# Fibration structure for Gromov h-principle

Koji Yamazaki

###### Abstract

The h-principle is a powerful tool for obtaining solutions to partial
differential inequalities and partial differential equations. Gromov
discovered the h-principle for the general partial differential relations to
generalize the results of Hirsch and Smale. In his book, Gromov generalizes
his theorem and discusses the sheaf theoretic h-principle, in which an object
called a flexible sheaf plays an important role. We show that a flexible sheaf
can be interpreted as a fibrant object with respect to a model structure.

###### Contents

  1. 0 Introduction
    1. 0.1 What’s h-principle?
    2. 0.2 Sheaf theoretic h-principle
    3. 0.3 Main results
    4. 0.4 History of h-principles
    5. 0.5 Advantages of the sheaf theoretic h-principle
  2. 1 Quasitopological spaces
    1. 1.1 Quasitopological spaces
      1. 1.1.1 Prequasitopological spaces
      2. 1.1.2 Quasitopological spaces
      3. 1.1.3 Geometric realizations
      4. 1.1.4 Filtered colimit trick
      5. 1.1.5 Patching Lemma
    2. 1.2 Some model structures
      1. 1.2.1 A model structure for prequasitopological spaces
      2. 1.2.2 A model structure for quasitopological spaces
      3. 1.2.3 Homotopy Theory in quasitopological spaces
  3. 2 Continuous sheaves
    1. 2.1 Continuous sheaves
    2. 2.2 Flexible sheaves
    3. 2.3 The sheaf of formal sections
  4. 3 Result 1 : Embedding to a model category
    1. 3.1 The base category
    2. 3.2 The compact flexible model structure
    3. 3.3 The pointwise projective model structure
    4. 3.4 The pointwise flexible model structure
  5. 4 Result 2 : The prefibration structure
    1. 4.1 The flexible track
    2. 4.2 A prefibration structure for continuous sheaves
  6. A Model categories
    1. A.1 Cofibrantly generated model categories
      1. A.1.1 Small object argument
      2. A.1.2 Cofibrantly generated model categories
    2. A.2 Cartesian closed model categories
    3. A.3 Proper model categories
    4. A.4 Mixed model categories

##  0 Introduction

The h-principle is a powerful tool for obtaining solutions to partial
differential inequalities and partial differential equations. Hirsch [12] and
Smale [24, 23] discovered the h-principle for the immersions, and Phillips
[19] discovered the h-principle for the submersion. Gromov [8] discovered
Gromov h-principle, the h-principle for the general partial differential
relations, to generalize the above results. The proof of Gromov h-principle is
favoured in modern times via Holonomic Approximation Theorem by Y. M.
Eliashberg and N. M. Mishachev [5, 4]. Their proof is rather elementary, as
they consider a space of functions to contain the Whitney topology. Before
that, Gromov himself has refined his proofs.

In his book, Gromov [9] generalizes his theorem and discusses the sheaf
theoretic h-principle, in which an object called the flexible sheaf plays an
important role. This proof is rather abstract. For this reason, the proof by
Eliashberg and Mishachev is more prefered to see the concrete images. However,
the abstract proof by Gromov also has some advantages (cf. Section 0.5).

We, therefore, wish to develop further the abstract proof by Gromov himself.
We first clarify what a flexible sheaf is. We show that a flexible sheaf can
be interpreted as a fibrant object with respect to a model structure.

###  0.1 What’s h-principle?

Consider the following partial differential inequality $R$:

$f(x,y,\frac{\partial y}{\partial x},\cdots,\frac{\partial^{r}y}{\partial
x^{r}})>0.$ ($R$)

In analysis, we often detect a weak solution and show later that it is a
strict solution. In differential topology, we detect a formal solution and
transform it into a strict solution by homotopy. A formal solution is a family
of continuous maps $s_{0}(x),s_{1}(x),\cdots,s_{r}(x)$ such that

$f(x,s_{0}(x),s_{1}(x),\cdots,s_{r}(x))>0.$

We say that the partial differential inequality $R$ satisfies the h-principle
or the homotopy principle if any formal solution can be homotopically
transformed into a strict solution. We say that it satisfies the parametric
h-principle or the parametric homotopy principle if the inclusion map
$\operatorname{Sol}(R)\hookrightarrow\Gamma(R)$ is a weak homotopy equivalence
where $\operatorname{Sol}(R)$ is the space of strict solutions of $R$ and
$\Gamma(R)$ is the space of formal solutions of $R$.

###  0.2 Sheaf theoretic h-principle

A quasitopological space is a generalization of a topological space, whose
homotopy groups can be defined (cf. Section 1). qTop is the category of
quasitopological spaces. qTop has a model structure (cf. Section 1.2).

Let $B$ be a topological space. A continuous sheaf on $B$ is a sheaf on $B$
with value in qTop. ${\bf Sh}(B;{\bf qTop})$ is the category of continuous
sheaves on $B$. For example, the spaces of solutions to a partial differential
relation give a continuous sheaf. A continuous sheaf $\mathcal{F}$ has the
sheaf of formal sections $\mathcal{F}^{\ast}$ and the diagonal map
$\Delta:\mathcal{F}\rightarrow\mathcal{F}^{\ast}$ (cf. Section 2.3). We say
that $\mathcal{F}$ satisfies the (sheaf theoretic) parametric h-principle if
the $\Delta$ is a sectionwise weak equivalence. The following fundamental
theorem is the basis of Gromov theory.

######  Theorem 0.1 (Gromov [9, page 76]).

Any flexible sheaf on a polyhedron satisfies the parametric h-principle.

This theorem follows immediately from the following two lemmas.

######  Lemma 0.2 (Gromov [9, page 76]).

Let $B$ be a polyhedron, and let $\mathcal{F}$ be a continuous sheaf on $B$.

(1)

The diagonal morphism $\Delta:\mathcal{F}\rightarrow\mathcal{F}^{\ast}$ is a
stalkwise weak equivalence.

(2)

The sheaf $\mathcal{F}^{\ast}$ of formal sections is flexible.

######  Lemma 0.3 (Gromov [9, page 77]).

Let $B$ be a polyhedron, and let $\mathcal{F}$ and $\mathcal{G}$ be flexble
sheaves on $B$. $f:\mathcal{F}\rightarrow\mathcal{G}$ is a sectionwise weak
equivalence if $f$ is a stalkwise weak equivalence.

The above Lemma 0.2 holds in more general settings (cf. Lemma 2.10 and Lemma
4.5).

###  0.3 Main results

We want to understand a flexible sheaf as a fibrant object. Furthermore, we
hope to understand $\Delta:\mathcal{F}\rightarrow\mathcal{F}^{\ast}$ as a
fibrant replacement of $\mathcal{F}$ by Lemma 0.2.

######  Question.

Is there a model structure on ${\bf Sh}(B;{\bf qTop})$ such that the
followings hold?

  * •

A morphism $f$ in ${\bf Sh}(B;{\bf qTop})$ is a weak equivalence
$\Leftrightarrow$ $f$ is a stalkwise weak equivalence.

  * •

An object $\mathcal{F}$ in ${\bf Sh}(B;{\bf qTop})$ is fibrant
$\Leftrightarrow$ $\mathcal{F}$ is flexible.

In this paper, we give two answers. The first result is that ${\bf Sh}(B;{\bf
qTop})$ can be embedded in a model category.

######  Theorem 3.1.

There exists a right proper model category ${\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ and a fully faithful and left
exact embedding $\iota:{\bf Sh}(B;{\bf qTop})\rightarrow{\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ such that the followings
hold.

Let $f:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism in ${\bf Sh}(B;{\bf
qTop})$.

  * •

$\iota(f)$ is a weak equivalence
$\Leftrightarrow$ $f$ is a stalkwise weak equivalence (i.e.
$f_{x}:\mathcal{F}_{x}\rightarrow\mathcal{G}_{x}$ is a weak equivalence for
any $x\in B$).

  * •

$\iota(f)$ is a fibration
$\Leftrightarrow$ $f$ is a flexible extension (cf. Definition 2.6).

In particular, for a continuous sheaf $\mathcal{F}$ on $B$,

  * •

$\iota(\mathcal{F})$ is fibrant
$\Leftrightarrow$ $\mathcal{F}$ is flexible (cf. Definition 2.6).

Unfortunately, ${\bf Sh}(B;{\bf qTop})$ itself cannot be expected to have
cofibratons. Therefore, ${\bf Sh}(B;{\bf qTop})$ itself cannot be expected to
have a model structure. However, ${\bf Sh}(B;{\bf qTop})$ has fibrations and
weak equivalences. An ABC (pre)fibration structure is known as a categorical
structure equipped with fibrations and weak equivalences (cf. [22] or Section
4). The second result is that ${\bf Sh}(B;{\bf qTop})$ has an ABC prefibration
structure.

######  Theorem 4.1.

Let $B$ be a strongly locally contractible normal Hausdorff space. There
exists an ABC prefibration structure on ${\bf Sh}(B;{\bf qTop})$ such that the
followings hold.

Let $f:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism in ${\bf Sh}(B;{\bf
qTop})$.

  * •

$f$ is a weak equivalence
$\Leftrightarrow$ $f$ is a stalkwise weak equivalence (i.e.
$f_{x}:\mathcal{F}_{x}\rightarrow\mathcal{G}_{x}$ is a weak equivalence for
any $x\in B$).

  * •

$f$ is a fibration
$\Leftrightarrow$ $f$ is a flexible extension (cf. Definition 2.6).

In particular, for a continuous sheaf $\mathcal{F}$ on $B$,

  * •

$\mathcal{F}$ is fibrant
$\Leftrightarrow$ $\mathcal{F}$ is flexible (cf. Definition 2.6).

The proof of the first result is almost purely category-theoretic. On the
other hand, the proof of the second result reflects geometric considerations
by Gromov. Most of the axioms of the ABC prefibration category ${\bf
Sh}(B;{\bf qTop})$ follow from Theorem 3.1. However, only the proof of the
axiom of the existence of a decomposition needs a generalisation of Lemma 0.2
(cf. Lemma 2.10 and Lemma 4.5).

###  0.4 History of h-principles

In the Cousin problem solution, Oka [18] showed that the solvability of
certain complex analytic problems can be attributed to the solvability of
topological problems. This is what is now called Oka principle, and it can be
said that the germ of the philosophy of the h-principle already existed here.

For a complex manifold $X$, let $\mathcal{O}_{X}^{\ast}$ be the sheaf of non-
zero holomorphic functions everywhere, and let $\mathcal{C}_{X}^{\ast}$ be the
sheaf of non-zero continuous functions everywhere. Oka principle can be stated
in the following format.

######  Theorem 0.4 (Oka [18]; cf. [6]).

Let $X$ be a Stein manifold. Then, the map
$H^{1}(X;\mathcal{O}_{X}^{\ast})\rightarrow H^{1}(X;\mathcal{C}_{X}^{\ast})$
induced by the inclusion $\mathcal{O}_{X}^{\ast}\subset\mathcal{C}_{X}^{\ast}$
is an isomorphism.

The sheaf $\mathcal{O}_{X}^{\ast}$ (resp. $\mathcal{C}_{X}^{\ast}$) is
regarded as the sheaf of holomorphic (resp. continuous) functions valued in
$\mathbb{C}^{\ast}(\cong GL_{1}(\mathbb{C}))$. The homology group
$H^{1}(X;\mathcal{O}_{X}^{\ast})$ (resp. $H^{1}(X;\mathcal{C}_{X}^{\ast})$) is
regarded as the set of isomorphism classes of holomorphic (resp. continuous)
complex line bundles over $X$. Then, the above theorem can be rephrased as
follow.

######  Corollary 0.5.

Let $X$ be a Stein manifold. For any $C^{0}$-almost complex line bundle, there
exists only one holomorphic complex line bundle which is isomorphic to it, up
to holomorphic isomorphic.

Because $GL_{1}(\mathbb{C})$ is abelian, extensive skills in sheaf
cohomologies can be used to prove Oka principle. On the other hand, the
general $GL_{n}(\mathbb{C})$ is not abelian. Grauert showed that the same
statement still holds true for $GL_{n}(\mathbb{C})$ in general.

######  Theorem 0.6 (Grauert [7]; cf. [6]).

Let $X$ be a Stein manifold. For any $C^{0}$-almost complex vector bundle,
there exists only one holomorphic complex vector bundle which is isomorphic to
it, up to holomorphic isomorphic.

I do not know whether Oka principle inspired Nash’s work, but the idea of
attributing the solvability of analytic problems to the solvability of
topological problems reappears in the proof of Nash embedding theorems [17].
This had a direct impact on the later work of Hirsch [12] and Smale [24, 23],
and the prototype of the h-principle was completed as the h-principle for
immersions of manifolds. This is now known as Hirsch-Smale h-principle.

Let $M$ and $N$ be manifolds. A formal immersion $M\rightarrow N$ is a pair
$(f,\tilde{f})$ of a continuous map $f:M\rightarrow N$ and a continuous bundle
map $\tilde{f}:TM\rightarrow TN$ over $f$ such that each
$\tilde{f}:T_{x}M\rightarrow T_{f(x)}N$ is injective. Let
$\operatorname{Imm}(M,N)$ be the set of immersions $M\looparrowright N$. Let
$\operatorname{Imm^{f}}(M,N)$ be the set of formal immersions $M\rightarrow
N$. Hirsch-Smale h-principle is stated in the following format.

######  Theorem 0.7 (Hirsch [12], Smale [24, 23]).

Let $M$ and $N$ be manifolds. If we have an inequality
$\operatorname{dim}(M)<\operatorname{dim}(N)$, then the inclusion map
$\operatorname{Imm}(M,N)\subset\operatorname{Imm^{f}}(M,N)$ is a weak homotopy
equivalence.

A similar theorem on submersions is given by Phillips. Let $M$ and $N$ be
manifolds. A formal submersion $M\rightarrow N$ is a pair $(f,\tilde{f})$ of a
continuous map $f:M\rightarrow N$ and a continuous bundle map
$\tilde{f}:TM\rightarrow TN$ over $f$ such that each
$\tilde{f}:T_{x}M\rightarrow T_{f(x)}N$ is surjective. Let
$\operatorname{Sub}(M,N)$ be the set of submersions $M\rightarrow N$. Let
$\operatorname{Sub^{f}}(M,N)$ be the set of formal submersions $M\rightarrow
N$. Phillips h-principle is stated in the following format.

######  Theorem 0.8 (Phillips [19]).

Let $M$ and $N$ be manifolds. If $M$ is an open manifold, then the inclusion
map $\operatorname{Sub}(M,N)\subset\operatorname{Sub^{f}}(M,N)$ is a weak
homotopy equivalence.

Let $B$ be a smooth manifold. For each open subset $U,V(\subset B)$, we define

$\operatorname{Diff}_{B}(U,V):=\\{\mbox{A diffeomorphism }U\rightarrow V\\}.$

$\operatorname{Diff}_{B}$ is a groupoid called a pseudogroup.

Let $p:E\rightarrow B$ be a smooth fibre bundle. Let $J^{r}p:J^{r}E\rightarrow
B$ be the $r$-jet bundle of $p$. A partial differential relation is a
subbundle of the $r$-jet bundle $J^{r}p$. A partial differential relation $R$
is open if the subbundle $R$ is an open subset of $J^{r}p$. For each open
subset $U,V(\subset B)$, we define

$\operatorname{Diff}_{E/B}(U,V):=\\{\mbox{A diffeomorphism
}\hat{\phi}:U\rightarrow V\,|\,p\circ\hat{\phi}=\phi\circ
p\,(^{\exists}\phi:p(U)\rightarrow p(V))\\}.$

$\operatorname{Diff}_{E/B}$ is a groupoid. The map $\hat{\phi}\mapsto\phi$
defines a groupoid morphism
$\operatorname{Diff}_{E/B}\rightarrow\operatorname{Diff}_{B}$. A fibre bundle
$E\rightarrow B$ is natural if the groupoid morphism
$\operatorname{Diff}_{E/B}\rightarrow\operatorname{Diff}_{B}$ has a section
$\operatorname{Diff}_{B}\rightarrow\operatorname{Diff}_{E/B}$.

A partial differential relation $R$ on a natural fibre bundle $E\rightarrow B$
is $\operatorname{Diff}_{B}$-invariant if a section
$\operatorname{Diff}_{B}\rightarrow\operatorname{Diff}_{E/B}$ as above
preserves $R$. Gromov h-principle is stated in the following format.

######  Theorem 0.9 (Gromov [8]; cf. [4, 9]).

Let $p:E\rightarrow B$ be a smooth natural fibre bundle. Let $R$ be an open,
$\operatorname{Diff}_{B}$-invariant partial differential relation on $p$. If
$M$ is an open manifold, $R$ satisfies the parametric h-principle.

###  0.5 Advantages of the sheaf theoretic h-principle

Hirsch-Smale h-principle holds even if the domain manifold $M$ is not open. To
show this, it is necessary to discuss the microextension trick. One advantage
of the sheaf theoretic h-principle is that we can state the argument for the
microextension trick as the microextension theorem. Another advantage is that
it has the potential to be applied to geometries (e.g. metric spaces) other
than smooth manifolds. In this section, another advantage, which is different
from the two above, is discussed in more detail. The advantage is that it has
the potential for more generalizations, such as including Haefliger
h-principle [11] and Oka-Grauert principle [18, 7].
Any integrable distribution can be realised locally as the kernel of a
submersion. On the basis of this observation, Phillips [20, 21] was thinking
of applying Phillips h-principle to foliations. Haefliger [11] took this a
step further and gave a classification of homotopy classes of integrable
distributions.

