nLab model structure on chain complexes

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 complexes 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

• In non-negative degree

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

• For unbounded chain complexes

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

• Injective structure on cochain complexes

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

• weak equivalences are the quasi-isomorphisms;

• fibrations are the morphisms that are underlying epimorphisms in $R$Mod in each positive degree;

• cofibrations are degreewise monomorphisms with degreewise projective cokernel;

called the projective model structure.

The projective model structure on $Ch_{\bullet \geq 0}$ is originally due to (Quillen 67, II.4, pages II.4.11, II.4.12). See also (Goerss-Schemmerhorn 06, Theorem 1.5, Dungan 10, 2.4.2, proof in section 2.5).

Proposition

The projective model structure on connective chain complexes is a proper model category.

(e.g. Jardine 03, Prop. 1.5)

Proposition

With respect to the tensor product of chain complexes this is a monoidal model category.

(e.g. Jardine 03, Prop. 1.5, Schwede & Shipley 2003, p. 312 (26 of 48)).

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

• weak equivalences are quasi-isomorphisms;

• cofibrations are the morphisms that are monomorphisms in $R Mod$ in each positive degree;

• fibrations are degreewise epimorphisms with injective kernel,

called the injective model structure.

(Dungan 10, Theorem 2.4.5)

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)

Definition

(…)

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.

See Bousfield2003, section 4.4.

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

• weak equivalences are cochain homotopy equivalences;

• fibrations are the morphisms that are degreewise split epimorphisms and whose kernels are injective objects;

• cofibration are the morphisms that are in positive degree monomorphisms.

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 that is used in rational homotopy theory.

Projective model structure on connective cochain complexes

We discuss a model structure on connective cochain complexes of abelian groups in which the fibrations are the degreewise epis. This follows an analogous proof in (Jardine 97).

Theorem

(projective model structure on connective cochain complexs )

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

• fibrations the degreewise surjections;

• weak equivalences the quasi-isomorphisms.

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.

The first part of this theorem is claimed, without proof, in Castiglioni-Cortinas 03, Def. 4.7.

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 10).

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 $R$ a commutative ring the category of unbounded chain complexes $Ch_\bullet(R Mod)$ of $R$-modules admits the structure of a

• proper

• cofibrantly generated

  with generating (acyclic) cofibrations being, for $n \in \mathbb{Z}$:

• combinatorial

model category with

• weak equivalences the quasi-isomorphisms

• fibrations the (degreewise) epimorphisms.

• cofibrations the degreewise split injections with cofibrant cokernel.

(For partial characterization of the cofibrant objects see further below.)

Proof

Properness and cofibrant generation are discussed in Hovey, Palmieri & Strickland (1997), remark after theorem 9.3.1 and Schwede & Shipley (1998), p. 7, see also Fauk (2006), Thm. 3.2. The characterization of the cofibrations is in Hovey (1999), Lem. 2.3.6 and that of the generating cofibrations are made explicit in Hovey (1999), Def. 2.3.3. Cf. also Muro & Roitzheim (2019), pp. 3.

It remains to see that the underlying category of chain complexes is locally presentable, so that the model structure is combinatorial. (This minor but important point must be clear to the above authors, but seems not to be made explicit in any of the references.) This follows because:

1. $R$Mod is a Grothendieck abelian category (by this example);

2. when $\mathcal{A}$ is a Grothendieck abelian category then so is $Ch_\bullet(\mathcal{A})$ (by this example);

3. all Grothendick abelian categories are locally presentable (by this example).

Remark

It is clear that every chain complex in the model structure of Prop. is fibrant. However, over general rings (not though over fields, see Prop. below) not every chain complex is cofibrant, not even those consisting of projective modules – a counterexample is given in Hovey (1999), Rem. 2.3.7:

For $\mathbb{K}$ any field, let $R \coloneqq \mathbb{K} \oplus \mathbb{K} \cdot x$ be its ring of dual numbers, i.e. with $x^2 = 0$.

