant. We show that
there is a zig-zag of weak equivalences in
$\widetilde{\mathscr{H}}_{\infty}^{p\ell}$ linking $G$ to
$\operatorname{hoRan}_{\iota}^{\prime}\circ\iota^{*}(G)$. First, since $G$ and
$\operatorname{hoRan}_{\iota}^{\prime}\circ\iota^{*}(G)$ are both sheaves on
${\mathscr{M}\mathrm{fd}}$, there are local weak equivalences

$G\overset{\sim}{\longrightarrow}\operatorname{Pl}G\qquad\text{and}\qquad\operatorname{hoRan}_{\iota}^{\prime}\circ\iota^{*}(G)\overset{\sim}{\longrightarrow}\operatorname{Pl}\big{(}\operatorname{hoRan}_{\iota}^{\prime}\circ\iota^{*}(G)\big{)}\,,$
(B.22)

where
$\operatorname{Pl}\colon\widetilde{\mathscr{H}}_{\infty}^{p\ell}\to\widetilde{\mathscr{H}}_{\infty}^{p\ell}$
is the functor from (B.19). Further, for $c\in{\mathscr{C}\mathrm{art}}$ we
have that

$\iota^{*}\big{(}\operatorname{hoRan}_{\iota}^{\prime}\circ\iota^{*}(G)\big{)}(c)=\underline{\mathscr{H}}\big{(}Q^{\prime}(c),\iota^{*}(G)\big{)}\,,$
(B.23)

so the natural weak equivalence
$Q^{\prime}\overset{\sim}{\longrightarrow}1_{\mathscr{H}_{\infty}}$ induces a
projective weak equivalence

$\iota^{*}(G)\overset{\sim}{\longrightarrow}\iota^{*}\big{(}\operatorname{hoRan}_{\iota}^{\prime}\circ\iota^{*}(G)\big{)}\,.$
(B.24)

For any manifold $M$ and any differentiably good open covering ${\mathscr{U}}$
of $M$, this induces a weak equivalence

$\underline{\mathscr{H}_{\infty}}({\check{C}}{\mathscr{U}},\iota^{*}G)\overset{\sim}{\longrightarrow}\underline{\mathscr{H}_{\infty}}\big{(}{\check{C}}{\mathscr{U}},\iota^{*}(\operatorname{hoRan}_{\iota}^{\prime}\circ\iota^{*}(G))\big{)}\,.$
(B.25)

Since ${\mathrm{GCov}}(M)^{\mathrm{op}}$ is filtered, this yields a weak
equivalence
$\operatorname{Pl}G\overset{\sim}{\longrightarrow}\operatorname{Pl}(\operatorname{hoRan}_{\iota}^{\prime}\circ\iota^{*}(G))$,
thus establishing the desired zig-zag of weak equivalences. ∎

Combining Lemmas B.6 and B.21, we obtain that the total derived functor
$\mathbb{R}\operatorname{hoRan}_{\iota}^{\prime}\colon\mathrm{h}\mathscr{H}_{\infty}^{p\ell}\to\mathrm{h}\widetilde{\mathscr{H}}_{\infty}^{p\ell}$
is an equivalence. Thus, we have shown

###### Theorem B.26

The adjunction

${Q^{\prime}\circ\iota^{*}:\widetilde{\mathscr{H}}_{\infty}^{p\ell}}$${\mathscr{H}_{\infty}^{p\ell}:\operatorname{hoRan}_{\iota}^{\prime}}$$\scriptstyle{\perp}$
(B.27)

is a Quillen equivalence.

##  Appendix C A modified two-sided bar construction

A very efficient and “unreasonably effective” [31, Sec. 4] tool for the
computation of homotopy colimits in simplicial model categories is given by
the two-sided (simplicial) bar construction. We recommend the book [31] as an
introduction and as a reference. Let $\mathscr{I}$ be a small category, and
consider $\mathscr{I}$-shaped diagrams in a simplicial category
${\mathscr{M}}$. Heuristically, the homotopical meaningfulness of the bar
construction stems from the fact that it introduces coherence data into
diagrams $D\colon\mathscr{I}\to{\mathscr{M}}$ by keeping track of composable
sequences of morphisms in $\mathscr{I}$. That is, it takes into account all
$n$-simplices of the nerve $N\mathscr{I}$.

For the purposes of Section 5 of this paper, however, we need a bar
construction for diagrams of simplicial presheaves that builds on cospans in
$\mathscr{I}$ rather than on ordinary morphisms in $\mathscr{I}$. More
concretely, in Section 5 we consider the category whose objects are the open
sets in an open covering of a manifold and all finite intersections of these
open sets. The morphisms in this category are inclusions of open subsets. In
the ordinary bar construction, we obtain a morphism for every inclusion.
However, geometrically, it is often more useful to view an overlap $U_{ab}$ of
two elements $U_{a}$ and $U_{b}$ of the cover as a morphism $U_{a}\to U_{b}$.
Including higher overlaps naturally leads us to considering subdivisions of
simplicial sets and the associated ${\operatorname{Ex}}$ functor (explained in
more detail below). This can be seen as a generalisation of constructions in
[11, Sec. 4].