Haefliger’s idea is that any integrable distribution can be expressed as a
kernel of submersion into a “universal $q$-dimensional manifold” $\Gamma^{q}$.
Here, the object $\Gamma^{q}$ is the étale groupoid corresponding to the
pseudogroup $\operatorname{Diff}_{\mathbb{R}^{q}}$ consisting of local
diffeomorphisms on $\mathbb{R}^{q}$. Haefliger h-principle is stated in the
following format.

######  Theorem 0.10 (Haefliger [11]).

Let $M$ be a smooth manifold. If $M$ is an open manifold, there exists a
bijection between the following.

  1. 1.

The integrable homotopy classes of integrable distribution on $M$ whose corank
is $q$.

  2. 2.

The integrable homotopy classes of the continuous bundle map $TM\rightarrow
N\Gamma^{q}$ such that it is surjective on each fibre, where $N\Gamma^{q}$ is
the classifying space of the tangent bundle $T\Gamma^{q}$ of $\Gamma^{q}$.

Haefliger h-principle means that
“$\pi_{0}(\operatorname{Sub}(M,\Gamma^{q}))\rightarrow\pi_{0}(\operatorname{Sub^{f}}(M,\Gamma^{q}))$
is bijection”.
Let $X$ and $Y$ be complex manifolds. Let $\mathcal{O}(X,Y)$ be the set of
holomorphic maps $X\rightarrow Y$. Let $\mathcal{C}(X,Y)$ be the set of
continuous maps $X\rightarrow Y$. The following properties are important in
the Oka-Grauert Theory.

######  Theorem 0.11 (cf. [6, Corollary 5.5.6]).

Let $X$ be a Stein manifold, and let $Y$ be an Oka manifold (e.g. complex Lie
group). Then, $\mathcal{O}(X,Y)\subset\mathcal{C}(X,Y)$ is a weak homotopy
equivalence.

A complex vector bundle on $X$ corresponds one-to-one to a
$GL_{n}(\mathbb{C})$-structure on $X$. In fact, a $G$-structure on $X$ is a
generalized map $X\rightarrow G$, where $G$ is any Lie groupoid. We consider a
Lie group to be a Lie groupoid whose object space is a singleton. Oka-Grauert
principle (Theorem 0.6) means that
“$\pi_{0}(\mathcal{O}(X,GL_{n}(\mathbb{C})))\rightarrow\pi_{0}(\mathcal{C}(X,GL_{n}(\mathbb{C})))$
is bijection”.
In order to unify Haefliger h-principle and Oka-Grauert principle into Gromov
h-principle, we need to develop the stack theoretic h-principle. To approach
this dream, first, need to develop the sheaf theoretic h-principle.

##  1 Quasitopological spaces

Some different topologies can be admitted into the space of maps, for example,
the Whitney topology. However, if we admit the quotient topology into the
stalk, the topologies will be collapsed. Here we want to deal with the germs
while keeping the local information. The solution to this problem is a
quasitopological space. The discussion in Section 1.1.4 solves this problem.

The idea of a quasitopological space is credited to Spanier and Whithead [26].
Spanier [25] has written a paper on quasitopological spaces. A
quasitopological space dealt with in this paper is not exactly the same as the
one they defined, but it is almost similar.

In this paper, a quasitopological space is a sheaf on a certain small site.
(For more information on sites and sheaves, see [15].) The category of
quasitopological spaces has a model structure.

###  1.1 Quasitopological spaces

We will define a quasitopological space and check its properties in this
subsection. The discussion in Section 1.1.4 is often used when we have a
stalkwise discussion on continuous sheaves. The discussion in Section 1.1.5
implies that finite polyhedra (e.g. simplexes, horns, their boundaries, etc.)
can be treated in the same way as topological spaces.

####  1.1.1 Prequasitopological spaces

Let Sing$\Delta$ be a small category defined as follows.

  * •

An object of Sing$\Delta$ is a closed subset of a simplex $\Delta^{n}$.

  * •

A morphism of Sing$\Delta$ is a continuous map.

######  Definition 1.1.

A prequasitopological space is a presheaf on the small category Sing$\Delta$.
A morphism between prequasitopological spaces is a morphism between
presheaves. pqTop is the category of prequasitopological spaces.

The category pqTop is the category ${\bf PSh}({\bf Sing\Delta})$ of
presheaves. This category is complete and cocomplete and Cartesian closed. For
a prequasitopological space $X$ and a closed subset $S$ of a simplex, denote
by $X[S]$ the set to which $S$ is mapped by a functor $X:{\bf
Sing\Delta}^{op}\rightarrow{\bf Set}$, where Set is the category of sets and
maps.

######  Example 1.2.

For a topological space $X$, we define a prequasitopological space
$\operatorname{Sing^{q}}(X)$ as the following:

$\operatorname{Sing^{q}}(X)[S]:=\\{\mbox{continuous map }S\rightarrow X\\}.$

The construction $X\mapsto\operatorname{Sing^{q}}(X)$ defines a functor
$\operatorname{Sing^{q}}:{\bf Top}\rightarrow{\bf pqTop}$. In particular, if
$X$ is a closed subset of a simplex, we obtain an equality
$\operatorname{Sing^{q}}(X)={\bf Sing\Delta}(-,X)$. If $X=\Delta^{n}$, and
there is no confusion, then $\operatorname{Sing^{q}}(\Delta^{n})$ is denoted
as $\Delta^{n}$.

######  Remark 1.3.

Each element $s\in X[S]$ corresponds one-to-one with a morphism
$s^{\prime}:\operatorname{Sing^{q}}(S)\rightarrow X$ by Yoneda’s Lemma. We
often equate $s$ with $s^{\prime}$, and use the same symbol.

####  1.1.2 Quasitopological spaces

The small category Sing$\Delta$ is a small site by defining a covering of an
object $S$ as a finite closed covering of the topological space $S$.

######  Definition 1.4.

A quasitopological space is a sheaf on the small site Sing$\Delta$. A morphism
between quasitopological spaces is a morphism between sheaves. qTop is the
category of quasitopological spaces.

The category qTop is the category ${\bf Sh}({\bf Sing\Delta})$ of sheaves.
This category is complete and cocomplete and Cartesian closed.

######  Example 1.5.

For a topological space $X$, the prequasitopological space
$\operatorname{Sing^{q}}(X)$ is a quasitopological space.

####  1.1.3 Geometric realizations

We define a functor $G_{2}:{\bf Top}\rightarrow{\bf qTop}$ as
$G_{2}(X):=\operatorname{Sing^{q}}(X)$. The functor $G_{2}$ is the left Kan
extension of Yoneda embedding $y:{\bf Sing\Delta}\rightarrow{\bf qTop}$ along
the inclusion functor ${\bf Sing\Delta}\hookrightarrow{\bf Top}$.

$\textstyle{{\bf qTop}}$$\textstyle{{\bf
Sing\Delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y}$$\textstyle{{\bf
Top}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{G_{2}}$

Let $F_{2}:{\bf qTop}\rightarrow{\bf Top}$ be the left Kan extension of the
inclusion functor ${\bf Sing\Delta}\hookrightarrow{\bf Top}$ along $y$. It is
classically known that there exists an adjunction $F_{2}\dashv G_{2}$. The
functor $F_{2}$ is called the geometric realizations. Specifically, $F_{2}$ is
defined as $F_{2}(X):=\int^{S}{\bf qTop}(S,X)\bullet S$ (cf. [16, Chapter 10,
Section 4]), where the integral symbol $\int$ is the coend and
“$(-)\bullet(-)$” is the copower object.

####  1.1.4 Filtered colimit trick

A limit on the category qTop can be calculated objectwise. (i.e. For any small
diagram $X_{\lambda}$ in qTop and any object $S$ in Sing$\Delta$, we have an
isomorphism
$(\displaystyle\lim_{\longleftarrow}X_{\lambda})[S]\cong\displaystyle\lim_{\longleftarrow}(X_{\lambda}[S])$.)
A colimit cannot be calculated in the same way in general. However, because
the covering on the site Sing$\Delta$ is finite, a filtered colimit can be
calculated objectwise.

######  Proposition 1.6.

For any closed subset $S$ of a simplex, the evaluation functor
$\operatorname{ev}_{S}:{\bf qTop}\rightarrow{\bf Set};X\mapsto X[S]$ preserves
any filtered colimit.

###### Proof.

Take any closed subset $S$ of a simplex and any filtered diagram $X_{\lambda}$
($\lambda\in\Lambda$) in qTop. We will show that the following canonical map
is an isomorphism:

$\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}(X_{\lambda}[S])\rightarrow(\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}X_{\lambda})[S].$ (1)

For any index $\lambda\in\Lambda$ and any finite closed covering
$\\{S_{i}\\}_{i}$ of $S$, the following diagram is an equalizer:

$X_{\lambda}[S]\rightarrow\prod_{i}X_{\lambda}[S_{i}]\rightrightarrows\prod_{i,j}X_{\lambda}[S_{i}\cap
S_{j}].$

Filtered colimits commute with finite limits in Set (cf. [16, Chapter 9,
Section 2]). Then, the following diagram is an equalizer:

$\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}(X_{\lambda}[S])\rightarrow\prod_{i}\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}(X_{\lambda}[S_{i}])\rightrightarrows\prod_{i,j}\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}(X_{\lambda}[S_{i}\cap S_{j}]).$ (2)

We define a prequasitopological space $Y$ as the following:

$Y[S]:=\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}(X_{\lambda}[S]).$

$Y$ is a quasitopological space because the diagram (2) is an equalizer. The
colimit of the diagram $X_{\lambda}$ is the sheafification of $Y$. However,
$Y$ is a sheaf, therefore the colimit of the diagram $X_{\lambda}$ is $Y$.
This means that the canonical map (1) is an isomorphism. Then, for any closed
subset $S$ of a simplex, the evaluation functor $\operatorname{ev}_{S}$
preserves any filtered colimit. ∎

The Hom-functor ${\bf qTop}(\operatorname{Sing^{q}}(S),-):{\bf
qTop}\rightarrow{\bf Set}$ is naturally isomorphic to the evaluation functor
$\operatorname{ev}_{S}$ by Yoneda’s Lemma. Proposition 1.6 means that, for any
filtered diagram $X_{\lambda}$ ($\lambda\in\Lambda$) in qTop, we have the
following isomorphism:

${\bf
qTop}(\operatorname{Sing^{q}}(S),\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}X_{\lambda})\cong\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}{\bf qTop}(\operatorname{Sing^{q}}(S),X_{\lambda}).$

We obtain the following proposition.

######  Proposition 1.7.

Let $S$ be a closed subset of a simplex, and let $X_{\lambda}$
($\lambda\in\Lambda$) be a filtered diagram in qTop. For any morphism
$f:\operatorname{Sing^{q}}(S)\rightarrow\displaystyle\lim_{\longrightarrow}X_{\lambda}$,
we have the followings.

  * •

There exist an index $\lambda$ and a morphism
$\tilde{f}:\operatorname{Sing^{q}}(S)\rightarrow X_{\lambda}$ such that the
following diagram is commutative:

$\textstyle{X_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Sing^{q}}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\tilde{f}}$$\textstyle{\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}X_{\lambda}.}$

  * •

If there exist an index $\lambda^{\prime}$ and a morphism
$\tilde{f^{\prime}}$, similar to the index $\lambda$ and the morphism
$\tilde{f}$ above, then there exist two arrows
$\lambda\rightarrow\mu\leftarrow\lambda^{\prime}$ between indexes such that
the following diagram is commutative:

$\textstyle{\operatorname{Sing^{q}}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f^{\prime}}}$$\scriptstyle{\tilde{f}}$$\textstyle{X_{\lambda^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{\mu}.}$

###### Proof.

The statement is a rephrasing of the definition of a filtered colimit in Set.
∎

We consider the following diagram:

$\textstyle{\operatorname{Sing^{q}}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}X_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Sing^{q}}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\lambda\end{subarray}}Y_{\lambda},}$

where $S$ and $T$ are closed subsets of simplexes, and $X_{\lambda}$ and
$Y_{\lambda}$ ($\lambda\in\Lambda$) are filtered diagrams in qTop. Then, there
exists a following commutative diagram:

$\textstyle{\operatorname{Sing^{q}}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Sing^{q}}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{\lambda}.}$

By the discussion in the later section, Section 1.1.5, we may accept $S$ and
$T$ as finite polyhedra.

####  1.1.5 Patching Lemma

The adjunction $F_{2}\dashv G_{2}:{\bf qTop}\rightarrow{\bf Top}$ is not an
equivalence. However, if $X$ is a finite polyhedron in Top (resp. qTop), ${\bf
Top}(X,Y)\rightarrow{\bf
qTop}(\operatorname{Sing^{q}}(X),\operatorname{Sing^{q}}(Y))$ (resp. ${\bf
qTop}(X,Y)\rightarrow{\bf Top}(F_{2}(X),F_{2}(Y))$) is a bijection. As such,
if $X$ is a finite polyhedron, then the quasitopological space
$\operatorname{Sing^{q}}(X)$ behaves in the almost same way as the topological
space $X$. These can be seen from the following proposition.

######  Lemma 1.8.

Let $X$ be a topological space, and let $\\{X_{i}\\}_{i}$ be a finite closed
covering of $X$. Then the following diagram is a coequalizer.

$\coprod_{i,j}\operatorname{Sing^{q}}(X_{i})\cap\operatorname{Sing^{q}}(X_{j})\rightrightarrows\coprod_{i}\operatorname{Sing^{q}}(X_{i})\rightarrow\operatorname{Sing^{q}}(X)$

###### Proof.

We will show that there exists $f^{\prime}$ in the following diagram.

$\textstyle{\displaystyle\coprod_{i,j}\operatorname{Sing^{q}}(X_{i})\cap\operatorname{Sing^{q}}(X_{j})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\coprod_{i}\operatorname{Sing^{q}}(X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\operatorname{Sing^{q}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{Y}$
(1)

Take any $s\in\operatorname{Sing^{q}}(X)[S](={\bf Top}(S,X))$. Let
$S_{i}:=s^{-1}(X_{i})$. Then $\\{S_{i}\\}_{i}$ is a finite closed covering of
$S$, and $s|_{S_{i}}\in\operatorname{Sing^{q}}(X_{i})[S_{i}](={\bf
Top}(S_{i},X_{i}))$. Let $t_{i}:=f(s|_{S_{i}})\in Y[S_{i}]$. Then, we have the
following equality:

$\begin{array}[]{rcl}t_{i}|_{S_{i}\cap S_{j}}&=&f(s|_{S_{i}})|_{S_{i}\cap
S_{j}}\\\ &=&f(s|_{S_{i}\cap S_{j}})\\\ &=&f(s|_{S_{j}})|_{S_{i}\cap S_{j}}\\\
&=&t_{j}|_{S_{i}\cap S_{j}}.\end{array}$

Then, the family of elements $(t_{i})_{i}\in\prod Y[S_{i}]$ lifts a unique
element $t\in Y[S]$. If we define $f^{\prime}(s):=t$, the above diagram (1) is
commutative. Conversely, if there exists $f^{\prime}$ such that the above
diagram (1) is commutative, this coinside with the above definition by
uniqueness of $t$. ∎

######  Proposition 1.9.

  1. 1.

If $X$ is a finite polyhedron in Top, ${\bf Top}(X,Y)\rightarrow{\bf
qTop}(\operatorname{Sing^{q}}(X),\operatorname{Sing^{q}}(Y))$ is a bijection.

  2. 2.

If $X$ is a finite polyhedron in qTop, ${\bf qTop}(X,Y)\rightarrow{\bf
Top}(F_{2}(X),F_{2}(Y))$ is a bijection.

###### Proof.

We will show the claim 1.. The proof of the claim 2. is similar.

If $X$ is a finite polyhedron, there exists a finite closed covering
$\\{X_{i}\\}_{i}$ of $X$ such that each $X_{i}$ is homeomorphic to a simplex.
Let $X^{\prime}$, $Y^{\prime}$ and $X^{\prime}_{i}$ be the quasitopological
spaces $\operatorname{Sing^{q}}(X)$, $\operatorname{Sing^{q}}(Y)$ and
$\operatorname{Sing^{q}}(X_{i})$ respectively. Then, ${\bf
Top}(X_{i},Y)\rightarrow{\bf qTop}(X^{\prime}_{i},Y^{\prime})$ and ${\bf
Top}(X_{i}\cap X_{j},Y)\rightarrow{\bf qTop}(X^{\prime}_{i}\cap
X^{\prime}_{j},Y^{\prime})$ are bijections. We consider the following diagram:

$\textstyle{{\bf
Top}(X,Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\prod_{i}{\bf
Top}(X_{i},Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\prod_{i,j}{\bf
Top}(X_{i}\cap
X_{j},Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
qTop}(X^{\prime},Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\prod_{i}{\bf
qTop}(X^{\prime}_{i},Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\prod_{i,j}{\bf
qTop}(X^{\prime}_{i}\cap X^{\prime}_{j},Y^{\prime}).}$

The upper row is an equalizer. The bottom row is an equalizer by Lemma 1.8.
The right two columns are isomorphisms. Then, the left column is an
isomorphism. ∎

${\bf Top^{f}}$ is the category of finite polyhedra in Top.

