# nLab model structure on chain complexes

Contents

model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## Idea

Model structures on chain complexes are model category structures on categories of chain complexes whose weak equivalences are quasi-isomorphisms. (There is also a Hurewicz model structure on chain complexes whose weak equivalences are chain homotopy equivalences.)

Via these model structures, all of the standard techniques in homological algebra, such as injective resolutions and projective resolutions, are special cases of constructions in homotopy theory, such as cofibrant resolutions and fibrant resolutions.

The existence of these model structures depends subtly on whether the chain complexes in question are bounded or not.

### In non-negative degree

Chain complexes in non-negative degree in an abelian category $A$ are special in that they may be identified via the Dold?Kan correspondence as simplicial objects in $A$.

$Ch_{\bullet \geq 0}(A) \simeq A^{\Delta^{op}} \,.$

Similarly, cochain complexs are identified with cosimplicial objects

$Ch^{\bullet \geq 0}(A) \simeq A^{\Delta} \,.$

At least if $A$ is the category of abelian groups, so that $A^{\Delta^{op}}$ is the category of abelian simplicial groups it inherits naturally a model category structure from the model structure on simplicial sets, which presents the (∞,1)-category of ∞-groupoids.

The model structure on chain complexes transports this presentation of the special $\infty$-groupoids given by abelian simplicial groups along the Dold-Kan correspondence to chain complexes.

Analogous statements apply to the category of unbounded chain complexes and the canonical stable (infinity,1)-category Spec of spectra.

This we discuss below in

### For unbounded chain complexes

Model structures on unbounded (co)chain complexes can be understood as presentations of spectrum objects in model structures of bounded (co)chain complexes.

See

## Definitions

### In non-negative degree

Let $C$ be an abelian category.

Recall that by the dual Dold-Kan correspondence the category $C^\Delta$ of cosimplicial objects in $C$ is equivalent to the category $Ch^\bullet_+(C)$ of cochain complexes in non-negative degree. This means that we can transfer results discussed at model structure on cosimplicial objects to cochain complexes (see Bousfield2003, section 4.4 for more).

#### The standard (Quillen) model structures

Let $R$ be a ring and write $\mathcal{A} \coloneqq R$Mod for its category of modules.

We discuss the

##### Projective structure on chain complexes
###### Theorem

There is a model category structure on the category of chain complexes $Ch_{\bullet \geq 0 }(\mathcal{A})$ (in non-negative degree) whose

called the projective model structure.

##### Injective structure on cochain complexes

Dually

###### Theorem

There is a model category structure on non-negatively graded cochain complexes $Ch^{\bullet \geq 0 }(\mathcal{A})$ whose

called the injective model structure.

The projective model structure on $Ch_{\bullet \geq 0}$ is originally due to (Quillen II, section 4). An account is given for instance in (Dungan, 2.4.2, proof in section 2.5).

###### Remark

This means that a chain complex $C_\bullet \in Ch_{\bullet}(\mathcal{A})$ is a cofibrant object in the projective model structure, theorem , precisely if it consists of projective modules. Accordingly, a cofibrant resolution in the projective model structure is precisely what in homological algebra is called a projective resolution. Dually for fibrant resolutions in the injective model structure, theorem , and injective resolutions in homological algebra.

This way the traditional definition of derived functor in homological algebra relates to the general construction of derived functors in model category theory. See there for more details. Similar comments apply to the various other model structures below.

#### Resolution model structures

There are resolution model structures on cosimplicial objects in a model category, due to (DwyerKanStover), reviewed in (Bousfield)

(…)

###### Theorem

Let $A$ be an abelian category and let $\mathcal{G} \in Obj(A)$ be a class of objects, such that $A$ has enough $\mathcal{G}$-injective objects.

Then there is a model category structure on non-negatively graded cochain complexes $Ch^{\bullet \geq 0}(A)_{\mathcal{G}}$ whose

• weak equivalences are maps $f : X \to Y$ such that for each $K \in A$ the induced map $A(Y,K) \to A(X,K)$ is a quasi-isomorphism of chain complexes of abelian groups;

• $f$ is a cofibration if it is $\mathcal{G}$-monic in positive degree;

• $f$ is a fibration if it is degreewise a split epimorphism with degreewise $\mathcal{G}$-injective kernel.