The purpose of this appendix is to introduce a modified two-sided (simplicial)
bar construction whose coherence data is encoded not by the nerve
$N\mathscr{I}$, but by the simplicial set ${\operatorname{Ex}}N\mathscr{I}$.
Again heuristically, since $N\mathscr{I}$ and
${\operatorname{Ex}}N\mathscr{I}$ are weakly equivalent in
${\mathscr{S}\mathrm{et}_{\Delta}}$, we should expect the modified bar
construction to be equivalent to the original version—we indeed prove this in
Proposition C.27.

Let $\mathscr{C}$ be a small category, and let ${\mathscr{K}}_{\infty}$ denote
the category of simplicial presheaves on $\mathscr{C}$. We denote by
${\mathscr{K}}_{\infty}^{p/i}$ the category ${\mathscr{K}}_{\infty}$ endowed
with the projective or the injective model structure, respectively. Let
$\mathscr{I}$ be a small category, and let
$E\colon\mathscr{I}\to{\mathscr{K}}_{\infty}$ be a diagram. We can
equivalently view $E$ as a functor
$E\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$.
Further, let
$F\colon\mathscr{I}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ be a
functor. Given these data, there is an associated _two-sided bar construction_
[31]

$B^{{\mathscr{K}}_{\infty}}(F,\mathscr{I},E)=\int^{n\in{\mathbbe{\Delta}}}\Delta^{n}\otimes
B^{{\mathscr{K}}_{\infty}}_{n}(F,\mathscr{I},E)\,,$ (C.1)

where the $n$-th level of the _two-sided simplicial bar construction_ in
${\mathscr{K}}_{\infty}$ reads as

$B^{{\mathscr{K}}_{\infty}}_{n}(F,\mathscr{I},E)=\coprod_{i_{0},\ldots,i_{n}\in\mathscr{I}}E(-,i_{0})\times\mathscr{I}(i_{0},i_{1})\times\cdots\times\mathscr{I}(i_{n-1},i_{n})\times
F(i_{n})\,.$ (C.2)

We will refer to the two-sided bar construction as the bar construction, for
short. We can use the bar construction to model the homotopy colimit in
${\mathscr{K}}_{\infty}$ as follows [31, Sec. 5]:

$\underset{\mathscr{I}}{{\mathrm{hocolim}}}^{{\mathscr{K}}_{\infty}}(E)\simeq
B^{{\mathscr{K}}_{\infty}}(*,\mathscr{I},Q^{{\mathscr{K}}_{\infty}}\circ
E)\,,$ (C.3)

where ${\mathscr{K}}_{\infty}$ is endowed with some simplicial model
structure, and where $Q^{{\mathscr{K}}_{\infty}}$ is a cofibrant replacement
functor for that model structure. In particular, if
$E\colon\mathscr{I}\to{\mathscr{K}}_{\infty}$ is objectwise cofibrant with
respect to that model structure, then we have

$\underset{\mathscr{I}}{{\mathrm{hocolim}}}^{{\mathscr{K}}_{\infty}}(E)\simeq
B^{{\mathscr{K}}_{\infty}}_{n}(*,\mathscr{I},E)\,.$ (C.4)

We record the following immediate results:

###### Lemma C.5

Let $F\in{\mathscr{K}}_{\infty}$,
$E\colon\mathscr{I}\to{\mathscr{K}}_{\infty}$, and view
${\mathscr{K}}_{\infty}$ as a simplicial category.

  1. (1)

For any $c\in\mathscr{C}$, there is a canonical isomorphism, natural in $c$,

$B^{{\mathscr{K}}_{\infty}}(F,\mathscr{I},E)(c)\cong
B^{\mathscr{S}\mathrm{et}_{\Delta}}\big{(}F,\mathscr{I},E(c,-)\big{)}\,.$
(C.6)

  2. (2)

The bar construction is naturally isomorphic to the composition

${({\mathscr{S}\mathrm{et}_{\Delta}})^{\mathscr{I}^{\mathrm{op}}}\times({\mathscr{K}}_{\infty})^{\mathscr{I}}}$${({\mathscr{K}}_{\infty})^{{\mathbbe{\Delta}}^{\mathrm{op}}}\cong(s{\mathscr{S}\mathrm{et}_{\Delta}})^{\mathscr{C}^{\mathrm{op}}}}$${{\mathscr{K}}_{\infty}\,.}$$\scriptstyle{B^{{\mathscr{K}}_{\infty}}_{\bullet}(-,\mathscr{I},-)}$$\scriptstyle{\delta^{*}\circ(-)}$
(C.7)