${\bf qTop^{f}}$ is the category of finite polyhedra in qTop.

######  Corollary 1.10.

The adjunction $F_{2}\dashv G_{2}$ indeces a categorical equivalence ${\bf
qTop^{f}}\simeq{\bf Top^{f}}$.

###### Proof.

It is obvious by Proposition 1.9. ∎

###  1.2 Some model structures

In this subsection, we will show that the category qTop of quasitopological
spaces has a model structure. Furthermore, we will review Homotopy Theory in
qTop.
Consider the following adjunctions sequence:

$\textstyle{{\bf
sSet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{0}}$$\scriptstyle{\top}$$\textstyle{{\bf
pqTop}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{0}}$$\scriptstyle{F_{1}}$$\scriptstyle{\top}$$\textstyle{{\bf
qTop}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{1}}$$\scriptstyle{F_{2}}$$\scriptstyle{\top}$$\textstyle{{\bf
Top}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{G_{2}}$

The above categories and functors are defined as follows.

  * •

sSet is the category of simplicial sets.

  * •

pqTop is the category of prequasitopological spaces (cf. Section 1.1.1).

  * •

qTop is the category of quasitopological spaces (cf. Section 1.1.2).

  * •

Top is the category of topological spaces and continuous maps.

  * •

$G_{0}$ is the functor ${\bf Set}^{{\bf Sing\Delta}^{op}}\rightarrow{\bf
Set}^{{\bf\Delta}^{op}}$ induced by the inclusion functor
${\bf\Delta}\hookrightarrow{\bf Sing\Delta}$, where ${\bf\Delta}$ is the
simplicial category.

  * •

$G_{1}$ is the forgetful functor.

  * •

$G_{2}$ is defined as $G_{2}(X):=\operatorname{Sing^{q}}(X)$ (cf. Section
1.1.3).

  * •

$F_{1}$ is the sheafification.

  * •

$F_{0}$ and $F_{2}$ are defined in the same way as the geometric realizations.
Specifically, these are defined as follows.

    * –

$F_{0}(X):=\int^{n}{\bf sSet}(\Delta^{n-1},X)\bullet\Delta^{n-1}$.

    * –

$F_{2}(X):=\int^{S}{\bf qTop}(S,X)\bullet S$ (cf. Section 1.1.3).

Each is the left Kan extension of the appropriate diagram.

We will show that pqTop (resp. qTop) has a model structure by Transfer Theorem
(cf. Theorem A.10). Furthermore, all the above adjunctions are Quillen
equivalences.

####  1.2.1 A model structure for prequasitopological spaces

We will show that the category pqTop has a model structure in this
subsubsection. We will prepare for this.
Let $I_{\bf sSet}$, $J_{\bf sSet}$ and $W_{\bf sSet}$ be the sets of some
morphisms in sSet defined as the followings:

$\begin{array}[]{rcl}I_{\bf
sSet}&:=&\\{\partial\Delta^{n}\hookrightarrow\Delta^{n}\,|\,n\\},\\\ J_{\bf
sSet}&:=&\\{\Lambda^{n-1}_{i}\hookrightarrow\Delta^{n}\,|\,n,i\\}.\end{array}$

It is classically known that the category sSet has a cofibrantly generated
model structure such that the class of cofibrations is generated by $I_{\bf
sSet}$ and the class of acyclic cofibrations is generated by $J_{\bf sSet}$
(cf. Definition A.8). Let $W_{\bf sSet}$ be the class of weak equivalences in
sSet.

######  Lemma 1.11.

The functor $F_{0}:{\bf sSet}\rightarrow{\bf pqTop}$ preserves any
monomorphism.

###### Proof.

Any monomorphism between simplicial sets is a cofibration. $F_{0}$ preserves
any push-out, any transfinite composition and any retract. Then, we obtain the
inclusion relation $F_{0}(\operatorname{Cof}(I_{\bf
sSet}))\subset\operatorname{Cof}(F_{0}I_{\bf sSet})$. Any morphism
$\partial\Delta^{n}\rightarrow\Delta^{n}$ in the set $F_{0}I_{\bf sSet}$ is a
monomorphism in pqTop. Any push-out, any transfinite composition and any
retract preserve monomorphisms in pqTop by the objectwise calculation. Then,
any morphism belonging to $\operatorname{Cof}(F_{0}I_{\bf sSet})$ is a
monomorphism. Therefore, the functor $F_{0}$ preserves any monomorphism. ∎

######  Lemma 1.12.

Let $\epsilon:F_{0}G_{0}\rightarrow\operatorname{Id}_{\bf pqTop}$ be the
counit of the adjunction $F_{0}\dashv G_{0}:{\bf sSet}\rightarrow{\bf pqTop}$.
For any prequasitopological space $X$, the morphism $\epsilon_{X}$ has the
right lifting property with respect to $F_{0}I_{\bf sSet}$.

###### Proof.

For any square in the following diagram, we will show that there is a morphism
$\gamma$ below.

$\textstyle{\partial\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{F_{0}(G_{0}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{X}}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\textstyle{X.}$

By definition, we obtain the following equality:

$\begin{array}[]{rcl}F_{0}(G_{0}(X))&=&\int^{k}{\bf
sSet}(\Delta^{k-1},G_{0}(X))\bullet\Delta^{k-1}\\\
&=&\int^{k}G_{0}(X)[\Delta^{k-1}]\bullet\Delta^{k-1}\\\
&=&\int^{k}X[\Delta^{k-1}]\bullet\Delta^{k-1}.\end{array}$

Then, we have the quotient morphism
$\coprod_{k}X[\Delta^{k}]\bullet\Delta^{k}\rightarrow F_{0}(G_{0}(X))$. Let
$\gamma$ be the composition of the sequence
$\Delta^{n}\overset{\beta}{\rightarrow}X[\Delta^{n}]\bullet\Delta^{n}\rightarrow\coprod_{k}X[\Delta^{k}]\bullet\Delta^{k}\rightarrow
F_{0}(G_{0}(X))$. Obviously, we have the equality
$\epsilon_{X}\circ\gamma=\beta$. For any embedding
$\sigma_{i}:\Delta^{n-1}\hookrightarrow\Delta^{n}$, the following diagram is
commutative:

$\textstyle{\Delta^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\circ\sigma_{i}}$$\scriptstyle{\sigma_{i}}$$\textstyle{F_{0}(G_{0}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{X}}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{X.}$

Then, the following diagram is commutative:

---
$\textstyle{\Delta^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\circ\sigma_{i}}$$\scriptstyle{\sigma_{i}}$$\textstyle{X[\Delta^{n-1}]\bullet\Delta^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id\bullet\sigma_{i}}$$\textstyle{X[\Delta^{n-1}]\bullet\Delta^{n}}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{X[\Delta^{n}]\bullet\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{X[\sigma_{i}]\bullet
id}$

This means that the following diagram is commutative:

$\textstyle{\Delta^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{i}}$$\scriptstyle{\sigma_{i}}$$\textstyle{\partial\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{F_{0}(G_{0}(X)).}$

We obtain the equality $\gamma|_{\partial\Delta^{n}}=\alpha$. Then, the
morphism $\epsilon_{X}$ has the right lifting property with respect to
$F_{0}I_{\bf sSet}$. ∎

######  Lemma 1.13.

Let $\eta:\operatorname{Id}_{\bf sSet}\rightarrow G_{0}F_{0}$ be the unit of
the adjunction $F_{0}\dashv G_{0}:{\bf sSet}\rightarrow{\bf pqTop}$.

(1)

$\eta_{\Delta^{n}}:\Delta^{n}\rightarrow G_{0}(F_{0}(\Delta^{n}))$ is a weak
equivalence.

(2)

$\eta_{\Lambda^{n}_{i}}:\Lambda^{n}_{i}\rightarrow
G_{0}(F_{0}(\Lambda^{n}_{i}))$ is a weak equivalence.

###### Proof.

The claim (1) is obvious by the equality
$G_{0}(F_{0}(\Delta^{n}))=\operatorname{Sing}(\Delta^{n})$.

We will show the claim (2) by induction on $n$. If $n=0$, the morphism
$\eta_{\Lambda^{0}_{0}}:{\bf 1}\rightarrow{\bf 1}$ is an isomorphism. This is
a weak equivalence.

Assume that for any $k$ such that $k<n$, the claim that $n$ is replaced by $k$
holds. Let $\Gamma^{k}$ be the simplicial set generated by $k$-surfaces of
$\Lambda^{n}_{i}$ containing the $i$-th vertex. We consider the following
diagram:

$\textstyle{\Gamma^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma^{n-1}=\Lambda^{n}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{\Lambda^{n}_{i}}}$$\textstyle{G_{0}(F_{0}(\Gamma^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{0}(F_{0}(\Gamma^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{0}(F_{0}(\Gamma^{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{0}(F_{0}(\Gamma^{n-1})=G_{0}(F_{0}(\Lambda^{n}_{i}).}$

We will show that $\Gamma^{k}\rightarrow G_{0}(F_{0}(\Gamma^{k})$ is a weak
equivalence by induction on $k$.

If $k=0$, this is obvious because of the equality $\Gamma^{0}={\bf
1}=G_{0}(F_{0}(\Gamma^{0})$.

Assume that for any $l$ such that $l<k$, $\Gamma^{l}\rightarrow
G_{0}(F_{0}(\Gamma^{l})$ is a weak equivalence. $\Gamma^{k-1}\rightarrow
G_{0}(F_{0}(\Gamma^{k-1})$ is a weak equivalence.
$\Gamma^{k-1}\rightarrow\Gamma^{k}$ is a finite relative
$\\{\Lambda^{k}_{j}\rightarrow\Delta^{k}\,|\,j\\}$-cell complex. i.e. There
exists a finite sequence:

$\Gamma^{k-1}=X_{0}\rightarrow X_{1}\rightarrow\cdots\rightarrow
X_{N}=\Gamma^{k},$

where each component $X_{s-1}\rightarrow X_{s}$ is a push-out of
$\Lambda^{k}_{j}\rightarrow\Delta^{k}$. $F_{0}$ preserves any colimit because
it is a left adjoint functor. $G_{0}$ preserves any colimit by the objectwise
calculation. $G_{0}(F_{0}(\Gamma^{k-1})\rightarrow G_{0}(F_{0}(\Gamma^{k})$ is
a finite relative $\\{G_{0}(F_{0}(\Lambda^{k}_{j})\rightarrow
G_{0}(F_{0}(\Delta^{k})\,|\,j\\}$-cell complex. If we show that
$G_{0}(F_{0}(\Lambda^{k}_{j})\rightarrow G_{0}(F_{0}(\Delta^{k})$ is an
acyclic cofibration, We obtain that $G_{0}(F_{0}(\Gamma^{k-1})\rightarrow
G_{0}(F_{0}(\Gamma^{k})$ is an acyclic cofibration. We will show that
$G_{0}(F_{0}(\Lambda^{k}_{j})\rightarrow G_{0}(F_{0}(\Delta^{k})$ is an
acyclic cofibration.

$\Lambda^{k}_{j}\rightarrow G_{0}(F_{0}(\Lambda^{k}_{j}))$ is a weak
equivalence by the assumption of induction.
$G_{0}(F_{0}(\Lambda^{k}_{j})\rightarrow G_{0}(F_{0}(\Delta^{k})$ is a weak
equivalence by the following diagram:

$\textstyle{\Lambda^{k}_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\textstyle{\Delta^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{G_{0}(F_{0}(\Lambda^{k}_{j}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{0}(F_{0}(\Delta^{k})).}$

$F_{0}$ preserves any monomorphism by Lemma 1.11. $G_{0}$ preserves any
monomorphism by the objectwise calculation.
$G_{0}(F_{0}(\Lambda^{k}_{j})\rightarrow G_{0}(F_{0}(\Delta^{k})$ is a
cofibration. Then, this is an acyclic cofibration.

Therefore, any relative $\\{G_{0}(F_{0}(\Lambda^{k}_{j})\rightarrow
G_{0}(F_{0}(\Delta^{k})\,|\,j\\}$-cell complex is an acyclic cofibration.
Then, $G_{0}(F_{0}(\Gamma^{k-1})\rightarrow G_{0}(F_{0}(\Gamma^{k})$ is a weak
equivalence because it is an acyclic cofibration. $\Gamma^{k}\rightarrow
G_{0}(F_{0}(\Gamma^{k})$ is a weak equivalence by the following diagram:

$\textstyle{\Gamma^{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\textstyle{\Gamma^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{0}(F_{0}(\Gamma^{k-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{G_{0}(F_{0}(\Gamma^{k})).}$

This completes the induction on $k$. Therefore, for any $k(\leq n)$,
$\Gamma^{k}\rightarrow G_{0}(F_{0}(\Gamma^{k})$ is a weak equivalence. In
particular, by considering the case $k=n$,
$\eta_{\Lambda^{n}_{i}}:\Lambda^{n}_{i}\rightarrow
G_{0}(F_{0}(\Lambda^{n}_{i}))$ is a weak equivalence. This completes the
induction on $n$. Therefore, for any $n$,
$\eta_{\Lambda^{n}_{i}}:\Lambda^{n}_{i}\rightarrow
G_{0}(F_{0}(\Lambda^{n}_{i}))$ is a weak equivalence. ∎

Let $I_{\bf pqTop}$, $J_{\bf pqTop}$ and $W_{\bf pqTop}$ be sets as the
followings:

$\begin{array}[]{rcl}I_{\bf pqTop}&:=&F_{0}I_{\bf sSet},\\\ J_{\bf
pqTop}&:=&F_{0}J_{\bf sSet},\\\ W_{\bf pqTop}&:=&G_{0}^{-1}W_{\bf
sSet},\end{array}$

Now we are ready to prove that pqTop has a model structure.

######  Theorem 1.14.

pqTop has a cofibrantly generated model structure satisfying the followings:

  * •

$I_{\bf pqTop}$ generates cofibrations,

  * •

$J_{\bf pqTop}$ generates acyclic cofibrations, and

  * •

$W_{\bf pqTop}$ is the class of weak equivalences.

Furthermore, $F_{0}\dashv G_{0}:{\bf sSet}\rightarrow{\bf pqTop}$ is a Quillen
equivalence.

###### Proof.

In order to apply Theorem A.10, we show the followings:

(1)

$I_{\bf pqTop}$ and $J_{\bf pqTop}$ permit the small object argument.

(2)

We have the inclusion relation $\operatorname{cell}(J_{\bf pqTop})\subset
W_{\bf pqTop}$.

We will show the condition (1). Any simplex $\Delta^{n}$ is small in pqTop by
Example A.3. $\partial\Delta^{n}$ and $\Lambda^{n}_{i}$ are finite polyhedra.
These are small in pqTop by Proposition A.5.

We will show the condition (2). $F_{0}$ preserves any monomorphism by Lemma
1.11. $G_{0}$ preserves any monomorphism by the objectwise calculation. Then,
$G_{0}(F_{0}(\Lambda^{n}_{i}\hookrightarrow\Delta^{n}))$ is a monomorphism.
This means that it is a cofibration in sSet. Furthermore, it is a weak
equivalence by Lemma 1.13. We obtain the inclusion relation $G_{0}(J_{\bf
pqTop})\subset\operatorname{Cof}_{\bf sSet}\cap W_{\bf sSet}$ because
$G_{0}(F_{0}(\Lambda^{n}_{i}\hookrightarrow\Delta^{n}))$ is an acyclic
cofibration. $G_{0}$ preserves any colimit by the objectwise calculation. We
obtain the sequence of the inclusion relations
$G_{0}(\operatorname{cell}(J_{\bf pqTop}))=\operatorname{cell}(G_{0}(J_{\bf
pqTop}))\subset\operatorname{Cof}_{\bf sSet}\cap W_{\bf sSet}\subset W_{\bf
sSet}$. Then, we obtain the inclusion relation $\operatorname{cell}(J_{\bf
pqTop})\subset G_{0}^{-1}(W_{\bf sSet})=W_{\bf pqTop}$.

Therefpre, pqTop has a cofibrantly generated model structure by Theorem A.10.