If $A$ has enough injective objects and $\mathcal{G}$ is the class of all of them, this reproduces the standard Quillen model structure discussed above:

###### Corollary

Let $A$ be an abelian category with enough injective objects. Then there is a model category structure on non-negatively graded cochain complexes $Ch^{\geq 0}(A)$ whose

• weak equivalences are the quasi-isomorphisms;

• fibrations are the morphisms that are epimorphisms with injective kernel in each degree;

• cofibrations are the morphisms which are monomorphisms in $A$ in each positive degree.

If we take $\mathcal{G}$ to be the class of all objects of $A$ this gives the following structure.

###### Corollary

There is a model structure on $Ch^{\bullet\geq 0}(A)_{tot}$ whose

###### Example

If $C =$ Vect is a category of vector spaces over some field, we have that every epi/mono splits and that every quasi-isomorphism is a homotopy equivalence. Moreover, in this case every chain complex is quasi-isomorphic to its homology (regarded as a chain complex with zero differentials).

This is the model structure which induces the transferred model structure on dg-algebras over a field.

#### Projective model structure on cochain complexes in non-negative degrees

We discuss a model structure on cochain complexes of abelian groups in which the fibrations are the degreewise epis. This follows an analogous proof in (Jardine).

This projective model structure on cochain complexes in non-negative degree is the one that induces via transfer the corresponding model structure on dg-algebras that plays the central role in the Sullivan model of rational homotopy theory.

###### Theorem

(projective model structure on cochain complexs in non-negative degree)

The category $Ch^{\bullet \geq 0}(Ab)$ of non-negatively graded cochain complexes of abelian groups becomes a model category with

Moreover this is a simplicial model category-structure with respect to the canonical structure of an sSet-enriched category induced from the dual Dold-Kan equivalence $Ch^\bullet_+(Ab) \simeq Ab^\Delta$ by the fact that $Ab^\Delta$ is a category of cosimplicial objects (see there) in a category with all limits and colimits.

###### Proof

We spell out a proof of the model structure below in a sequence of lemmas. The proof that this is a simplicial model category is at model structure on cosimplicial abelian groups.

We record a detailed proof of the model structure on $Ch^{\bullet \geq 0}(Ab)$ with fibrations the degreewise surjections, following the appendix of (Stel).

As usual, for $n \in \mathbb{N}$ write $\mathbb{Z}[n]$ for the complex concentrated on the additive group of integers in degree $n$, and for $n \geq 1$ write $\mathbb{Z}[n-1,n]$ for the cochain complex $(0 \to \cdots 0 \to \mathbb{Z} \stackrel{Id}{\to} \mathbb{Z} \to 0 \cdots)$ with the two copies of $\mathbb{Z}$ in degree $n-1$ and $n$.

For $n = 0$ let $\mathbb{Z}[-1,0] = 0$, for convenience.

###### Lemma

For all $n \in \mathbb{N}$ the canonical maps $0 \to \mathbb{Z}[n]$ and $\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]$ are cofibrations, in that they have the left lifting property against acyclic fibrations.

###### Proof

Let $p : A \stackrel{\simeq}{\to} B$ be degreewise surjective and an isomorphism on cohomology.

First consider $\mathbb{Z}[0]\to \mathbb{Z}[-1,0] = 0$. We need to construct lifts

$\array{ \mathbb{Z}[0] &\stackrel{f}{\to}& A \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{p} \\ 0 &\stackrel{}{\to}& B } \,.$

Since $p(f_0(1)) = 0$ we have by using that $p$ is a quasi-iso that $f_0(1) = 0 \; mod\; im d_A$. But in degree 0 this means that $f_0(1) = 0$. And so the unique possible lift in the above diagram does exist.