Let
${\mathscr{Y}}_{(-)}\colon\mathscr{I}\to{\mathscr{C}\mathrm{at}}(\mathscr{I}^{\mathrm{op}},{\mathscr{S}\mathrm{et}})$
denote the Yoneda embedding of $\mathscr{I}$, and let
${\mathscr{Y}}^{(-)}\colon\mathscr{I}^{\mathrm{op}}\to{\mathscr{C}\mathrm{at}}(\mathscr{I},{\mathscr{S}\mathrm{et}})$
denote the co-Yoneda embedding of $\mathscr{I}$, i.e. for $i\in\mathscr{I}$ we
have ${\mathscr{Y}}^{i}=\mathscr{I}(i,-)$. Recall that there are canonical
natural isomorphisms

${\mathscr{S}\mathrm{et}}^{\mathscr{I}}({\mathscr{Y}}^{i},Z)\overset{\cong}{\longrightarrow}Z(i)$
(C.8)

for any $i\in\mathscr{I}$ and $Z\in{\mathscr{S}\mathrm{et}}^{\mathscr{I}}$.

###### Definition C.9

Consider a pair of functors
$X\colon\mathscr{I}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}}$ and
$Y\colon\mathscr{I}\to{\mathscr{S}\mathrm{et}}$. We define the following
categories:

  1. (1)

The category $\mathscr{I}_{/X}$ has as objects the morphisms
$\chi\colon{\mathscr{Y}}_{i}\to X$ in
${\mathscr{S}\mathrm{et}}^{\mathscr{I}^{\mathrm{op}}}$. A morphism
$\chi_{0}\to\chi_{1}$ consists of a morphism $f\colon i_{0}\to i_{1}$ such
that the following triangle commutes:

${{\mathscr{Y}}_{i_{0}}}$${X}$${{\mathscr{Y}}_{i_{1}}}$$\scriptstyle{{\mathscr{Y}}_{f}}$$\scriptstyle{\chi_{0}}$$\scriptstyle{\chi_{1}}$
(C.10)

  2. (2)

The category $\mathscr{I}_{Y/}$ has as objects the morphisms
$\nu\colon{\mathscr{Y}}^{i}\to Y$ in ${\mathscr{S}\mathrm{et}}^{\mathscr{I}}$.
A morphism $\nu_{0}\to\nu_{1}$ consists of a morphism $f\colon i_{0}\to i_{1}$
such that the following triangle commutes:

${{\mathscr{Y}}^{i_{0}}}$${Y}$${{\mathscr{Y}}^{i_{1}}}$$\scriptstyle{\nu_{0}}$$\scriptstyle{{\mathscr{Y}}^{f}}$$\scriptstyle{\nu_{1}}$
(C.11)

  3. (3)

We define the category $\mathscr{I}_{Y{//}X}$ as the strict pullback of
categories

$\mathscr{I}_{Y{//}X}\coloneqq\mathscr{I}_{Y/}\underset{\mathscr{I}}{\times}\mathscr{I}_{/X}\,.$
(C.12)

###### Example C.13

Let $i\in\mathscr{I}$ and $Y={\mathscr{Y}}^{i}$. Then, the category
$\mathscr{I}_{{\mathscr{Y}}^{i}/}$ can be described as follows: an object
$\nu\colon{\mathscr{Y}}^{i_{0}}\to{\mathscr{Y}}^{i}$ is equivalent to a
morphism $\nu\colon i\to i_{0}$, and a morphism $f\colon\nu_{0}\to\nu_{1}$ is
equivalent to a morphism $f\colon i_{0}\to i_{1}$ such that
$f\circ\nu_{0}=\nu_{1}$. In other words, there is an isomorphism of categories

$\mathscr{I}_{{\mathscr{Y}}^{i}/}\cong\mathscr{I}_{i/}\,,$ (C.14)

where the category on the right-hand side is the usual slice category under
the object $i\in\mathscr{I}$. This also justifies our notation
$\mathscr{I}_{Y/}$. $\triangleleft$

We obtain the following lemma directly from Definition C.9:

###### Lemma C.15

Let ${\mathscr{C}\mathrm{at}}_{1}$ be the strict (1-)category of categories
and functors. The constructions in Definition C.9 give rise to functors

$\displaystyle\mathscr{I}_{/(-)}\colon{\mathscr{S}\mathrm{et}}^{\mathscr{I}^{\mathrm{op}}}\longrightarrow{\mathscr{C}\mathrm{at}}_{1}\,,\qquad\mathscr{I}_{(-)/}\colon{\mathscr{S}\mathrm{et}}^{\mathscr{I}}\longrightarrow{\mathscr{C}\mathrm{at}}_{1}\,,\qquad\text{and}\qquad\mathscr{I}_{(-){//}(-)}\colon{\mathscr{S}\mathrm{et}}^{\mathscr{I}}\times{\mathscr{S}\mathrm{et}}^{\mathscr{I}^{\mathrm{op}}}\longrightarrow{\mathscr{C}\mathrm{at}}_{1}\,.$
(C.16)