We will show that the Quillen adjunction $F_{0}\dashv G_{0}$ is a Quillen
equivalence. We consider the following diagram:

$\textstyle{{\bf
pqTop}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}F_{1}\dashv
G_{1}G_{2}}$$\textstyle{{\bf
sSet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{0}\dashv
G_{0}}$$\scriptstyle{|-|\dashv\operatorname{Sing}}$$\scriptstyle{\simeq}$$\textstyle{{\bf
Top},}$

where $|-|$ is the geometric realization and $\operatorname{Sing}(-)$ is the
singular simplicial complex. It is classically known that
$|-|\dashv\operatorname{Sing}$ is a Quillen equivalence. Then, the Quillen
adjunction $F_{0}\dashv G_{0}$ is a “split monomorphism”. Strictly speaking,
the unit of the derived adjunction $\mathbb{L}F_{0}\dashv\mathbb{R}G_{0}$ is a
natural isomorphism. The counit of the adjunction $F_{0}\dashv G_{0}$ is a
weak equivalence by Lemma 1.12. All objects in sSet are cofibrant. Then, the
counit of the derived adjunction $\mathbb{L}F_{0}\dashv\mathbb{R}G_{0}$ is a
natural isomorphism. $\mathbb{L}F_{0}\dashv\mathbb{R}G_{0}$ is a categorical
equivalence. Therefore, $F_{0}\dashv G_{0}$ is a Quillen equivalence. ∎

######  Remark 1.15.

In fact, any acyclic cofibratoin in sSet is a relative $J_{\bf sSet}$-cell
complex. However, I think, this is not well known. Using this fact, the proof
of Theorem 1.14 can be simplified slightly by Corollary A.11, which we will
not explain here.

####  1.2.2 A model structure for quasitopological spaces

We will show that the category qTop has a model structure in this
subsubsection. We will prepare for this.

######  Lemma 1.16.

Let $\eta$ be the unit of adjunction $F_{1}\dashv G_{1}$. For any
prequasitopological space $X$, $\eta_{X}:X\rightarrow G_{1}(F_{1}(X))$ is a
weak equivalence in pqTop.

###### Proof.

$F_{0}\dashv G_{0}$ and $|-|\dashv\operatorname{Sing}$ are Quillen
equivalences. Then, $F_{2}F_{1}\dashv G_{1}G_{2}$ is a Quillen equivalence.
All objects in Top are fibrant. Then, the unit map of the adjunction
$F_{2}F_{1}\dashv G_{1}G_{2}$ is a weak equivalence. We have an isomorphism
$F_{1}(G_{1}(F_{1}(X)))\cong F_{1}(X)$ because the counit of the adjunction
$F_{1}\dashv G_{1}$ is a natural isomorphism. We consider the following
diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\eta_{X}}$$\textstyle{G_{1}(F_{1}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{G_{1}(G_{2}(F_{2}(F_{1}(X))))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{G_{1}(G_{2}(F_{2}(F_{1}(G_{1}(F_{1}(X)))))).}$

Then, $\eta_{X}$ is a weak equivalence. ∎

######  Lemma 1.17.

Let $\epsilon$ be the counit of adjunction $F_{1}F_{0}\dashv G_{0}G_{1}:{\bf
sSet}\rightarrow{\bf qTop}$. For any quasitopological space $X$,
$G_{1}(\epsilon_{X}):G_{1}(F_{1}(F_{0}(G_{0}(G_{1}(X)))))\rightarrow G_{1}(X)$
is a weak equivalence in pqTop.

###### Proof.

We consider the following diagram:

$\textstyle{F_{0}(G_{0}(G_{1}(X)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{G_{1}(F_{1}(F_{0}(G_{0}(G_{1}(X)))))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{1}(\epsilon_{X})}$$\textstyle{G_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{1}(X).}$

$\alpha$ is a weak equivalence by Lemma 1.12. $\beta$ is a weak equivalence by
Lemma 1.16. Then, $G_{1}(\epsilon_{X})$ is a weak equivalence. ∎

Let $I_{\bf qTop}$, $J_{\bf qTop}$ and $W_{\bf qTop}$ be sets as the
followings:

$\begin{array}[]{rcl}I_{\bf qTop}&:=&F_{1}I_{\bf pqTop},\\\ J_{\bf
qTop}&:=&F_{1}J_{\bf pqTop},\\\ W_{\bf qTop}&:=&G_{1}^{-1}W_{\bf
pqTop},\end{array}$

Now we are ready to prove that qTop has a model structure.

######  Theorem 1.18.

qTop has a cofibrantly generated model structure satisfying the followings:

  * •

$I_{\bf qTop}$ generates cofibrations,

  * •

$J_{\bf qTop}$ generates acyclic cofibrations, and

  * •

$W_{\bf qTop}$ is the class of weak equivalences.

Furthermore, $F_{1}\dashv G_{1}:{\bf pqTop}\rightarrow{\bf qTop}$ is a Quillen
equivalence.

###### Proof.

In order to apply Theorem A.10, we show the followings:

(1)

$I_{\bf qTop}$ and $J_{\bf qTop}$ permit the small object argument.

(2)

We have the inclusion relation $\operatorname{cell}(J_{\bf qTop})\subset
W_{\bf qTop}$.

We will show the condition (1). Any simplex $\Delta^{n}$ is small in pqTop by
Example A.4. $\partial\Delta^{n}$ and $\Lambda^{n}_{i}$ are finite polyhedra.
These are small in qTop by Proposition A.5.

We will show the condition (2). Any element $G_{1}(f)\in G_{1}(J_{\bf qTop})$
is isomorphic to
$\Delta^{n-1}\overset{0}{\hookrightarrow}\Delta^{n-1}\times[0,1]$ by Corollary
1.10. This is isomorphic to a acyclic cofibration
$F_{0}(\Delta^{n-1}\hookrightarrow\Delta^{n})$ in pqTop. Then, we obtain the
inclusion relation $\operatorname{cell}(G_{1}(J_{\bf qTop}))\subset W_{\bf
pqTop}$. We have the sequence of the inclusion relation
$G_{1}(\operatorname{cell}(J_{\bf qTop}))\subset
G_{1}(F_{1}(\operatorname{cell}(G_{1}(J_{\bf qTop}))))\subset W_{\bf pqTop}$
by Lemma 1.16. Then, we obtain the inclusion relation
$\operatorname{cell}(J_{\bf qTop})\subset W_{\bf qTop}$.

Therefpre, pqTop has a cofibrantly generated model structure by Theorem A.10.

We will show that the Quillen adjunction $F_{1}\dashv G_{1}$ is a Quillen
equivalence. We consider the following diagram:

$\textstyle{{\bf
qTop}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}\dashv
G_{2}}$$\textstyle{{\bf
sSet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{1}F_{0}\dashv
G_{0}G_{1}}$$\scriptstyle{|-|\dashv\operatorname{Sing}}$$\scriptstyle{\simeq}$$\textstyle{{\bf
Top}.}$

It is classically known that $|-|\dashv\operatorname{Sing}$ is a Quillen
equivalence. Then, the Quillen adjunction $F_{1}F_{0}\dashv G_{0}G_{1}$ is a
“split monomorphism”. Strictly speaking, the unit of the derived adjunction
$\mathbb{L}(F_{1}F_{0})\dashv\mathbb{R}(G_{0}G_{1})$ is a natural isomorphism.
The counit of the adjunction $F_{1}F_{0}\dashv G_{0}G_{1}$ is a weak
equivalence in pqTop by Lemma 1.17. This is a weak equivalence in qTop by
Definition of $W_{\bf qTop}$. All objects in sSet are cofibrant. Then, the
counit of the derived adjunction
$\mathbb{L}(F_{1}F_{0})\dashv\mathbb{R}(G_{0}G_{1})$ is a natural isomorphism.
$\mathbb{L}(F_{1}F_{0})\dashv\mathbb{R}(G_{0}G_{1})$ is a categorical
equivalence. Therefore, $F_{1}F_{0}\dashv G_{0}G_{1}$ is a Quillen
equivalence. $F_{0}\dashv G_{0}$ is a Quillen equivalence by Theorem 1.14.
Then, $F_{1}\dashv G_{1}$ is a Quillen equivalence. ∎

####  1.2.3 Homotopy Theory in quasitopological spaces

In this subsubsection, we will discuss Homotopy Theory in qTop.

Any element $f\in J_{\bf qTop}$ is isomorphic to
$\Delta^{n-1}\overset{0}{\hookrightarrow}\Delta^{n-1}\times[0,1]$ by Corollary
1.10. We call a fibration in qTop a Serre fibration. The model category qTop
has the following properties.

######  Proposition 1.19.

  * •

Any quasitopological space $X$ is fibrant in qTop.

  * •

qTop is a right proper model category.

  * •

qTop is a Cartesian closed model category.

###### Proof.

We will show that any quasitopological space $X$ is fibrant in qTop. Any
element $f\in J_{\bf qTop}$ is isomorphic to
$\Delta^{n-1}\overset{0}{\hookrightarrow}\Delta^{n-1}\times[0,1]$ by Corollary
1.10. This is a retract. Then, the morphism $X\rightarrow{\bf 1}$ has the
right lifting property with respect to
$\Delta^{n-1}\overset{0}{\hookrightarrow}\Delta^{n-1}\times[0,1]$. This means
that $X$ is fibrant in qTop. Furthermore, qTop is a right proper model
category by Proposition A.16.

We will show that qTop is a Cartesian closed model category. Any morphism
$(\partial\Delta^{n}\times\Delta^{m})\coprod_{(\partial\Delta^{n}\times\partial\Delta^{m})}(\Delta^{n}\times\partial\Delta^{m})\rightarrow\Delta^{n}\times\Delta^{m}$
is isomorphic to $\partial\Delta^{n+m}\rightarrow\Delta^{n+m}$ by Corollary
1.10. This is a cofibration. Any morphism
$(\Lambda^{n}_{i}\times\Delta^{m})\coprod_{(\Lambda^{n}_{i}\times\partial\Delta^{m})}(\Delta^{n}\times\partial\Delta^{m})\rightarrow\Delta^{n}\times\Delta^{m}$
is isomorphic to $\Lambda^{n+m}_{i}\rightarrow\Delta^{n+m}$ by Corollary 1.10.
This is an acyclic cofibration. Therefore, qTop is a Cartesian closed model
category by Proposition A.14. ∎

For any quasitopological space $X$, the simplicial set $G_{0}G_{1}(X)$ is a
Kan complex because a right Quillen functor preserves fibrant objects. We can
define the homotopy groups of $X$ as the homotopy groups of the Kan complex
$G_{0}G_{1}(X)$. A morphism in qTop is a weak equivalence if and only if all
the maps induced between the homotopy groups are isomorphisms. We call a weak
equivalence in qTop a weak homotopy equivalence.
Recall some other concepts and properties in Homotopy Theory in qTop, as an
analogue of that in Top.

For any quasitopological space $X$, the exponential object $X^{\Delta^{1}}$ is
a path object of $X$.

######  Proposition 1.20.

Let $X$ be a quasitopological space.

  * •

The morphism $X^{\Delta^{1}}\rightarrow X^{\partial\Delta^{1}}\cong X\times X$
induced by $\partial\Delta^{1}\rightarrow\Delta^{1}$ is a fibration.

  * •

The morphism $X^{\Delta^{1}}\overset{i}{\rightarrow}X^{\Delta^{0}}\cong
X$($i=0,1$) induced by $\Delta^{0}\overset{i}{\rightarrow}\Delta^{1}$ is an
acyclic fibration.

###### Proof.

Let $f$ be the morphism $\partial\Delta^{1}\rightarrow\Delta^{1}$, and let $g$
be the morphism $X\rightarrow{\bf 1}$. $f$ is a cofibration. $g$ is a
fibration by Proposition 1.19. Then the morphism
$g^{\\#f}:X^{\Delta^{1}}\rightarrow X^{\partial\Delta^{1}}$ is a fibration
because qTop is a Cartesian closed model category by Proposition 1.19 (cf.
Definition A.13).

Let $f^{\prime}$ be the morphism
$\Delta^{0}\overset{i}{\rightarrow}\Delta^{1}$. $f^{\prime}$ is an acyclic
cofibration.
$g^{\\#f^{\prime}}:X^{\Delta^{1}}\overset{i}{\rightarrow}X^{\Delta^{0}}$ is an
acyclic fibration because qTop is a Cartesian closed model category by
Proposition 1.19 (cf. Definition A.13). ∎

######  Proposition 1.21.

Any acyclic cofibration in qTop is a strong deformation retract.

###### Proof.

Take any acyclic cofibration $f:X\rightarrow Y$ in qTop. $f$ has a retraction
$g:Y\rightarrow X$ because $X$ is fibrant (cf. Proposition 1.19).

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{{\bf
1}.}$

There exists a homotopy $H$ from $f\circ g$ to $id_{Y}$ up to $X$ by
Proposition 1.20.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\Delta^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{Y}\times(f\circ
g)}$$\scriptstyle{H}$$\textstyle{Y\times Y.}$

This means that $f$ is a strong deformation retract. ∎

##### The mapping track

Any morphism $f$ in qTop has a decomposition $f=f^{\prime}\circ e_{f}$ such
that the morphism $e$ is a weak equivalence and the morphism $p$ is a
fibration. The existence of such a decomposition is required by the axiom of a
model category. One of such decompositions, called the mapping track, can also
be constructed in a specific way.

Let $f:X\rightarrow Y$ be a morphism in qTop. We define a quasitopological
space $X_{f}$ as the following pull-back diagram:

$\textstyle{X_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{[0,1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y.}$

We call the quasitopological space $X_{f}$ the mapping track of $f$. We define
two morphisms $e_{f}:X\rightarrow X_{f}$ and $f^{\prime}:X_{f}\rightarrow Y$
as the following diagram:

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{f}}$$\scriptstyle{f}$$\textstyle{X_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{[0,1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{X_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{Y^{[0,1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$$\textstyle{Y.}$

Then, we have the equality $f=f^{\prime}\circ e_{f}$.

######  Lemma 1.22.

Let $\Sigma:=([0,1]\times\\{1\\})\cup(\\{0\\}\times[0,1])$. The inclusion map
$\Sigma\hookrightarrow[0,1]^{2}$ has a strong deformation retraction
$R_{u}:[0,1]^{2}\rightarrow[0,1]^{2}$ satisfying the followings:

  * •

$R_{0}=id$ and $Im(R_{1})\subset\Sigma$,

  * •

$R_{u}|_{\Sigma}=id$, and

  * •

$R_{u}(s,0)\in[0,1]\times\\{0\\}$.

###### Proof.

We define two homotopies $R^{1}_{u}$ and $R^{2}_{u}$ as the following:

$\begin{array}[]{l}R^{1}_{u}(s,t):=((1-(1-t)u)s,t),\\\
R^{2}_{u}(s,t):=(s,t)+\frac{(1-|s+t-1|)(1-|s-t|)}{2}u(-1,1).\end{array}$

These are the deformations illustrated in the following figure.

$\begin{array}[]{ccc}\includegraphics[width=142.26378pt]{deform1.eps}&&\includegraphics[width=142.26378pt]{deform2.eps}\\\
R^{1}_{u}&&R^{2}_{u}\end{array}$

We define a homotopy $R_{u}$ as the following:

$R_{u}(s,t):=\left\\{\begin{array}[]{ll}R^{1}_{2u}(s,t)&(\mbox{if }0\leq
u\leq\frac{1}{2})\\\ R^{2}_{2u-1}(R^{1}_{1}(s,t))&(\mbox{if }\frac{1}{2}\leq
u\leq 1).\end{array}\right.$

The above $R_{u}$ is the homotopy we want. ∎

######  Proposition 1.23.

  * •

The above morphism $e_{f}$ is a weak equivalence.

  * •

The above morphism $f^{\prime}$ is a fibration.

###### Proof.

$X_{f}\rightarrow X$ is an acyclic fibration because it is the pull-back of an
acyclic fibration $Y^{[0,1]}\overset{0}{\rightarrow}Y$ (cf. Proposition 1.20).
In particular, $X_{f}\rightarrow X$ is a weak equivalence. $e_{f}$ is a weak
equivalence because the weak equivalence $X_{f}\rightarrow X$ is a left
inverse of $e_{f}$.

In order to show that $f^{\prime}$ is a fibration, we consider the following
diagram:

$\textstyle{\Delta^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{X_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{\Delta^{n-1}\times[0,1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{Y.}$
(1)

We define a morphism $\gamma_{1}$ as the following composition:

$\gamma_{1}:\Delta^{n-1}\times[0,1]\overset{pr_{1}}{\rightarrow}\Delta^{n-1}\overset{\alpha}{\rightarrow}X_{f}\rightarrow
X$

Take the subset $\Sigma\hookrightarrow[0,1]^{2}$ and the strong deformation
retraction $R_{u}:[0,1]^{2}\rightarrow[0,1]^{2}$ in Lemma 1.22. Let
$\alpha^{\prime}:\Delta^{n-1}\times[0,1]\rightarrow Y$ be the adjunct of the
composite morphism $\Delta^{n-1}\overset{\alpha}{\rightarrow}X_{f}\rightarrow
Y^{[0,1]}$. We define a morphism
$\gamma^{\prime}_{2}:\Delta^{n-1}\times\Sigma\rightarrow Y$ as the following:

$\begin{array}[]{l}\gamma^{\prime}_{2}|_{\Delta^{n-1}\times([0,1]\times\\{1\\})}:\Delta^{n-1}\times([0,1]\times\\{1\\})\cong\Delta^{n-1}\times[0,1]\overset{\beta}{\rightarrow}Y,\\\
\gamma^{\prime}_{2}|_{\Delta^{n-1}\times(\\{0\\}\times[0,1])}:\Delta^{n-1}\times(\\{0\\}\times[0,1])\cong\Delta^{n-1}\times[0,1]\overset{\alpha^{\prime}}{\rightarrow}Y.\end{array}$

Let $\gamma_{2}:\Delta^{n-1}\times[0,1]\rightarrow Y^{[0,1]}$ be the adjunct
of the composite morphism $\gamma^{\prime}_{2}\circ R_{1}$. We obtain a
morphism
$\gamma(=\gamma_{1}\times\gamma_{2}):\Delta^{n-1}\times[0,1]\rightarrow
X_{f}$. The morphism $\gamma$ is a lift of the diagram (1). Therefore, the
morphism $f^{\prime}$ has the right lifting property with respect to $J_{\bf
qTop}$. This means that $f^{\prime}$ is a fibration. ∎

##### HELP Lemma

We will characterise a weak equivalence by HELP (Homotopy Extension Lifting
Property).