Consider now $\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]$ for $n \geq 1$. We need to construct a lift in all diagrams of the form

$\array{ \mathbb{Z}[n] &\stackrel{f}{\to}& A \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{p} \\ \mathbb{Z}[n-1,n] &\stackrel{g}{\to}& B } \,.$

Such a lift is equivalently an element $\sigma \in A_{n-1}$ such that

• $d_A \sigma = f_n(1)$

• $p_{n-1}(\sigma) = g_{n-1}(1)$.

Since $p$ is a quasi-isomorphism, and since it takes the closed element $f_n(1) \in A_n$ to the exact element $p_n(f_n(1)) = d_B g_{n-1}(1)$ it follows that $f_n(1)$ itself must be exact in that there is $z \in A_{n-1}$ with $d_A z = f_n(1)$. Pick such.

So then $d_B ( p(z) - g_{n-1}(1) ) = 0$ and again using that $p$ is a quasi-isomorphism this means that there must be a closed $a \in A_{n-1}$ such that $p(a) = p(z)- g_{n-1}(1) + d_B b$ for some $b \in B_{n-2}$. Choose such $a$ and $b$.

Since $p$ is degreewise onto, there is $a'$ with $p(a') = b$. Choosing this the above becomes $p(a) = p(z) - g_{n-1}(1) + p(d_A a')$.

Set then

$\sigma := z - a + d_A a' \,.$

It follows with the above that this satisfies the two conditions on $\sigma$:

\begin{aligned} d_A \sigma &= d_A z - d_A a + d_A d_A a' \\ & = d_A z \\ & = f_n(1) \end{aligned}
\begin{aligned} p( \sigma ) &= p(z) - p(a) + p(d_A a') \\ & = g_{(n-1)}(1) \end{aligned} \,.

Finally consider $0 \to \mathbb{Z}[n]$ for all $n$. We need to produce lifts in

$\array{ 0 &\stackrel{}{\to}& A \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{p} \\ \mathbb{Z}[n] &\stackrel{g}{\to}& B } \,.$

Such a lift is a choice of element $\sigma \in A_n$ such that

• $d_A \sigma = 0$;

• $p(\sigma) = g_n(1)$;

Since $g_n(1)$ is closed and $p$ a surjective quasi-isomorphism, we may find a closed $a \in A_n$ and an $a' \in A_{n-1}$ such that $p (a) = g_{n}(1) + d_B(p(a'))$. Set then

$\sigma := a - d_A a' \,.$
###### Lemma

For all $n \in \mathbb{N}$, the morphism $0 \to \mathbb{Z}[n-1,n]$ are acyclic cofibrations, in that they have the left lifting property again all degreewise surjections.

###### Proof

For $n = 0$ this is trivial. For $n \geq 1$ a diagram

$\array{ 0 &\to& A \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \mathbb{Z}[n-1,n] &\stackrel{g}{\to}& B }$

is equivalently just any element $g_{n-1}(1) \in B$ and a lift $\sigma$ accordingly just any element $\sigma \in A$ with $p(\sigma) = g_{n-1}(1)$. Such exists because $p$ is degreewise surjctive by assumption.

###### Lemma

A morphism $f : A \to B$ is an acyclic fibration precisely if it has the right lifting property against $0 \to \mathbb{Z}[n]$ and $\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]$ for all $n$.

###### Proof

By the above lemmas, it remains to show only one direction: if $f$ has the RLP, then it is an acyclic fibration.

So assume $f$ has the RLP. Then from the existence of the lifts

$\array{ 0 &\to& A \\ \downarrow && \downarrow \\ \mathbb{Z}[n] &\stackrel{g}{\to}& B }$

one deduces that $f$ is degreewise surjective on closed elements. In particular this means it is surjective in cohomology.

With that, it follows from the existence of all the lifts

$\array{ \mathbb{Z}[n] &\stackrel{f}{\to}& A \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow \\ \mathbb{Z}[n-1,n] &\stackrel{g}{\to}& B }$

for $f$ a lift of the closed element $g_n(1)$ that $f$ is degreewise surjective on all elements.