Now consider again the functors
$F\colon\mathscr{I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ and
$E\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$.
We can equivalently view these as functors

$F\colon{\mathbbe{\Delta}}^{\mathrm{op}}\times\mathscr{I}\longrightarrow{\mathscr{S}\mathrm{et}}\qquad\text{and}\qquad
E\colon{\mathbbe{\Delta}}^{\mathrm{op}}\times\mathscr{C}^{\mathrm{op}}\times\mathscr{I}\longrightarrow{\mathscr{S}\mathrm{et}}\,.$
(C.17)

Given any $[k]\in{\mathbbe{\Delta}}$ and $c\in\mathscr{C}$, we see from (C.2)
and Definition C.9 that there is a canonical isomorphism of sets (or, more
formally, of discrete simplicial sets)

$B^{\mathscr{S}\mathrm{et}_{\Delta}}_{n}\big{(}F_{k},\mathscr{I},E_{k}(c,-)\big{)}\cong
N(\mathscr{I}_{E_{k}(c,-){//}F_{k}})_{n}\,,$ (C.18)

which extends to an isomorphism of simplicial sets

$B^{\mathscr{S}\mathrm{et}_{\Delta}}_{\bullet}\big{(}F_{k},\mathscr{I},E_{k}(c,-)\big{)}\cong
N(\mathscr{I}_{E_{k}(c,-){//}F_{k}})\,.$ (C.19)

Because of the functoriality of $\mathscr{I}_{(-){//}(-)}$, we even obtain a
natural isomorphism

$B^{{\mathscr{K}}_{\infty}}_{\bullet}\big{(}F_{k},\mathscr{I},E_{k}\big{)}\cong
N(\mathscr{I}_{E_{k}{//}F_{k}})\colon\mathscr{C}^{\mathrm{op}}\longrightarrow{\mathscr{S}\mathrm{et}_{\Delta}}$
(C.20)

of functors $\mathscr{C}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$,
i.e. of simplicial presheaves. Letting $[k]\in{\mathbbe{\Delta}}$ vary, and
again using the functoriality of $\mathscr{I}_{(-){//}(-)}$, we obtain an
isomorphism

$B^{{\mathscr{K}}_{\infty}}_{\bullet}\big{(}F_{\star},\mathscr{I},E_{\star}\big{)}\cong
N(\mathscr{I}_{E_{\star}{//}F_{\star}})\colon{\mathbbe{\Delta}}^{\mathrm{op}}\longrightarrow{\mathscr{K}}_{\infty}$
(C.21)

of simplicial objects in ${\mathscr{K}}_{\infty}$.

We introduce the following auxiliary construction, which is interesting in its
own right. Let
${\operatorname{Ex}}\colon{\mathscr{S}\mathrm{et}_{\Delta}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
denote the right adjoint to the simplicial subdivision functor
$\operatorname{Sd}$—recall that $\operatorname{Sd}$ sends $\Delta^{n}$ to the
nerve of the category of totally ordered subsets of $[n]$. For instance,
$\operatorname{Sd}\Delta^{1}$ can be sketched as the cospan
$\\{0\\}\rightarrow\\{0,1\\}\leftarrow\\{1\\}$. Our main reference on
${\operatorname{Ex}}$ is [7, Sec. 3.1]. The functor ${\operatorname{Ex}}$
comes with a natural weak equivalence $b\colon
1_{\mathscr{S}\mathrm{et}_{\Delta}}\overset{\sim}{\longrightarrow}{\operatorname{Ex}}$
(see, for instance, [7, Prop. 3.1.21]).

###### Definition C.22

Let $F\colon\mathscr{I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ and
$E\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{I}\to{\mathscr{S}\mathrm{et}_{\Delta}}$
be functors. Given any $[k]\in{\mathbbe{\Delta}}$ and $c\in\mathscr{C}$, we
set

$B^{\mathscr{S}\mathrm{et}_{\Delta}}_{{\operatorname{Ex}},\bullet,k}\big{(}F,\mathscr{I},E(c,-)\big{)}\coloneqq{\operatorname{Ex}}\circ
N(\mathscr{I}_{E_{k}(c,-){//}F_{k}})\,.$ (C.23)

By the functoriality of ${\operatorname{Ex}}$ and $\mathscr{I}_{(-){//}(-)}$,
we obtain a functor

$B^{{\mathscr{K}}_{\infty}}_{{\operatorname{Ex}},\bullet,\star}(F,\mathscr{I},E)\coloneqq{\operatorname{Ex}}\circ
N(\mathscr{I}_{E_{\star}{//}F_{\star}})\colon{\mathbbe{\Delta}}^{\mathrm{op}}\times{\mathbbe{\Delta}}^{\mathrm{op}}\longrightarrow{\mathscr{S}\mathrm{et}}^{\mathscr{C}^{\mathrm{op}}}\,.$
(C.24)

