# Model Category Structure on Simplicial Algebras
via Dold-Kan Correspondence

Hossein Faridian  Hossein Faridian, School of Mathematical and Statistical
Sciences, Clemson University, SC 29634, USA<EMAIL_ADDRESS>
###### Abstract.

This expository article sets forth a self-contained and purely algebraic proof
of a deep result of Quillen stating that the category of simplicial
commutative algebras over a commutative ring is a model category. This is
accomplished by starting from the model structure on the category of
connective chain complexes, transferring it to the category of simplicial
modules via Dold-Kan Correspondence, and further transferring it to the
category of simplicial commutative algebras through Quillen-Kan Transfer
Machine. The subtlety of overcoming the acyclicity condition is addressed by
introducing and studying the shuffle product of connective chain complexes,
establishing a variant of Eilenberg-Zilber Theorem, and carefully scrutinizing
the subtle structures under study.

###### Key words and phrases:

model category; connective chain complex; simplicial module; simplicial
algebra

###### 2010 Mathematics Subject Classification:

18N40; 18G31; 18G50; 18N50.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on Model Categories
  3. 3 Model Structure on Connective Chain Complexes
  4. 4 Simplicial Objects and Dold-Kan Correspondence Theorem
  5. 5 Shuffle Product and a Variant of Eilenberg-Zilber Theorem
  6. 6 Model Structure on Simplicial Commutative Algebras

##  1\. Introduction

Homological ideas grew out of attempts to investigate deeper properties of
topological spaces by means of studying certain algebraic objects, namely
homology and cohomology groups. These ideas took abstract forms later and
planted the early seeds of homological algebra in the framework of category
theory. In the grand scheme of things, the main goal of homological algebra is
to construct derived functors of a given functor which are in some sense the
best approximations of the functor at hand.

To get a flavor of derived functors in a classical setting, let $R$ and $S$ be
two rings, and let
$\mathcal{F}:\mathcal{M}\mathpzc{od}(R)\rightarrow\mathcal{M}\mathpzc{od}(S)$
be an additive covariant functor between module categories. For any $i\geq 0$,
the $i$th left derived functor of $\mathcal{F}$ is a functor
$\textrm{L}_{i}\mathcal{F}:\mathcal{M}\mathpzc{od}(R)\rightarrow\mathcal{M}\mathpzc{od}(S)$,
whose effect on a left $R$-module $M$ is given by
$\textrm{L}_{i}\mathcal{F}(M)=H_{i}\left(\mathcal{F}(P)\right)$ where $P$ is a
chosen projective resolution of $M$, and the $i$th right derived functor of
$\mathcal{F}$ is a functor
$\textrm{R}^{i}\mathcal{F}:\mathcal{M}\mathpzc{od}(R)\rightarrow\mathcal{M}\mathpzc{od}(S)$,
whose effect on a left $R$-module $M$ is given by
$\textrm{R}^{i}\mathcal{F}(M)=H_{-i}\left(\mathcal{F}(I)\right)$ where $I$ is
a chosen injective resolution of $M$. To go one step further, one can assemble
all these derived functors into a single total derived functor. More
specifically, let $\mathcal{F}:\mathcal{C}(R)\rightarrow\mathcal{C}(S)$ be a
covariant functor between categories of chain complexes that preserves
homotopy. Then the total left derived functor of $\mathcal{F}$ is a functor
$\textrm{L}\mathcal{F}:\mathcal{D}(R)\rightarrow\mathcal{D}(S)$ between
derived categories, whose effect on an $R$-complex $X$ is given by
$\textrm{L}\mathcal{F}(X)=\mathcal{F}(P)$ where $P$ is a chosen semi-
projective resolution of $X$, and the total right derived functor of
$\mathcal{F}$ is a functor
$\textrm{R}\mathcal{F}:\mathcal{D}(R)\rightarrow\mathcal{D}(S)$, whose effect
on an $R$-complex $X$ is given by $\textrm{R}\mathcal{F}(X)=\mathcal{F}(I)$
where $I$ is a chosen semi-injective resolution of $X$. This machinery can be
developed in abelian categories and their corresponding derived categories.
However, there are certain contexts in which one requires to have some sort of
derived functors in a non-abelian setting where the notion of a chain complex
is missing.

One such situation arises in the context of commutative algebras over
commutative rings. Let $R$ be a commutative ring, $A$ a commutative
$R$-algebra, and $M$ an $A$-module. A derivation of $A$ over $R$ with
coefficients in $M$ is an $R$-homomorphism $D:A\rightarrow M$ that satisfies
the Leibniz rule, i.e. $D(ab)=aD(b)+bD(a)$ for every $a,b\in A$. The set of
all derivations of $A$ over $R$ with coefficients in $M$, denoted by
$\operatorname{Der}_{R}(A,M)$, is an $A$-submodule of
$\operatorname{Hom}_{R}(A,M)$. In addition, the induced functor
$\operatorname{Der}_{R}(A,-):\mathcal{M}\mathpzc{od}(A)\rightarrow\mathcal{M}\mathpzc{od}(A)$
is representable, i.e. there exists an $A$-module $\Omega_{A|R}$, the so-
called module of differentials of $A$ over $R$, and a universal derivation
$d_{A|R}:A\rightarrow\Omega_{A|R}$ that induces a natural $A$-isomorphism
$\operatorname{Der}_{R}(A,M)\cong\operatorname{Hom}_{A}\left(\Omega_{A|R},M\right)$.
The module of differentials $\Omega_{A|R}$ can be concretely described as
$\Omega_{A|R}=\mathfrak{k}/\mathfrak{k}^{2}$ where
$\mathfrak{k}=\operatorname{Ker}(\mu)$ in which $\mu:A\otimes_{R}A\rightarrow
A$ is given by $\mu(a\otimes b)=ab$ for every $a,b\in A$. In addition, the
universal derivation $d_{A|R}:A\rightarrow\Omega_{A|R}$ is given by
$d_{A|R}(a)=(1\otimes a-a\otimes 1)+\mathfrak{k}^{2}$ for every $a\in A$. Now
assume that $R\rightarrow S\rightarrow T$ are two homomorphisms of commutative
rings, and $M$ is a $T$-module. Then one has the following exact sequences:

$\Omega_{S|R}\otimes_{S}M\rightarrow\Omega_{T|R}\otimes_{T}M\rightarrow\Omega_{T|S}\otimes_{T}M\rightarrow
0$

and

$0\rightarrow\operatorname{Der}_{S}(T,M)\rightarrow\operatorname{Der}_{R}(T,M)\rightarrow\operatorname{Der}_{R}(S,M)$

Historically speaking, people grew interested in extending these sequences
beyond the above-displayed three terms. Viewing derivations and differentials
as functors of commutative algebras, one requires to have some sort of derived
functors in order to extend the sequences. However, the category of
commutative algebras is not abelian; it fails to have a zero object or a zero
morphism. As a consequence, the machinery of homological algebra as we know it
fails in this situation.

The solution comes from Quillen’s model category theory. The ideas behind the
theory stem from homotopy theory in algebraic topology that are suitably
adapted to the language of category theory. This has led to the emergence of
non-abelian homological algebra. A model category is a locally small
bicomplete category equipped with three distinguished classes of morphisms
called fibrations, cofibrations, and weak equivalences that satisfy certain
axioms. A model category supports a notion of replacement and a notion of
homotopy that allow one to construct derived functors in a more general
setting than the derived categories of abelian categories. We outline the
general theory of derived functors as follows. If
$\mathcal{F}:\mathcal{C}\rightarrow\mathcal{D}$ and
$\mathcal{G}:\mathcal{C}\rightarrow\mathcal{E}$ are two functors between
arbitrary categories, and
$\mathcal{G}^{\ast}:\mathcal{D}^{\mathcal{E}}\rightarrow\mathcal{D}^{\mathcal{C}}$
denotes the right composition functor, then a left Kan extension of
$\mathcal{F}$ along $\mathcal{G}$ is a functor
$\operatorname{LKan}_{\mathcal{G}}(\mathcal{F}):\mathcal{E}\rightarrow\mathcal{D}$
together with a natural transformation
$\sigma:\mathcal{F}\rightarrow\operatorname{LKan}_{\mathcal{G}}(\mathcal{F})\mathcal{G}$
such that the pair
$\left(\operatorname{LKan}_{\mathcal{G}}(\mathcal{F}),\sigma\right)$ is an
initial object in the comma category
$\left(\mathcal{F}\downarrow\mathcal{G}^{\ast}\right)$, and a right Kan
extension of $\mathcal{F}$ along $\mathcal{G}$ is a functor
$\operatorname{RKan}_{\mathcal{G}}(\mathcal{F}):\mathcal{E}\rightarrow\mathcal{D}$
together with a natural transformation
$\varsigma:\operatorname{RKan}_{\mathcal{G}}(\mathcal{F})\mathcal{G}\rightarrow\mathcal{F}$
such that the pair
$\left(\operatorname{RKan}_{\mathcal{G}}(\mathcal{F}),\varsigma\right)$ is a
terminal object in the comma category
$\left(\mathcal{G}^{\ast}\downarrow\mathcal{F}\right)$. Now if
$\mathcal{F}:\mathcal{C}\rightarrow\mathcal{D}$ is a functor between model
categories, then the left derived functor of $\mathcal{F}$ is
$\textrm{L}\mathcal{F}=\operatorname{RKan}_{\mathcal{L}_{\mathcal{C}}}(\mathcal{\mathcal{L}_{\mathcal{D}}F}):\operatorname{Ho}(\mathcal{C})\rightarrow\operatorname{Ho}(\mathcal{D})$,
and the right derived functor of $\mathcal{F}$ is
$\textrm{R}\mathcal{F}=\operatorname{LKan}_{\mathcal{L}_{\mathcal{C}}}(\mathcal{\mathcal{L}_{\mathcal{D}}F}):\operatorname{Ho}(\mathcal{C})\rightarrow\operatorname{Ho}(\mathcal{D})$,
where
$\mathcal{L}_{\mathcal{C}}:\mathcal{C}\rightarrow\operatorname{Ho}(\mathcal{C})$
and
$\mathcal{L}_{\mathcal{D}}:\mathcal{D}\rightarrow\operatorname{Ho}(\mathcal{D})$
are localization functors. Derived functors at this level of generality can be
shown to exist under mild conditions by leveraging fibrant and cofibrant
replacements which in turn can be thought of as generalizations of
resolutions. It is also worth noting that if $R$ is a ring, then the category
$\mathcal{C}(R)$ of $R$-complexes has two model structures under any of which,
one has $\operatorname{Ho}\left(\mathcal{C}(R)\right)\cong\mathcal{D}(R)$, and
the general theory of derived functors reduces to the special abelian case
described before.

In order to effectively apply Quillen’s theory to the situation of interest,
i.e. commutative algebras over commutative rings, one needs a non-abelian
substitute for chain complexes. The Dold-Kan Correspondence is inspiring in
this direction. It states that the category of connective chain complexes over
a ring is equivalent to the category of simplicial modules over the ring.
Although chain complexes are not available in non-abelian categories,
simplicial objects are always at our disposal in any category. The punch line
is that simplicial objects can serve as the non-abelian analogues of chain
complexes. In particular, one should consider the category of simplicial
commutative algebras in this situation and equip it with a model structure.
Unfortunately, this task is by far harder than it seems. In addition, all the
existing arguments in the literature rely one way or another on the model
structure of simplicial sets; see for example [Qu2, Chapter II, Section 3].
Simplicial sets can be regarded as combinatorial models for topological spaces
up to homotopy. Both topological spaces and simplicial sets are primordial
prototypes of model categories. Let us look at the classical model structure
of simplicial sets. Fibrations are Kan fibrations, cofibrations are injective
morphisms, and weak equivalences are morphisms whose geometric realizations
induce isomorphisms on topological homotopy groups. As it stands, the weak
equivalences lack a purely algebraic description, so the proof of model
structure of simplicial sets and hence simplicial commutative algebras is
based partly on topological foundations. From an algebraist’s point of view,
it can be reasonable to look for a purely algebraic proof of the model
structure of simplicial commutative algebras. This is in fact the motivation
of writing this article.

Back to the commutative algebra setting, we observe that after equipping the
category of simplicial commutative algebras with a model structure, one can
apply the general theory of derived functors to construct the cotangent
complex $\mathbb{L}^{A|R}\in\operatorname{Obj}\left(\mathcal{D}(A)\right)$ as
the image of the identity map $1^{A}:A\rightarrow A$ under the left derived
functor of the abelianization functor followed by the right derived functor of
the normalization functor. The cotangent complex is in turn used to define the
André-Quillen homology and cohomology modules as
$H_{i}^{AQ}\left(A|R;M\right)=\operatorname{Tor}_{i}^{A}\left(\mathbb{L}^{A|R},M\right)$
and
$H_{AQ}^{i}\left(A|R;M\right)=\operatorname{Ext}_{A}^{i}\left(\mathbb{L}^{A|R},M\right)$
for every $i\geq 0$. These modules enable one to extend the above-mentioned
sequences to the following exact sequences:

$\displaystyle\cdots\rightarrow H_{2}^{AQ}\left(S|R;M\right)\rightarrow
H_{2}^{AQ}\left(T|R;M\right)\rightarrow
H_{2}^{AQ}\left(T|S;M\right)\rightarrow
H_{1}^{AQ}\left(S|R;M\right)\rightarrow$ $\displaystyle
H_{1}^{AQ}\left(T|R;M\right)\rightarrow
H_{1}^{AQ}\left(T|S;M\right)\rightarrow\Omega_{S|R}\otimes_{S}M\rightarrow\Omega_{T|R}\otimes_{T}M\rightarrow\Omega_{T|S}\otimes_{T}M\rightarrow
0$

and

$\displaystyle
0\rightarrow\operatorname{Der}_{S}(T,M)\rightarrow\operatorname{Der}_{R}(T,M)\rightarrow\operatorname{Der}_{R}(S,M)\rightarrow
H_{AQ}^{1}\left(T|S;M\right)\rightarrow
H_{AQ}^{1}\left(T|R;M\right)\rightarrow$ $\displaystyle
H_{AQ}^{1}\left(S|R;M\right)\rightarrow
H_{AQ}^{2}\left(T|S;M\right)\rightarrow
H_{AQ}^{2}\left(T|R;M\right)\rightarrow
H_{AQ}^{2}\left(S|R;M\right)\rightarrow\cdots$

As it turns out, André-Quillen homology and cohomology modules are powerful
tools in commutative algebra that can characterize various classes of rings
and algebras such as regular rings, complete intersection rings, smooth
algebras, étale algebras, etc. They have numerous applications in number
theory and deformation theory as well; see for example [An], [Av], [BI],
[BIK], [DI], [Fa], [Il], [Iy], [MR], [Qu1], [Qu2], and [Qu3]. Despite being a
desirable object of study for algebraists, André-Quillen homology and
cohomology modules have not received the kind of attention they merit from the
algebra community. This could be mostly due to its foundation that has not
been properly developed in an algebraic language. The surprising fact is that
the foundation of the theory, i.e. the model structure of simplicial
commutative algebras, is rooted in topology, while its applications, i.e. the
characterization of various classes of rings and algebras, fit completely
within the realm of commutative algebra. In this article, we strive to provide
a purely algebraic proof of the model structure of simplicial commutative
algebras that does not rely on the model structure of simplicial sets or
topological spaces.