Moreover, these lifts say that if $f_n(1)$ is any closed element such that under $p$ it becomes exact ($d_B g_{n-1}(1) = p(f_n(1))$), then it must already be exact itself ($d_A \sigma_{n-1}(1) = f_n(1)$). Hence $f$ is also injective on cohomology and hence by the above is an isomorphism on cohomology.

###### Lemma

Every morphism $f : A \to B$ can be factored as a morphism with left lifting property against all fibrations followed by a fibration.

###### Proof

Apply the small object argument-reasoning to the maps in $J = \{0 \to \mathbb{Z}[n-1,n]\}$.

Since for $n \in \mathbb{N}$ a morphism $\mathbb{Z}[n,n+1]\to B$ corresponds to an element $b \in B_n$. From the commuting diagram

$\array{ 0 &\to& A \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \coprod_{{n \in \mathbb{n}} \atop {b \in B_n}} \mathbb{Z}[n,n+1] &\stackrel{}{\to}& B }$

one obtains a factorization through its pushout

$\array{ && A \\ &{}^{\mathllap{j}}\swarrow& \downarrow \\ A \coprod \coprod_{{n \in \mathbb{n}} \atop {b \in B_n}} \mathbb{Z}[n,n+1] && \downarrow^{\mathrlap{f}} \\ &\searrow_{p}& \downarrow \\ && B } \,.$

Since $j$ is the pushout of an acyclic cofibration, it is itself an acyclic cofibration. Moreover, since the cohomology of $\coprod_{{n \in \mathbb{n}} \atop {b \in B_n}} \mathbb{Z}[n,n+1]$ clearly vanishes, it is a quasi-isomorphism.

The map $p$ is manifestly degreewise onto and hence a fibration.

###### Lemma

Every morphism $f : A \to B$ may be factored as a cofibration followed by an acyclic fibration.

###### Proof

By a lemma above acyclic fibrations are precisely the maps with the right lifting property against morphisms in $I = \{0 \to \mathbb{Z}[n], \mathbb{Z}[n]\to \mathbb{Z}[n-1,n]\}$, which by the first lemma above are cofibrations.

The claim then follows again from the small object argument apllied to $I$.

###### Lemma

A morphism $f : A \to B$ that is both a cofibration (:= LLP against acyclic fibrations ) and a weak equivalence has the left lifting property against all fibrations.

###### Proof

By a standard argument, this follows from the factorization lemma proven above, which says that we may find a factorization

$\array{ A &\stackrel{j}{\to}& \hat B \\ & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{p}} \\ && B }$

with $j$ having LLP against all fibrations and being a weak equivalence, and $p$ a fibration. Since $f$ is assumed to be a weak equivalence, it follows that $p$ is an acyclic fibration. By definition of cofibrations as $LLP(Fib \cap W)$ this implies that we have the lift in

$\array{ A &\stackrel{j}{\to}& \hat B \\ {}^{\mathllap{f}}\downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{\mathrlap{p}} \\ B &\stackrel{Id}{\to}& B } \,.$

Equivalently redrawing this as

$\array{ A &\stackrel{Id}{\to}& A &\stackrel{Id}{\to}& A \\ {}^{\mathllap{f}}\downarrow && {}^{\mathllap{p}} \downarrow && {}^{\mathllap{i}}\downarrow \\ B &\stackrel{\sigma}{\to}& \hat B & \stackrel{p}{\to} & B }$

makes manifest that this exhibts $f$ as a retract of $j$ and as such inherits its left lefting properties.

This series of lemmas establishes the claimed model structure on $Ch^\bullet_+(Ab)$.

### In unbounded degree

There are several approaches to defining model structures on the category of unbounded chain complexes $Ch(\mathcal{A})$ -

#### Standard projective model structure on unbounded chain complexes

###### Proposition

For $k$ a commutative ring the category of unbounded chain complexes of $k$-modules $Ch_\bullet(k Mod)$ carries the structure of a

model category with

The cofibrations are all in particular degreewise split injections, but not every degreewise split injection is a cofibration.