Finally, we set

$B^{{\mathscr{K}}_{\infty}}_{\operatorname{Ex}}(F,\mathscr{I},E)\coloneqq\int^{n}\Delta^{n}\otimes
B^{{\mathscr{K}}_{\infty}}_{{\operatorname{Ex}},n,\star}(F,\mathscr{I},E)\cong\int^{n}\Delta^{n}\otimes\big{(}{\operatorname{Ex}}\circ
N(\mathscr{I}_{E_{\star}{//}F_{\star}})\big{)}_{n}\,.$ (C.25)

This defines a functor

$B^{{\mathscr{K}}_{\infty}}_{\operatorname{Ex}}(-,\mathscr{I},-)\colon{\mathscr{S}\mathrm{et}}^{\mathscr{I}^{\mathrm{op}}}\times({\mathscr{K}}_{\infty})^{\mathscr{I}}\longrightarrow{\mathscr{K}}_{\infty}\,.$
(C.26)

###### Proposition C.27

The natural weak equivalence $b\colon
1_{\mathscr{S}\mathrm{et}_{\Delta}}\overset{\sim}{\longrightarrow}{\operatorname{Ex}}$
indues a natural transformation

$b^{{\mathscr{K}}_{\infty}}\colon
B^{{\mathscr{K}}_{\infty}}(-,\mathscr{I},-)\longrightarrow
B^{{\mathscr{K}}_{\infty}}_{\operatorname{Ex}}(-,\mathscr{I},-)\,,$ (C.28)

all of whose components $b^{{\mathscr{K}}_{\infty}}_{F,E}\colon
B^{{\mathscr{K}}_{\infty}}(F,\mathscr{I},E)\overset{\sim}{\longrightarrow}B^{{\mathscr{K}}_{\infty}}_{\operatorname{Ex}}(F,\mathscr{I},E)$
are objectwise weak equivalences in ${\mathscr{K}}_{\infty}$.

###### Proof.

The naturality of $b\colon
1_{\mathscr{S}\mathrm{et}_{\Delta}}\to{\operatorname{Ex}}$ readily implies
that the morphisms

$b_{|N(\mathscr{I}_{E_{k}(c,-){//}F_{k}})}\colon
N(\mathscr{I}_{E_{k}(c,-){//}F_{k}})\longrightarrow{\operatorname{Ex}}\circ
N(\mathscr{I}_{E_{k}(c,-){//}F_{k}})$ (C.29)

are weak equivalences of simplicial sets and that they are natural in
$[k]\in{\mathbbe{\Delta}}$ and $c\in\mathscr{C}$, as well as in $F$ and $E$.
Letting $[k]$ vary, we obtain a morphism

$b_{|N(\mathscr{I}_{E_{\star}(c,-){//}F_{\star}})}\colon
N(\mathscr{I}_{E_{\star}(c,-){//}F_{\star}})\longrightarrow{\operatorname{Ex}}\circ
N(\mathscr{I}_{E_{\star}(c,-){//}F_{\star}})$ (C.30)

of bisimplicial sets which is natural in $c$, and which is a _horizontal_ weak
equivalence. We know that the diagonal $\delta^{*}\colon
s{\mathscr{S}\mathrm{et}_{\Delta}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ is
homotopical, i.e. that it sends all _vertical_ weak equivalences of
bisimplicial sets to weak equivalences in ${\mathscr{S}\mathrm{et}_{\Delta}}$.
However, since $\delta^{*}(X_{\bullet,\star})=\delta^{*}(X_{\star,\bullet})$,
it follows that $\delta^{*}$ also sends horizontal weak equivalences in
$s{\mathscr{S}\mathrm{et}_{\Delta}}$ to weak equivalences in
${\mathscr{S}\mathrm{et}_{\Delta}}$. Therefore, the natural isomorphisms

$B^{{\mathscr{K}}_{\infty}}_{\operatorname{Ex}}(F,\mathscr{I},E)(c)\cong
B^{\mathscr{S}\mathrm{et}_{\Delta}}_{\operatorname{Ex}}\big{(}F,\mathscr{I},E(c,-)\big{)}\cong\delta^{*}\big{(}B^{\mathscr{S}\mathrm{et}_{\Delta}}_{{\operatorname{Ex}},\bullet,\star}\big{(}F,\mathscr{I},E(c,-)\big{)}\big{)}$
(C.31)

complete the proof. ∎

###### Corollary C.32

Let $E\colon\mathscr{I}\to{\mathscr{K}}_{\infty}$ be an $\mathscr{I}$-shaped
diagram in ${\mathscr{K}}_{\infty}$. The modified bar construction models the
homotopy colimit:

$\underset{\mathscr{I}}{{\mathrm{hocolim}}}^{{\mathscr{K}}_{\infty}}(E)\simeq
B^{{\mathscr{K}}_{\infty}}_{\operatorname{Ex}}(*,\mathscr{I},Q^{{\mathscr{K}}_{\infty}}\circ
E)\,.$ (C.33)