Denote its augmentation by

$$\array{ \mathllap{ \epsilon \;\colon\; } \mathbb{K} \oplus \mathbb{K} \cdot x &\longrightarrow& \mathbb{K} \\ a + b x &\mapsto& a }$$

Via this ring homomorphism we regard $\mathbb{K}$ as an $R$-module.

Now in the category $Ch_\bullet(R Mod)$, consider the following unbounded chain complex:

$$\mathcal{A} \;\coloneqq\; \left( \cdots \to \mathbb{K} \oplus \mathbb{K}x \xrightarrow{ \;\; \cdot x \;\; } \mathbb{K} \oplus \mathbb{K}x \xrightarrow{ \;\; \cdot x \;\; } \mathbb{K} \oplus \mathbb{K}x \to \cdots \right) \,.$$

Since its chain homology clearly vanishes in every degree, the morphism it receives out of the zero object is a quasi-isomorphism and hence a weak equivalence

$$0 \underset{\in \mathrm{W}}{\longrightarrow} \mathcal{A}$$

and hence would be an acyclic cofibration if $\mathcal{A}$ were cofibrant.

But consider then the following lifting problem with this morphism

$$\array{ &\longrightarrow& \big( \cdots \to 0 \to 0 \to R \to 0 \to 0 \to \cdots \big) \\ \Bigg\downarrow && \Bigg\downarrow {}^{ \epsilon } \\ \mathcal{A} & \overset{ \;\; \epsilon \;\;\; }{ \longrightarrow } & \big( \cdots \to 0 \to 0 \to \mathbb{K} \to 0 \to 0 \to \cdots \big) \mathrlap{\,,.} }$$

Since the morphism on the right is clearly degreewise surjective and hence a fibration in the model structure, cofibrancy of $\mathcal{A}$ would imply that a lift in this diagram exists. But to be even a lift of the underlying graded modules this lift would have to be the identity morphism on $R$ in degree 0, in order to make (in degree 0), this diagram of $R$-modules commute:

$$\array{ && R \\ & \mathllap{^{id}}\nearrow & \downarrow \mathrlap{^\epsilon} \\ R &\underset{\epsilon}{\longrightarrow}& \mathbb{K} }$$

But that underlying lift fails to be a chain map in degrees (-1,0), where the following diagram does not commute

$$\array{ 0 &\longrightarrow& \mathbb{K} \oplus \mathbb{K}x \\ \mathllap{^{id}} \Big\uparrow &\color{red}\times& \Big\uparrow \mathrlap{^{id}} \\ \mathbb{K} \oplus \mathbb{K}x &\underset{\cdot x}{\longrightarrow}& \mathbb{K} \oplus \mathbb{K}x \mathrlap{\,.} }$$

It follows that the lift does not exist, hence that we have found an object $\mathcal{A}$ in the model structure from Prop. which is not cofibrant.

On the other hand:

Proposition

(bounded-below chain complexes of projective modules are projectively cofibrant)
Every bounded-below chain complex of projective modules is cofibrant in the model structure of Prop. .

(Hovey (1999), Lem. 2.3.6)

This implies:

Proposition

For $k$ a field:

1. every object in the projective model structure $Ch_\bullet(k Mod)$ (Prop. ) is cofibrant.

2. the cofibrations are exactly the monomorphisms.

Proof

By the assumption that $k$ is a field, every $k$-module (i.e. every $k$-vector space) is projective (this Prop.). Therefore Prop. says, in this situation, that every bounded-below chain complex is cofibrant Moreover, since every injection of vector spaces splits (here) the characterization of cofibrations in Prop. says that every injection into a bounded-below chain complex of vector spaces is a cofibration (since its cokernel is clearly itself bounded-below and hence cofibrant by the previous statement) .

Now every chain complex $V_\bullet$ is the colimit of its stages of lower connective covers:

$$cn_0 V_\bullet \hookrightarrow cn_{-1} V_\bullet \hookrightarrow cn_{-2} V_\bullet \hookrightarrow \cdots \,.$$

By the previous paragraph, $cn_0 V_\bullet$ is cofibrant and each morphism in this cotower is a cofibration. Therefore $cn_0 V_\bullet \hookrightarrow V_\bullet$ is a transfinite composition of cofibrations, hence a cofibration, and therefore $V_\bullet$ is cofibrant.