###### Proposition

The category of simplicial objects $(Ch_\bullet(k Mod))^{\Delta^{op}}$ in the category of unbounded chain complexes carries the structure of a simplicial model category whose

This is (Rezk-Schwede-Shipley 01, cor 4.6), using the methods discussed at simplicial model category – Simplicial Quillen equivalent models.

Below this model structure is recovered as one example of the Christensen-Hovey projective class construction, as example .

#### Christensen-Hovey model structures using projective classes

Let $\mathcal{A}$ be an abelian category with all limits and colimits.

Christensen-Hovey construct a family of model category structures on $Ch(\mathcal{A})$ parameterized by a choice of projective class . The cofibrations, fibrations and weak equivalences all depend on the projective class.

###### Definition

A projective class on $\mathcal{A}$ is a collection $\mathcal{P} \subset ob \mathcal{A}$ of objects and a collection $\mathcal{E} \subset mor \mathcal{A}$ of morphisms, such that

• $\mathcal{E}$ is precisely the collection of $\mathcal{P}$-epic maps;

• $\mathcal{P}$ is precisely the collection of all objects $P$ such that each map in $\mathcal{E}$ is $P$-epic;

• for each object $X$ in $\mathcal{A}$, there is a morphism $P \to X$ in $\mathcal{E}$ with $P$ in $\mathcal{P}$.

###### Example

Taking $\mathcal{P} := ob \mathcal{A}$ to be the class of all objects yields a projective class – called the trivial projective class . The corresponding morphisms are the class $\mathcal{E}$ of all split epimorphisms in $\mathcal{A}$.

###### Example

Let $R$ be a ring and $\mathcal{A} =$ $R$-Mod be the category of $R$-modules. Choosing $\mathcal{P}$ to be the class of all summands of direct sums of finitely presented modules yields a projective class.

###### Example

Given a pair of adjoint functors

$(F \dashv U) : \mathcal{A} \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} \mathcal{B}$

between abelian categories and given $(\mathcal{P}, \mathcal{E})$ a projective class in $\mathcal{B}$ then its pullback projective class $(U * \mathcal{P}, U^* \mathcal{E})$ along $U$ on $\mathcal{A}$ is defined by

• $U^* \mathcal{P} := \{retracts\;of\; F P | P \in \mathcal{P}\}$
###### Definition/Theorem

Given a projective class $\mathcal{P}$ in $\mathcal{A}$ (def. ), call a morphism $f \in Ch(\mathcal{A})$

• a fibration if $\mathcal{A}(P,f)$ is a surjection in Ab for all $P \in \mathcal{P}$;

• a weak equivalence if $\mathcal{A}(P,f)$ is a quasi-isomorphism in $Ch(Ab)$ for all $P \in \mathcal{P}$.

Then this constitutes a model category structure precisely if cofibrant resolutions exist, which is the case in particular if

1. $\mathcal{P}$ is the pullback projective class (def. ) of a trivial projective class (def. ) along a functor $U$ that preserves countable direct sums;

2. 1

When the structure exists, it is a proper model category.

This is theorem 2.2 in Christensen-Hovey.

We shall write $Ch(\mathcal{A})_{\mathcal{P}}$ for this model category structure.

#### Examples

We list some examples for the model structures on chain complexes in unbounded degree discussed above.

Let $R$ be an associative ring and $\mathcal{A} = R$Mod.

##### Categorical projective class structure
###### Example

The categorical projective class on $\mathcal{A}$ is the projective class (def. ) with $\mathcal{P}$ the class of direct summands of free modules. The $\mathcal{P}$-model structure on $Ch(\mathcal{A})$ has

• as fibrations the degreewise surjections.

So this reproduces the standard projective model structure from prop. .

##### Pure projective class structure
###### Example

The pure projective class on $\mathcal{A}$ has as $\mathcal{P}$ summands of sums of finitely presented modules. Fibrations in the corresponding model structure are the maps that are degreewise those epimorphisms that appear in $\mathcal{P}$-exact sequences.

#### Gillespie’s approach using cotorsion pairs

Hovey has shown that, roughly speaking, model structures on abelian categories correspond to cotorsion pairs. See abelian model structure.