If $E$ is pointwise cofibrant in ${\mathscr{K}}_{\infty}$, then we have

$\underset{\mathscr{I}}{{\mathrm{hocolim}}}^{{\mathscr{K}}_{\infty}}(E)\simeq
B^{{\mathscr{K}}_{\infty}}_{\operatorname{Ex}}(*,\mathscr{I},E)\,.$ (C.34)

We conclude this section by considering the case $\mathscr{I}=\mathscr{C}$.
Let $F\colon\mathscr{C}^{\mathrm{op}}\to{\mathscr{S}\mathrm{et}_{\Delta}}$ be
an object in ${\mathscr{K}}_{\infty}$. We recall Dugger’s cofibrant
replacement functor for the projective model structure
${\mathscr{K}}_{\infty}^{p}$: it reads as

$Q^{p}F=B^{{\mathscr{K}}_{\infty}}(F,\mathscr{C},{\mathscr{Y}})\,.$ (C.35)

More explicitly, for $c\in\mathscr{C}$, and $[n]\in{\mathbbe{\Delta}}$ we have

$\displaystyle Q^{p}F(c)_{n}$
$\displaystyle=\coprod_{\vec{c}\in(N\mathscr{C})_{n}}{\mathscr{Y}}_{c_{0}}(c)\times
F_{n}(c_{n})$ (C.36)
$\displaystyle\cong\big{(}N(\mathscr{C}_{c{//}F_{n}})\big{)}_{n}\,.$ (C.37)

Here, $\vec{c}=(c_{0}\to\cdots\to c_{n})$ is equivalently a functor
$[n]\to\mathscr{C}$ (or, equivalently, a morphism $\Delta^{n}\to
N\mathscr{C}$). Using the modified two-sided bar construction, we define

$Q^{p}_{\operatorname{Ex}}F\coloneqq
B^{{\mathscr{K}}_{\infty}}_{\operatorname{Ex}}(F,\mathscr{C},{\mathscr{Y}})\,.$
(C.38)

That is, for $c\in\mathscr{C}$, and $[n]\in{\mathbbe{\Delta}}$ we have

$\big{(}(Q^{p}_{\operatorname{Ex}}F)(c)\big{)}_{n}={\operatorname{Ex}}\circ
N(\mathscr{C}_{{\mathscr{Y}}_{(-)}(c){//}F_{n}})_{n}\cong\big{(}{\operatorname{Ex}}\circ
N(\mathscr{C}_{c{//}F_{n}})\big{)}_{n}\,.$ (C.39)

The map $N(\mathscr{C}_{c{//}F_{n}})_{n}\longrightarrow F_{n}(c)$, which sends
an element of $N(\mathscr{C}_{c{//}F_{n}})_{n}$ to the unique composition
${\mathscr{Y}}_{c}\to F_{n}$, induces a natural augmentation map
$q^{p}_{\operatorname{Ex}}\colon Q^{p}_{\operatorname{Ex}}\to
1_{{\mathscr{K}}_{\infty}}$ and we obtain a commutative diagram

${Q^{p}F}$${F}$${Q^{p}_{\operatorname{Ex}}F}$$\scriptstyle{q^{p}_{F}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{q^{p}_{{\operatorname{Ex}},F}}$
(C.40)

Hence, $q^{p}_{\operatorname{Ex}}\colon
Q^{p}_{\operatorname{Ex}}\overset{\sim}{\longrightarrow}1_{{\mathscr{K}}_{\infty}}$
is a natural weak equivalence. Since every level of
$Q^{p}_{\operatorname{Ex}}F$ is a coproduct of representables, we infer

###### Proposition C.41

$(Q^{p}_{\operatorname{Ex}},q^{p}_{\operatorname{Ex}})$ is a cofibrant
replacement functor in ${\mathscr{K}}_{\infty}^{p}$.

##  Appendix D Comparing adjunctions and detecting Quillen equivalences

Here we recall and collect some basic background on how to detect Quillen
equivalences. We first recall the following well-known fact and definition:

###### Proposition D.1

Let $\mathscr{C}$ and $\mathscr{D}$ be categories, and let
$F:\mathscr{C}\rightleftarrows\mathscr{D}:G$ be an adjoint pair. The following
are equivalent:

  1. (1)

The counit and unit of the adjunction are isomorphisms.

  2. (2)

$F$ is an equivalence.

  3. (3)

$G$ is an equivalence.