This proves the first statement. From this the second follows by the characterization of the cofibrations in Prop. and using again that all injections here are split.

(Alternatively one may argue via the generating cofibration, cf. MO:a/2457259.)

Proposition

The tensor product of chain complexes makes the projective model structure on unbounded chain complexes $Ch_\bullet(R Mod)$ (Prop. ) a monoidal model category.

For the special case that all submodules of free modules are again free (such as over $R = \mathbb{Z}$ the integers, by this Prop., or for $R = k$ a field by this Prop, and in general for $R$ a principal ideal domain, by this Prop.) a short proof is given in Strickland (2020), Prop. 25. The general statement is also a special case of Hovey (2001), Cor. 3.7 Fausk (2006), Thm. 6.1 (who state an even more general result about sheaves of chain complexes).

Proposition

The category of simplicial objects $sCh(R Mod)_\bullet$ in the projective model structure on unbounded chain complexes (from Prop. ) carries the structure of a combinatorial simplicial model category (obtained as a left Bousfield localization of the Reedy model structure), whose weak equivalences are the maps that are quasi-isomorphisms under the total chain complex functor, and such that the underlying model category is Quillen equivalent via :

(1)$$const \,\colon\, Ch_\bullet(R Mod) \rightleftarrows sCh_\bullet(R Mod) \,\colon\, ev_0 \,.$$

Proof

The existence as a simplicial model category and Quillen equivalence of the underlying categories is due to Rezk, Schwede & Shipley (2001), cor. 4.6, using methods like those discussed at simplicial model category – Simplicial Quillen equivalent models.

Moreover, general facts imply that

1. a Reedy model structure with coefficients in a combinatorial model category is itself combinatorial (see here)

2. the left Bousfield localization of a combinatorial model category is itself combinatorial (see here).

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

Proposition

Over a field $k$, every object in the model structure on $sCh_\bullet(k Mod)$ (from Prop. ) is cofibrant.

Proof

Since left Bousfield localization does not change the class of cofibrations, we need to show that every object $V_\bullet^\bullet \in sCh_\bullet(k Mod)$ is Reedy cofibrant, hence (cf. this Remark) that the comparison morphisms from the latching objects $L_r V^r_\bullet \to V^r_\bullet$ are monomorphisms for all $r \in \mathbb{R}$. But since $Ch_\bullet(k Mod)$ is an abelian category (cf. here), this Prop. at Reedy model structure says that these are monomorphisms and hence the claim follows by Prop. .

Proposition

At least over a field $k$, the local model structure on $sCh_\bullet(k)$ from Prop. becomes a monoidal model category via the $\Delta$-object-wise tensor product of chain complexes, and the Quillen equivalence (1) is a compatibly monoidal Quillen adjunction with respect to the corresponding monoidal model structure on $Ch_\bullet(k)$ from Prop. , Prop. .

This ought to be compatible with the simplicial structure such as to give a simplicial monoidal model category.

Proof

First, the plain Reedy model structure in $sCh_\bullet(k)$ becomes a monoidal model category under the objectwise tensor product of chain complexes, by Barwick (2010), Thm. 3.51 (beware that the notation “$\mathbf{M}(A)$” there does refer to the Reedy model structure on presheaves, $Func(A^{op}, \mathbf{M})$ (cf. p. 265), which means that the condition that $A^{\leftarrow}$ consists of epimorphisms is satisfied for our case where, under this notational convention, $A = \Delta$).

Next, to see that this monoidal model structure passes to the left Bousfield localization of $sCh_\bullet(k)$ at the realization equivalences:

Observing that every object in $sCh(k)$ is a simplicial homotopy colimit of simplicially constant objects (by an argument as in this Prop.) and recalling that the local objects in $sCh(k)$ are the homotopically constant simplicial objects, it is sufficient to check — by Barwick (2010), Prop. 4.47 — that for $V \,\in\, Ch(k)$ and $\mathscr{W} \,\in\, sCh(k)$ homotopically constant, also their internal hom $[const(V),\,\mathscr{W}] \,\in\, sCh(k)$ is homotopically constant.