The article is designed as follows. In section two, we collect all we need
from the theory of model categories with suitable references in order to set
up the language and notation. We further prove a few lemmas that will be used
later. In section three, we provide a proof of the model structure of
connective chain complexes. The direct proof of this fact is only outlined in
the literature in a disperse and incomplete manner; see for example [GS,
Theorem 1.5] or [DS, Theorem 7.2]. We give a neat and complete proof for the
sake of reference. This model structure is described completely algebraically
and serves as our archetypal example of a model category. Furthermore, we show
that this model structure is cofibrantly generated which allows us to transfer
it along adjunctions. In section four, we review simplicial objects, the
normalization and Dold-Kan functors, and the Dold-Kan Correspondence. In
addition, we transfer the model structure of connective chain complexes to
simplicial modules. In section five, we take advantage of the Dold-Kan functor
to introduce and study the shuffle product of connective chain complexes. The
shuffle product of elements of chain complexes has been studied in the
literature; see [Mc2, Theorem 8.8] and [Lu, Proposition 2.5.7.1 and Notation
2.5.7.2]. However, the shuffle product of connective chain complexes is a new
perspective. We then prove a variant of the Eilenberg-Zilber Theorem. The
contents of this section are of grave significance in the sequel. In section
six, we deploy the materials of section five and the Quillen-Kan Transfer
Machine to transport the model structure of connective chain complexes to
simplicial commutative algebras. One should note that [GS] adopts the same
approach, but at the most crucial step, i.e. the establishment of the
acyclicity condition, it resorts to the model structure of simplicial sets;
see [GS, Proposition 4.12, Corollary 4.14, and Theorem 4.17] and [GJ,
Proposition 11.5, Axiom 3.1, Theorem 4.1, and Example 5.2]. We mainly follow
the ideas of [GS], but leverage the shuffle product to avoid the model
structure of simplicial sets altogether. As it happens, we wind up with an
organized, self-contained, and completely algebraic proof with no reliance on
topology.

##  2\. Preliminaries on Model Categories

Quillen’s theory of model categories provides a general framework to practice
homological algebra far beyond the realm of abelian categories and their
corresponding derived categories. In this section, we collect some basic
definitions and facts from the theory of model categories and also prove a few
lemmas that will be needed later. For general references on model categories,
refer to [Ci], [DS], [GJ], [Hi], [Ho], [Qu2], and [Ri].

######  Definition 2.1.

A model category is a locally small bicomplete category $\mathcal{C}$ endowed
with three classes of morphisms called fibrations, cofibrations, and weak
equivalences. A morphism that is a fibration as well as a weak equivalence is
called a trivial fibration, and a morphism that is a cofibration as well as a
weak equivalence is called a trivial cofibration. Moreover, these morphisms
should satisfy the following conditions:

  1. (i)

Lifting: Any commutative diagram on the left in which $f$ is a trivial
cofibration and $g$ is a fibration, or $f$ is a cofibration and $g$ is a
trivial fibration, can be completed to the commutative diagram on the right:

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  2. (ii)

Retract: If $f$ is a retract of $g$ in the arrow category
$\operatorname{Arr}(\mathcal{C})$, i.e. there is a commutative diagram

${A}$${C}$${A}$${B}$${D}$${B}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{l}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{q}$

in which $lh=1^{A}$ and $qp=1^{B}$, and if $g$ is a fibration (cofibration, or
weak equivalence), then $f$ is a fibration (cofibration, or weak equivalence).

  3. (iii)

Two-Out-of-Three: If $f=gh$, and any two of $f$, $g$, or $h$ are weak
equivalences, then so is the third.

  4. (iv)

Factorization: For any morphism $f$, there is a factorization $f=gh$ for a
fibration $g$ and a trivial cofibration $h$, and also a factorization
$f=g^{\prime}h^{\prime}$ for a trivial fibration $g^{\prime}$ and a
cofibration $h^{\prime}$.

A model category supports a notion of homotopy which is the bedrock of
constructing derived functors. In order to define this notion, we need the
definitions of cylinder and path objects which could be shown to exist in
every model category.

######  Definition 2.2.

Let $\mathcal{C}$ be a model category, and
$A\in\operatorname{Obj}(\mathcal{C})$. Then:

  1. (i)

A cylinder object for $A$ is an object $\operatorname{Cyl}(A)$ through which
the codiagonal morphism $\nabla_{A}:A\sqcup A\rightarrow A$ factors as

${A\sqcup
A}$${A}$${\operatorname{Cyl}(A)}$$\scriptstyle{\nabla_{A}}$$\scriptstyle{\kappa_{A}}$$\scriptstyle{\xi_{A}}$

in which $\kappa_{A}$ is a cofibration and $\xi_{A}$ is a weak equivalence.

  2. (ii)

A path object for $A$ is an object $\operatorname{Path}(A)$ through which the
diagonal morphism $\Delta_{A}:A\rightarrow A\sqcap A$ factors as

${A}$${A\sqcap
A}$${\operatorname{Path}(A)}$$\scriptstyle{\Delta_{A}}$$\scriptstyle{\zeta_{A}}$$\scriptstyle{\nu_{A}}$

in which $\nu_{A}$ is a fibration and $\zeta_{A}$ is a weak equivalence.

######  Definition 2.3.

Let $\mathcal{C}$ be a model category, and $f,g:A\rightarrow B$ two morphisms
in $\mathcal{C}$. Then:

  1. (i)

$f$ is said to be left homotopic to $g$, denoted by $f\sim_{l}g$, if there is
a cylinder object $\operatorname{Cyl}(A)$ for $A$, as in Definition 2.2, such
that $f$ and $g$ factor through $\operatorname{Cyl}(A)$ as

${A}$${B}$${A}$${\operatorname{Cyl}(A)}$$\scriptstyle{f}$$\scriptstyle{\kappa_{A}\iota_{1}^{A}}$$\scriptstyle{g}$$\scriptstyle{\kappa_{A}\iota_{2}^{A}}$$\scriptstyle{\phi}$

in which $\iota_{1}^{A}:A\rightarrow A\sqcup A$ and
$\iota_{2}^{A}:A\rightarrow A\sqcup A$ are canonical injections. Moreover in
this case, $\phi$ is said to be a left homotopy from $f$ to $g$.

  2. (ii)

$f$ is said to be right homotopic to $g$, denoted by $f\sim_{r}g$, if there is
a path object $\operatorname{Path}(B)$ for $B$, as in Definition 2.2, such
that $f$ and $g$ factor through $\operatorname{Path}(B)$ as

${B}$${A}$${B}$${\operatorname{Path}(B)}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\psi}$$\scriptstyle{\pi_{1}^{B}\nu_{B}}$$\scriptstyle{\pi_{2}^{B}\nu_{B}}$

in which $\pi_{1}^{B}:B\sqcap B\rightarrow B$ and $\pi_{2}^{B}:B\sqcap
B\rightarrow B$ are canonical projections. Moreover in this case, $\psi$ is
said to be a right homotopy from $f$ to $g$.

  3. (iii)

$f$ is said to be homotopic to $g$, denoted by $f\sim g$, if $f\sim_{l}g$ and
$f\sim_{r}g$.

Recall that a model category is bicomplete, i.e. it has all small direct and
inverse limits. In particular, a model category has initial and terminal
objects. If $\mathcal{C}$ is a model category, $\circ$ is its initial object,
and $\ast$ is its terminal object, then an object $A$ in $\mathcal{C}$ is said
to be fibrant if the unique morphism $A\rightarrow\ast$ is a fibration, and is
said to be cofibrant if the unique morphism $\circ\rightarrow A$ is a
cofibration. Now we can see that homotopy is an equivalence relation on
morphisms.

######  Proposition 2.4.

Let $\mathcal{C}$ be a model category, and $f,g:A\rightarrow B$ two morphisms
in $\mathcal{C}$. Then the following assertions hold:

  1. (i)

If $f\sim_{l}g$ and $A$ is cofibrant, then $f\sim_{r}g$ and $\sim_{l}$ is an
equivalence relation on $\operatorname{Mor}_{\mathcal{C}}(A,B)$.

  2. (ii)

If $f\sim_{r}g$ and $B$ is fibrant, then $f\sim_{l}g$ and $\sim_{r}$ is an
equivalence relation on $\operatorname{Mor}_{\mathcal{C}}(A,B)$.

  3. (iii)

If $A$ is cofibrant and $B$ is fibrant, then $f\sim_{l}g$ if and only if
$f\sim_{r}g$, so $\sim$ is an equivalence relation on
$\operatorname{Mor}_{\mathcal{C}}(A,B)$.

###### Proof.

See [Hi, Proposition 7.4.5]. ∎

A model category enjoys a concept of replacement which can be thought of as a
generalization as well as a non-abelian analogue of a resolution in abelian
homological algebra. If $\mathcal{C}$ is a model category, and $A$ is an
object in $\mathcal{C}$, then a fibrant replacement of $A$ is a trivial
cofibration $\varrho_{A}:A\rightarrow A_{{\textrm{f}}}$ in which
$A_{{\textrm{f}}}$ is fibrant, and a cofibrant replacement of $A$ is a trivial
fibration $\rho_{A}:A_{{\textrm{c}}}\rightarrow A$ in which $A_{{\textrm{c}}}$
is cofibrant. Now we have a comparison lemma for replacements as expected.

######  Proposition 2.5.

Let $\mathcal{C}$ be a model category. Then the following assertions hold:

  1. (i)

Every object $A$ of $\mathcal{C}$ has a fibrant replacement
$\varrho_{A}:A\rightarrow A_{\emph{f}}$. Choosing a fibrant replacement for
each object of $\mathcal{C}$, if $f:A\rightarrow B$ is a morphism in
$\mathcal{C}$, then there is a morphism $f_{\emph{f}}:A_{\emph{f}}\rightarrow
B_{\emph{f}}$ which is unique up to homotopy and makes the following diagram
commutative:

${A}$${A_{\emph{f}}}$${B}$${B_{\emph{f}}}$$\scriptstyle{\varrho_{A}}$$\scriptstyle{f}$$\scriptstyle{f_{\emph{f}}}$$\scriptstyle{\varrho_{B}}$

Moreover, if $g:B\rightarrow C$ is another morphism in $\mathcal{C}$, then
$(gf)_{\emph{f}}\sim g_{\emph{f}}f_{\emph{f}}$, and $(1^{A})_{\emph{f}}\sim
1^{A_{\emph{f}}}$.

  2. (ii)

Every object $A$ of $\mathcal{C}$ has a cofibrant replacement
$\rho_{A}:A_{\emph{c}}\rightarrow A$. Choosing a cofibrant replacement for
each object of $\mathcal{C}$, if $f:A\rightarrow B$ is a morphism in
$\mathcal{C}$, then there is a morphism $f_{\emph{c}}:A_{\emph{c}}\rightarrow
B_{\emph{c}}$ which is unique up to homotopy and makes the following diagram
commutative:

${A_{\emph{c}}}$${A}$${B_{\emph{c}}}$${B}$$\scriptstyle{\rho_{A}}$$\scriptstyle{f_{\emph{c}}}$$\scriptstyle{f}$$\scriptstyle{\rho_{B}}$

Moreover, if $g:B\rightarrow C$ is another morphism in $\mathcal{C}$, then
$(gf)_{\emph{c}}\sim g_{\emph{c}}f_{\emph{c}}$, and $(1^{A})_{\emph{c}}\sim
1^{A_{\emph{c}}}$.

###### Proof.

See [Hi, Corollary 8.1.8]. ∎

The notions of fibrant and cofibrant replacements also allow us to define the
homotopy category of a model category which can be thought of as a
generalization as well as a non-abelian analogue of the derived category of an
abelian category.

######  Definition 2.6.

Let $\mathcal{C}$ be a model category. Choose a fibrant replacement
$\varrho_{A}:A\rightarrow A_{{\textrm{f}}}$ as well as a cofibrant replacement
$\rho_{A}:A_{{\textrm{c}}}\rightarrow A$ for every
$A\in\operatorname{Obj}(\mathcal{C})$. Then the homotopy category
$\operatorname{Ho}(\mathcal{C})$ of $\mathcal{C}$ is defined as follows:

  1. (i)

$\operatorname{Obj}\left({\textrm{Ho}}(\mathcal{C})\right):=\operatorname{Obj}(\mathcal{C})$.

  2. (ii)

$\operatorname{Mor}_{{\textrm{Ho}}(\mathcal{C})}(A,B):=\operatorname{Mor}_{\mathcal{C}}(A_{{\textrm{fc}}},B_{{\textrm{fc}}})/\sim$
for every $A,B\in\operatorname{Obj}\left({\textrm{Ho}}(\mathcal{C})\right)$.

We next have the universal property of the homotopy category which in turn
shows that the homotopy category is the localization of the model category
with respect to the class of weak equivalences.

######  Proposition 2.7.

Let $\mathcal{C}$ be a model category. Choose a fibrant replacement
$\varrho_{A}:A\rightarrow A_{\emph{f}}$ as well as a cofibrant replacement
$\rho_{A}:A_{\emph{c}}\rightarrow A$ for every
$A\in\operatorname{Obj}(\mathcal{C})$. Then there is a covariant functor
$\mathcal{L}_{\mathcal{C}}:\mathcal{C}\rightarrow\operatorname{Ho}(\mathcal{C})$
defined as $\mathcal{L}_{\mathcal{C}}(A):=A$ for every
$A\in\operatorname{Obj}(\mathcal{C})$, and
$\mathcal{L}_{\mathcal{C}}(f):=[f_{\emph{fc}}]$ for every
$f\in\operatorname{Mor}_{\mathcal{C}}(A,B)$. Moreover,
$\mathcal{L}_{\mathcal{C}}$ has the following properties:

  1. (i)

A morphism $f$ in $\mathcal{C}$ is a weak equivalence if and only if
$\mathcal{L}_{\mathcal{C}}(f)$ is an isomorphism in
$\operatorname{Ho}(\mathcal{C})$.

  2. (ii)

If $\mathcal{F}:\mathcal{C}\rightarrow\mathcal{D}$ is a functor that maps weak
equivalences in $\mathcal{C}$ to isomorphisms in $\mathcal{D}$, then
$\mathcal{F}$ factors uniquely through $\mathcal{L}_{\mathcal{C}}$, i.e. there
is a unique functor
$\mathcal{\check{F}}:\operatorname{Ho}(\mathcal{C})\rightarrow\mathcal{D}$
that makes the following diagram commutative:

${\mathcal{C}}$${\mathcal{D}}$${\operatorname{Ho}(\mathcal{C})}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\mathcal{L}_{\mathcal{C}}}$$\scriptstyle{\mathcal{\check{F}}}$

###### Proof.

See [Hi, Proposition 8.3.7 and Theorem 8.3.10]. ∎

We next focus on cofibrantly generated model categories whose importance stems
from the transferability of their model structure along adjunctions.

Let $\mathcal{C}$ be a category, and $f:A\rightarrow B$ and $g:C\rightarrow D$
two morphisms in $\mathcal{C}$. If any commutative diagram as the one on the
left can be completed to the commutative diagram on the right

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then we say that $f$ has the left lifting property against $g$ and $g$ has the
right lifting property against $f$. Given a class $\mathcal{X}$ of morphisms
in $\mathcal{C}$, let $\operatorname{LLP}(\mathcal{X})$ denote the class of
morphisms in $\mathcal{C}$ that have the left lifting property against all
morphisms in $\mathcal{X}$, and $\operatorname{RLP}(\mathcal{X})$ denote the
class of morphisms in $\mathcal{C}$ that have the right lifting property
against all morphisms in $\mathcal{X}$. If $\mathcal{C}$ is a model category,
then by [Hi, Proposition 7.2.3], $g$ is a fibration if and only if it has the
right lifting property against all trivial cofibrations; $g$ is a trivial
fibration if and only if it has the right lifting property against all
cofibrations; $f$ is a cofibration if and only if it has the left lifting
property against all trivial fibrations; and $f$ is a trivial cofibration if
and only if it has the left lifting property against all fibrations.