######  Proposition 1.24 (HELP Lemma; cf. [27]).

For any morphism $f:X\rightarrow Y$ in qTop, the following conditions are
equivalent.

(1)

$f$ is a weak equivalence.

(2)

$f$ has the right Homotopy Extension Lifting Property with respact to $I_{\bf
qTop}$. i.e. For a following commutative square, there exists a morphism
$\gamma$ in the following diagram such that the upper triangle in the diagram
is commutative and that the lower triangle in the diagram is commutative up to
homotopy fixing $\partial\Delta^{n}$:

$\textstyle{\partial\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{\cdot.}$

###### Proof.

(1) $\Rightarrow$ (2)
Take a decomposition $f=p_{f}\circ i_{f}$ of $f$ such that $p_{f}$ is a
fibration and that $i_{f}$ is an acyclic cofibration. If $f$ is a weak
equivalence, $p_{f}$ is acyclic. Any morphism
$\partial\Delta^{n}\hookrightarrow\Delta^{n}$ has the left lifting property
with respect to $p_{f}$. $i_{f}$ has a strong deformation retraction by
Proposition 1.21. Then, we obtain the condition (2).
(2) $\Rightarrow$ (1)
In order to show that $f$ is a weak equivalence, we will show that
$f^{\prime}:X_{f}\rightarrow Y$ is an acyclic fibration. We consider the
following diagram:

$\textstyle{\partial\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{X_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{pr_{1}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{Y}$

By the condition (2), there exist a morphism
$\gamma^{\prime}:\Delta^{n}\rightarrow X$ and a homotopy
$H:\Delta^{n}\rightarrow Y^{[0,1]}$ such that $H$ fixes $\partial\Delta^{n}$
and that the following diagram is commutative:

$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\circ\gamma^{\prime}}$$\scriptstyle{\beta}$$\scriptstyle{H}$$\textstyle{Y}$$\textstyle{Y^{[0,1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{1}$$\textstyle{Y}$

Let $\gamma:=\gamma^{\prime}\times H:\Delta^{n}\rightarrow X_{f}$. $\gamma$ is
a lift of the diagram $(\alpha,\beta)$. Therefore, the morphism $f^{\prime}$
has the right lifting property with respect to $I_{\bf qTop}$. This means that
$f^{\prime}$ is an acyclic fibration. Then, $f$ is a weak equivalence. ∎

##  2 Continuous sheaves

In this section, we discuss continuous sheaves. The sheaf of solutions of a
partial differential relation is one of the most typical examples of a
continuous sheaf. A flexible sheaf is an important class of continuous
sheaves. Every continuous sheaf has a flexible sheaf callled the “sheaf of
formal sections” (cf. Lemma 4.5).

###  2.1 Continuous sheaves

First, we define a continuous sheaf.

######  Definition 2.1.

Let $B$ be a topological space.

A continuous presheaf on $B$ is a presheaf with value in qTop, where qTop is
the category of quasitopological spaces (cf. Definition 1.4). ${\bf
PSh}(B;{\bf qTop})$ is the category of continuous presheaves on $B$.

A continuous sheaf on $B$ is a sheaf with value in qTop. ${\bf Sh}(B;{\bf
qTop})$ is the category of continuous sheaves on $B$.

For any compact subset $K$ of $B$, we define a quasitopological space
$\mathcal{F}(K)$ as

$\mathcal{F}(K):=\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
U\supset K\end{subarray}}\mathcal{F}(U).$

Let $B_{top}$ be the site of open subsets of $B$. Let $B_{top}^{tri}$ be the
category, forgetting the Grothendieck topology of $B_{top}$. $B_{top}^{tri}$
can be regarded as a site with the trivial topology. Then, we obtain the
following isomorphisms:

$\begin{array}[]{lclcl}{\bf PSh}(B;{\bf qTop})&\cong&{\bf
Sh}(B_{top}^{tri}\times{\bf Sing\Delta};{\bf Set})&\cong&{\bf Sh}({\bf
Sing\Delta};{\bf PSh}(B;{\bf Set})),\\\ {\bf Sh}(B;{\bf qTop})&\cong&{\bf
Sh}(B_{top}\times{\bf Sing\Delta};{\bf Set})&\cong&{\bf Sh}({\bf
Sing\Delta};{\bf Sh}(B;{\bf Set})).\end{array}$

The category ${\bf Sh}(B_{top}^{tri}\times{\bf Sing\Delta};{\bf Set})$ is
embeded in the category ${\bf PSh}(B_{top}\times{\bf Sing\Delta};{\bf Set})$
of presheaves. Then, the embedding functor ${\bf Sh}(B_{top}\times{\bf
Sing\Delta};{\bf Set})\rightarrow{\bf Sh}(B_{top}^{tri}\times{\bf
Sing\Delta};{\bf Set})$ has a left adjoint functor. To repeat the same
meaning, the forgetful functor ${\bf Sh}(B;{\bf qTop})\rightarrow{\bf
PSh}(B;{\bf qTop})$ has a left adjoint functor. This left adjoint functor is
called the sheafification.

Let $\mathcal{F}$ be a continuous (pre)sheaf. For any open subset $U$ of $B$,
$\mathcal{F}(U)$ is a quasitopological space. For any closed subset $S$ of a
simplex, $\mathcal{F}(U)[S]$ is a set. We define a (pre)sheaf $\mathcal{F}[S]$
as $\mathcal{F}[S](U):=\mathcal{F}(U)[S]$.

######  Proposition 2.2.

Let $\mathcal{F}$ be a continuous (pre)sheaf. For any point $x\in B$ and any
closed subset $S$ of a simplex, we have an isomorphism
$\mathcal{F}_{x}[S]\cong\mathcal{F}[S]_{x}$ of stalks.

###### Proof.

It is obvious by Proposition 1.6. ∎

######  Proposition 2.3.

Let $\mathcal{F}$ be a continuous presheaf, and let $\widehat{\mathcal{F}}$ be
the sheafification of $\mathcal{F}$. For any closed subset $S$ of a simplex,
the sheaf $\widehat{\mathcal{F}}[S]$ is the sheafification of the presheaf
$\mathcal{F}[S]$.

###### Proof.

For any point $x\in B$, we have an isomorphism
$\mathcal{F}_{x}\cong\widehat{\mathcal{F}}_{x}$ of stalks. This means that,
for any closed subset $S$ of a simplex, we have an isomorphism
$\mathcal{F}[S]_{x}\cong\widehat{\mathcal{F}}[S]_{x}$ of stalks, by
Proposition 2.2. Then, the sheaf $\widehat{\mathcal{F}}[S]$ is the
sheafification of the presheaf $\mathcal{F}[S]$. ∎

######  Proposition 2.4.

Let $f:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism between continuous
presheaves. If $f$ is a sectionwise weak equivalence (i.e.
$f_{U}:\mathcal{F}(U)\rightarrow\mathcal{G}(U)$ is a weak equivalence for any
open set $U$), then $f$ is a stalkwise weak equivalence (i.e.
$f_{x}:\mathcal{F}_{x}\rightarrow\mathcal{G}_{x}$ is a weak equivalence for
any point $x$),

###### Proof.

Suppose that $f$ is a sectionwise weak equivalence. We will show that $f_{x}$
has the right Homotopy Extension Lifting Property with respect to $I_{\bf
qTop}$ (cf. Proposition 1.24). i.e. We will show that, for a following
commutative square, there exists a morphism $\gamma$ in the following diagram
such that the upper triangle in the diagram is commutative and that the lower
triangle in the diagram is commutative up to homotopy fixing
$\partial\Delta^{n}$:

$\textstyle{\partial\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{x}}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{\mathcal{G}_{x}.}$

By the discussion in Section 1.1.4, there exists a small enough open
neighbourhood $U$ of $x$ such that the following diagram is commutative:

$\textstyle{\partial\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{U}}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}(U).}$

$f_{U}$ has the right Homotopy Extension Lifting Property with respect to
$I_{\bf qTop}$ by Proposition 1.24. Then, $f_{x}$ also has the right Homotopy
Extension Lifting Property with respect to $I_{\bf qTop}$. $f_{x}$ is a weak
equivalence by Proposition 1.24. Therefore, $f$ is a stalkwise weak
equivalence. ∎

###  2.2 Flexible sheaves

Before defining a flexible sheaf, we prepare a notation.

######  Definition 2.5.

Let $B$ be a topological space. Let $f:\mathcal{F}\rightarrow\mathcal{G}$ be a
morphism between continuous sheaves on $B$. For compact subsets $K$ and $L$ of
$B$ with $K\subset L$, denote the following morphism
$\mathcal{F}(L)\rightarrow\mathcal{F}(K)\times_{\mathcal{G}(K)}\mathcal{G}(L)$
by $f_{K\subset L}$:

$\textstyle{\mathcal{F}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{L}}$$\scriptstyle{f_{K\subset
L}}$$\textstyle{\mathcal{F}(K)\times_{\mathcal{G}(K)}\mathcal{G}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{K}}$$\textstyle{\mathcal{G}(K).}$

######  Definition 2.6.

Let $B$ be a topological space. A morphism
$f:\mathcal{F}\rightarrow\mathcal{G}$ between continuous sheaves on $B$ is a
flexible extension if, for any compact subsets $K$ and $L$ of $B$ with
$K\subset L$, the induced morphism $f_{K\subset L}$ is a fibration. A
continuous sheaf $\mathcal{F}$ is flexible if the unique morphism
$\mathcal{F}\rightarrow{\bf 1}$ is a flexible extension, where 1 is the
terminal object.

A continuous sheaf $\mathcal{F}$ is a flexible sheaf if and only if, for any
compact subsets $K$ and $L$ of $B$ with $K\subset L$, the restriction morphism
$\mathcal{F}(L)\rightarrow\mathcal{F}(K)$ is a fibration.

###  2.3 The sheaf of formal sections

Recall that a partial differential relation is a subbundle of a jet bundle and
that a formal solution of a partial differential relation $R$ is a continuous
section of the fibre bundle $R$ (cf. [4, 9]). In this subsection, we define
the sheaf of formal sections of a continuous sheaf.

######  Definition 2.7.

For a continuous sheaf $\mathcal{F}$ on a topological space$B$, we define a
continuous presheaf $\mathcal{F}^{\square}$ on $B$ as the following:

$\mathcal{F}^{\square}(U):=\mathcal{F}(U)^{\operatorname{Sing^{q}}(U)}.$

The sheaf of formal sections of $\mathcal{F}$ is the sheafification
$\mathcal{F}^{\ast}$ of $\mathcal{F}^{\square}$.

The morphisms
$\mathcal{F}(U)\rightarrow\mathcal{F}(U)^{\operatorname{Sing^{q}}(U)}$ induced
the canonnical projections
$\mathcal{F}(U)\times\operatorname{Sing^{q}}(U)\rightarrow\mathcal{F}(U)$
define a morphism $\mathcal{F}\rightarrow\mathcal{F}^{\square}$. The diagonal
morphism is the induced morphism
$\Delta:\mathcal{F}\rightarrow\mathcal{F}^{\ast}$ by the above morphism
$\mathcal{F}\rightarrow\mathcal{F}^{\square}$.

######  Remark 2.8.

Compared with a formal solution of a partial differential relation, the above
definition may seem strange. In fact, it is more appropriate to call the above
$\mathcal{F}^{\ast}$ a sheaf of fibrewise holonomic sections (cf. [4, page
24]).

We can prove the claim (1) of Lemma 0.2. First, we define the properties that
$B$ must satisfy.

######  Definition 2.9.

A topological space $B$ is strongly locally contractible if any point $x\in B$
has an open neighbourhood basis $\mathcal{N}$ such that, for any open
neighbourhood $U\in\mathcal{N}$, the inclusion map $\\{x\\}\hookrightarrow U$
is a strong deformation retract.

######  Lemma 2.10 (Gromov [9]).

Let $B$ be a strongly locally contractible space. For any continuous sheaf
$\mathcal{F}$ on $B$, the diagonal morphism
$\Delta:\mathcal{F}\rightarrow\mathcal{F}^{\ast}$ is a stakwise weak
equivalence.

###### Proof.

We will show that $\Delta_{x}$ has the right Homotopy Extension Lifting
Property with respect to $I_{\bf qTop}$ (cf. Proposition 1.24). i.e. We will
show that, for a following commutative square, there exists a morphism
$\gamma$ in the following diagram such that the upper triangle in the diagram
is commutative and that the lower triangle in the diagram is commutative up to
homotopy fixing $\partial\Delta^{n}$:

$\textstyle{\partial\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{x}}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{\mathcal{F}^{\ast}_{x}\cong\mathcal{F}^{\square}_{x}.}$

By the discussion in Section 1.1.4, there exists a small enough open
neighbourhood $U$ of $x$ such that the following diagram is commutative:

$\textstyle{\partial\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathcal{F}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\mathcal{F}(U)^{U}.}$

We can assume that the inclusion map $\\{x\\}\hookrightarrow U$ has a strong
deformation retraction $p_{u}:U\rightarrow U$ because $B$ is strongly locally
contractible. $p_{u}$ induces a strong deformation retraction
$H_{u}:\mathcal{F}(U)^{U}\rightarrow\mathcal{F}(U)^{U}$ of the morphism
$\mathcal{F}(U)\rightarrow\mathcal{F}(U)^{U}$ in the above diagram. Let
$\gamma:=H_{1}\circ\beta:\Delta^{n}\rightarrow\mathcal{F}(U)$. Then, the upper
triangle is commutative, and the lower triangle is commutative up to homotopy
$H_{u}\circ\beta$. The homotopy $H_{u}\circ\beta$ fixes $\partial\Delta^{n}$.
Therefore, $\Delta_{x}$ has the right Homotopy Extension Lifting Property with
respect to $I_{\bf qTop}$. $\Delta_{x}$ is a weak equivalence by Proposition
1.24. ∎

Let $B$ be a topological space, and let $\mathcal{F}$ be a continuous sheaf on
$B$. Let $\phi:S\times B\rightarrow T$ be a continuous map, where $S$ and $T$
are closed subsets of simplexes. We will define a morphism
$\phi^{\ast}:\mathcal{F}^{\ast}[T]\rightarrow\mathcal{F}^{\ast}[S]$ between
sheaves of sets. For any open subset $U$ of $B$, we define a map
$\phi^{\dagger}_{U}:\mathcal{F}^{\square}[T](U)\rightarrow\mathcal{F}^{\square}[S](U)$
as the following diagram:

$\textstyle{\mathcal{F}^{\square}[T](U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\dagger}_{U}}$$\textstyle{\mathcal{F}^{\square}[S](U)}$$\textstyle{{\bf
qTop}(\operatorname{Sing^{q}}(T),\mathcal{F}(U)^{\operatorname{Sing^{q}}(U)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$Yoneda
iso.$\textstyle{{\bf
qTop}(\operatorname{Sing^{q}}(S),\mathcal{F}(U)^{\operatorname{Sing^{q}}(U)}),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$Yoneda
iso.$\textstyle{{\bf qTop}(\operatorname{Sing^{q}}(T\times
U),\mathcal{F}(U))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$adjunction$\scriptstyle{\operatorname{Sing^{q}}(\hat{\phi})^{\ast}}$$\textstyle{{\bf
qTop}(\operatorname{Sing^{q}}(S\times
U),\mathcal{F}(U)),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$adjunction

where $\hat{\phi}:S\times U\rightarrow T\times U$ is a continuous map defined
as $\hat{\phi}(p,x):=(\phi(p,x),x)$. These maps $\phi^{\dagger}_{U}$ define a
morphism
$\phi^{\dagger}:\mathcal{F}^{\square}[T]\rightarrow\mathcal{F}^{\square}[S]$
between presheaves of sets. This morphism $\phi^{\dagger}$ induces a morphism
$\phi^{\ast}:\mathcal{F}^{\ast}[T]\rightarrow\mathcal{F}^{\ast}[S]$ between
sheaves of sets (cf. Proposition 2.3).

For any open subset $U$ of $B$, the map
$\phi^{\ast}_{U}:\mathcal{F}^{\ast}[T](U)\rightarrow\mathcal{F}^{\ast}[S](U)$
induces the map ${\bf
qTop}(\operatorname{Sing^{q}}(T),\mathcal{F}^{\ast}(U))\rightarrow{\bf
qTop}(\operatorname{Sing^{q}}(S),\mathcal{F}^{\ast}(U))$ as the following
diagram:

$\textstyle{\mathcal{F}^{\ast}[T](U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\ast}_{U}}$$\textstyle{\mathcal{F}^{\ast}[S](U)}$$\textstyle{{\bf
qTop}(\operatorname{Sing^{q}}(T),\mathcal{F}^{\ast}(U))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$Yoneda
iso.$\textstyle{{\bf
qTop}(\operatorname{Sing^{q}}(S),\mathcal{F}^{\ast}(U)).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$Yoneda
iso.