Now, on general grounds, the internal hom in $sCh(k)$ is given by an end over the internal hom in $Ch(k)$ (to be denoted by the same angular bracket notation), as follows:

$$[\mathscr{V},\,\mathscr{W}] \;\colon\; [s] \;\mapsto\; \int_{[s'] \in \Delta} \big[ (\Delta[s] \cdot \mathscr{V})_{s'} ,\, \mathscr{W}_{s'} \big] \,,$$

where $(-)\cdot(-)$ denotes the canonical tensoring of $sCh(k)$ over sSet.

So in the case at hand, where $\mathscr{V} \,=\, const(V)$, we find this to be:

$$\begin{array}{l} [const(V),\, \mathscr{W}] \,\colon\, [s] \;\mapsto\; \\ \int_{s'} \big[ \Delta(s',s) \cdot V ,\, \mathscr{W}_{s'} \big] \\ \;\simeq\; \Big[ V ,\, \int_{s'} (\mathscr{W}_{s'})^{\Delta(s',s)} \Big] \\ \;\simeq\; \Big[ V ,\, \mathscr{W}_s \Big] \,, \end{array}$$

where we passed from tensoring $S\cdot (-)$ to its right adjoint powering $(-)^S$, used that an internal hom preserves limits and then the enriched Yoneda lemma in its end-form (discussed at co-Yoneda lemma).

But since the objects $V,\,\mathscr{W}_{k'} \,\in\,Ch(k)$ are cofibrant and fibrant (as all objects of $Ch(k)$, by the above discussion), the functor $[V,-] \,\colon\, Ch(k) \to Ch(k)$ is a right Quillen functor and as such preserves weak equivalences between the fibrant objects $\mathscr{W}_{\bullet}$ (by Ken Brown’s lemma). This means that if the simplicial chain complex $\mathscr{W}_\bullet$ is homotopically constant then so is the simplicial chain complex $[const(V),\mathscr{W}] \,\colon\, [s] \,\mapsto\, [V,\, \mathscr{W}_s]$, which was to be shown.

It remains to observe that the Quillen equivalence to $Ch_\bullet(k)$ is a monoidal Quillen adjunction, but this is immediate since $const$ is by aconstruction a strong monoidal functor and the tensor unit is cofibrant (as all objects).

Standard injective model structure on unbounded chain complexes

Proposition

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

• cofibrantly generated

• combinatorial

model category with

• weak equivalences the quasi-isomorphisms

• cofibrations the (degreewise) injections.

This is Hovey (1999), Thm. 2.3.13.

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. (…)

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

• Pure projective class structure

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

Beware that this entry has evolved in a way that deserves re-organization now: What follows are mainly properties of or arguments for the model structures on not-unbounded chain complexes.

Properness

Model categories of chain complexes tend to be proper model categories.

Proposition

(pushout along degreewise injections presrves quasi-isomorphism)
Let

$$\array{ A &\overset{\; i \;}{\longrightarrow}& B \\ {}^{\mathllap{f}} \big\downarrow &{}^{{}_{(po)}}& \big\downarrow {}^{\mathrlap{g}} \\ C &\underset{\; j \;}{\longrightarrow}& D }$$

be a pushout square of chain maps between (unbounded) chain complexes, such that

• $i$ is a degreewise injection;

• $f$ is a quasi-isomorphism.

Then also $g$ is a quasi-isomorphism.

Dually, the pullback of a quasi-isomorphism along a degreewise surjection is again a quasi-isomorphism.