Recall that if $\mathcal{C}$ is a locally small cocomplete category and
$\mathcal{X}$ is a class of morphisms in $\mathcal{C}$, then an object $A$ of
$\mathcal{C}$ is said to be small with respect to $\mathcal{X}$ if the functor
$\operatorname{Mor}_{\mathcal{C}}(A,-):\mathcal{C}\rightarrow\mathcal{S}\mathpzc{et}$
preserves sequential direct limits of direct systems in $\mathcal{X}$, i.e.
for any sequence $\left\\{g_{n}:B_{n}\rightarrow B_{n+1}\right\\}_{n\geq 1}$
of morphisms in $\mathcal{X}$, the natural morphism

$\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{C}}(A,B_{n})\rightarrow\operatorname{Mor}_{\mathcal{C}}\left(A,\underset{n\geq
1}{\varinjlim}B_{n}\right)$

is an isomorphism.

Quillen devised a method to transfer model structures along adjunctions which
is primarily based on his well-known ”small object argument”. A set
$\mathcal{X}$ of morphisms in a locally small cocomplete category
$\mathcal{C}$ is said to permit the small object argument if the source of any
morphism in $\mathcal{X}$ is small with respect to
$\operatorname{LLP}\left(\operatorname{RLP}(\mathcal{X})\right)$. This
condition is necessary in order to be able to run the small object argument.
We now recall the notion of a cofibrantly generated model category.

######  Definition 2.8.

A model category $\mathcal{C}$ is said to be cofibrantly generated if there
exist two sets $\mathcal{X}$ and $\mathcal{Y}$ of morphisms in $C$ that
satisfy the following conditions:

  1. (i)

$\mathcal{X}$ and $\mathcal{Y}$ permit the small object argument.

  2. (ii)

$\operatorname{RLP}(\mathcal{X})$ is the class of trivial fibrations.

  3. (iii)

$\operatorname{RLP}(\mathcal{Y})$ is the class of fibrations.

The next theorem is known as Quillen-Kan Transfer Machine.

######  Theorem 2.9.

Let $\mathcal{C}$ be a cofibrantly generated model category with the
corresponding sets of morphisms $\mathcal{X}$ and $\mathcal{Y}$, $\mathcal{D}$
a locally small bicomplete category, and
$(\mathcal{F},\mathcal{G}):\mathcal{C}\leftrightarrows\mathcal{D}$ an adjoint
pair of functors. Then:

  1. (i)

Let a morphism $f:A\rightarrow B$ in $\mathcal{D}$ be a fibration if
$\mathcal{G}(f):\mathcal{G}(A)\rightarrow\mathcal{G}(B)$ is a fibration in
$\mathcal{C}$.

  2. (ii)

Let a morphism $f:A\rightarrow B$ in $\mathcal{D}$ be a weak equivalence if
$\mathcal{G}(f):\mathcal{G}(A)\rightarrow\mathcal{G}(B)$ is a weak equivalence
in $\mathcal{C}$.

  3. (iii)

Let a morphism $f:A\rightarrow B$ in $\mathcal{D}$ be a cofibration if it has
the left lifting property against trivial fibrations defined in (i) and (ii).

With the above definitions of fibrations, cofibrations, and weak equivalences
in place, if the sets of morphisms $\mathcal{F}(\mathcal{X})$ and
$\mathcal{F}(\mathcal{Y})$ permit the small object argument, and any
cofibration in $\mathcal{D}$ that has the left lifting property against
fibrations is also a weak equivalence, then $\mathcal{D}$ becomes a
cofibrantly generated model category.

###### Proof.

See [Hi, Theorem 11.3.2]. ∎

The last condition in Theorem 2.9 is commonly known as the “acyclicity
condition” which is in practice notoriously hard to establish. We next prove a
lemma that facilitates the permission of small object argument in our cases of
interest.

######  Lemma 2.10.

Let $\mathcal{C}$ and $\mathcal{D}$ be two locally small cocomplete
categories, and
$(\mathcal{F},\mathcal{G}):\mathcal{C}\leftrightarrows\mathcal{D}$ an adjoint
pair of functors in which $\mathcal{G}$ preserves sequential direct limits.
Suppose that $\mathcal{X}$ is a set of morphisms in $\mathcal{C}$ such that
the source of any morphism in $\mathcal{X}$ is small with respect to
$\mathcal{G}\left(\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)\right)$.
Then $\mathcal{F}(\mathcal{X})=\left\\{F(f)\;|\;f\in\mathcal{X}\right\\}$
permits the small object argument.

###### Proof.

Let $f:A\rightarrow B$ be a morphism in $\mathcal{X}$, and consider the
morphism $\mathcal{F}(f):\mathcal{F}(A)\rightarrow\mathcal{F}(B)$ in
$\mathcal{F}(\mathcal{X})$. We need to show that $\mathcal{F}(A)$ is small
with respect to
$\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)$.
To this end, let $\left\\{h_{n}:C_{n}\rightarrow C_{n+1}\right\\}_{n\geq 1}$
be a sequence of morphisms in
$\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)$.
Let $\phi_{m}:C_{m}\rightarrow\underset{n\geq 1}{\varinjlim}C_{n}$ be the
canonical injection for every $m\geq 1$. Then there is a unique morphism
$\eta:\underset{n\geq
1}{\varinjlim}\mathcal{G}(C_{n})\rightarrow\mathcal{G}\left(\underset{n\geq
1}{\varinjlim}C_{n}\right)$ that makes the following diagram commutative for
every $m\geq 1$:

${\mathcal{G}(C_{m})}$${\mathcal{G}(C_{m+1})}$${\underset{n\geq
1}{\varinjlim}\mathcal{G}(C_{n})}$${\mathcal{G}\left(\underset{n\geq
1}{\varinjlim}C_{n}\right)}$$\scriptstyle{\mathcal{G}(h_{m})}$$\scriptstyle{\psi_{m}}$$\scriptstyle{\mathcal{G}(\phi_{m})}$$\scriptstyle{\psi_{m+1}}$$\scriptstyle{\mathcal{G}(\phi_{m+1})}$$\scriptstyle{\eta}$

Since $\mathcal{G}$ preserves sequential direct limits, $\eta$ is an
isomorphism. Moreover, there is a unique morphism $\theta:\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{n})\right)\rightarrow\operatorname{Mor}_{\mathcal{C}}\left(A,\underset{n\geq
1}{\varinjlim}\mathcal{G}(C_{n})\right)$ that makes the following diagram
commutative for every $m\geq 1$:

${\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{m})\right)}$${\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{m+1})\right)}$${\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{n})\right)}$${\operatorname{Mor}_{\mathcal{C}}\left(A,\underset{n\geq
1}{\varinjlim}\mathcal{G}(C_{n})\right)}$$\scriptstyle{\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(h_{m})\right)}$$\scriptstyle{\chi_{m}}$$\scriptstyle{\operatorname{Mor}_{\mathcal{C}}(A,\psi_{m})}$$\scriptstyle{\chi_{m+1}}$$\scriptstyle{\operatorname{Mor}_{\mathcal{C}}(A,\psi_{m+1})}$$\scriptstyle{\theta}$

Also, there is a unique morphism $\vartheta:\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{n})\right)\rightarrow\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}\left(\underset{n\geq
1}{\varinjlim}C_{n}\right)\right)$ that makes the following diagram
commutative for every $m\geq 1$:

${\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{m})\right)}$${\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{m+1})\right)}$${\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{n})\right)}$${\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}\left(\underset{n\geq
1}{\varinjlim}C_{n}\right)\right)}$$\scriptstyle{\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(h_{m})\right)}$$\scriptstyle{\chi_{m}}$$\scriptstyle{\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(\phi_{m})\right)}$$\scriptstyle{\chi_{m+1}}$$\scriptstyle{\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(\phi_{m+1})\right)}$$\scriptstyle{\vartheta}$

Now we see that the following diagram is commutative:

${\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{n})\right)}$${\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}\left(\underset{n\geq
1}{\varinjlim}C_{n}\right)\right)}$${\operatorname{Mor}_{\mathcal{C}}\left(A,\underset{n\geq
1}{\varinjlim}\mathcal{G}(C_{n})\right)}$$\scriptstyle{\vartheta}$$\scriptstyle{\theta}$$\scriptstyle{\operatorname{Mor}_{\mathcal{C}}(A,\eta)}$

Indeed, we have for every $m\geq 1$:

$\operatorname{Mor}_{\mathcal{C}}(A,\eta)\theta\chi_{m}=\operatorname{Mor}_{\mathcal{C}}(A,\eta)\operatorname{Mor}_{\mathcal{C}}(A,\psi_{m})=\operatorname{Mor}_{\mathcal{C}}(A,\eta\psi_{m})=\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(\phi_{m})\right)=\vartheta\chi_{m}$

It follows that $\operatorname{Mor}_{\mathcal{C}}(A,\eta)\theta=\vartheta$ as
desired. We next note that for any $n\geq 1$, we have
$h_{n}\in\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)$,
so
$\mathcal{G}(h_{n})\in\mathcal{G}\left(\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)\right)$.
By the hypothesis, $A$ is small with respect to
$\mathcal{G}\left(\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)\right)$,
so we conclude that $\theta$ is an isomorphism. We had also observed above
that $\eta$ was an isomorphism, so $\operatorname{Mor}_{\mathcal{C}}(A,\eta)$
is an isomorphism. Therefore,
$\vartheta=\operatorname{Mor}_{\mathcal{C}}(A,\eta)\theta$ is an isomorphism.
On the other hand, there is a unique morphism $\lambda:\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),C_{n}\right)\rightarrow\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),\underset{n\geq
1}{\varinjlim}C_{n}\right)$ that makes the following diagram commutative for
every $m\geq 1$:

${\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),C_{m}\right)}$${\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),C_{m+1}\right)}$${\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),C_{n}\right)}$${\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),\underset{n\geq
1}{\varinjlim}C_{n}\right)}$$\scriptstyle{\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),h_{m}\right)}$$\scriptstyle{\rho_{m}}$$\scriptstyle{\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),\phi_{m}\right)}$$\scriptstyle{\rho_{m+1}}$$\scriptstyle{\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),\phi_{m+1}\right)}$$\scriptstyle{\lambda}$

Let
$\sigma_{U,V}:\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(U),V\right)\rightarrow\operatorname{Mor}_{\mathcal{C}}\left(U,\mathcal{G}(V)\right)$
be the natural bijection of the adjunction for every
$U\in\operatorname{Obj}(\mathcal{C})$ and
$V\in\operatorname{Obj}(\mathcal{D})$. Then the following diagram is
commutative:

${\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),C_{n}\right)}$${\underset{n\geq
1}{\varinjlim}\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(C_{n})\right)}$${\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),\underset{n\geq
1}{\varinjlim}C_{n}\right)}$${\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}\left(\underset{n\geq
1}{\varinjlim}C_{n}\right)\right)}$$\scriptstyle{\underset{n\geq
1}{\varinjlim}\sigma_{A,C_{n}}}$$\scriptstyle{\lambda}$$\scriptstyle{\vartheta}$$\scriptstyle{\sigma_{A,\underset{n\geq
1}{\varinjlim}C_{n}}}$

Indeed, the commutativity of the above diagram follows from the naturality of
the vertical morphisms applied to the natural transformation
$\sigma_{A,-}:\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),-\right)\rightarrow\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(-)\right)$.
Now since all the morphisms in the above diagram are isomorphisms except
possibly for $\lambda$, we infer that $\lambda$ must also be an isomorphism.
This means that $\mathcal{F}(A)$ is small with respect to
$\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)$.
Therefore, $\mathcal{F}(\mathcal{X})$ permits the small object argument. ∎

######  Corollary 2.11.

Let $\mathcal{C}$ and $\mathcal{D}$ be two locally small cocomplete
categories, and
$(\mathcal{F},\mathcal{G}):\mathcal{C}\leftrightarrows\mathcal{D}$ an adjoint
pair of functors in which $\mathcal{G}$ preserves sequential direct limits.
Suppose that $\mathcal{X}$ is a set of morphisms in $\mathcal{C}$ such that
the source of any morphism in $\mathcal{X}$ is small with respect to
$\operatorname{Mor}(\mathcal{C})$. Then the source of any morphism in
$\mathcal{F}(\mathcal{X})$ is small with respect to
$\operatorname{Mor}(\mathcal{D})$. In particular, $\mathcal{F}(\mathcal{X})$
permits the small object argument.

###### Proof.

Follows from the proof of Lemma 2.10. ∎

The following lemma might be of independent interest.

######  Lemma 2.12.

Let $\mathcal{C}$ and $\mathcal{D}$ be two locally small categories, and
$(\mathcal{F},\mathcal{G}):\mathcal{C}\leftrightarrows\mathcal{D}$ an adjoint
pair of functors. Let
$\sigma_{A,B}:\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),B\right)\rightarrow\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(B)\right)$
be the natural bijection of the adjunction for every
$A\in\operatorname{Obj}(\mathcal{C})$ and
$B\in\operatorname{Obj}(\mathcal{D})$. Then the following assertions hold:

  1. (i)

For any $f\in\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),B\right)$
and $g\in\operatorname{Mor}_{\mathcal{D}}(B,B^{\prime})$, we have
$\sigma_{A,B^{\prime}}(gf)=\mathcal{G}(g)\sigma_{A,B}(f)$.

  2. (ii)

For any $f\in\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(B)\right)$
and $g\in\operatorname{Mor}_{\mathcal{D}}(B,B^{\prime})$, we have
$\sigma_{A,B^{\prime}}^{-1}\left(\mathcal{G}(g)f\right)=g\sigma_{A,B}^{-1}(f)$.

  3. (iii)

For any $g\in\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(B)\right)$
and $f\in\operatorname{Mor}_{\mathcal{C}}(A^{\prime},A)$, we have
$\sigma_{A^{\prime},B}^{-1}(gf)=\sigma_{A,B}^{-1}(g)\mathcal{F}(f)$.

  4. (iv)

For any $g\in\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),B\right)$
and $f\in\operatorname{Mor}_{\mathcal{C}}(A^{\prime},A)$, we have
$\sigma_{A^{\prime},B}\left(g\mathcal{F}(f)\right)=\sigma_{A,B}(g)f$.

###### Proof.

(i): Let $f\in\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(A),B\right)$
and $g\in\operatorname{Mor}_{\mathcal{D}}(B,B^{\prime})$. Let
$u_{A}:A\rightarrow\mathcal{G}\left(\mathcal{F}(A)\right)$ be the unit
morphism of $A$. Then using [Mc1, Page 82, Eq. (5)], we have
$\sigma_{A,B^{\prime}}(gf)=\mathcal{G}(gf)u_{A}=\mathcal{G}(g)\mathcal{G}(f)u_{A}=\mathcal{G}(g)\sigma_{A,B}(f)$.

(ii): Let $f\in\operatorname{Mor}_{\mathcal{C}}\left(A,\mathcal{G}(B)\right)$
and $g\in\operatorname{Mor}_{\mathcal{D}}(B,B^{\prime})$. Set
$h=\sigma_{A,B^{\prime}}^{-1}\left(\mathcal{G}(g)f\right)$ and
$l=\sigma_{A,B}^{-1}(f)$. Using (i), we get
$\sigma_{A,B^{\prime}}(gl)=\mathcal{G}(g)\sigma_{A,B}(l)=\mathcal{G}(g)f=\sigma_{A,B^{\prime}}(h)$.
Since $\sigma_{A,B^{\prime}}$ is bijective, we conclude that
$g\sigma_{A,B}^{-1}(f)=gl=h=\sigma_{A,B^{\prime}}^{-1}\left(\mathcal{G}(g)f\right)$.