Gillespie shows that if $\mathcal{A}$ is a Grothendieck abelian category, then a cotorsion pair induces an abelian model structure on the category of (unbounded) complexes $Ch(\mathcal{A})$, where the weak equivalences are quasi-isomorphisms.

###### Theorem

Let $\mathcal{A}$ be a Grothendieck abelian category. Suppose $(\mathcal{D}, \mathcal{E})$ is a hereditary cotorsion pair that is cogenerated by a set, such that $\mathcal{D}$ is a Kaplansky class? on $\mathcal{A}$ and $\mathcal{A}$ has enough $\mathcal{D}$-objects.

Then there is an abelian model structure on the category of complexes $Ch(\mathcal{A})$ such that the trivial objects are the acyclic complexes.

Gillespie uses this result to get a monoidal model structure on $Ch(Qcoh(X))$, the category of complexes of quasi-coherent sheaves on a quasi-compact separated scheme $X$. This gives a better understanding of the derived category of quasi-coherent sheaves $D(Qcoh(X))$, and in particular gives immediately the derived functor $\cdot \otimes^{\mathbf{L}} \cdot$ (which is usually a problem due to sheaves not having enough projectives).

#### Cisinski-Deglise approach using descent structures

A third approach is due to Cisinski-Deglise.

Let $\mathcal{A}$ be a Grothendieck abelian category. We will define a notion of descent structures on $\mathcal{A}$.

###### Definition

For each object $E$ of $\mathcal{A}$ and integer $n \in \mathbf{Z}$, we define the complexes $S^n E$ and $D^n E$ as follows: let $(S^n E)^n = E$ in degree $n$ and 0 elsewhere; and let $(D^n E)^n = (D^n E)^{n+1} = E$ and 0 elsewhere. There are canonical morphisms $S^{n+1} E \hookrightarrow D^n E$.

###### Definition

Let $\mathcal{G}$ be an essentially small set of objects of $\mathcal{A}$. A morphism in $Ch(\mathcal{A})$ is called a $\mathcal{G}$-cofibration if it is contained in the smallest class of morphisms in $Ch(\mathcal{A})$ that is closed under pushouts, transfinite compositions and retracts, generated by the inclusions $S^{n+1} E \to D^n E$, for any integer $n$ and any $E \in \mathcal{G}$. A complex $C$ in $Ch(\mathcal{A})$ is called $\mathcal{G}$-cofibrant if the morphism $0 \to C$ is a $\mathcal{G}$-cofibration.

###### Definition

A chain complex $C$ in $Ch(\mathcal{A})$ is called $\mathcal{G}$-local if for all $E \in \mathcal{G}$ and $n \in \mathbf{Z}$, the canonical morphism

$Hom_{\mathbf{K}(\mathcal{A})}(E[n], C) \to Hom_{\mathbf{D}(\mathcal{A})}(E[n], C)$

is an isomorphism. Here $\mathbf{K}(\mathcal{A})$ and $\mathbf{D}(\mathcal{A})$ denote the homotopy category of complexes? and the derived category of $\mathcal{A}$, respectively.

###### Definition

Let $\mathcal{H}$ be a small family of complexes in $Ch(\mathcal{A})$. An complex $C$ in $Ch(\mathcal{A})$ is called $\mathcal{H}$-flasque if for all $n \in \mathbf{Z}$ and $H \in \mathcal{H}$,

$Hom_{\mathbf{K}(\mathcal{A})}(H, C[n]) = 0.$

Finally we define:

###### Definition

A descent structure on $\mathcal{A}$ is a pair $(\mathcal{G},\mathcal{H})$, where $\mathcal{G}$ is an essentially small set of generators of $\mathcal{A}$, and $\mathcal{H}$ is an essentially small set of $\mathcal{G}$-cofibrant acyclic complexes such that any $\mathcal{H}$-flasque complex is $\mathcal{G}$-local.

Now one defines a model structure associated to any such descent structure.

###### Theorem

Let $(\mathcal{G},\mathcal{H})$ be a descent structure on the Grothendieck abelian category $\mathcal{A}$. There is a proper cellular model structure on the category $Ch(\mathcal{A})$, where the weak equivalences are quasi-isomorphisms of complexes, and cofibrations are $\mathcal{G}$-cofibrations.