We denote this map by the same symbol, $\phi^{\ast}_{U}$.

##  3 Result 1 : Embedding to a model category

The goal of this section is to prove the following theorem.

######  Theorem 3.1.

There exists a right proper model category ${\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ and a fully faithful and left
exact embedding $\iota:{\bf Sh}(B;{\bf qTop})\rightarrow{\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ such that the followings
hold.

Let $f:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism in ${\bf Sh}(B;{\bf
qTop})$.

  * •

$\iota(f)$ is a weak equivalence
$\Leftrightarrow$ $f$ is a stalkwise weak equivalence (i.e.
$f_{x}:\mathcal{F}_{x}\rightarrow\mathcal{G}_{x}$ is a weak equivalence for
any $x\in B$).

  * •

$\iota(f)$ is a fibration
$\Leftrightarrow$ $f$ is a flexible extension (cf. Definition 2.6).

In particular, for a continuous sheaf $\mathcal{F}$ on $B$,

  * •

$\iota(\mathcal{F})$ is fibrant
$\Leftrightarrow$ $\mathcal{F}$ is flexible (cf. Definition 2.6).

###### Proof.

This follows immediately from Proposition 3.4 and Theorem 3.7. ∎

As a first step, we want to characterise flexible extensions by the right
lifting property. First, suppose that, for any compact subset $K$ of $B$, the
inverse functor $i^{\ast}:{\bf Sh}(K;{\bf qTop})\rightarrow{\bf Sh}(B;{\bf
qTop})$ had a left adjoint functor $i_{!}:{\bf Sh}(B;{\bf
qTop})\rightarrow{\bf Sh}(K;{\bf qTop})$. The unique map $\mu:K\rightarrow{\bf
1}$ induces an adjunction $\mu^{\ast}\dashv\mu_{\ast}:{\bf
qTop}\rightarrow{\bf Sh}(K;{\bf qTop})$. We obtain an adjunction
$i_{!}\mu^{\ast}\dashv\mu_{\ast}i^{\ast}:{\bf qTop}\rightarrow{\bf Sh}(B;{\bf
qTop})$. For a quasitopological space $X$, we define a continuous sheaf
$X_{!K}$ as $i_{!}\mu^{\ast}(X)$. For a continuous sheaf $\mathcal{F}$, we
have the equality $\mu_{\ast}i^{\ast}(\mathcal{F})=\mathcal{F}(K)$. Then, for
any morphism $f:\mathcal{F}\rightarrow\mathcal{G}$ between continuous sheaves
and any compact subsets $K$ and $L$ of $B$ with $K\subset L$, the following
conditions are equivalent.

  * •

The following square has a lift:

$\textstyle{\Lambda^{n}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{K\subset
L}}$$\textstyle{\Delta^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}(K)\times_{\mathcal{G}(K)}\mathcal{G}(L).}$

  * •

The following square has a lift:

$\textstyle{(\Lambda^{n}_{i})_{!L}\cup_{(\Lambda^{n}_{i})_{!K}}\Delta^{n}_{!K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\Delta^{n}_{!L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}.}$

Then, a morphism $f$ between continuous sheaves is a flexible extension if and
only if it has the reight lifting property with respect to
$(\Lambda^{n}_{i})_{!L}\cup_{(\Lambda^{n}_{i})_{!K}}\Delta^{n}_{!K}\rightarrow\Delta^{n}_{!L}$.
We have characterized flexible extensions by the right lifting property in
this situation.

However, the supposition of the existence of a left adjoint functor $i_{!}$ is
too strong. To avoid this problem, we define a category
$\tilde{\mathscr{U}}_{B}$ (cf. Section 3.1) by extending the site associated
with the topological space $B$, and consider the category of presheaves on
$\tilde{\mathscr{U}}_{B}$. Furthermore, for a technical reason (cf.
Proposition 3.3), we require the condition $\mathcal{F}(\emptyset)={\bf 1}$ on
a presheaf. We call such a presheaf an augmented presheaf (cf. Definition
3.2). ${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ is the category
of augmented presheaves on $\tilde{\mathscr{U}}_{B}$.
The next step is to define a model structure $\mathcal{M}$ on ${\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$, satisfying the following two
conditions.

(W)

$f$ is a weak equivalence
$\Leftrightarrow$ $f$ is a stalkwise weak equivalence.

(F)

$f$ is a fibration
$\Leftrightarrow$ $f$ is a flexible extension.

To construct $\mathcal{M}$, we define two model structures, $\mathcal{M}_{1}$
and $\mathcal{M}_{2}$. $\mathcal{M}_{1}$ satisfies the above condition (F).
$\mathcal{M}_{2}$ satisfies the above condition (W). These are mixed to make
$\mathcal{M}$. We call the model structure $\mathcal{M}_{1}$ the compact
flexible model structure (cf. Section 3.2). We call the model structure
$\mathcal{M}_{2}$ the pointwise projective model structure (cf. Section 3.3).
We call the model structure $\mathcal{M}$ the pointwise flexible model
structure (cf. Section 3.4).

###  3.1 The base category

We define a set $\tilde{\mathscr{U}}_{B}$ as the following:

$\tilde{\mathscr{U}}_{B}:=\\{U\cap K\,|\,\mbox{$U(\subset B)$ is open, and
$K(\subset B)$ is compact.}\\}.$

$\tilde{\mathscr{U}}_{B}$ is an ordered set. Then, $\tilde{\mathscr{U}}_{B}$
is a small category.

We define an augmented presheaf.

######  Definition 3.2.

A presheaf $\mathcal{F}$ is augmented if $\mathcal{F}(\emptyset)={\bf 1}$,
where $\emptyset$ is the initial object and 1 is the terminal object.

Grothendieck’s original definition of presheaf is a contravariant functor from
the set of non-empty open sets to a category (cf. [10]). A presheaf
$\mathcal{F}$ in Grothendieck’s sense is regarded as an augmented presheaf by
defining $\mathcal{F}(\emptyset):={\bf 1}$. (Compare the definition of the
augmented simplicial set.)

The advantage of a presheaf being augmented is the following property.

######  Proposition 3.3.

Let $f:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism between continuous
sheaves on $B$. If $\mathcal{F}$ and $\mathcal{G}$ are augmented, then
$f_{\emptyset\subset K}$ is isomorphic to
$f_{K}:\mathcal{F}(K)\rightarrow\mathcal{G}(K)$ for any compact subset $K$ of
$B$.

###### Proof.

It is obvious by definition of $f_{\emptyset\subset K}$ (cf. Definition 2.5).
∎

${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ is the category of
augmented presheaves with value in qTop on $\tilde{\mathscr{U}}_{B}$. We
define a functor $\iota:{\bf Sh}(B;{\bf qTop})\rightarrow{\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ as the following:

$\iota(\mathcal{F})(A):=\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
U\supset A\end{subarray}}\mathcal{F}(U).$

######  Proposition 3.4.

The above functor $\iota$ is fully faithful and left exact.

###### Proof.

Let $\iota_{1}:{\bf Sh}(B;{\bf qTop})\rightarrow{\bf PSh}_{\ast}(B;{\bf
qTop})$ be the forgetful functor, and let $\iota_{2}:{\bf PSh}_{\ast}(B;{\bf
qTop})\rightarrow{\bf PSh}_{\ast}(B_{cpt};{\bf qTop})$ be a functor defined as
the following:

$\iota_{2}(\mathcal{F})(A):=\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
U\supset A\end{subarray}}\mathcal{F}(U).$

We have the equality $\iota=\iota_{2}\circ\iota_{1}$. $\iota_{1}$ is fully
faithful and left exact. We will show that $\iota_{2}$ is fully faithful and
left exact. We define a functor $\nu:{\bf PSh}_{\ast}(B_{cpt};{\bf
qTop})\rightarrow{\bf PSh}_{\ast}(B;{\bf qTop})$ as the following:

$\nu(\mathcal{F})(U):=\mathcal{F}(U).$

Then, there exists an adjunction $\iota_{2}\dashv\nu$. The unit of
$\iota_{2}\dashv\nu$ is a natural isomorphism. Then, $\iota_{2}$ is fully
faithful. Any finite limit and any filtered colimit are commutative in qTop by
the objectwise calculation (cf. Proposition 1.6). Then, $\iota_{2}$ is left
exact. Therefore, $\iota(=\iota_{2}\circ\iota_{1})$ is fully faithful and left
exact. ∎

###  3.2 The compact flexible model structure

In this subsection, we will prove the following theorem.

######  Theorem 3.5 (the compact flexible model structure).

The category ${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ has a
model structure satisfying the followings:

  * •

$f$ is a weak equivalence
$\Leftrightarrow$ $f_{K}:\mathcal{F}(K)\rightarrow\mathcal{G}(K)$ is a weak
equivalence for any compact subset $K$ of $B$.

  * •

$f$ is a fibration
$\Leftrightarrow$ $f$ is a flexible extension (cf. Definition 2.6).

To prove the above theorem, we apply Recognition Theorem (cf. Theorem A.9) of
a cofibrantly generated model structure.

First, we prepare a notation. For any quasitopological space $X$ and any
compact subset $K$ of $B$, we define an object $X_{!K}$ in ${\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ as the following:

$X_{!K}(A):=\left\\{\begin{array}[]{ll}X&(\mbox{if }\emptyset\neq A\subset
K)\\\ \emptyset&(\mbox{if }A\not\subset K)\\\ {\bf 1}&(\mbox{if
}A=\emptyset).\end{array}\right.$

For any morphism $f:X\rightarrow Y$ in qTop and any compact subsets $K$ and
$L$ of $B$ with $K\subset L$, denote the following morphism
$X_{!L}\cup_{X_{!K}}Y_{!K}\rightarrow Y_{!L}$ by $f_{!(K\subset L)}$:

---
$\textstyle{Y_{!L}}$$\textstyle{X_{!L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{!L}}$$\textstyle{X_{!L}\cup_{X_{!K}}Y_{!K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{!(K\subset
L)}}$$\textstyle{X_{!K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{!K}^{\prime}}$$\textstyle{Y_{!K}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We define two index sets $\mathfrak{K}$ and $\mathfrak{K}^{(2)}$ as the
followings:

$\begin{array}[]{lcl}\mathfrak{K}&:=&\\{K\subset B:\mbox{compact}\\},\\\
\mathfrak{K}^{(2)}&:=&\\{(K,L)\,|\,K,L\in\mathfrak{K},K\subset
L\\}.\end{array}$

We define two sets $I_{cpt}$ and $J_{cpt}$ and a class $W_{cpt}$ of some
morphisms in ${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ as the
followings:

$\begin{array}[]{lcl}I_{cpt}&:=&\\{f_{!(K\subset L)}\,|\,f\in I_{\bf
qTop},(K,L)\in\mathfrak{K}^{(2)}\\},\\\ J_{cpt}&:=&\\{f_{!(K\subset
L)}\,|\,f\in J_{\bf qTop},(K,L)\in\mathfrak{K}^{(2)}\\},\\\
W_{cpt}&:=&\\{f:\mathcal{F}\rightarrow\mathcal{G}\,|\,f_{K}:\mathcal{F}(K)\rightarrow\mathcal{G}(K)\mbox{
is a weak equivalence for }^{\forall}K\in\mathfrak{K}\\}.\end{array}$

Then, a morphism $f$ in ${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$
is a flexible extension if and only if $f$ has the right lifting property with
respect to $J_{cpt}$ (cf. the preamble to Section 3).

###### Proof of Theorem 3.5.

In order to apply Theorem A.9, we show the followings:

(1)

We have the inclusion relation $\operatorname{Cof}(J_{cpt})\subset
W_{cpt}\cap\operatorname{Cof}(I_{cpt})$. i.e. We have the followings:

(1.1)

$\operatorname{Cof}(J_{cpt})\subset\operatorname{Cof}(I_{cpt})$, and

(1.2)

$\operatorname{Cof}(J_{cpt})\subset W_{cpt}$.

(2)

We have the equality
$\operatorname{rlp}(I_{cpt})=\operatorname{rlp}(J_{cpt})\cap W_{cpt}$. i.e. We
have the followings:

(2.1)

$\operatorname{rlp}(I_{cpt})\subset\operatorname{rlp}(J_{cpt})$,

(2.2)

$\operatorname{rlp}(I_{cpt})\subset W_{cpt}$, and

(2.3)

$\operatorname{rlp}(I_{cpt})\supset\operatorname{rlp}(J_{cpt})\cap W_{cpt}$.

(3)

If two of the three morphisms $f$, $g$ and $g\circ f$ are contained in
$W_{cpt}$, then the third is also contained in $W_{cpt}$.

(4)

$I_{cpt}$ and $J_{cpt}$ permit the small object argument.

We will prove these in order.
Proof of the condition (1.1)
We have the inclusion relation $J_{cpt}\subset\operatorname{cell}(I_{cpt})$ by
the decomposition of $f_{!(K\subset L)}\in J_{cpt}$represented by the
following diagram:

|
---|---
$\textstyle{(\Lambda^{n}_{i})_{!L}\cup_{(\Lambda^{n}_{i})_{!K}}\Delta^{n}_{!K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{!(K\subset
L)}}$$\textstyle{\partial\Delta^{n}_{!L}\cup_{\partial\Delta^{n}_{!K}}\Delta^{n}_{!K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{!(K\subset
L)}}$$\textstyle{\Delta^{n}_{!L}}$$\textstyle{\partial\Delta^{n-1}_{!L}\cup_{\partial\Delta^{n-1}_{!K}}\Delta^{n-1}_{!K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{!(K\subset
L)}}$$\textstyle{\Delta^{n-1}_{!L},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $g_{!(K\subset L)}$ and $h_{!(K\subset L)}$ belong to $I_{cpt}$ and the
left square is a push-out. Then, we obtain the inclusion relation
$\operatorname{cell}(J_{cpt})\subset\operatorname{cell}(I_{cpt})$. Then, we
obtain the inclusion relation
$\operatorname{Cof}(J_{cpt})\subset\operatorname{Cof}(I_{cpt})$.
Proof of the condition (1.2)
For any quasitopological space $X$ and any compact subset $K$ of $B$, we have
the equality $X_{!K}=X_{!B}\times{\bf 1}_{!K}$. For any morphism $X\rightarrow
Y$ belonging to $J_{\bf qTop}$, $X_{!B}\rightarrow Y_{!B}$ is a sectionwise
acyclic cofibration. (i.e. For any object $A$ in $\tilde{\mathscr{U}}_{B}$,
the morphism $X_{!B}(A)\rightarrow Y_{!B}(A)$ in qTop is an acyclic
cofibration.) For any compact subsets $K$ and $L$ of $B$ with $K\subset L$,
the morphism ${\bf 1}_{!K}\rightarrow{\bf 1}_{!L}$ is a sectionwise
cofibration. Then, the morphism $X_{!L}\coprod_{X_{!K}}Y_{!K}\rightarrow
Y_{!L}$ is a sectionwise acyclic cofibration because qTop is a Cartesian
closed model category. This means that any morphism $f_{!(K\subset L)}$
belonging to $J_{cpt}$ is a sectionwise acyclic cofibration. Any push-out, any
transfinite composition and any retract preserve a sectionwise acyclic
cofibration by the sectionwise calculation. Then, any morphism $f$ belonging
to $\operatorname{Cof}(J_{cpt})$ is a sectionwise acyclic cofibration. In
particular, the morphism $f_{K}$ is a weak equivalence for any
$K\in\mathfrak{K}$. Therefore, we have the inclusion relation
$\operatorname{Cof}(J_{cpt})\subset W_{cpt}$.
Proof of the condition (2.1)
We have the followings:

$\begin{array}[]{rl}&f\in\operatorname{rlp}(I_{cpt})\\\
\Leftrightarrow&\mbox{$f_{K\subset L}$ is an acyclic fibration for
${}^{\forall}(K,L)\in\mathfrak{K}^{(2)}$}\\\ \Rightarrow&\mbox{$f_{K\subset
L}$ is a fibration for ${}^{\forall}(K,L)\in\mathfrak{K}^{(2)}$}\\\
\Leftrightarrow&f\in\operatorname{rlp}(J_{cpt}).\end{array}$

We obtain the inclusion relation
$\operatorname{rlp}(I_{cpt})\subset\operatorname{rlp}(J_{cpt})$.
Proof of the condition (2.2)
Take any element $f\in\operatorname{rlp}(I_{cpt})$. The induced morphism
$f_{\emptyset\subset K}$ is an acyclic fibration for any compact subset $K$ of
$B$. In particular, this is a weak equivalence. Then, $f_{K}$ is a weak
equivalence by Proposition 3.3. Therefore, we obtain the inclusion relation
$\operatorname{rlp}(I_{cpt})\subset W_{cpt}$.
Proof of the condition (2.3)
Take any element $f\in\operatorname{rlp}(J_{cpt})\cap W_{cpt}$. The induced
morphism $f_{K\subset L}$ is a fibration because $f$ belongs to
$\operatorname{rlp}(J_{cpt})$. In particular, $f_{\emptyset\subset
K}:\mathcal{F}(K)\rightarrow\mathcal{G}(K)$ is a fibration. Then, $f_{K}$ is a
fibration by Proposition 3.3. Furthermore, $f_{K}$ is an acyclic fibration
because $f$ belongs to $W_{cpt}$. We consider the following diagram:

$\textstyle{\mathcal{F}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{L}}$$\scriptstyle{\simeq}$$\scriptstyle{f_{K\subset
L}}$$\textstyle{\mathcal{F}(K)\times_{\mathcal{G}(K)}\mathcal{G}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{G}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{K}}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{G}(K).}$

The above
$\mathcal{F}(K)\times_{\mathcal{G}(K)}\mathcal{G}(L)\rightarrow\mathcal{G}(L)$
is an acyclic fibration because a pull-back of an acyclic fibration is an
acyclic fibration. In particular, this is a weak equivalence. Then,
$f_{K\subset L}$ is a weak equivalence. Recall that $f_{K\subset L}$ is a
fibration. Then, $f_{K\subset L}$ is an acyclic fibration. This means that $f$
belongs to $\operatorname{rlp}(I_{cpt})$. Therefore, we obtain the inclusion
relation $\operatorname{rlp}(I_{cpt})\subset W_{cpt}$.
Proof of the condition (3)
This is obvious because the same property holds for weak equivalences in qTop.
Proof of the condition (4)
We consider the following adjunction:

$\textstyle{{\bf
qTop}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X\mapsto
X_{!K}}$$\scriptstyle{\top}$$\textstyle{{\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf
qTop}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}(K)\mapsfrom\mathcal{F}}$

The functor $\mathcal{F}\mapsto\mathcal{F}(K)$ preserves any colimit by the
sectionwise calculation. The functor $X\mapsto X_{!K}$ preserves any small
object by Proposition A.6. For any morphism $X\rightarrow Y$ belonging $I_{\bf
qTop}$ and any morphism $X^{\prime}\rightarrow Y^{\prime}$ belonging $J_{\bf
qTop}$, the objects $X$, $X^{\prime}$, $Y$ and $Y^{\prime}$ are small by
Example A.4. The objects $X_{!K}$, $X^{\prime}_{!K}$, $Y_{!K}$ and
$Y^{\prime}_{!K}$ are small for any compact subset $K$ of $B$. Then,
$X_{!L}\coprod_{X_{!K}}Y_{!K}$ and
$X^{\prime}_{!L}\coprod_{X^{\prime}_{!K}}Y^{\prime}_{!K}$ are small by
Proposition A.5. Therefore, $I_{cpt}$ and $J_{cpt}$ permit the small object
argument. ∎

###  3.3 The pointwise projective model structure

In this subsection, we will prove the following theorem.

######  Theorem 3.6 (the pointwise projective model structure).

The category ${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ has a
model structure satisfying the followings:

  * •

$f$ is a weak equivalence
$\Leftrightarrow$ $f$ is a pointwise weak equivalence (i.e. $f_{\\{x\\}}$ is a
weak equivalence for any point $x\in B$, where $\\{x\\}$ is a singleton).

  * •

$f$ is a fibration
$\Leftrightarrow$ $f$ is a pointwise fibration.

Furthermore, this model structure is right proper.

To prove the above theorem, we apply Corollary A.11, the corollary of Transfer
Theorem (cf. Theorem A.10) of a cofibrantly generated model structure.

For each point $x\in B$, let ${\bf qTop}_{x}$ be a copy of the category qTop.
There exist two sets $I_{x}$ and $J_{x}$ and a class $W_{x}$ as copies of the
sets $I_{\bf qTop}$ and $J_{\bf qTop}$ and the class $W_{\bf qTop}$
respectively. The category $\prod_{x}{\bf qTop}_{x}$ has a model structure.
Furthermore, the model category $\prod_{x}{\bf qTop}_{x}$ is cofibrantly
generated. We will explain this cofibrantly model structure. For each point
$x\in B$, we define two sets $\overline{I_{x}}$ and $\overline{J_{x}}$ of some
morphisms in $\prod_{x}{\bf qTop}_{x}$ as the following:

$\begin{array}[]{lcl}\overline{I_{x}}&:=&\left(\displaystyle\prod_{y\neq
x}\\{id_{\emptyset}\\}\right)\times I_{x},\\\
\overline{J_{x}}&:=&\left(\displaystyle\prod_{y\neq
x}\\{id_{\emptyset}\\}\right)\times J_{x},\end{array}$

where $\emptyset$ is the initial object in each category ${\bf qTop}_{y}$. The
model category $\prod_{x}{\bf qTop}_{x}$ is cofibrantly generated satisfying
the followings:

  * •

$\coprod_{x}\overline{I_{x}}$ generates cofibrations,

  * •

$\coprod_{x}\overline{J_{x}}$ generates acyclic cofibrations, and

  * •

$\prod_{x}W_{x}$ is the class of weak equivalences.

Next, we construct the following adjunction:

$\textstyle{\displaystyle\prod_{x\in B}{\bf
qTop}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{pt}}$$\scriptstyle{\top}$$\textstyle{{\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf
qTop}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{pt}}$

We define the above two functors $F_{pt}$ and $G_{pt}$ as the followings:

  * •

$G_{pt}(\mathcal{F}):=(\mathcal{F}(\\{x\\}))_{x\in B}$, and

  * •

$[F_{pt}((X^{x})_{x\in B})](U):=\left\\{\begin{array}[]{ll}\emptyset&(|U|\geq
2)\\\ X^{x}&(U=\\{x\\})\\\ {\bf 1}&(U=\emptyset).\end{array}\right.$

Then, there exists an adjunction $F_{pt}\dashv G_{pt}$.

We define two sets $I_{pt}$ and $J_{pt}$ and a class $W_{pt}$ of some
morphisms in ${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ as the
followings:

$\begin{array}[]{lcl}I_{pt}&:=&F_{pt}(\coprod_{x}\overline{I_{x}}),\\\
J_{pt}&:=&F_{pt}(\coprod_{x}\overline{J_{x}}),\\\
W_{pt}&:=&\\{f\,|\,f_{\\{x\\}}\mbox{ is a weak equivalence for }^{\forall}x\in
B\\}.\end{array}$

Then, a morphism $g$ has the right lifting property with respect to $J_{pt}$
if and only if $G_{pt}(g)$ has the right lifting property with respect to
$\coprod_{x}\overline{J_{x}}$. This means that $g$ has the right lifting
property with respect to $J_{pt}$ if and only if $g$ is a pointwise fibration.

###### Proof of Theorem 3.6.

In order to apply Corollary A.11, we show the followings:

(1)

$G_{pt}$ preserves any colimit.

(2)

We have the inclusion relations
$G_{pt}(F_{pt}(I))\subset\operatorname{cell}(I_{pt})$ and
$G_{pt}(F_{pt}(J_{pt}))\subset\operatorname{cell}(J_{pt})$.

The condition (1) holds by the sectionwise calculation. The condition (2)
holds by the isomorphism $G_{pt}F_{pt}\cong Id$. Therefore, ${\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ has the cofibrantly generated
model structure we want, by Corollary A.11. This model structure is right
proper by Proposition A.16. ∎

###  3.4 The pointwise flexible model structure

In this subsection, we will prove the following theorem.

######  Theorem 3.7 (the pointwise flexible model structure).

The category ${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ has a
model structure satisfying the followings:

  * •

$f$ is a weak equivalence
$\Leftrightarrow$ $f$ is a pointwise weak equivalence.

  * •

$f$ is a fibration
$\Leftrightarrow$ $f$ is a flexible extension.

Furthermore, this model structure is right proper.

###### Proof.

We construct the model structure we want by mixing the compact flexible model
structure (cf. Theorem 3.5) and the pointwise projective model structure (cf.
Theorem 3.6). In order to apply Theorem A.17, we show that we have the
following inclusion relations:

(1)

$W_{cpt}\subset W_{pt}$, and

(2)

$\operatorname{rlp}(J_{cpt})\subset\operatorname{rlp}(J_{pt})$.

For any point $x\in B$, the singleton $\\{x\\}$ is compact. Then, we obtain
the inclusion relation $W_{cpt}\subset W_{pt}$. If a morphism $f$ belongs to
$\operatorname{rlp}(J_{cpt})$, $f_{\emptyset\subset K}$ is a fibration for any
compact subset $K$ of $B$. Then, $f_{K}$ is a fibration by Proposition 3.3. In
particular, by considering the case where $K$ is a singleton $\\{x\\}$, $f$
belongs to $\operatorname{rlp}(J_{pt})$. We obtain the inclusion relation
$\operatorname{rlp}(J_{cpt})\subset\operatorname{rlp}(J_{pt})$. Therefore,
${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ has the model structure
we want, by Theorem A.17. Furthermore, this model structure is right proper
because the pointwise projective model structure is right proper (cf. Theorem
3.6 and Proposition A.18). ∎

##  4 Result 2 : The prefibration structure

The goal of this section is to prove the following theorem.

######  Theorem 4.1.

Let $B$ be a strongly locally contractible normal Hausdorff space. There
exists an ABC prefibration structure on ${\bf Sh}(B;{\bf qTop})$ such that the
followings hold.

Let $f:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism in ${\bf Sh}(B;{\bf
qTop})$.

  * •

$f$ is a weak equivalence
$\Leftrightarrow$ $f$ is a stalkwise weak equivalence (i.e.
$f_{x}:\mathcal{F}_{x}\rightarrow\mathcal{G}_{x}$ is a weak equivalence for
any $x\in B$).

  * •

$f$ is a fibration
$\Leftrightarrow$ $f$ is a flexible extension (cf. Definition 2.6).

In particular, for a continuous sheaf $\mathcal{F}$ on $B$,

  * •

$\mathcal{F}$ is fibrant
$\Leftrightarrow$ $\mathcal{F}$ is flexible (cf. Definition 2.6).

The proof will be in Section 4.2. The axioms of an ABC prefibration structure
are a generalisation of the axioms of a model category. Most of them on ${\bf
Sh}(B;{\bf qTop})$ follow from Theorem 3.1. However, only the axiom 6. of
Definition 4.2, of the existence of a decomposition, needs to be proved
independently. To prove this, we define the flexible track and prove Lemma
4.5, a generalisation of Lemma 0.2 (cf. Section 4.1).
Review the definition of an ABC prefibration structure.

######  Definition 4.2 (A. Radulescu-Banu [22]).

Let $\mathscr{C}$ be a category. An ABC (Anderson-Brown-Cisinski) prefibration
structure $(W,\operatorname{Fib})$ on $\mathscr{C}$ is a pair of two classes
$W$ and $\operatorname{Fib}$ of some morphisms, called weak equivalences and
fibration respectively, satisfying the following axioms:

  1. 1.

$\mathscr{C}$ has a terminal object 1. (An object $X$ is fibrant if the unique
morphism $X\rightarrow{\bf 1}$ is a fibration.)

  2. 2.

$W$ and $\operatorname{Fib}$ are closed under composition.

  3. 3.

Any isomorphism is a weak equivalence. Furthermore, if the codomain is
fibrant, any isomorphism is an acyclic fibration. (A morphism $f$ is an
acyclic fibration if $f$ is a fibration and a weak equivalence.)

  4. 4.

If two of the three morphisms $f$, $g$ and $g\circ f$ are weak equivalences,
then the third is also a weak equivalence.

  5. 5.

Let $Y$ and $Y^{\prime}$ be fibrant objects, and let $f:X\rightarrow Y$ be a
fibration. Then, there exists a pull-back $f^{\prime}:X^{\prime}\rightarrow
Y^{\prime}$ of $f$ along any morphism $Y^{\prime}\rightarrow Y$. And the
morphism $f^{\prime}$ is a fibration. Furthermore, if $f$ is acyclic, then
$f^{\prime}$ is also acyclic.

  6. 6.

Any morphism $f$ has a decomposition $f=p\circ e$ such that the morphism $e$
is a weak equivalence and the morphism $p$ is a fibration.

“ABC (Anderson-Brown-Cisinski)” comes from names of D. W. Anderson [1], K. S.
Brown [2] and D.-C. Cisinski.

###  4.1 The flexible track

Let $f:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism between continuous
sheaves on $B$. $f$ induces a morphism
$\tilde{f}:\mathcal{F}^{\ast}\rightarrow\mathcal{G}^{\ast}$ between the
sheaves of formal sections. We denote the mapping track of $\tilde{f}$ by
$\tilde{f^{\prime}}:\mathcal{F}^{\ast}_{f}\rightarrow\mathcal{G}^{\ast}$.

######  Definition 4.3.

We call the above $\tilde{f^{\prime}}$ the flexible track of $f$.

######  Remark 4.4.

If $\mathcal{G}={\bf 1}$, then $\mathcal{G}^{\ast}={\bf 1}$. Then, we obtain
the equality $\mathcal{F}^{\ast}=\mathcal{F}^{\ast}_{f}$.

Recall that the claim (1) of Lemma 0.2 has been proved (cf. Lemma 2.10). We
can prove the claim (2) of Lemma 0.2. Here we prove a more general statement.

######  Lemma 4.5.

Let $B$ be a normal Hausdorff space. For any morphism
$f:\mathcal{F}\rightarrow\mathcal{G}$ between continuous sheaves on $B$, the
flexible track $\tilde{f^{\prime}}$ of $f$ is a flexible extension. In
prticular, for any continuous sheaf $\mathcal{F}$ on $B$, the sheaf
$\mathcal{F}^{\ast}$ of formal sections of $\mathcal{F}$ is flexible.

###### Proof.

For any two compact subsets $K$ and $L$ of $B$ with $K\subset L$, we will show
that the morphism $\tilde{f^{\prime}}_{K\subset
L}:\mathcal{F}^{\ast}_{f}(L)\rightarrow\mathcal{F}^{\ast}_{f}(K)\times_{\mathcal{G}^{\ast}(K)}\mathcal{G}^{\ast}(L)$
is a fibration. i.e. We will show that, for a following commutative square,
there exists a lift $\gamma$:

$\textstyle{\Delta^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathcal{F}^{\ast}_{f}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f^{\prime}}_{K\subset
L}}$$\textstyle{\Delta^{n-1}\times[0,1]_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\textstyle{\mathcal{F}^{\ast}_{f}(K)\times_{\mathcal{G}^{\ast}(K)}\mathcal{G}^{\ast}(L).}$
(1)

(The above $[0,1]_{1}$ is a copy of the closed interval $[0,1]$. To
distinguish it from the closed interval $[0,1]_{2}$ which comes later, we mark
it with a subscript.) By the discussion in Section 1.1.4, there exist small
enough open neighbourhoods $U$ and $V$ of $K$ and $L$ respectively with
$U\subset V$ such that the following diagram is commutative:

$\textstyle{\Delta^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\alpha}}$$\textstyle{\mathcal{F}^{\ast}_{f}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f^{\prime}}_{U\subset
V}}$$\textstyle{\Delta^{n-1}\times[0,1]_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\beta}}$$\textstyle{\mathcal{F}^{\ast}_{f}(U)\times_{\mathcal{G}^{\ast}(U)}\mathcal{G}^{\ast}(V).}$
(2)

We have the equality
$\mathcal{F}^{\ast}_{f}(V)=\mathcal{F}^{\ast}(V)\times_{\mathcal{G}^{\ast}(V)}\mathcal{G}^{\ast}(V)^{[0,1]_{2}}$.
(The above $[0,1]_{2}$ is a copy of the closed interval $[0,1]$. The above
morphism $\tilde{\alpha}:\Delta^{n-1}\rightarrow\mathcal{F}^{\ast}_{f}(V)$ can
be represented as $\tilde{\alpha}=\alpha^{1}\times\alpha^{2}$ using the
following $\alpha^{1}$ and $\alpha^{2}$:

$\begin{array}[]{l}\alpha^{1}:\Delta^{n-1}\rightarrow\mathcal{F}^{\ast}(V),\\\
\alpha^{2}:\Delta^{n-1}\rightarrow\mathcal{G}^{\ast}(V)^{[0,1]_{2}}.\end{array}$

To distinguish it from the closed interval $[0,1]_{1}$ which appeared first,
we mark it with a subscript.) Similarly, we decompose the above morphism
$\tilde{\beta}$. $\tilde{\beta}$ can be represented as
$\tilde{\beta}=\beta^{1}\times\beta^{2}$ using the following $\beta^{1}$ and
$\beta^{2}$:

$\begin{array}[]{l}\beta^{1}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{F}^{\ast}_{f}(U),\\\
\beta^{2}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{G}^{\ast}(V).\end{array}$

Furthermore, $\beta^{1}$ can be represented as
$\beta^{1}=\beta^{11}\times\beta^{12}$ using the following $\beta^{11}$ and
$\beta^{12}$:

$\begin{array}[]{l}\beta^{11}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{F}^{\ast}(U),\\\
\beta^{12}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{G}^{\ast}(U)^{[0,1]_{2}}.\end{array}$

Take a support function $\phi:V\rightarrow[0,1]$ such that we have the
inclusion relations $\operatorname{supp}(\phi)\subset U$ and the equality
$\phi\equiv 1$ on an open neighbourhood of $K$. (This is possible because $B$
is a normal Hausdorff space.) Let $W:=V-\operatorname{supp}(\phi)$.