(iii): Similar to (i) using the counit morphism.

(iv): Similar to (ii). ∎

We can now compare the lifting properties along an adjunction. Note that the
next lemma is in [Hi, Theorem 7.2.17], but the proof there skips some crucial
steps which we include here for clarity.

######  Lemma 2.13.

Let $\mathcal{C}$ and $\mathcal{D}$ be two locally small categories, and
$(\mathcal{F},\mathcal{G}):\mathcal{C}\leftrightarrows\mathcal{D}$ an adjoint
pair of functors. Let $f:A\rightarrow B$ be a morphism in $\mathcal{C}$, and
$g:C\rightarrow D$ a morphism in $\mathcal{D}$. Then $\mathcal{F}(f)$ has the
left lifting property against $g$ if and only if $f$ has the left lifting
property against $\mathcal{G}(g)$.

###### Proof.

Let
$\sigma_{U,V}:\operatorname{Mor}_{\mathcal{D}}\left(\mathcal{F}(U),V\right)\rightarrow\operatorname{Mor}_{\mathcal{C}}\left(U,\mathcal{G}(V)\right)$
be the natural bijection of the adjunction for every
$U\in\operatorname{Obj}(\mathcal{C})$ and
$V\in\operatorname{Obj}(\mathcal{D})$. Suppose that $\mathcal{F}(f)$ has the
left lifting property against $g$. Consider a commutative diagram as follows:

${A}$${\mathcal{G}(C)}$${B}$${\mathcal{G}(D)}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{\mathcal{G}(g)}$$\scriptstyle{l}$

Then the following diagram is commutative:

${\mathcal{F}(A)}$${C}$${\mathcal{F}(B)}$${D}$$\scriptstyle{\sigma_{A,C}^{-1}(h)}$$\scriptstyle{\mathcal{F}(f)}$$\scriptstyle{g}$$\scriptstyle{\sigma_{B,D}^{-1}(l)}$

Indeed, using Lemma 2.12 (ii) and (iii), we have:

$\sigma_{B,D}^{-1}(l)\mathcal{F}(f)=\sigma_{A,D}^{-1}(lf)=\sigma_{A,D}^{-1}\left(\mathcal{G}(g)h\right)=g\sigma_{A,C}^{-1}(h)$

By the assumption, there is a morphism $u:\mathcal{F}(B)\rightarrow C$ that
makes the following diagram commutative:

${\mathcal{F}(A)}$${C}$${\mathcal{F}(B)}$${D}$$\scriptstyle{\sigma_{A,C}^{-1}(h)}$$\scriptstyle{\mathcal{F}(f)}$$\scriptstyle{g}$$\scriptstyle{\sigma_{B,D}^{-1}(l)}$$\scriptstyle{u}$

Then the following diagram is commutative:

${A}$${\mathcal{G}(C)}$${B}$${\mathcal{G}(D)}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{\mathcal{G}(g)}$$\scriptstyle{l}$$\scriptstyle{\sigma_{B,C}(u)}$

Indeed, using Lemma 2.12 (i) and (iv), we have:

$\sigma_{B,C}(u)f=\sigma_{A,C}\left(u\mathcal{F}(f)\right)=\sigma_{A,C}\left(\sigma_{A,C}^{-1}(h)\right)=h$

and

$\mathcal{G}(g)\sigma_{B,C}(u)=\sigma_{B,D}(gu)=\sigma_{B,D}\left(\sigma_{B,D}^{-1}(l)\right)=l$

This shows that $f$ has the left lifting property against $\mathcal{G}(g)$.
The converse is similarly established. ∎

######  Corollary 2.14.

Let $\mathcal{C}$ and $\mathcal{D}$ be two locally small categories, and
$(\mathcal{F},\mathcal{G}):\mathcal{C}\leftrightarrows\mathcal{D}$ an adjoint
pair of functors. Let $\mathcal{X}$ be a class of morphisms in $\mathcal{C}$,
and $\mathcal{Y}$ a class of morphisms in $\mathcal{D}$. Then the following
assertions hold:

  1. (i)

For any morphism $f$ in $\mathcal{C}$, we have
$\mathcal{F}(f)\in\operatorname{LLP}(\mathcal{Y})$ if and only if
$f\in\operatorname{LLP}\left(\mathcal{G}(\mathcal{Y})\right)$.

  2. (ii)

For any morphism $g$ in $\mathcal{D}$, we have
$g\in\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)$ if and only if
$\mathcal{G}(g)\in\operatorname{RLP}\left(\mathcal{X}\right)$.

###### Proof.

(i): Let $f$ be a morphism in $\mathcal{C}$. By Lemma 2.13,
$\mathcal{F}(f)\in\operatorname{LLP}(\mathcal{Y})$ if and only if
$\mathcal{F}(f)$ has the left lifting property against every $g\in\mathcal{Y}$
if and only if $f$ has the left lifting property against every
$\mathcal{G}(g)\in\mathcal{G}(\mathcal{Y})$ if and only if
$f\in\operatorname{LLP}\left(\mathcal{G}(\mathcal{Y})\right)$.

(ii): Similar to (i). ∎

The next proposition transfers model structures along equivalences.

######  Proposition 2.15.

Let $\mathcal{C}$ be a model category, $\mathcal{D}$ a category, and
$\mathcal{G}:\mathcal{D}\rightarrow\mathcal{C}$ a covariant equivalence of
categories. Then $\mathcal{D}$ becomes a model category by letting fibrations
(cofibrations, or weak equivalences) be morphisms $f:A\rightarrow B$ in
$\mathcal{D}$ such that
$\mathcal{G}(f):\mathcal{G}(A)\rightarrow\mathcal{G}(B)$ is a fibration
(cofibration, or weak equivalence) in $\mathcal{C}$, respectively.

###### Proof.

One can directly check the conditions bearing in mind that $\mathcal{G}$ is
fully faithful. ∎

######  Corollary 2.16.

Let $\mathcal{C}$ be a cofibrantly generated model category, $\mathcal{D}$ a
category, and
$(\mathcal{F},\mathcal{G}):\mathcal{C}\leftrightarrows\mathcal{D}$ an adjoint
equivalence of categories. Then $\mathcal{D}$ becomes a cofibrantly generated
model category by letting fibrations (cofibrations, or weak equivalences) be
morphisms $f:A\rightarrow B$ in $\mathcal{D}$ such that
$\mathcal{G}(f):\mathcal{G}(A)\rightarrow\mathcal{G}(B)$ is a fibration
(cofibration, or weak equivalence) in $\mathcal{C}$, respectively.

###### Proof.

By Proposition 2.15, $\mathcal{D}$ is a model category with the specified
classes of fibrations, cofibrations, and weak equivalences. Moreover, since
$\mathcal{G}$ is a covariant equivalence of categories, $\mathcal{G}$
preserves direct limits. Now let $\mathcal{X}$ and $\mathcal{Y}$ be sets of
morphisms as in Definition 2.8. Then $\operatorname{RLP}(\mathcal{X})$ is the
class of trivial fibrations and $\operatorname{RLP}(\mathcal{Y})$ is the class
of fibrations in $\mathcal{C}$. Since $\mathcal{C}$ is a model category, [Hi,
Proposition 7.2.3] implies that
$\operatorname{LLP}\left(\operatorname{RLP}(\mathcal{X})\right)$ is the class
of cofibrations and
$\operatorname{LLP}\left(\operatorname{RLP}(\mathcal{Y})\right)$ is the class
of trivial cofibrations in $\mathcal{C}$. By Corollary 2.14 (ii), for a given
morphism $f$ in $\mathcal{D}$, we have
$f\in\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)$ if and only if
$\mathcal{G}(f)\in\operatorname{RLP}(\mathcal{X})$ if and only if
$\mathcal{G}(f)$ is a trivial fibration in $C$ if and only if $f$ is a trivial
fibration in $\mathcal{D}$. As a result,
$\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)$ is the class of
trivial fibrations in $\mathcal{D}$. A similar argument shows that
$\operatorname{RLP}\left(\mathcal{F}(\mathcal{Y})\right)$ is the class of
fibrations in $\mathcal{D}$. As $\mathcal{D}$ is a model category,
$\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)$
is the class of cofibrations and
$\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{Y})\right)\right)$
is the class of trivial cofibrations in $\mathcal{D}$. By definition, if $f$
is a cofibration in $\mathcal{D}$, then $\mathcal{G}(f)$ is a cofibration in
$\mathcal{C}$, meaning that
$\mathcal{G}\left(\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)\right)\subseteq\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{X}\right)\right)$.
But $\mathcal{C}$ is cofibrantly generated, so $\mathcal{X}$ permits the small
object argument, whence the source of any morphism in $\mathcal{X}$ is small
with respect to
$\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{X}\right)\right)$,
hence with respect to
$\mathcal{G}\left(\operatorname{LLP}\left(\operatorname{RLP}\left(\mathcal{F}(\mathcal{X})\right)\right)\right)$.
By Lemma 2.10, $\mathcal{F}(\mathcal{X})$ permits the small object argument. A
similar argument shows that $\mathcal{F}(\mathcal{Y})$ permits the small
object argument. Therefore, $\mathcal{D}$ is cofibrantly generated with the
corresponding sets $\mathcal{F}(\mathcal{X})$ and $\mathcal{F}(\mathcal{Y})$.
∎

##  3\. Model Structure on Connective Chain Complexes

In this section, we use some ideas from [Ho, Chapter 2, Section 2.3] and [DS,
Section 7] to cook up a straightforward and clear-cut proof of the model
structure of connective chain complexes.

Let $R$ be a ring. Recall that an $R$-complex is a $\mathbb{Z}$-indexed
sequence

$X:\cdots\rightarrow
X_{i+1}\xrightarrow{\partial_{i+1}^{X}}X_{i}\xrightarrow{\partial_{i}^{X}}X_{i-1}\rightarrow\cdots$

of left $R$-modules and $R$-homomorphisms such that
$\partial_{i}^{X}\partial_{i+1}^{X}=0$ for every $i\in\mathbb{Z}$. Given an
$R$-complex $X$, we utilize the conventional notations
$Z_{i}(X)=\operatorname{Ker}\left(\partial_{i}^{X}\right)$ and
$B_{i}(X)=\operatorname{Im}\left(\partial_{i+1}^{X}\right)$ for every
$i\in\mathbb{Z}$. An $R$-complex $X$ is said to be connective if $X_{i}=0$ for
every $i<0$. The category of connective $R$-complexes is denoted by
$\mathcal{C}_{\geq 0}(R)$ which is a full subcategory of the category
$\mathcal{C}(R)$ of unbounded $R$-complexes. In this section, we directly show
that $\mathcal{C}_{\geq 0}(R)$ is a cofibrantly generated model category. The
cofibrant generation property of this model structure allows us to transfer it
along adjunctions.

We first recall the Eilenberg-MacLane complexes as follows. For any
$n\in\mathbb{Z}$, we set $S(n):=\Sigma^{n}R$, i.e. the $R$-complex

$\cdots\rightarrow 0\rightarrow R\rightarrow 0\rightarrow\cdots$

where $R$ is located in degree $n$. Then it is easy to verify that if $X$ is
an $R$-complex, then the map
$\phi_{n}^{X}:\operatorname{Mor}_{\mathcal{C}(R)}\left(S(n),X\right)\rightarrow
Z_{n}(X)$, given by $f\mapsto f_{n}(1)$, is a natural isomorphism. Similarly,
for any $n\in\mathbb{Z}$, we set
$D(n):=\Sigma^{n-1}\operatorname{Cone}\left(1^{R}\right)$, i.e. the
$R$-complex

$\cdots\rightarrow 0\rightarrow R\xrightarrow{1^{R}}R\rightarrow
0\rightarrow\cdots$

where $R$ is located in degrees $n$ and $n-1$. Then it is easy to verify that
if $X$ is an $R$-complex, then the map
$\psi^{X}_{n}:\operatorname{Mor}_{\mathcal{C}(R)}\left(D(n),X\right)\rightarrow
X_{n}$, given by $f\mapsto f_{n}(1)$, is a natural isomorphism. We make
repeated use of these facts in the proof of the following theorem.

######  Theorem 3.1.

Let $R$ be a ring. Then $\mathcal{C}_{\geq 0}(R)$ is a cofibrantly generated
model category with the model structure given as follows:

  1. (i)

Fibrations are morphisms $f:X\rightarrow Y$ in $\mathcal{C}_{\geq 0}(R)$ such
that $f_{i}:X_{i}\rightarrow Y_{i}$ is surjective for every $i\geq 1$.

  2. (ii)

Cofibrations are morphisms $f:X\rightarrow Y$ in $\mathcal{C}_{\geq 0}(R)$
such that $f_{i}:X_{i}\rightarrow Y_{i}$ is injective and
$\operatorname{Coker}(f_{i})$ is a projective left $R$-module for every $i\geq
0$.

  3. (iii)

Weak equivalences are quasi-isomorphisms, i.e. morphisms $f:X\rightarrow Y$ in
$\mathcal{C}_{\geq 0}(R)$ such that $H_{i}(f):H_{i}(X)\rightarrow H_{i}(Y)$ is
an isomorphism for every $i\geq 0$.

###### Proof.

We first note that $\mathcal{C}_{\geq 0}(R)$ is a locally small bicomplete
category. Let

$\mathcal{X}=\left\\{\iota^{n}:S(n-1)\rightarrow D(n)\;|\;n\geq
1\right\\}\cup\left\\{\lambda:0\rightarrow S(0)\right\\}$

where $\iota^{n}$ is the inclusion morphism for every $n\geq 1$. Also, let

$\mathcal{Y}=\left\\{\kappa^{n}:0\rightarrow D(n)\;|\;n\geq 1\right\\}.$

We show that $\mathcal{C}_{\geq 0}(R)$ is a cofibrantly generated model
category with the corresponding sets $\mathcal{X}$ and $\mathcal{Y}$ of
morphisms as in Definition 2.8. For any $n\geq 0$, we have
$\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(S(n),-\right)\cong
Z_{n}(-)$, so since the functor $Z_{n}(-)$ preserves direct limits of direct
systems, we conclude that the functor $\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(S(n),-\right)$ preserves direct limits of direct systems as well,
thereby $S(n)$ is small with respect to
$\operatorname{Mor}\left(\mathcal{C}_{\geq 0}(R)\right)$. Also, the source of
$\lambda$ is $0$, so it is clearly small with respect to
$\operatorname{Mor}\left(\mathcal{C}_{\geq 0}(R)\right)$. In particular, the
sources of morphisms in $\mathcal{X}$ are small with respect to
$\operatorname{LLP}\left(\operatorname{RLP}(\mathcal{X})\right)$, so
$\mathcal{X}$ permits the small object argument. Similarly, since the source
of any morphism in $\mathcal{Y}$ is $0$, we see that $\mathcal{Y}$ permits the
small object argument.