Also, a complex $C$ in $Ch(\mathcal{A})$ is fibrant if and only if it is $\mathcal{H}$-flasque or equivalently $\mathcal{G}$-local.

We call this the $\mathcal{G}$-model structure on $Ch(\mathcal{A})$. As in Gillespie’s approach we can sometimes get a monoidal model structure. We refer to Cisinski-Deglise for the notion of a weakly flat descent structure.

###### Theorem

Suppose $(\mathcal{G}, \mathcal{H})$ is a weakly flat descent structure on $\mathcal{A}$. Then the $\mathcal{G}$-model structure is further monoidal.

## Properties

### Left/right exact functors and Quillen adjunctions

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let the categories of chain complexes $Ch_\bullet^+(\mathcal{A})$ and $Ch_\bullet^+(\mathcal{B})$ be equipped with the model structure described above where fibrations are the degreewise split monomorphisms with injective kernels.

###### Proposition

If

$(L \dashv R) : \mathcal{A} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} \mathcal{B}$

is a pair of adjoint functors where $L$ preserves monomorphisms, then

$(Ch_\bullet^+(L) \dashv Ch_\bullet(R) : Ch_\bullet^+(\mathcal{A}) \stackrel{\overset{Ch_\bullet^+(L)}{\leftarrow}}{\underset{Ch_\bullet(R)}{\to}} Ch_\bullet^+(\mathcal{B})$

###### Proof

Every functor preserves split epimorphism. Being a right adjoint in particular $R$ is a left exact functor and hence preserves kernels. Using the characterization of injective objects as those $I$ for which $Hom(-,I)$ sends monomorphisms to epimorphisms, we have that $R$ preserves injectives because $L$ preserves monomorphisms, by the adjunction isomorphism.

Hence $L$ preserves all cofibrations and $R$ all fibrations.

### Cofibrant generation

###### Proposition

The injective model structure on $Ch_{\geq 0}(R Mod)$ (from theorem ) is a cofibrantly generated model category.

This appears for instance as Hovey, theorem 2.3.13.

For results on model structures on chain complexes that are provably not cofibrantly generated see section 5.4 of Christensen, Hovey.

### Inclusion into simplicial objects

Let $\mathcal{A} =$ Ab be the category of abelian groups. The Dold-Kan correspondence provides a Quillen equivalence

$(N \dashv \Gamma) : Ch_\bullet^+ \stackrel{\overset{N}{\leftarrow}}{\underset{\Gamma}{\to}} sAb$

between the projective model structure on connective chain complexes and the model structure on simplicial abelian groups. This in turns sits as a transferred model structure along the forgetful functor over the model structure on simplicial sets

$(F \dashv U) : sAb \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet \,.$

$(N F \dashv U \Gamma) : Ch_\bullet \stackrel{\leftarrow}{\to} sSet$

prolongs to a Quillen adjunction on the projective model structure on simplicial presheaves on any site $C$ , which we denote by the same symbols

$(N F \dashv U \Gamma) : [C,Ch_\bullet]_{proj} \stackrel{\leftarrow}{\to} [C, sSet]_{proj} \,.$

With due care this descends to the local model structure on simplicial presheaves which presents the (∞,1)-sheaf (∞,1)-topos on $C$. Then the above Quillen adjunction serves to embed abelian sheaf cohomology on $C$ into the larger context of nonabelian cohomology on $C$. See cohomology for more on this.

### Cofibrations

We discuss cofibrations in the model structures on unbounded complexes.

Let $\mathcal{P}$ be a given projective class on an abelian category $\mathcal{A}$, def. and write $Ch(\mathcal{A})_{\mathcal{P}}$ for the corresponding model structure on unbounded chain complexes, theorem .

###### Proposition

An object $C \in Ch(\mathcal{A})_{\mathcal{P}}$ is cofibrant precisely if

1. in each degree $n \in \mathbb{Z}$ the object $C_n$ is relatively projective in $\mathcal{A}$;

2. every morphism from $C$ into a weakly contractible complex in $Ch(\mathcal{A})_{\mathcal{P}}$ is chain homotopic to the zero morphism.