Using these data $\phi$ and $W$ and the morphisms $\alpha^{1}$, $\alpha^{2}$,
$\beta^{11}$, $\beta^{12}$ and $\beta^{2}$, we will construct the following
morphisms $\gamma^{1}_{1}$, $\gamma^{1}_{2}$, $\gamma^{2}_{1}$ and
$\gamma^{2}_{2}$:

$\begin{array}[]{l}\gamma^{1}_{1}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{F}^{\ast}(U),\\\
\gamma^{1}_{2}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{F}^{\ast}(W),\\\
\gamma^{2}_{1}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{G}^{\ast}(U)^{[0,1]_{2}},\\\
\gamma^{2}_{2}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{G}^{\ast}(W)^{[0,1]_{2}}.\end{array}$

Take the subset $\Sigma$ of $[0,1]_{1}\times[0,1]_{2}$ and the strong
deformation retraction
$R_{u}:[0,1]_{1}\times[0,1]_{2}\rightarrow[0,1]_{1}\times[0,1]_{2}$ in Lemma
1.22. A strong deformation retraction $r_{u}:[0,1]_{1}\rightarrow[0,1]_{1}$ of
the inclusion $\\{0\\}\hookrightarrow[0,1]_{1}$ is defdined by the equality
$R_{u}(s,0)=(r_{u}(s),0)$. A continuous map $\psi:\Delta^{n-1}\times
V\times[0,1]_{1}\rightarrow\Delta^{n-1}\times[0,1]_{1}$ is defined as
$\psi(p,x,s):=(p,r_{1-\phi(x)}(s))$. Identifying $\Delta^{n-1}\times[0,1]_{1}$
with $\Delta^{n}$ by a homeomorphism, we regard $\Delta^{n-1}\times[0,1]_{1}$
as an object in ${\bf Sing\Delta}$. By the discussion in Section 2.3, the
continuous map $\psi$ induces a morphism

$\psi^{\ast}:\mathcal{F}^{\ast}[\Delta^{n-1}\times[0,1]_{1}]|_{V}\rightarrow\mathcal{F}^{\ast}[\Delta^{n-1}\times[0,1]_{1}]|_{V}$

between sheaves of sets.

Let $\gamma^{1}_{1}:=\psi^{\ast}(\beta^{11})$. Let
$\gamma^{1}_{2}:=\alpha^{1}|_{W}\circ\operatorname{pr}$, where the morphism
$\alpha^{1}|_{W}$ is the composite morphism
$\Delta^{n-1}\overset{\alpha^{1}}{\rightarrow}\mathcal{F}^{\ast}(V)\rightarrow\mathcal{F}^{\ast}(W)$
and $\operatorname{pr}$ is the canonical projection $\Delta^{n-1}\times
W\times[0,1]_{1}\rightarrow\Delta^{n-1}\times W$.

A continuous map $\Psi:\Delta^{n-1}\times
V\times[0,1]_{1}\times[0,1]_{2}\rightarrow\Delta^{n-1}\times[0,1]_{1}\times[0,1]_{2}$
is defined as $\Psi(p,x,s,t):=(p,R_{1-\phi(x)}(s,t))$. By the discussion in
Section 2.3, the continuous map $\Psi$ induces a morphism

$\Psi^{\ast}:\mathcal{G}^{\ast}[\Delta^{n-1}\times[0,1]_{1}\times[0,1]_{2}]|_{V}\rightarrow\mathcal{G}^{\ast}[\Delta^{n-1}\times[0,1]_{1}\times[0,1]_{2}]|_{V}$

between sheaves of sets.

Let $\gamma^{2}_{1}:=\Psi^{\ast}(\beta^{12})$. To define $\gamma^{2}_{2}$, we
define $\nu_{1}$ and $\nu_{2}$ as the following compositions:

$\begin{array}[]{l}\nu_{1}:\Delta^{n-1}\times\\{0\\}\times[0,1]_{2}\overset{\cong}{\rightarrow}\Delta^{n-1}\times[0,1]_{2}\overset{\alpha^{2}}{\rightarrow}\mathcal{G}^{\ast}(W),\\\
\nu_{2}:\Delta^{n-1}\times[0,1]_{1}\times\\{1\\}\overset{\cong}{\rightarrow}\Delta^{n-1}\times[0,1]_{1}\overset{\beta^{2}}{\rightarrow}\mathcal{G}^{\ast}(W).\end{array}$

We have the equality $\nu_{1}=\nu_{2}$ on
$\Delta^{n-1}\times\\{0\\}\times\\{1\\}$ because the diagram (2) is
commutative. Then, we obtain a morphism $\nu:\Delta^{n-1}\times
W\times\Sigma\rightarrow\mathcal{G}^{\ast}(W)$ by pasting $\nu_{1}$ and
$\nu_{2}$. Let $\gamma^{2}_{2}:=\nu\circ(id\times R_{1})$.

$\gamma^{1}_{1}$ and $\gamma^{2}_{1}$ induce a morphism
$\gamma_{1}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{F}^{\ast}_{f}(U)$.
$\gamma^{1}_{2}$ and $\gamma^{2}_{2}$ induce a morphism
$\gamma_{2}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{F}^{\ast}_{f}(W)$.
$\gamma_{1}$ and $\gamma_{2}$ induce a morphism
$\tilde{\gamma}:\Delta^{n-1}\times[0,1]_{1}\rightarrow\mathcal{F}^{\ast}_{f}(V)$
with the sheaf condition of $\mathcal{F}^{\ast}_{f}$ with respect to the open
covering $\\{U,W\\}$ of $V$. $\tilde{\gamma}$ induces a lift $\gamma$ of the
diagram (1). Therefore, the morphism $\tilde{f^{\prime}}_{K\subset L}$ has the
right lifting property with respect to $J_{\bf qTop}$. This means that
$\tilde{f^{\prime}}$ is a flexible extension. ∎

###  4.2 A prefibration structure for continuous sheaves

We can prove Theorem 4.1.

###### proof of Theorem 4.1.

We will check the axioms 1. to 6. of Definition 4.2. The axioms 1. to 5. is
obvious by Theorem 3.1. We will check the axiom 6..

We consider the following diagram on ${\bf Sh}(B;{\bf qTop})$, embedded in
${\bf PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$:

---
$\textstyle{\mathcal{F}^{\ast}_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f^{\prime}}}$$\textstyle{\mathcal{F}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathcal{G}^{\ast}}$$\textstyle{\mathcal{F}^{\ast}_{f}\times_{\mathcal{G}^{\ast}}\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{f}$$\textstyle{\mathcal{G}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$

The diagonal morphisms $\mathcal{F}\rightarrow\mathcal{F}^{\ast}$ and
$\mathcal{G}\rightarrow\mathcal{G}^{\ast}$ are stalkwise weak equivalences by
Lemma 2.10. The flexible track $\tilde{f^{\prime}}$ of $f$ is a flexible
extension by Lemma 4.5. The above morphism
$\mathcal{F}^{\ast}\rightarrow\mathcal{F}^{\ast}_{f}$ is a sectionwise weak
equivalence. Then, this is a stalkwise weak equivalence by Proposition 2.4.
The above morphism
$\mathcal{F}^{\ast}_{f}\times_{\mathcal{G}^{\ast}}\mathcal{G}\rightarrow\mathcal{F}^{\ast}_{f}$
is a stalkwise weak equivalence because the model category ${\bf
PSh}_{\ast}(\tilde{\mathscr{U}}_{B};{\bf qTop})$ is right proper. Then, the
above morphism
$\mathcal{F}\rightarrow\mathcal{F}^{\ast}_{f}\times_{\mathcal{G}^{\ast}}\mathcal{G}$
is a stalkwise weak equivalence. The above morphism
$\mathcal{F}^{\ast}_{f}\times_{\mathcal{G}^{\ast}}\mathcal{G}\rightarrow\mathcal{G}$
is a flexible extension because it is the pull-back of a flexible extension.
Then, the decomposition
$\mathcal{F}\rightarrow\mathcal{F}^{\ast}_{f}\times_{\mathcal{G}^{\ast}}\mathcal{G}\rightarrow\mathcal{G}$
of $f$ is what we want. The axiom 6. is satisfied. Therefore, the category
${\bf Sh}(B;{\bf qTop})$ of continuous sheaves has an ABC prefibration
structure we want. ∎

##  Appendix A Model categories

A model category is a category in which Homotopy Theory can be discussed. In
this paper, basic concepts such as a model category, a Quillen adjunction,
etc. are assumed to be known. See [14] or [13] for more information. We review
some of the additional properties.
Note the following notation used in the model category theory.

######  Definition A.1.

  * •

$f\pitchfork g$ $\Leftrightarrow$ $g$ has the right lifting property with
respect to $f$.

For a class $I$ of some morphisms, we define the followings.

  * •

$\operatorname{rlp}(I)$ is the class of morphisms having the right lifting
property with respect to $I$.

  * •

$\operatorname{cell}(I)$ is the class of relative $I$-cell complexes.

  * •

$\operatorname{Cof}(I)$ is the class of retracts of relative $I$-cell
complexes.

###  A.1 Cofibrantly generated model categories

It is difficult to check the axioms of a model category. It is easier to check
that the necessary and sufficient conditions of a cofibrantly generated model
category.

####  A.1.1 Small object argument

######  Definition A.2 (small objects).

Let $\mathscr{C}$ be a cocomplete category, and let $\mathscr{D}$ be a
subcategory of $\mathscr{C}$.

  * •

Let $\kappa$ be a cardinal. An object $W$ in $\mathcal{C}$ is $\kappa$-small
relative to $\mathscr{D}$ if, for any regular cardinal $\lambda(\geq\kappa)$
and any $\lambda$-sequence $X$ in $\mathscr{D}$, the following map is a
bijection:

$\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}\mathscr{C}(W,X_{\beta})\cong\mathscr{C}(W,\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}X_{\beta}).$

(Remark that $\displaystyle\lim_{\longrightarrow}X$ is a colimit in
$\mathscr{C}$.)

  * •

In particular, if $\mathscr{D}=\mathscr{C}$, $W$ is $\kappa$-small.

  * •

An object $W$ in $\mathcal{C}$ is small relative to $\mathscr{D}$ if there
exists a cardinal $\kappa$ such that the object $W$ is $\kappa$-small relative
to $\mathscr{D}$.

  * •

In particular, if $\mathscr{D}=\mathscr{C}$, $W$ is small.

######  Example A.3.

Let $\mathscr{U}$ be a small category. For any object $u\in Ob(\mathscr{U})$,
the presheaf $\mathscr{U}(-,u)\in{\bf PSh}(\mathscr{U};{\bf Set})$ is
$\aleph_{0}$-small. In fact, by Yoneda’s Lemma, there exists the following
bijection:

$\begin{array}[]{rcl}\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}{\bf PSh}(\mathscr{U};{\bf
Set})(\mathscr{U}(-,u),X_{\beta})&\cong&\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}(X_{\beta}[u])\\\
&\cong&(\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}X_{\beta})[u]\\\ &\cong&{\bf PSh}(\mathscr{U};{\bf
Set})(\mathscr{U}(-,u),\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}X_{\beta}).\end{array}$

######  Example A.4.

Let $\mathscr{U}$ be a small site such that every covering has a finite
refinement. If a sheaf $\mathcal{F}$ on $\mathscr{U}$ is $\kappa$-small as a
presheaf, the sheaf $\mathcal{F}$ is $\kappa$-small as a sheaf. There exists
the following bijection:

$\begin{array}[]{rcl}\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}{\bf Sh}(\mathscr{U};{\bf
Set})(\mathcal{F},X_{\beta})&\cong&\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}{\bf PSh}(\mathscr{U};{\bf Set})(\mathcal{F},X_{\beta})\\\
&\cong&{\bf PSh}(\mathscr{U};{\bf
Set})(\mathscr{U}(-,u),\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}X_{\beta})\\\ &\cong&{\bf Sh}(\mathscr{U};{\bf
Set})(\mathscr{U}(-,u),\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}X_{\beta}).\end{array}$

(Remark that the colimit
$\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\end{subarray}}X_{\beta}$
as a sheaf is the colimit as a presheaf because every covering has a finite
refinement.)

######  Proposition A.5 (cf. [13, Proposition 10.4.8]).

Let $W:\mathscr{P}\rightarrow\mathscr{C}$ be a small diagram (i.e. $W$ is a
functor from a finite category $\mathscr{P}$). If each
$W_{p}$($p\in\operatorname{Ob}(\mathscr{P})$) is small relative to
$\mathscr{D}$, the colimit
$\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\end{subarray}}W_{p}$ is
also small relative to $\mathscr{D}$.

######  Proposition A.6.

Let $F\dashv G:\mathscr{C}\rightarrow\mathscr{C^{\prime}}$ be an adjunction,
and let $\mathscr{D}$ be a subcategory of $\mathscr{C}$. Suppose that the
functor $G$ preserves any filtered colimit. If an object $c\in
Ob(\mathscr{C})$ is $\kappa$-small relative to $\mathscr{D}$, the object
$F(c)\in Ob(\mathscr{C})$ is $\kappa$-small relative to $G^{-1}(\mathscr{D})$.

###### Proof.

$\begin{array}[]{rcl}\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}\mathscr{C^{\prime}}(F(c),X_{\beta})&\cong&\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}\mathscr{C}(c,G(X_{\beta}))\\\
&\cong&\mathscr{C}(c,\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}G(X_{\beta}))\\\
&\cong&\mathscr{C}(c,G(\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}X_{\beta}))\\\
&\cong&\mathscr{C^{\prime}}(F(c),\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
\beta\end{subarray}}X_{\beta}).\end{array}$

∎

######  Definition A.7 (small object argument).

Let $\mathscr{C}$ be a cocomplete category, and let $I$ be a set of some
morphisms in $\mathscr{C}$.

  * •

Let $\kappa$ be a cardinal. $I$ permits the $\kappa$-small object argument if,
for any morphism $x\rightarrow y$ belonging $I$, the domain $x$ is
$\kappa$-small relative to the subcategory $\operatorname{cell}(I)$.

  * •

$I$ permits the small object argument if there exists a cardinal $\kappa$ such
that $I$ permits the $\kappa$-small object argument.

####  A.1.2 Cofibrantly generated model categories

######  Definition A.8 (cofibrantly generated model categories).

A model category $\mathscr{C}$ is cofibrantly generated if there exist two
sets $I$ and $J$ of some morphisms in $\mathscr{C}$ satisfying the following:

  * •

The class of cofibrations of $\mathscr{C}$ is $\operatorname{Cof}(I)$ (cf.
Definition A.1).

  * •

The class of acyclic cofibrations of $\mathscr{C}$ is $\operatorname{Cof}(J)$
(cf. Definition A.1).

  * •

$I$ and $J$ permit the small object argument (cf. Definition A.7).

Then, we say that $I$ (resp. $J$) generates cofibrations (resp. acyclic
cofibrations).

######  Theorem A.9 (Recognition Theorem; cf. [13, Theorem 11.3.1]).

Let $\mathscr{C}$ be a complete and cocomplete category. Let $W$ be a class of
some morphisms in $\mathscr{C}$, and let $I$ and $J$ be two sets of some
morphisms in $\mathscr{C}$. There exists a model structure on $\mathscr{C}$
such that the class of weak equivalences is $W$, the class of cofibrations is
generated by $I$ and the class of acyclic cofibrations is generated by $J$ if
and only if the following conditions hold:

  1. 1.

We have the inclusion relation
$\operatorname{Cof}(J)\subset\operatorname{Cof}(I)\cap W$ (cf. Definition
A.1).

  2. 2.

We have the inclusion relation
$\operatorname{rlp}(I)\subset\operatorname{rlp}(J)\cap W$ (cf. Definition
A.1).

  3. 3.

The reverse inclusion holds for one of the two inclusions above. i.e. We have
one of the following inclusion relations:

    1. (a)

$\operatorname{Cof}(J)\supset\operatorname{Cof}(I)\cap W$, or

    2. (b)

$\operatorname{rlp}(I)\supset\operatorname{rlp}(J)\cap W$.

  4. 4.

If two of the three morphisms $f$, $g$ and $g\circ f$ are contained in $W$,
then the third is also contained in $W$.

  5. 5.

$I$ and $J$ permit the small object argument (cf. Definition A.7).

######  Theorem A.10 (Transfer Theorem; cf. [13, Theorem 11.3.2]).

Let $\mathscr{C}$ be a cofibrantly generated model category such that a set
$I$ generates cofibrations and that a set $J$ generates acyclic cofibrations.
Let $W$ be the class of weak equivalences in $\mathscr{C}$. For a category
$\mathscr{D}$ and adjunction $F\dashv G:\mathscr{C}\rightarrow\mathscr{D}$,
suppose the followings:

  1. 1.

$F(I)$ and $F(J)$ permit the small object argument (cf. Definition A.7).

  2. 2.

We have the inclusion relation $\operatorname{cell}(F(J))\subset G^{-1}(W)$
(cf. Definition A.1).

Then, $\mathscr{D}$ has a cofibrantly generated model structure such that the
class of weak equivalences is $G^{-1}(W)$, the class of cofibrations is