We now show that $\operatorname{RLP}(\mathcal{X})$ is the class of trivial
fibrations in $\mathcal{C}_{\geq 0}(R)$. Let $f:X\rightarrow Y$ be a morphism
in $\mathcal{C}_{\geq 0}(R)$. Suppose that
$f\in\operatorname{RLP}(\mathcal{X})$. We show that $f$ is a trivial
fibration, i.e. $f$ is a quasi-isomorphism and $f_{n}$ is surjective for every
$n\geq 1$. Let $n\geq 0$. We first show that $H_{n}(f):H_{n}(X)\rightarrow
H_{n}(Y)$ is an isomorphism. To see that $H_{n}(f)$ is injective, let $x\in
Z_{n}(X)$ be such that $H_{n}(f)\left(x+B_{n}(X)\right)=f_{n}(x)+B_{n}(Y)=0$.
Then $f_{n}(x)\in B_{n}(Y)$, so $f_{n}(x)=\partial^{Y}_{n+1}(y)$ for some
$y\in Y_{n+1}$. Consider the natural isomorphisms
$\phi_{n}^{X}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(S(n),X\right)\rightarrow Z_{n}(X)$ and
$\psi^{Y}_{n+1}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(D(n+1),Y\right)\rightarrow Y_{n+1}$. There is a
$g\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(S(n),X\right)$ such
that $\phi_{n}^{X}(g)=g_{n}(1)=x$, and an
$h\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(D(n+1),Y\right)$ such
that $\psi^{Y}_{n+1}(h)=h_{n+1}(1)=y$. Then the following diagram is
commutative:

${S(n)}$${X}$${D(n+1)}$${Y}$$\scriptstyle{g}$$\scriptstyle{\iota^{n+1}}$$\scriptstyle{f}$$\scriptstyle{h}$

Indeed, it is clear from the structure of $S(n)$ that
$\iota^{n+1}_{i}=0=g_{i}$ for every $i\neq n$. Moreover, we have:

$h_{n}\left(\iota_{n}^{n+1}(1)\right)=h_{n}(1)=\partial^{Y}_{n+1}\left(h_{n+1}(1)\right)=\partial^{Y}_{n+1}(y)=f_{n}(x)=f_{n}\left(g_{n}(1)\right)$

By the choice of $f$, the above diagram can be completed to the following
commutative diagram:

${S(n)}$${X}$${D(n+1)}$${Y}$$\scriptstyle{g}$$\scriptstyle{\iota^{n+1}}$$\scriptstyle{f}$$\scriptstyle{h}$$\scriptstyle{l}$

Thus
$\partial_{n+1}^{X}\left(l_{n+1}(1)\right)=l_{n}(1)=l_{n}\left(\iota_{n}^{n+1}(1)\right)=g_{n}(1)=x$,
so $x\in B_{n}(X)$. This means that $H_{n}(f)$ is injective. To see that
$H_{n}(f)$ is surjective, let $z\in Z_{n}(Y)$. If $n=0$, then consider the
isomorphism $\phi_{0}^{Y}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(S(0),Y\right)\rightarrow Z_{0}(Y)$. There is an
$r\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(S(0),Y\right)$ such
that $\phi_{0}^{Y}(r)=r_{0}(1)=z$. Then it is clear that the following diagram
is commutative:

${0}$${X}$${S(0)}$${Y}$$\scriptstyle{\lambda}$$\scriptstyle{f}$$\scriptstyle{r}$

By the choice of $f$, the above diagram can be completed to the following
commutative diagram:

${0}$${X}$${S(0)}$${Y}$$\scriptstyle{\lambda}$$\scriptstyle{f}$$\scriptstyle{r}$$\scriptstyle{s}$

Set $w=s_{0}(1)\in X_{0}=Z_{0}(X)$. Then
$f_{0}(w)=f_{0}\left(s_{0}(1)\right)=r_{0}(1)=z$. If $n\geq 1$, then consider
the isomorphism $\psi^{Y}_{n}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(D(n),Y\right)\rightarrow Y_{n}$. There is an
$r\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(D(n),Y\right)$ such
that $\psi^{Y}_{n}(r)=r_{n}(1)=z$. Then the following diagram is commutative:

${S(n-1)}$${X}$${D(n)}$${Y}$$\scriptstyle{0}$$\scriptstyle{\iota^{n}}$$\scriptstyle{f}$$\scriptstyle{r}$

Indeed, it is clear that $\iota_{i}^{n}=0$ for every $i\neq n-1$. Moreover,
since $z\in Z_{n}(Y)$, we have
$r_{n-1}\left(\iota_{n-1}^{n}(1)\right)=r_{n-1}(1)=\partial_{n}^{Y}\left(r_{n}(1)\right)=\partial_{n}^{Y}(z)=0$.
By the choice of $f$, the above diagram can be completed to the following
commutative diagram:

${S(n-1)}$${X}$${D(n)}$${Y}$$\scriptstyle{0}$$\scriptstyle{\iota^{n}}$$\scriptstyle{f}$$\scriptstyle{r}$$\scriptstyle{s}$

Set $w=s_{n}(1)\in X_{n}$. Then
$f_{n}(w)=f_{n}\left(s_{n}(1)\right)=r_{n}(1)=z$. Moreover,
$\partial_{n}^{X}(w)=\partial_{n}^{X}\left(s_{n}(1)\right)=s_{n-1}(1)=s_{n-1}\left(\iota_{n-1}^{n}(1)\right)=0$,
so $w\in Z_{n}(X)$. In other words, for any $n\geq 0$ and $z\in Z_{n}(Y)$, we
have found an element $w\in Z_{n}(X)$ such that $f_{n}(w)=z$. In particular,
we have $H_{n}(f)\left(w+B_{n}(X)\right)=f_{n}(w)+B_{n}(Y)=z+B_{n}(Y)$. This
shows that $H_{n}(f)$ is surjective. Therefore, $H_{n}(f)$ is an isomorphism.
Next let $n\geq 1$. We show that $f_{n}$ is surjective. Let $v\in Y_{n}$. Then
$\partial_{n}^{Y}(v)\in B_{n-1}(Y)\subseteq Z_{n-1}(Y)$, so as noted above,
there is an element $u\in Z_{n-1}(X)$ such that
$f_{n-1}(u)=\partial_{n}^{Y}(v)$. Consider the isomorphisms
$\phi_{n-1}^{X}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(S(n-1),X\right)\rightarrow Z_{n-1}(X)$ and
$\psi^{Y}_{n}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(D(n),Y\right)\rightarrow Y_{n}$. There is a
$p\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(S(n-1),X\right)$ such
that $\phi_{n-1}^{X}(p)=p_{n-1}(1)=u$, and a
$q\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(D(n),Y\right)$ such
that $\psi^{Y}_{n}(q)=q_{n}(1)=v$. Then the following diagram is commutative:

${S(n-1)}$${X}$${D(n)}$${Y}$$\scriptstyle{p}$$\scriptstyle{\iota^{n}}$$\scriptstyle{f}$$\scriptstyle{q}$

Indeed, it is clear that $\iota_{i}^{n}=0=p_{i}$ for every $i\neq n-1$.
Moreover, we have:

$q_{n-1}\left(\iota_{n-1}^{n}(1)\right)=q_{n-1}(1)=\partial^{Y}_{n}\left(q_{n}(1)\right)=\partial^{Y}_{n}(v)=f_{n-1}(u)=f_{n-1}\left(p_{n-1}(1)\right)$

By the choice of $f$, the above diagram can be completed to the following
commutative diagram:

${S(n-1)}$${X}$${D(n)}$${Y}$$\scriptstyle{p}$$\scriptstyle{\iota^{n}}$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{t}$

Then $f_{n}\left(t_{n}(1)\right)=q_{n}(1)=v$. Thus $f_{n}$ is surjective. As a
result, $f$ is a trivial fibration.

Conversely, suppose that $f$ is a trivial fibration, i.e. $f$ is a quasi-
isomorphism and $f_{n}$ is surjective for every $n\geq 1$. We first show that
$f_{0}$ is also surjective. Consider the following commutative diagram:

${X_{1}}$${B_{0}(X)}$${Y_{1}}$${B_{0}(Y)}$$\scriptstyle{\overline{\partial_{1}^{X}}}$$\scriptstyle{f_{1}}$$\scriptstyle{B_{0}(f)}$$\scriptstyle{\overline{\partial_{1}^{Y}}}$

As $f_{1}$ is surjective, the above diagram implies that $B_{0}(f)$ is
surjective. Now consider the following commutative diagram:

${0}$${B_{0}(X)}$${X_{0}}$${H_{0}(X)}$${0}$${0}$${B_{0}(Y)}$${Y_{0}}$${H_{0}(Y)}$${0}$$\scriptstyle{B_{0}(f)}$$\scriptstyle{f_{0}}$$\scriptstyle{H_{0}(f)}$

Since $B_{0}(f)$ is surjective and $H_{0}(f)$ is an isomorphism, the Short
Five Lemma implies that $f_{0}$ is surjective. Therefore, $f$ is a surjective
quasi-isomorphism. We next show that $f\in\operatorname{RLP}(\mathcal{X})$.
Let $n\geq 1$, and consider a commutative diagram as follows:

${S(n-1)}$${X}$${D(n)}$${Y}$$\scriptstyle{p}$$\scriptstyle{\iota^{n}}$$\scriptstyle{f}$$\scriptstyle{q}$

Let $x=p_{n-1}(1)\in X_{n-1}$ and $y=q_{n}(1)\in Y_{n}$. Then we have:

$f_{n-1}(x)=f_{n-1}\left(p_{n-1}(1)\right)=q_{n-1}\left(\iota_{n-1}^{n}(1)\right)=q_{n-1}(1)=\partial_{n}^{Y}\left(q_{n}(1)\right)=\partial_{n}^{Y}(y)$

Moreover, $\partial_{n-1}^{X}(x)=\partial_{n-1}^{X}\left(p_{n-1}(1)\right)=0$,
so $x\in Z_{n-1}(X)$. But then
$H_{n-1}(f)\left(x+B_{n-1}(X)\right)=f_{n-1}(x)+B_{n-1}(Y)=\partial_{n}^{Y}(y)+B_{n-1}(Y)=0$,
so since $H_{n-1}(f)$ is an isomorphism, we get $x+B_{n-1}(X)=0$, i.e. $x\in
B_{n-1}(X)$, so $x=\partial_{n}^{X}(u)$ for some $u\in X_{n}$. As $f$ is a
morphism of $R$-complexes, we have
$\partial_{n}^{Y}(y)=f_{n-1}(x)=f_{n-1}\left(\partial_{n}^{X}(u)\right)=\partial_{n}^{Y}\left(f_{n}(u)\right)$,
so $y-f_{n}(u)\in Z_{n}(Y)$. But $f$ is a surjective quasi-isomorphism, so it
is easy to see that $Z(f):Z(X)\rightarrow Z(Y)$ is surjective. In particular,
there is an element $v\in Z_{n}(X)$ such that $f_{n}(v)=y-f_{n}(u)$. Set
$z=u+v\in X_{n}$. Then
$\partial_{n}^{X}(z)=\partial_{n}^{X}(u)+\partial_{n}^{X}(v)=\partial_{n}^{X}(u)=x$
and $f_{n}(z)=f_{n}(u)+f_{n}(v)=y$. Consider the isomorphism
$\psi^{X}_{n}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(D(n),X\right)\rightarrow X_{n}$. There is an
$l\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(D(n),X\right)$ such
that $\psi^{X}_{n}(l)=l_{n}(1)=z$. Then the following diagram is commutative:

${S(n-1)}$${X}$${D(n)}$${Y}$$\scriptstyle{p}$$\scriptstyle{\iota^{n}}$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{l}$

Indeed, $\iota_{i}^{n}=0$ for every $i\neq n-1$. Furthermore, we have:

$l_{n-1}\left(\iota_{n-1}^{n}(1)\right)=l_{n-1}(1)=\partial_{n}^{X}\left(l_{n}(1)\right)=\partial_{n}^{X}(z)=x=p_{n-1}(1)$

On the other hand, $q_{i}=0=l_{i}$ for every $i\neq n,n-1$. Moreover, we have
$f_{n}\left(l_{n}(1)\right)=f_{n}(z)=y=q_{n}(1)$ and

$f_{n-1}\left(l_{n-1}(1)\right)=f_{n-1}\left(\partial_{n}^{X}\left(l_{n}(1)\right)\right)=f_{n-1}\left(\partial_{n}^{X}(z)\right)=f_{n-1}(x)=\partial_{n}^{Y}(y)=\partial_{n}^{Y}\left(q_{n}(1)\right)=q_{n-1}(1).$

Next consider a commutative diagram as follows:

${0}$${X}$${S(0)}$${Y}$$\scriptstyle{\lambda}$$\scriptstyle{f}$$\scriptstyle{r}$

By the surjectivity of $f_{0}$, there is an element $w\in X_{0}=Z_{0}(X)$ such
that $f_{0}(w)=r_{0}(1)$. Consider the isomorphism
$\phi_{0}^{X}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(S(0),X\right)\rightarrow Z_{0}(X)$. There is an
$s\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(S(0),X\right)$ such
that $\phi_{0}^{X}(s)=s_{0}(1)=w$. Then the following diagram is commutative:

${0}$${X}$${S(0)}$${Y}$$\scriptstyle{\lambda}$$\scriptstyle{f}$$\scriptstyle{r}$$\scriptstyle{s}$

Indeed, it is clear that the upper triangle is commutative. Moreover, for the
lower triangle, we note that $s_{i}=0=r_{i}$ for every $i\neq 0$. Also,
$f_{0}\left(s_{0}(1)\right)=f_{0}(w)=r_{0}(1)$. This means that
$f\in\operatorname{RLP}(\mathcal{X})$. Therefore,
$\operatorname{RLP}(\mathcal{X})$ is the class of trivial fibrations.

We next show that $\operatorname{RLP}(\mathcal{Y})$ is the class of fibrations
in $\mathcal{C}_{\geq 0}(R)$. Let $f:X\rightarrow Y$ be a morphism in
$\mathcal{C}_{\geq 0}(R)$. Suppose that $f\in\operatorname{RLP}(\mathcal{Y})$.
We show that $f$ is a fibration, i.e. $f_{n}$ is surjective for every $n\geq
1$. Let $n\geq 1$ and $v\in Y_{n}$. Consider the isomorphism
$\psi^{Y}_{n}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(D(n),Y\right)\rightarrow Y_{n}$. There is a
$q\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(D(n),Y\right)$ such
that $\psi^{Y}_{n}(q)=q_{n}(1)=v$. By the choice of $f$, the commutative
diagram

${0}$${X}$${D(n)}$${Y}$$\scriptstyle{\kappa^{n}}$$\scriptstyle{f}$$\scriptstyle{q}$

can be completed to the following commutative diagram:

${0}$${X}$${D(n)}$${Y}$$\scriptstyle{\kappa^{n}}$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{t}$

Then $f_{n}\left(t_{n}(1)\right)=q_{n}(1)=v$. This shows that $f_{n}$ is
surjective. As a result, $f$ is a fibration.

Conversely, suppose that $f$ is a fibration, i.e. $f_{n}$ is surjective for
every $n\geq 1$. Let $n\geq 1$, and consider a commutative diagram as follows:

${0}$${X}$${D(n)}$${Y}$$\scriptstyle{\kappa^{n}}$$\scriptstyle{f}$$\scriptstyle{q}$

Since $f_{n}$ is surjective, there is an element $x\in X_{n}$ such that
$f_{n}(x)=q_{n}(1)$. Consider the isomorphism
$\psi^{X}_{n}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(D(n),X\right)\rightarrow X_{n}$. There is an
$l\in\operatorname{Mor}_{\mathcal{C}_{\geq 0}(R)}\left(D(n),X\right)$ such
that $\psi^{X}_{n}(l)=l_{n}(1)=x$. Then the following diagram is commutative:

${0}$${X}$${D(n)}$${Y}$$\scriptstyle{\kappa^{n}}$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{l}$

Indeed, it is obvious that the upper triangle is commutative. For the lower
triangle, we note that $q_{i}=0=l_{i}$ for every $i\neq n,n-1$. Moreover,
using the fact that $f$ is a morphism of $R$-complexes, we have
$f_{n}\left(l_{n}(1)\right)=f_{n}(x)=q_{n}(1)$ and

$f_{n-1}\left(l_{n-1}(1)\right)=f_{n-1}\left(\partial_{n}^{X}\left(l_{n}(1)\right)\right)=f_{n-1}\left(\partial_{n}^{X}(x)\right)=\partial_{n}^{Y}\left(f_{n}(x)\right)=\partial_{n}^{Y}\left(q_{n}(1)\right)=q_{n-1}(1).$

This implies that $f\in\operatorname{RLP}(\mathcal{Y})$. Therefore,
$\operatorname{RLP}(\mathcal{Y})$ is the class of fibrations.