This appears as (ChristensenHovey, lemma 2.4).

###### Proposition

A morphism $f : A \to B$ in $Ch(\mathcal{A})_{\mathcal{P}}$ is a cofibration precisely if it is degreewise

1. with cofibrant cokernel.

This appears as (ChristensenHovey, prop. 2.5).

### Relation to module spectra

For $R$ any ring, there is the Eilenberg-MacLane spectrum $H R$. This is an algebra spectrum, hence there is a notion of $H R$-module spectra. These are Quillen equivalent to chain complexes of $R$-modules. See module spectrum for details.

## References

### For bounded chain complexes

An original source for the standard model structure on $Ch^{\bullet \geq 0}(A)$ with $A$ having enough injectives is

• Dan Quillen, Homotopical Algebra , Lecture Notes in Mathematics, vol. 43, Springer-Verlag, 1967

Of course the description of model categories of chain complexes as (presentations of) special cases of (stable) $(\infty,1)$-categories is exactly opposite to the historical development of these ideas.

While the homotopical treatment of weak equivalences of chain complexes (quasi-isomorphisms) in homological algebra is at the beginning of all studies of higher categories and a “folk theorem” ever since

• Andre Joyal, Letter to Alexander Grothendieck. April 11, 1984

it seems that the injective model structure on chain complexes has been made fully explicit in print only in proposition 3.13 of

• Tibor Beke, Sheafifiable homotopy model categories (arXiv, pdf)

(at least according to the remark below that).

The projective model structure is discussed after that in

• Mark Hovey, Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353, 6 (pdf)

An explicit proof of the injective model structure with monos in positive degree is spelled out in

• Gregory Dungan, Model categories (pdf)

An explicit proof of the model structure on cochain complexes of abelian group with fibrations the degreewise surjections is recorded in the appendix of

The resolution model structures on cofibrant objects go back to

• William Dwyer, Dan Kan, C. Stover, An $E_2$ model category structure for pointed simplicial spaces, J. Pure and Applied Algebra 90 (1993) 137–152

and are reviewed in

• A. Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories Geometry and Topology, volume 7 (2003)

A general textbook account is in chapter 2 of

### For unbounded chain complexes

Work specifically on model structures on unbounded complexes includes the following.

Spaltenstein wrote a famous paper

• N. Spaltenstein, Resolutions of unbounded complexes, Compositio Mathematica, 65 no. 2 (1988), p. 121-154 (numdam)

on how to do homological algebra with unbounded complexes (in both sides) where he introduced notions like K-projective and K-injective complexes. Later,

• Vladimir Hinich, Homological algebra of homotopy algebras, Comm. Algebra, vol. 25 (1997), no. 10, 3291-3323

shows that there is a model category structure on the category of unbounded chain complexes, reproduces Spaltenstein’s results from that perspective and extends them.

The model structure on unbounded chain complexs with fibrations the degreewise surjections is noted in the remark after theorem 9.3.1 in

and noticed as cofibrantly generated model structure on p. 7 of

That the corresponding category of simplicial objects in unbounded chain complexes is thus a Quillen equivalent simplicial model category is cor. 4.6 in

The article

discusses model structures on unbounded chain complexes with generalized notions of epimorphisms induced from “projective classes”.

Another approach is due to James Gillespie?, using cotorsion pairs. An overview of this work is in

Some generalizations and simplifications of the original approach are discussed in

• James Gillespie?, Kaplansky classes and derived categories, 2007, pdf

Finally a third approach to the unbounded case is discussed in

• Denis-Charles Cisinski, F. Déglise, Local and stable homologial algebra in Grothendieck abelian categories, Homology, Homotopy and Applications, vol. 11 (1) (2009) (url)

A discussion of the homotopy theory of presheaves of unbounded chain complex is in

A model structure on noncommutative dg-algebras whose proof strategy is useful also for cochain complexes is in

1. Some one finish this part.

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