(e.g. Strickland 2020, Prop. 24)

Proof

The pushout of chain complexes is degreewise a pushout in the underlying abelian category. Since pushout in abelian categories preserves monomorphisms (this Prop.) it follows that also $j_n$ is a monomorphism. Finally, the pasting law implies that the induced morphism of cokernels of $i$ and $j$ is an isomorphism. In summary, this means that we have a commuting diagram of chain complexes as follows:

$$\array{ A &\overset{\;\;\; i \;\;\;}{\hookrightarrow}& B &\longrightarrow& cok(i) \\ {}^{\mathllap{f}} \big\downarrow &{}^{{}_{(po)}}& \big\downarrow {}^{\mathrlap{g}} && \big\downarrow {}^{\mathrlap{\simeq}} \\ C &\underset{\;\;\; j \;\;\;}{\hookrightarrow}& D &\longrightarrow& cok(j) \,. }$$

This implies a morphism of the corresponding long exact sequences in chain homology of the form:

$$\array{ \cdots &\to& H_{n+1} \big( cok(i) \big) & \xrightarrow{\; \delta \;} & H_n(A) & \xrightarrow{\; i_\ast\;} & H_n(B) & \xrightarrow{\;\;\;} & H_n\big(cok(i)\big) & \xrightarrow{\;\delta\;} & H_{n-1}(A) & \to & \cdots \\ && {}^{\mathllap{}} \big\downarrow {}^{\mathrlap{\simeq}} && {}^{\mathllap{f_\ast}} \big\downarrow {}^{\mathrlap{\simeq}} && {}^{\mathllap{g_\ast}} \big\downarrow && \big\downarrow {}^{\mathrlap{\simeq}} && {}^{\mathllap{f_\ast}} \big\downarrow {}^{\mathrlap{\simeq}} \\ \cdots &\to& H_{n+1} \big( cok(j) \big) & \xrightarrow{\; \delta \;} & H_n(C) & \xrightarrow{\; j_\ast\;} & H_n(D) & \xrightarrow{\;\;\;} & H_n\big(cok(j)\big) & \xrightarrow{\;\delta\;} & H_{n-1}(C) & \to & \cdots }$$

From this the five lemma implies that $g_\ast$ is an isomorphism on chain homology, hence that $g$ is a quasi-isomorphism.

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

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

is a Quillen adjunction.

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, where it is stated for unbounded (in both directions) chain complexes.

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

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

The combined Quillen adjunction

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

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. a split monomorphism;

2. 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.

Related concepts

• model structure on equivariant chain complexes

• model structure on chain complexes of super vector spaces

• model structure on dg-modules

• model structure on dg-algebras

• stable model category

• (∞,1)-category of chain complexes

References

For bounded chain complexes

The projective model structure on connective chain complexes (Theorem ) is due to

• Daniel Quillen, Section II.4 item 5 in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

see also:

• William Dwyer, Jan Spalinski, Section 7 of: Homotopy theories and model categories (pdf)

  in: I. M. James, Handbook of Algebraic Topology, North Holland 1995 (ISBN:9780080532981, doi:10.1016/B978-0-444-81779-2.X5000-7)

• Paul Goerss, Kirsten Schemmerhorn, Theorem 1.5 in: Model categories and simplicial methods, Notes from lectures given at the University of Chicago, August 2004, in: Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007(arXiv:math.AT/0609537, doi:10.1090/conm/436)

The projective model structure on connective cochain complexes (Theorem ) is claimed, without proof, in:

• Kathryn Hess, p. 6 of Rational homotopy theory: a brief introduction, contribution to Summer School on Interactions between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago (arXiv:math.AT/0604626), chapter in Luchezar Lavramov, Dan Christensen, William Dwyer, Michael Mandell, Brooke Shipley (eds.), Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007 (doi:10.1090/conm/436)

• J. L. Castiglioni, G. Cortiñas, Def. 4.7 of: Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence, J. Pure Appl. Algebra 191 (2004), no. 1-2, 119–142, (arXiv:math.KT/0306289, doi:10.1016/j.jpaa.2003.11.009)

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 (according to remark 3.14 there) of

• Tibor BekeSheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 3 (2000) 447-475 [arXiv:math/0102087, doi:10.1017/S0305004100004722]