We are now ready to check the axioms of the model category as follows:

Retract: Suppose that $f$ is a retract of $g$, i.e. there is a commutative
diagram

${X}$${Z}$${X}$${Y}$${W}$${Y}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{l}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{q}$

in $\mathcal{C}_{\geq 0}(R)$ in which $lh=1^{X}$ and $qp=1^{Y}$. If $g$ is a
fibration, then $g_{i}$ is surjective for every $i\geq 1$. On the other hand,
the relation $q_{i}p_{i}=1^{Y_{i}}$ implies that $q_{i}$ is surjective for
every $i\geq 0$. Therefore, $q_{i}g_{i}$ is surjective for every $i\geq 1$.
Now the relation $f_{i}l_{i}=q_{i}g_{i}$ yields that $f_{i}$ is surjective for
every $i\geq 1$. That means that $f$ is a fibration.

If $g$ is a cofibration, then $g_{i}$ is injective and
$\operatorname{Coker}(g_{i})$ is projective for every $i\geq 0$. The relation
$l_{i}h_{i}=1^{X_{i}}$ implies that $h_{i}$ is injective for every $i\geq 0$.
Therefore, $g_{i}h_{i}$ is injective for every $i\geq 0$. Now the relation
$p_{i}f_{i}=g_{i}h_{i}$ yields that $f_{i}$ is injective for every $i\geq 0$.
Furthermore, we get the following induced commutative diagram:

${X}$${Z}$${X}$${Y}$${W}$${Y}$${\operatorname{Coker}(f)}$${\operatorname{Coker}(g)}$${\operatorname{Coker}(f)}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{l}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{\bar{p}}$$\scriptstyle{\bar{q}}$

It is also evident that $\bar{q}\bar{p}=1^{\operatorname{Coker}(f)}$. It
follows that $\operatorname{Coker}(f)$ is isomorphic to a direct summand of
$\operatorname{Coker}(g)$. Therefore, $\operatorname{Coker}(f_{i})$ is
projective for every $i\geq 0$. Thus $f$ is a cofibration.

Finally, if $g$ is a weak equivalence, then $g$ is a quasi-isomorphism. Fix
$i\geq 0$. The functoriality of homology yields the following commutative
diagram:

${H_{i}(X)}$${H_{i}(Z)}$${H_{i}(X)}$${H_{i}(Y)}$${H_{i}(W)}$${H_{i}(Y)}$$\scriptstyle{H_{i}(h)}$$\scriptstyle{H_{i}(f)}$$\scriptstyle{H_{i}(g)}$$\scriptstyle{H_{i}(l)}$$\scriptstyle{H_{i}(f)}$$\scriptstyle{H_{i}(p)}$$\scriptstyle{H_{i}(q)}$

Setting $\phi=H_{i}(l)H_{i}(g)^{-1}H_{i}(p):H_{i}(Y)\rightarrow H_{i}(X)$, we
observe from the above commutative diagram that $\phi$ is the inverse of
$H_{i}(f)$, so $H_{i}(f)$ is an isomorphism. That is to say, $f$ is a quasi-
isomorphism, hence a weak equivalence.

Two-Out-of-Three: Suppose that we have morphisms $f=gh$ in $\mathcal{C}_{\geq
0}(R)$ in which two of $f$, $g$, or $h$ are quasi-isomorphisms. Then by the
functoriality of homology, it is clear that the third morphism must be a
quasi-isomorphism as well.

First Half of Lifting: Consider a commutative diagram

${X}$${Z}$${Y}$${W}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{l}$

in $\mathcal{C}_{\geq 0}(R)$ in which $f$ is a cofibration and $g$ is a
trivial fibration. By definition, $f_{i}:X_{i}\rightarrow Y_{i}$ is injective
and $\operatorname{Coker}(f_{i})$ is projective for every $i\geq 0$. As a
result, for any $i\geq 0$, the short exact sequence

$0\rightarrow
X_{i}\xrightarrow{f_{i}}Y_{i}\rightarrow\operatorname{Coker}(f_{i})\rightarrow
0$

is split, resulting in a decomposition $Y_{i}=\operatorname{Im}(f_{i})\oplus
P_{i}$ for some projective left $R$-module
$P_{i}\cong\operatorname{Coker}(f_{i})$. Moreover, as noted before,
$g:X\rightarrow Y$ is a surjective quasi-isomorphism, so there is a short
exact sequence

$0\rightarrow\operatorname{Ker}(g)\xrightarrow{\iota}Z\xrightarrow{g}W\rightarrow
0$

of $R$-complexes in which $\operatorname{Ker}(g)$ is exact. Now we construct a
morphism $\phi=(\phi_{i})_{i\geq 0}:Y\rightarrow Z$ of $R$-complexes
inductively. Let $i=0$, and consider the following commutative diagram:

${X_{0}}$${Z_{0}}$${Y_{0}}$${W_{0}}$$\scriptstyle{h_{0}}$$\scriptstyle{f_{0}}$$\scriptstyle{g_{0}}$$\scriptstyle{l_{0}}$

Since $g_{0}$ is surjective and $P_{0}$ is projective, there is an
$R$-homomorphism $\rho_{0}:P_{0}\rightarrow Z_{0}$ that makes the following
diagram commutative:

${P_{0}}$${Z_{0}}$${W_{0}}$$\scriptstyle{l_{0}\mid_{P_{0}}}$$\scriptstyle{\rho_{0}}$$\scriptstyle{g_{0}}$

If $y\in Y_{0}=\operatorname{Im}(f_{0})\oplus P_{0}$, then we can uniquely
write $y=f_{0}(x)+z$ for some $x\in X_{0}$ and $z\in P_{0}$. Define a map
$\phi_{0}:Y_{0}\rightarrow Z_{0}$ by setting
$\phi_{0}(y)=h_{0}(x)+\rho_{0}(z)$. Then it is clear that $\phi_{0}$ is an
$R$-homomorphism. Moreover, the following diagram is commutative:

${X_{0}}$${Z_{0}}$${Y_{0}}$${W_{0}}$$\scriptstyle{h_{0}}$$\scriptstyle{f_{0}}$$\scriptstyle{g_{0}}$$\scriptstyle{l_{0}}$$\scriptstyle{\phi_{0}}$

Indeed, if $x\in X_{0}$, then using the definition, we have
$\phi_{0}\left(f_{0}(x)\right)=h_{0}(x)$. Further, if $y=f_{0}(x)+z\in
Y_{0}=\operatorname{Im}(f_{0})\oplus P_{0}$, then we have:

$g_{0}\left(\phi_{0}(y)\right)=g_{0}\left(h_{0}(x)\right)+g_{0}\left(\rho_{0}(z)\right)=l_{0}\left(f_{0}(x)\right)+l_{0}(z)=l_{0}\left(f_{0}(x)+z\right)=l_{0}(y)$

Now suppose that $n\geq 1$, and for any $0\leq i\leq n-1$, the
$R$-homomorphism $\phi_{i}:Y_{i}\rightarrow Z_{i}$ is constructed in a way
that the following diagrams are commutative:

${X_{i}}$${Z_{i}}$${Y_{i}}$${W_{i}}$$\scriptstyle{h_{i}}$$\scriptstyle{f_{i}}$$\scriptstyle{g_{i}}$$\scriptstyle{l_{i}}$$\scriptstyle{\phi_{i}}$
and
${Y_{i}}$${Y_{i-1}}$${Z_{i}}$${Z_{i-1}}$$\scriptstyle{\partial_{i}^{Y}}$$\scriptstyle{\phi_{i}}$$\scriptstyle{\phi_{i-1}}$$\scriptstyle{\partial_{i}^{Z}}$

That is, $\phi_{i}f_{i}=h_{i}$, $g_{i}\phi_{i}=l_{i}$, and
$\partial_{i}^{Z}\phi_{i}=\phi_{i-1}\partial_{i}^{Y}$ for every $0\leq i\leq
n-1$. By the same argument as in the case $i=0$, we can obtain an
$R$-homomorphism $\rho_{n}:P_{n}\rightarrow Z_{n}$ that makes the diagram

${P_{n}}$${Z_{n}}$${W_{n}}$$\scriptstyle{l_{n}\mid_{P_{n}}}$$\scriptstyle{\rho_{n}}$$\scriptstyle{g_{n}}$

commutative, and we can construct an $R$-homomorphism
$\psi_{n}:Y_{n}\rightarrow Z_{n}$, given by $\psi_{n}(y)=h_{n}(x)+\rho_{n}(z)$
for every $y=f_{n}(x)+z\in Y_{n}=\operatorname{Im}(f_{n})\oplus P_{n}$, that
makes the following diagram commutative:

${X_{n}}$${Z_{n}}$${Y_{n}}$${W_{n}}$$\scriptstyle{h_{n}}$$\scriptstyle{f_{n}}$$\scriptstyle{g_{n}}$$\scriptstyle{l_{n}}$$\scriptstyle{\psi_{n}}$

That is, $\psi_{n}f_{n}=h_{n}$ and $g_{n}\psi_{n}=l_{n}$. However, the
following diagram need not be commutative:

${Y_{n}}$${Y_{n-1}}$${Z_{n}}$${Z_{n-1}}$$\scriptstyle{\partial_{n}^{Y}}$$\scriptstyle{\psi_{n}}$$\scriptstyle{\phi_{n-1}}$$\scriptstyle{\partial_{n}^{Z}}$

To fix this, we set
$\zeta_{n}=\partial_{n}^{Z}\psi_{n}-\phi_{n-1}\partial_{n}^{Y}:Y_{n}\rightarrow
Z_{n-1}$. Using the fact that $l:Y\rightarrow W$ and $g:Z\rightarrow W$ are
morphisms of $R$-complexes, we have for every $z\in P_{n}\subseteq Y_{n}$:

$\begin{split}g_{n-1}\left(\zeta_{n}(z)\right)&=g_{n-1}\left(\partial_{n}^{Z}\left(\psi_{n}(z)\right)\right)-g_{n-1}\left(\phi_{n-1}\left(\partial_{n}^{Y}(z)\right)\right)=g_{n-1}\left(\partial_{n}^{Z}\left(\rho_{n}(z)\right)\right)-l_{n-1}\left(\partial_{n}^{Y}(z)\right)\\\
&=\partial_{n}^{W}\left(g_{n}\left(\rho_{n}(z)\right)\right)-\partial_{n}^{W}\left(l_{n}(z)\right)=\partial_{n}^{W}\left(l_{n}(z)\right)-\partial_{n}^{W}\left(l_{n}(z)\right)=0\end{split}$

That is, $\zeta_{n}(z)\in\operatorname{Ker}(g_{n-1})$. Therefore, we have for
every $z\in P_{n}\subseteq Y_{n}$:

$\begin{split}\partial_{n-1}^{\operatorname{Ker}(g)}\left(\zeta_{n}(z)\right)&=\partial_{n-1}^{Z}\left(\zeta_{n}(z)\right)=\partial_{n-1}^{Z}\left(\partial_{n}^{Z}\left(\psi_{n}(z)\right)\right)-\partial_{n-1}^{Z}\left(\phi_{n-1}\left(\partial_{n}^{Y}(z)\right)\right)\\\
&=\partial_{n-1}^{Z}\left(\partial_{n}^{Z}\left(\psi_{n}(z)\right)\right)-\phi_{n-2}\left(\partial_{n-1}^{Y}\left(\partial_{n}^{Y}(z)\right)\right)=0\end{split}$

That is, $\zeta_{n}(z)\in Z_{n-1}\left(\operatorname{Ker}(g)\right)$. But
$\operatorname{Ker}(g)$ is exact, so $\zeta_{n}(z)\in
B_{n-1}\left(\operatorname{Ker}(g)\right)$. Therefore, $\zeta_{n}$ induces an
$R$-homomorphism $\overline{\zeta_{n}\mid_{P_{n}}}:P_{n}\rightarrow
B_{n-1}\left(\operatorname{Ker}(g)\right)$. By the projectivity of $P_{n}$,
there is an $R$-homomorphism
$\mu_{n}:P_{n}\rightarrow\operatorname{Ker}(g_{n})$ that makes the following
diagram commutative:

${P_{n}}$${\operatorname{Ker}(g_{n})}$${B_{n-1}\left(\operatorname{Ker}(g)\right)}$$\scriptstyle{\overline{\zeta_{n}\mid_{P_{n}}}}$$\scriptstyle{\mu_{n}}$$\scriptstyle{\overline{\partial_{n}^{\operatorname{Ker}(g)}}}$

Let $\pi_{n}:Y_{n}=\operatorname{Im}(f_{n})\oplus P_{n}\rightarrow P_{n}$ be
the canonical projection. Also, let $\chi_{n}:Y_{n}\rightarrow Z_{n}$ be the
following composition:

$Y_{n}\xrightarrow{\pi_{n}}P_{n}\xrightarrow{\mu_{n}}\operatorname{Ker}(g_{n})\xrightarrow{\iota_{n}}Z_{n}$

Set $\phi_{n}=\psi_{n}-\chi_{n}:Y_{n}\rightarrow Z_{n}$. Then the following
diagram is commutative:

${X_{n}}$${Z_{n}}$${Y_{n}}$${W_{n}}$$\scriptstyle{h_{n}}$$\scriptstyle{f_{n}}$$\scriptstyle{g_{n}}$$\scriptstyle{l_{n}}$$\scriptstyle{\phi_{n}}$

Indeed, we have for every $x\in X_{n}$:

$\phi_{n}\left(f_{n}(x)\right)=\psi_{n}\left(f_{n}(x)\right)-\chi_{n}\left(f_{n}(x)\right)=h_{n}(x)-\iota_{n}\left(\mu_{n}\left(\pi_{n}\left(f_{n}(x)\right)\right)\right)=h_{n}(x)$

Also, we have for every $y=f_{n}(x)+z\in Y_{n}=Im(f_{n})\oplus P_{n}$:

$g_{n}\left(\phi_{n}(y)\right)=g_{n}\left(\psi_{n}(y)\right)-g_{n}\left(\chi_{n}(y)\right)=l_{n}(y)-g_{n}\left(\iota_{n}\left(\mu_{n}\left(\pi_{n}(y)\right)\right)\right)=l_{n}(y)-g_{n}\left(\mu_{n}(z)\right)=l_{n}(y)$

In addition, the following diagram is commutative:

${Y_{n}}$${Y_{n-1}}$${Z_{n}}$${Z_{n-1}}$$\scriptstyle{\partial_{n}^{Y}}$$\scriptstyle{\phi_{n}}$$\scriptstyle{\phi_{n-1}}$$\scriptstyle{\partial_{n}^{Z}}$