###### Proof.

It is clear that (1) implies both (2) and (3). We will show that (2) implies
(1)—the proof that (3) implies (1) is analogous. Thus, suppose that $F$ is an
equivalence, i.e. suppose that it is fully faithful and essentially
surjective. It follows that the coevaluation morphism $co\colon
1_{\mathscr{C}}\to GF$ is a natural isomorphism. The triangle identity then
implies that the component $e_{|Fc}\colon FG(Fc)\to F(c)$ of the evaluation
morphism $e\colon FG\to 1_{\mathscr{D}}$ is an isomorphism for every
$c\in\mathscr{C}$. Since $F$ is essentially surjective, it follows from the
naturality of $e$ that $e$ is a natural isomorphism as well. ∎

###### Definition D.2

In any of the equivalent cases of Proposition D.1, the adjunction $F\dashv G$
is called an _adjoint equivalence_.

A similar characterisation exists for Quillen equivalences:

###### Proposition D.3

_[ 24, A.2.5.1]_ Let $F:\mathscr{C}\rightleftarrows\mathscr{D}:G$ be a Quillen
adjunction between model categories. The following are equivalent:

  1. (1)

The total left derived functor
$\mathbb{L}F\colon\mathrm{h}\mathscr{C}\to\mathrm{h}\mathscr{D}$ is an
equivalence of categories.

  2. (2)

The total right derived functor
$\mathbb{R}G\colon\mathrm{h}\mathscr{D}\to\mathrm{h}\mathscr{C}$ is an
equivalence of categories.

  3. (3)

The pair $F\dashv G$ is a Quillen equivalence.

Let $(F,G,\varphi)$ be an adjunction $\mathscr{C}\rightleftarrows\mathscr{D}$,
where $F\colon\mathscr{C}\to\mathscr{D}$ is the left adjoint,
$G\colon\mathscr{D}\to\mathscr{C}$ is the right adjoint, and
$\varphi\colon\mathscr{D}(F(-),-)\longrightarrow\mathscr{C}(-,G(-))$ is the
binatural isomorphism that establishes the adjunction. A morphism of
adjunctions $(F,G,\varphi)\to(F^{\prime},G^{\prime},\varphi^{\prime})$ is a
pair $(f,g)$ of natural transformations $f\colon F\to F^{\prime}$, $g\colon
G^{\prime}\to G$, such that, for every $c\in\mathscr{C}$ and
$d\in\mathscr{D}$, the diagram

${\mathscr{D}(F^{\prime}c,d)}$${\mathscr{D}(Fc,d)}$${\mathscr{C}(c,G^{\prime}d)}$${\mathscr{C}(c,Gd)}$$\scriptstyle{f_{|c}^{*}}$$\scriptstyle{\varphi^{\prime}_{c,d}}$$\scriptstyle{\cong}$$\scriptstyle{\varphi_{c,d}}$$\scriptstyle{\cong}$$\scriptstyle{{(g_{|d})}_{*}}$
(D.4)

commutes. As an equation, this amounts to demanding that

$\varphi_{c,d}(\chi\circ f_{|c})=g_{|d}\circ\varphi^{\prime}_{c,d}(\chi)$
(D.5)

for every morphism $\chi\colon F^{\prime}c\to d$ in $\mathscr{D}$.

###### Definition D.6

Let $\mathscr{C},\mathscr{D}$ be categories. We define the following
categories:

  1. (1)

Let
${\mathscr{F}\mathrm{un}}^{L}(\mathscr{C},\mathscr{D})\subset{\mathscr{C}\mathrm{at}}(\mathscr{C},\mathscr{D})$
be the full subcategory on the left adjoint functors.

  2. (2)

Let ${\mathscr{A}\mathrm{dj}}(\mathscr{C},\mathscr{D})$ be the category whose
objects are adjunctions $(F,G,\varphi)$, and whose morphisms are morphisms
$(f,g)$ of adjunctions.

  3. (3)

Let
${\mathscr{F}\mathrm{un}}^{L,AE}(\mathscr{C},\mathscr{D})\subset{\mathscr{F}\mathrm{un}}^{L}(\mathscr{C},\mathscr{D})$
be the full subcategory on the left adjoint functors which are equivalences of
categories.

  4. (4)

Let
${\mathscr{A}\mathrm{d}\mathscr{E}\mathrm{q}}(\mathscr{C},\mathscr{D})\subset{\mathscr{A}\mathrm{dj}}(\mathscr{C},\mathscr{D})$
be the full subcategory on the adjoint equivalences, i.e. on those adjunctions
$(F,G,\varphi)$ where both $F$ and $G$ are equivalences.

  5. (5)

Suppose both $\mathscr{C}$ and $\mathscr{D}$ are endowed with model
structures. Let
${\mathscr{F}\mathrm{un}}^{LQ}(\mathscr{C},\mathscr{D})\subset{\mathscr{F}\mathrm{un}}^{L}(\mathscr{C},\mathscr{D})$
be the full subcategory on the left Quillen functors.

  6. (6)

Finally, let
${\mathscr{QA}\mathrm{dj}}(\mathscr{C},\mathscr{D})\subset{\mathscr{A}\mathrm{dj}}(\mathscr{C},\mathscr{D})$
be the full subcategory on the Quillen adjunctions.