The projective model structure is discussed after that (and shown to be a monoidal model category) in:

• Mark Hovey, Model category structures on chain complexes of sheaves (1999) [K-theory:0366, pdf, pdf]

• Mark Hovey, Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353 6 (2001) [ams:S0002-9947-01-02721-0, jstor:221954]

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

• Gregory Dungan, Review of model categories, 2010 (pdf, pdf)

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

• Herman Stel, ∞-Stacks and their Function Algebras – with Applications to ∞-Lie Theory (2010)

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

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

For unbounded chain complexes

Influential precusor discussion of homotopy theory of unbounded chain complexes (introducing notions like K-projective and K-injective complexes):

• Nicolas Spaltenstein, Resolutions of unbounded complexes, Compositio Mathematica 65 2 (1988) 121-154 [numdam:CM_1988__65_2_121_0, eudml:89885]

The observation that from this one obtains a model category structure on unbounded chain complexes is due to:

• Vladimir Hinich, Homological algebra of homotopy algebras, Communications in Algebra 25 10 (1997) 3291-3323 [arXiv:q-alg/9702015, doi:10.1080/00927879708826055, Erratum: arXiv:math/0309453]

and maybe independently due to:

• Mark Hovey, John Palmieri, Neil Strickland, after theorem 9.3.1 in: Axiomatic stable homotopy theory, Memoirs Amer. Math. Soc. 610 (1997) [ISBN:978-1-4704-0195-5, pdf]

(where the relevant insights are credited to Weibel (1994))

and shown to be cofibrantly generated in:

• Stefan Schwede, Brooke Shipley, p. 7 of: Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 2 (2000) 491-511 [arXiv:math/9801082, doi:10.1112/S002461150001220X]

• Mark Hovey, Thm 2.3.1 in: Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books]

and shown to be (cofibrantly generated and in addition) proper and monoidal in:

• Neil Strickland, The model structure for chain complexes [arXiv:2001.08955]

In the context of the Dold-Kan correspondence with the model structure on simplicial abelian groups:

• Stefan Schwede, Brooke Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003) 287-334 [arXiv:math.AT/0209342, euclid:euclid.agt/1513882376]

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

• Charles Rezk, Stefan Schwede, Brooke Shipley, Cor. 4.6 in: Simplicial structures on model categories and functors, American Journal of Mathematics 123 3 (2001) 551-575 [arXiv:0101162, jstor:25099071]

Generalization to presheaves of chain complexes:

• Halvard Fausk, T-model structures on chain complexes of presheaves [arXiv:math/0612414]

The article

• Dan Christensen, Mark Hovey, Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 2 (2002) 261-293 [arXiv:math/0011216, doi:10.1017/S0305004102006126]

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

• Dan Christensen, Derived categories and projective classes , (2005) (hopf archive)

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

• Mark Hovey, Cotorsion pairs and model categories, 2006 (pdf)

Some generalizations and simplifications of the original approach are discussed in

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

Generalization to bicomplexes:

• Fernando Muro, Constanze Roitzheim, Homotopy Theory of Bicomplexes, Journal of Pure and Applied Algebra 223 5 (2019) 1913-1939 [arXiv:1802.07610, doi:10.1016/j.jpaa.2018.08.007]

Finally a third approach to the unbounded case is discussed (in a context of mixed motives) in:

• Denis-Charles Cisinski, Frédéric Déglise, Local and stable homological algebra in Grothendieck abelian categories, Homology, Homotopy and Applications 11 1 (2009) 219–260 [arXiv:0712.3296, hha:1251832567]

• Denis-Charles Cisinski, Frédéric Déglise, Triangulated categories of mixed motives, Springer Monographs in Mathematics, Springer (2019) [arXiv:0912.2110, doi:10.1007/978-3-030-33242-6]

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

• J. F. Jardine, Presheaves of chain complexes, K-theory 30.4 (2003): 365-420 (pdf, pdf)

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

• J. F. Jardine, A Closed Model Structure for Differential Graded Algebras, Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications 17, AMS (1997) 55-58.