Indeed, noting that $f:X\rightarrow Y$ and $h:X\rightarrow Z$ are morphisms of
$R$-complexes, we have for every $y=f_{n}(x)+z\in
Y_{n}=\operatorname{Im}(f_{n})\oplus P_{n}$:

$\begin{split}\partial_{n}^{Z}\left(\phi_{n}(y)\right)&=\partial_{n}^{Z}\left(\psi_{n}(y)\right)-\partial_{n}^{Z}\left(\chi_{n}(y)\right)\\\
&=\partial_{n}^{Z}\left(\psi_{n}(y)\right)-\partial_{n}^{Z}\left(\iota_{n}\left(\mu_{n}\left(\pi_{n}(y)\right)\right)\right)\\\
&=\partial_{n}^{Z}\left(\psi_{n}(y)\right)-\partial_{n}^{Z}\left(\mu_{n}(z)\right)\\\
&=\partial_{n}^{Z}\left(\psi_{n}(y)\right)-\partial_{n}^{\operatorname{Ker}(g)}\left(\mu_{n}(z)\right)\\\
&=\partial_{n}^{Z}\left(\psi_{n}(y)\right)-\zeta_{n}(z)\\\
&=\partial_{n}^{Z}\left(\psi_{n}(y)\right)-\partial_{n}^{Z}\left(\psi_{n}(z)\right)+\phi_{n-1}\left(\partial_{n}^{Y}(z)\right)\\\
&=\partial_{n}^{Z}\left(\psi_{n}(y-z)\right)+\phi_{n-1}\left(\partial_{n}^{Y}\left(y-f_{n}(x)\right)\right)\\\
&=\partial_{n}^{Z}\left(\psi_{n}\left(f_{n}(x)\right)\right)+\phi_{n-1}\left(\partial_{n}^{Y}(y)\right)-\phi_{n-1}\left(\partial_{n}^{Y}\left(f_{n}(x)\right)\right)\\\
&=\partial_{n}^{Z}\left(\psi_{n}\left(f_{n}(x)\right)\right)+\phi_{n-1}\left(\partial_{n}^{Y}(y)\right)-\phi_{n-1}\left(f_{n-1}\left(\partial_{n}^{X}(x)\right)\right)\\\
&=\partial_{n}^{Z}\left(h_{n}(x)\right)+\phi_{n-1}\left(\partial_{n}^{Y}(y)\right)-h_{n-1}\left(\partial_{n}^{X}(x)\right)\\\
&=\partial_{n}^{Z}\left(h_{n}(x)\right)+\phi_{n-1}\left(\partial_{n}^{Y}(y)\right)-\partial_{n}^{Z}\left(h_{n}(x)\right)\\\
&=\phi_{n-1}\left(\partial_{n}^{Y}(y)\right)\end{split}$

As a consequence, we can continue this construction inductively and find a
morphism $\phi=(\phi_{i})_{i\geq 0}:Y\rightarrow Z$ of $R$-complexes that
makes the following diagram commutative:

${X}$${Z}$${Y}$${W}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{l}$$\scriptstyle{\phi}$

First Half of Factorization: Let $f:X\rightarrow Y$ be a morphism in
$\mathcal{C}_{\geq 0}(R)$. For any $n\geq 1$, consider the isomorphism
$\psi_{n}^{Y}:\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(D(n),Y\right)\rightarrow Y_{n}$. For any $n\geq 1$ and $x\in
Y_{n}$, there is a $p^{n,x}\in\operatorname{Mor}_{\mathcal{C}_{\geq
0}(R)}\left(D(n),Y\right)$ such that
$\psi_{n}^{Y}\left(p^{n,x}\right)=p^{n,x}_{n}(1)=x$. Let
$P=\bigoplus_{n=1}^{\infty}\bigoplus_{x\in Y_{n}}D(n)$. By the universal
property of direct sum, there is a unique morphism $p:P\rightarrow Y$ of
$R$-complexes that makes the diagram

${D(n)}$${P}$${Y}$$\scriptstyle{\iota^{n,x}}$$\scriptstyle{p^{n,x}}$$\scriptstyle{p}$

commutative in which $\iota^{n,x}$ is the canonical injection for every $n\geq
1$ and $x\in Y_{n}$. Thus if $n\geq 1$ and $x\in Y_{n}$, then
$p_{n}\left(\iota_{n}^{n,x}(1)\right)=p_{n}^{n,x}(1)=x$. This shows that
$p_{n}$ is surjective for every $n\geq 1$. In addition, there is a unique
morphism $\eta:X\oplus P\rightarrow U$ of $R$-complexes that makes the diagram

${X}$${X\oplus
P}$${P}$${Y}$$\scriptstyle{\kappa}$$\scriptstyle{f}$$\scriptstyle{\eta}$$\scriptstyle{\nu}$$\scriptstyle{p}$

commutative in which $\kappa$ and $\nu$ are canonical injections. For any
$i\geq 1$, $\eta_{i}\nu_{i}=p_{i}$ is surjective, so $\eta_{i}$ is surjective.
This implies that $\eta$ is a fibration. On the other hand, for any $i\geq 0$,
$\kappa_{i}$ is injective, and $\operatorname{Coker}(\kappa_{i})\cong P_{i}$
is a direct sum of copies of $R$, so $\operatorname{Coker}(\kappa_{i})$ is
projective. That is, $\kappa$ is a cofibration. Also, for any $i\geq 0$, we
have

$H_{i}(P)=H_{i}\left(\textstyle\bigoplus_{n=1}^{\infty}\textstyle\bigoplus_{x\in
Y_{n}}D(n)\right)\cong\textstyle\bigoplus_{n=1}^{\infty}\bigoplus_{x\in
Y_{n}}H_{i}\left(D(n)\right)=0,$

which implies that $H_{i}(\kappa):H_{i}(X)\rightarrow H_{i}(X\oplus P)$ is an
isomorphism, so $\kappa$ is a quasi-isomorphism. Therefore, $\kappa$ is a
trivial cofibration. Now $f=\eta\kappa$ is a factorization of $f$ in which
$\eta$ is a fibration and $\kappa$ is a trivial cofibration. We further show
that $\kappa$ has the left lifting property against all fibrations. To this
end, let $t:U\rightarrow V$ be a fibration in $\mathcal{C}_{\geq 0}(R)$.
Consider a commutative diagram as follows:

${X}$${U}$${X\oplus
P}$${V}$$\scriptstyle{r}$$\scriptstyle{\kappa}$$\scriptstyle{t}$$\scriptstyle{s}$

For any $n\geq 1$ and $x\in Y_{n}$, let $u^{n,x}:D(n)\rightarrow V$ be the
following composition:

$D(n)\xrightarrow{\iota^{n,x}}P\xrightarrow{\nu}X\oplus P\xrightarrow{s}V$

As we observed above, $\operatorname{RLP}(\mathcal{Y})$ is the class of
fibrations, so $t\in\operatorname{RLP}(\mathcal{Y})$. It follows that for any
$n\geq 1$ and $x\in Y_{n}$, there is a morphism $q^{n,x}:D(n)\rightarrow U$
that makes the following diagram commutative:

${0}$${U}$${D(n)}$${V}$$\scriptstyle{\kappa^{n}}$$\scriptstyle{t}$$\scriptstyle{u^{n,x}}$$\scriptstyle{q^{n,x}}$

By the universal property of direct sum, there is a unique morphism
$q:P\rightarrow U$ of $R$-complexes that makes the following diagram
commutative for every $n\geq 1$ and $x\in Y_{n}$:

${D(n)}$${P}$${U}$$\scriptstyle{\iota^{n,x}}$$\scriptstyle{q^{n,x}}$$\scriptstyle{q}$

In addition, there is a unique morphism $\theta:X\oplus P\rightarrow U$ of
$R$-complexes that makes the following diagram commutative:

${X}$${X\oplus
P}$${P}$${U}$$\scriptstyle{\kappa}$$\scriptstyle{r}$$\scriptstyle{\theta}$$\scriptstyle{\nu}$$\scriptstyle{q}$

Then the following diagram is commutative:

${X}$${U}$${X\oplus
P}$${V}$$\scriptstyle{r}$$\scriptstyle{\kappa}$$\scriptstyle{t}$$\scriptstyle{s}$$\scriptstyle{\theta}$

Clearly, $\theta\kappa=r$. Also,
$tq\iota^{n,x}=tq^{n,x}=u^{n,x}=s\nu\iota^{n,x}$ for every $n\geq 1$ and $x\in
Y_{n}$, implying that $tq=s\nu$. Further, $t\theta\kappa=tr=s\kappa$ and
$t\theta\nu=tq=s\nu$, showing that $t\theta=s$. As a consequence, $\kappa$ has
the left lifting property against all fibrations.

Second Half of Lifting: Consider a commutative diagram

${X}$${Z}$${Y}$${W}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{l}$

in $\mathcal{C}_{\geq 0}(R)$ in which $f$ is a trivial cofibration and $g$ is
a fibration. Consider the factorization

${X}$${Y}$${X\oplus
P}$$\scriptstyle{f}$$\scriptstyle{\kappa}$$\scriptstyle{\eta}$

of $f$ that was established above in which
$P=\bigoplus_{n=1}^{\infty}\bigoplus_{x\in Y_{n}}D(n)$, $\eta$ is a fibration,
and $\kappa$ is a trivial cofibration that further has the left lifting
property against all fibrations. Since $f$ and $\kappa$ are quasi-
isomorphisms, we deduce that $\eta$ is also a quasi-isomorphism, so $\eta$ is
a trivial fibration. Consider the following commutative diagram:

${X}$${X\oplus
P}$${Y}$${Y}$$\scriptstyle{\kappa}$$\scriptstyle{f}$$\scriptstyle{\eta}$$\scriptstyle{1^{Y}}$

As $f$ is a cofibration and $\eta$ is a trivial fibration, the first half of
lifting that was established above provides a morphism $r:Y\rightarrow X\oplus
P$ that makes the following diagram commutative:

${X}$${X\oplus
P}$${Y}$${Y}$$\scriptstyle{\kappa}$$\scriptstyle{f}$$\scriptstyle{\eta}$$\scriptstyle{1^{Y}}$$\scriptstyle{r}$

This yields the following commutative diagram:

${X}$${X}$${X}$${Y}$${X\oplus
P}$${Y}$$\scriptstyle{1^{X}}$$\scriptstyle{f}$$\scriptstyle{1^{X}}$$\scriptstyle{\kappa}$$\scriptstyle{f}$$\scriptstyle{r}$$\scriptstyle{\eta}$

This shows that $f$ is a retract of $\kappa$. But $\kappa$ has the left
lifting property against all fibrations, so the proof of [Hi, Lemma 7.2.8]
that makes no use of the model category assumption, shows that $f$ has the
left lifting property against all fibrations. This implies that the above
diagram can be completed as follows:

${X}$${Z}$${Y}$${W}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{l}$

Second Half of Factorization: Let $f:X\rightarrow Y$ be a morphism in
$\mathcal{C}_{\geq 0}(R)$. We construct the desired factorization inductively.
Let $\rho_{0}:P_{0}\rightarrow Y_{0}$ be an $R$-epimorphism in which $P_{0}$
is a projective left $R$-module, and set $Q_{0}=X_{0}\oplus P_{0}$. By the
universal property of direct sum, there is a unique morphism
$\eta_{0}:Q_{0}\rightarrow Y_{0}$ of $R$-complexes that makes the diagram

${X_{0}}$${Q_{0}}$${P_{0}}$${Y_{0}}$$\scriptstyle{\kappa_{0}}$$\scriptstyle{f_{0}}$$\scriptstyle{\eta_{0}}$$\scriptstyle{\nu_{0}}$$\scriptstyle{\rho_{0}}$

commutative in which $\kappa_{0}$ and $\nu_{0}$ are canonical injections. Then
$f_{0}=\eta_{0}\kappa_{0}$ and $\rho_{0}=\eta_{0}\nu_{0}$. It is clear that
$\kappa_{0}$ is injective whose $\operatorname{Coker}(\kappa_{0})\cong P_{0}$
is projective. On the other hand, since $\rho_{0}$ is surjective, we infer
that $\eta_{0}$ is surjective. Now suppose that $n\geq 1$, and for any $0\leq
i\leq n-1$, the $R$-module $Q_{i}$ and the $R$-homomorphisms
$\partial_{i}^{Q}:Q_{i}\rightarrow Q_{i-1}$, $\kappa_{i}:X_{i}\rightarrow
Q_{i}$, and $\eta_{i}:Q_{i}\rightarrow Y_{i}$ are constructed in a way that
$\partial_{i-1}^{Q}\partial_{i}^{Q}=0$, $f_{i}=\eta_{i}\kappa_{i}$,
$\kappa_{i}$ is injective whose $\operatorname{Coker}(\kappa_{i})$ is
projective, $\eta_{i}$ is surjective with $H_{i-1}(\eta)$ an isomorphism, and
the following diagram is commutative:

${X_{i}}$${X_{i-1}}$${Q_{i}}$${Q_{i-1}}$${Y_{i}}$${Y_{i-1}}$$\scriptstyle{\partial_{i}^{X}}$$\scriptstyle{\kappa_{i}}$$\scriptstyle{\kappa_{i-1}}$$\scriptstyle{\partial_{i}^{Q}}$$\scriptstyle{\eta_{i}}$$\scriptstyle{\eta_{i-1}}$$\scriptstyle{\partial_{i}^{Y}}$

Using the above diagram, we see that if $x\in X_{n}$, then
$\partial_{n-1}^{Q}\left(\kappa_{n-1}\left(\partial_{n}^{X}(x)\right)\right)=\kappa_{n-2}\left(\partial_{n-1}^{X}\left(\partial_{n}^{X}(x)\right)\right)=0$,
so $\kappa_{n-1}\left(\partial_{n}^{X}(x)\right)\in Z_{n-1}(Q)$. Therefore, we
get an induced $R$-homomorphism
$\overline{\kappa_{n-1}\partial_{n}^{X}}:X_{n}\rightarrow Z_{n-1}(Q)$.
Consider the following commutative diagram:

${X_{n}}$${Y_{n}}$${Z_{n-1}(Q)}$${Z_{n-1}(Y)}$$\scriptstyle{f_{n}}$$\scriptstyle{\overline{\kappa_{n-1}\partial_{n}^{X}}}$$\scriptstyle{\overline{\partial_{n}^{Y}}}$$\scriptstyle{Z_{n-1}(\eta)}$

Indeed, using the fact that $f$ is a morphism of $R$-complexes, we have
$\eta_{n-1}\left(\kappa_{n-1}\left(\partial_{n}^{X}(x)\right)\right)=f_{n-1}\left(\partial_{n}^{X}(x)\right)=\partial_{n}^{Y}\left(f_{n}(x)\right)$
for every $x\in X_{n}$. Set:

$L_{n}=Z_{n-1}(Q)\textstyle\mathop{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle\bigsqcup$}}{\rotatebox[origin={c}]{180.0}{$\textstyle\bigsqcup$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle\bigsqcup$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle\bigsqcup$}}}_{Z_{n-1}(Y)}Y_{n}=\left\\{(x,y)\in
Z_{n-1}(Q)\oplus Y_{n}\;|\;\eta_{n-1}(x)=\partial_{n}^{Y}(y)\right\\}$

Then there is a unique morphism $\varrho_{n}:X_{n}\rightarrow L_{n}$ that
makes the following pullback diagram commutative:

${X_{n}}$${L_{n}}$${Y_{n}}$${Z_{n-1}(Q)}$${Z_{n-1}(Y)}$$\scriptstyle{\varrho_{n}}$$\scriptstyle{\overline{\kappa_{n-1}\partial_{n}^{X}}}$$\scriptstyle{f_{n}}$$\scriptstyle{\zeta_{n}}$$\scriptstyle{\xi_{n}}$$\scriptstyle{\overline{\partial_{n}^{Y}}}$$\scriptstyle{Z_{n-1}(\eta)}$

As in the case $i=0$, let $\rho_{n}:P_{n}\rightarrow L_{n}$ be an
$R$-epimorphism in which $P_{n}$ is a projective left $R$-module, and set
$Q_{n}=X_{n}\oplus P_{n}$. By the universal property of direct sum, there is a
unique morphism $\theta_{n}:Q_{n}\rightarrow L_{n}$ of $R$-complexes that
makes the diagram

${X_{n}}$${Q_{n}}$${P_{n}}$${L_{n}}$$\scriptstyle{\kappa_{n}}$$\scriptstyle{\varrho_{n}}$$\scriptstyle{\theta_{n}}$$\scriptstyle{\nu_{n}}$$\scriptstyle{\rho_{n}}$

commutative in which $\kappa_{n}$ and $\nu_{n}$ are canonical injections. Set
$\eta_{n}=\zeta_{n}\theta_{n}:Q_{n}\rightarrow Y_{n}$. Then
$f_{n}=\zeta_{n}\varrho_{n}=\zeta_{n}\theta_{n}\kappa_{n}=\eta_{n}\kappa_{n}$.
It is also clear that $\kappa_{n}$ is injective whose
$\operatorname{Coker}(\kappa_{n})\cong P_{n}$ is projective.