There are canonical projection functors

$\displaystyle\pi\colon{\mathscr{A}\mathrm{dj}}(\mathscr{C},\mathscr{D})$
$\displaystyle\longrightarrow{\mathscr{F}\mathrm{un}}^{L}(\mathscr{C},\mathscr{D})\,,$
(D.7)
$\displaystyle\pi_{AE}\colon{\mathscr{A}\mathrm{d}\mathscr{E}\mathrm{q}}(\mathscr{C},\mathscr{D})$
$\displaystyle\longrightarrow{\mathscr{F}\mathrm{un}}^{L,AE}(\mathscr{C},\mathscr{D})\,,$
(D.8)
$\displaystyle\pi_{Q}\colon{\mathscr{QA}\mathrm{dj}}(\mathscr{C},\mathscr{D})$
$\displaystyle\longrightarrow{\mathscr{F}\mathrm{un}}^{LQ}(\mathscr{C},\mathscr{D})\,.$
(D.9)

###### Proposition D.10

The functors $\pi$, $\pi_{Q}$, and $\pi_{AE}$ are fibred and cofibred in
groupoids, with contractible fibres. They are equivalences and surjective on
objects.

###### Proof.

Consider the functor
$\pi\colon{\mathscr{A}\mathrm{dj}}(\mathscr{C},\mathscr{D})\longrightarrow{\mathscr{F}\mathrm{un}}^{L}(\mathscr{C},\mathscr{D})$,
$(F,G,\varphi)\mapsto F$. This is surjective on objects. Let
$F_{0},F_{1}\in{\mathscr{F}\mathrm{un}}^{L}(\mathscr{C},\mathscr{D})$, and let
$f\colon F_{0}\to F_{1}$ be a natural transformation. Suppose we are given
lifts $(F_{i},G_{i},\varphi_{i})$ of $F_{i}$ to
${\mathscr{A}\mathrm{dj}}(\mathscr{C},\mathscr{D})$, for $i=0,1$. Then, for
any $d\in\mathscr{D}$, we define a morphism $g_{|d}\colon G_{1}d\to G_{0}d$ by
the Yoneda Lemma and by demanding commutativity of diagram (D.4). Thus, once
the lifts $(F_{i},G_{i},\varphi_{i})$ are specified, there exists a unique
lift of $f$ to a morphism
$(f,g)\colon(F_{0},G_{0},\varphi_{0})\longrightarrow(F_{1},G_{1},\varphi_{1})$
of adjunctions. In particular, $\pi$ is an equivalence of categories. This
also shows that ${\mathscr{A}\mathrm{dj}}(\mathscr{C},\mathscr{D})$ is both
fibred and cofibred in groupoids over
${\mathscr{F}\mathrm{un}}^{L}(\mathscr{C},\mathscr{D})$.

The functors $\pi_{AE}$ and $\pi_{Q}$ can be seen as restrictions of $\pi$ to
full subcategories; the same reasoning applies. ∎

Consequently, given adjunctions $(F_{0},G_{0},\varphi_{0})$ and
$(F_{1},G_{1},\varphi_{1})$, specifying a morphism of adjunctions is
equivalent to specifying a natural transformation $f\colon F_{0}\to F_{1}$.
Recalling that any functor which is naturally isomorphic to an equivalence is
an equivalence itself, and combining this with Proposition D.1, we obtain

###### Proposition D.11

Let $(F_{0},G_{0},\varphi_{0})$ and $(F_{1},G_{1},\varphi_{1})$ be two
adjunctions $\mathscr{C}\rightleftarrows\mathscr{D}$ such that there exists a
natural isomorphism $f\colon F_{0}\to F_{1}$. Then,
$(F_{0},G_{0},\varphi_{0})$ is an adjoint equivalence if and only if
$(F_{1},G_{1},\varphi_{1})$ is so.

###### Corollary D.12

Let $F_{0},F_{1}\colon\mathscr{C}\to\mathscr{D}$ be left Quillen functors such
that there exists a natural weak equivalence
$f\colon\mathbb{L}F_{0}\to\mathbb{L}F_{1}$ as functors
$\mathrm{h}\mathscr{C}\to\mathrm{h}\mathscr{D}$. Then, $F_{0}$ is a Quillen
equivalence if and only if $F_{1}$ is a Quillen equivalence.

###### Proof.

Let $(F_{i},G_{i},\varphi_{i})$ be lifts of $F_{i}$ to Quillen adjunctions
$\mathscr{C}\rightleftarrows\mathscr{D}$. We can apply Proposition D.11 to the
induced adjunctions $(\mathbb{L}F_{i},\mathbb{R}G_{i},\mathrm{h}\varphi)$ on
homotopy categories
$\mathrm{h}\mathscr{C}\rightleftarrows\mathrm{h}\mathscr{D}$. The claim then
follows from Proposition D.3. ∎

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