Consider the following compositions of $R$-homomorphisms:

$X_{n}\xrightarrow{\partial_{n}^{X}}X_{n-1}\xrightarrow{\kappa_{n-1}}Q_{n-1}\quad\text{and}\quad
P_{n}\xrightarrow{\rho_{n}}L_{n}\xrightarrow{\xi_{n}}Z_{n-1}(Q)\xrightarrow{\iota_{n-1}^{Q}}Q_{n-1}$

By the universal property of direct sum, there is a unique morphism
$\partial_{n}^{Q}:Q_{n}\rightarrow Q_{n-1}$ of $R$-complexes that makes the
following diagram commutative:

${X_{n}}$${Q_{n}}$${P_{n}}$${Q_{n-1}}$$\scriptstyle{\kappa_{n}}$$\scriptstyle{\kappa_{n-1}\partial_{n}^{X}}$$\scriptstyle{\partial_{n}^{Q}}$$\scriptstyle{\nu_{n}}$$\scriptstyle{\iota_{n-1}^{Q}\xi_{n}\rho_{n}}$

Then we have
$\partial_{n-1}^{Q}\partial_{n}^{Q}\kappa_{n}=\partial_{n-1}^{Q}\kappa_{n-1}\partial_{n}^{X}=\kappa_{n-2}\partial_{n-1}^{X}\partial_{n}^{X}=0$
and
$\partial_{n-1}^{Q}\partial_{n}^{Q}\nu_{n}=\partial_{n-1}^{Q}\iota_{n-1}^{Q}\xi_{n}\rho_{n}=0$
as $\operatorname{Im}(\xi_{n})\subseteq Z_{n-1}(Q)$. It follows that
$\partial_{n-1}^{Q}\partial_{n}^{Q}=0$.

We next show that the following diagram is commutative:

${X_{n}}$${X_{n-1}}$${Q_{n}}$${Q_{n-1}}$${Y_{n}}$${Y_{n-1}}$$\scriptstyle{\partial_{n}^{X}}$$\scriptstyle{\kappa_{n}}$$\scriptstyle{\kappa_{n-1}}$$\scriptstyle{\partial_{n}^{Q}}$$\scriptstyle{\eta_{n}}$$\scriptstyle{\eta_{n-1}}$$\scriptstyle{\partial_{n}^{Y}}$

Indeed, we have $\partial_{n}^{Q}\kappa_{n}=\kappa_{n-1}\partial_{n}^{X}$ from
the previous commutative diagram. Moreover, we have:

$\partial_{n}^{Y}\eta_{n}\kappa_{n}=\partial_{n}^{Y}\zeta_{n}\theta_{n}\kappa_{n}=\partial_{n}^{Y}\zeta_{n}\varrho_{n}=\partial_{n}^{Y}f_{n}=f_{n-1}\partial_{n}^{X}=\eta_{n-1}\kappa_{n-1}\partial_{n}^{X}=\eta_{n-1}\partial_{n}^{Q}\kappa_{n}$

and

$\partial_{n}^{Y}\eta_{n}\nu_{n}=\partial_{n}^{Y}\zeta_{n}\theta_{n}\nu_{n}=\partial_{n}^{Y}\zeta_{n}\rho_{n}=\iota_{n-1}^{Y}\overline{\partial_{n}^{Y}}\zeta_{n}\rho_{n}=\iota_{n-1}^{Y}Z_{n-1}(\eta)\xi_{n}\rho_{n}=\eta_{n-1}\iota_{n-1}^{Q}\xi_{n}\rho_{n}=\eta_{n-1}\partial_{n}^{Q}\nu_{n}$

It follows that $\partial_{n}^{Y}\eta_{n}=\eta_{n-1}\partial_{n}^{Q}$.

We now show that $H_{n-1}(\eta)$ is injective. Suppose that
$H_{n-1}(\eta)\left(x+B_{n-1}(Q)\right)=\eta_{n-1}(x)+B_{n-1}(Y)=0$ for some
$x\in Z_{n-1}(Q)$. Then $\eta_{n-1}(x)\in B_{n-1}(Y)$, so
$\eta_{n-1}(x)=\partial_{n}^{Y}(y)$ for some $y\in Y_{n}$. This implies that
$(x,y)\in L_{n}$. On the other hand, $\rho_{n}=\theta_{n}\nu_{n}$ and
$\rho_{n}$ is surjective, so $\theta_{n}:Q_{n}\rightarrow L_{n}$ is
surjective. Therefore, there is an element $(z,w)\in Q_{n}=X_{n}\oplus P_{n}$
such that $\theta_{n}(z,w)=(x,y)$. Then we have:

$\begin{split}x&=\xi_{n}(x,y)=\xi_{n}\left(\theta_{n}(z,w)\right)=\xi_{n}\left(\varrho_{n}(z)\right)+\xi_{n}\left(\rho_{n}(w)\right)=\kappa_{n-1}\left(\partial_{n}^{X}(z)\right)+\partial_{n}^{Q}\left(\nu_{n}(w)\right)\\\
&=\partial_{n}^{Q}\left(\kappa_{n}(z)\right)+\partial_{n}^{Q}\left(\nu_{n}(w)\right)=\partial_{n}^{Q}\left(\kappa_{n}(z)+\nu_{n}(w)\right)=\partial_{n}^{Q}(z,w)\end{split}$

Hence $x\in B_{n-1}(Q)$, so $x+B_{n-1}(Q)=0$. This shows that $H_{n-1}(\eta)$
is injective.

We next show that $H_{n-1}(\eta)$ is surjective. Let $y\in Z_{n}(Y)$. Then
$\eta_{n-1}(0)=0=\partial_{n}^{Y}(y)$, so $(0,y)\in L_{n}$. By the
surjectivity of $\theta_{n}:Q_{n}\rightarrow L_{n}$, there is an element
$(z,w)\in Q_{n}=X_{n}\oplus P_{n}$ such that $\theta_{n}(z,w)=(0,y)$. Then
$y=\zeta_{n}(0,y)=\zeta_{n}\left(\theta_{n}(z,w)\right)=\eta_{n}(z,w)$. Also,
a similar calculation as in the previous paragraph shows that
$\partial_{n}^{Q}(z,w)=0$, so $(z,w)\in Z_{n}(Q)$. This shows that
$Z_{n}(\eta):Z_{n}(Q)\rightarrow Z_{n}(Y)$ is surjective. Since we have
constructed all these $R$-homomorphisms in the same way in each step, we can
infer that $Z_{n-1}(\eta)$ is surjective. It follows that $H_{n-1}(\eta)$ is
surjective. Therefore, $H_{n-1}(\eta)$ is an isomorphism.

We finally show that $\eta_{n}$ is surjective. Let $y\in Y_{n}$. Then
$\partial_{n}^{Y}(y)\in B_{n-1}(Y)\subseteq Z_{n-1}(Y)$, so by the
surjectivity of $Z_{n-1}(\eta)$, there is an element $x\in Z_{n-1}(Q)$ such
that $\eta_{n-1}(x)=\partial_{n}^{Y}(y)$. Then $(x,y)\in L_{n}$, so by the
surjectivity of $\theta_{n}$, there is an element $(z,w)\in Q_{n}=X_{n}\oplus
P_{n}$ such that $\theta_{n}(z,w)=(x,y)$. It follows that
$y=\zeta_{n}(x,y)=\zeta_{n}\left(\theta_{n}(z,w)\right)=\eta_{n}(z,w)$. Thus
$\eta_{n}$ is surjective.

As a consequence, we can inductively construct the morphisms
$\kappa:X\rightarrow Q$ and $\eta:Q\rightarrow Y$ of $R$-complexes such that
$f=\eta\kappa$, $\eta$ is a trivial fibration, and $\kappa$ is a cofibration.

All in all, we have a cofibrantly generated model structure on
$\mathcal{C}_{\geq 0}(R)$. ∎

##  4\. Simplicial Objects and Dold-Kan Correspondence Theorem

Simplicial objects are nowadays widely used in modern mathematics. Simplicial
sets in particular are combinatorial models for topological spaces. They also
serve as the bedrock of the language of $\infty$-categories. Our central focus
of attention is on simplicial modules and simplicial commutative algebras
which can be thought of as generalizations as well as non-abelian analogues of
chain complexes. In this section, we recall the basic definitions and examples
of simplicial objects. Moreover, we fix our notations on the normalization and
Dld-Kan functors. Deploying the Dold-Kan Correspondence, we transfer the model
structure on connective chain complexes to simplicial modules.

######  Definition 4.1.

Let $\mathcal{C}$ be a category. A simplicial object over $\mathcal{C}$ is a
family $A=\\{A_{n}\\}_{n\geq 0}$ of objects of $\mathcal{C}$ together with
face morphisms $d_{n,i}^{A}:A_{n}\rightarrow A_{n-1}$ for every $n\geq 1$ and
$0\leq i\leq n$, and degeneracy morphisms $s_{n,i}^{A}:A_{n}\rightarrow
A_{n+1}$ for every $n\geq 0$ and $0\leq i\leq n$, that satisfy the following
relations for every $n\geq 0$ and $0\leq i,j\leq n$:

  1. (i)

$d_{n,i}^{A}d_{n+1,j}^{A}=d_{n,j-1}^{A}d_{n+1,i}^{A}$ if $i<j$.

  2. (ii)

$s_{n,i}^{A}s_{n-1,j}^{A}=s_{n,j+1}^{A}s_{n-1,i}^{A}$ if $i\leq j$.

  3. (iii)

$d_{n,i}^{A}s_{n-1,j}^{A}=\begin{cases}s_{n-2,j-1}^{A}d_{n-1,i}^{A}&\text{if
}i<j\\\ 1^{A_{n-1}}&\text{if }i=j\text{ or }j+1\\\
s_{n-2,j}^{A}d_{n-1,i-1}^{A}&\text{if }i>j+1\end{cases}$

Given two simplicial objects $A=\\{A_{n}\\}_{n\geq 0}$ and
$B=\\{B_{n}\\}_{n\geq 0}$ over $\mathcal{C}$, a morphism $\tau:A\rightarrow B$
of simplicial objects is a collection $\tau=(\tau_{n})_{n\geq 0}$ of morphisms
$\tau_{n}:A_{n}\rightarrow B_{n}$ of $\mathcal{C}$ for every $n\geq 0$, such
that the following diagram is commutative for every $n\geq 1$ and $0\leq i\leq
n$:

${A_{n-1}}$${A_{n}}$${A_{n-1}}$${B_{n-1}}$${B_{n}}$${B_{n-1}}$$\scriptstyle{s_{n-1,i}^{A}}$$\scriptstyle{\tau_{n-1}}$$\scriptstyle{d_{n,i}^{A}}$$\scriptstyle{\tau_{n}}$$\scriptstyle{\tau_{n-1}}$$\scriptstyle{s_{n-1,i}^{B}}$$\scriptstyle{d_{n,i}^{B}}$

We speak of simplicial sets, simplicial modules, or simplicial commutative
algebras when we consider simplicial objects over the categories of sets,
modules, or commutative algebras.

######  Remark 4.2.

It is immediate from Definition 4.1 that we have a category
$\mathpzc{s}\mathcal{C}$ of simplicial objects over $\mathcal{C}$. As a matter
of fact, $\mathpzc{s}\mathcal{C}$ is a functor category. Let $\Delta$ denote
the simplex category described by

$\operatorname{Obj}(\Delta)=\left\\{[n]=\\{0,1,2,...,n\\}\;|\;n\geq
0\right\\}$

and

$\operatorname{Mor}_{\Delta}\left([n],[m]\right)=\left\\{f\in\operatorname{Mor}_{\mathcal{S}\mathpzc{et}}\left([n],[m]\right)\;|\;f\textrm{
is order-preserving}\right\\}$

for every $[n],[m]\in\operatorname{Obj}(\Delta)$. Then it is folklore that
$\mathpzc{s}\mathcal{C}=\mathcal{C}^{\Delta^{\operatorname{op}}}$. In other
words, a simplicial object over $\mathcal{C}$ is nothing but a contravariant
functor $\Delta\rightarrow\mathcal{C}$. Accordingly, we sometimes use the
functorial point of view when defining a simplicial object.

For any $n\geq 0$ and $0\leq i\leq n$, the face map
$\delta_{n,i}:[n-1]\rightarrow[n]$ is the unique injective map that ”misses”
$i$, i.e.

$\delta_{n,i}(k)=\begin{dcases}k&\textrm{if }k<i\\\ k+1&\textrm{if }k\geq
i\end{dcases}$

for every $0\leq k\leq n-1$, and the degeneracy map
$\sigma_{n,i}:[n+1]\rightarrow[n]$ is the unique surjective map that ”hits”
$i$ twice, i.e.

$\sigma_{n,i}(k)=\begin{dcases}k&\textrm{if }k\leq i\\\ k-1&\textrm{if
}k>i\end{dcases}$

for every $0\leq k\leq n+1$. It is clear that the face and the degeneracy maps
are morphisms in the simplex category. Moreover, $\delta_{n,n}$ is the
inclusion map for every $n\geq 0$. One can show by brute force that the
following face-degeneracy relations hold for every $0\leq i,j\leq n$:

  1. (i)

$\delta_{n+1,j}\delta_{n,i}=\delta_{n+1,i}\delta_{n,j-1}$ if $i<j$.

  2. (ii)

$\sigma_{n-1,j}\sigma_{n,i}=\sigma_{n-1,i}\sigma_{n,j+1}$ if $i\leq j$.

  3. (iii)

$\sigma_{n-1,j}\delta_{n,i}=\begin{cases}\delta_{n-1,i}\sigma_{n-2,j-1}&\text{if
}i<j\\\ 1^{[n-1]}&\text{if }i=j\text{ or }j+1\\\
\delta_{n-1,i-1}\sigma_{n-2,j}&\text{if }i>j+1.\end{cases}$

Suppose that $n,m\geq 0$ and
$f\in\operatorname{Mor}_{\Delta}\left([n],[m]\right)$. If $f$ is bijective,
then $n=m$ and $f=1^{[n]}$. Now assume that $f$ is not bijective. If $f$ is