nLab model structure on chain complexes

Contents

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

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 AA are special in that they may be identified via the Dold-Kan correspondence as simplicial objects in AA.

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

Similarly, cochain complexes are identified with cosimplicial objects

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

At least if AA is the category of abelian groups, so that A Δ opA^{\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 CC be an abelian category.

Recall that by the dual Dold-Kan correspondence the category C ΔC^\Delta of cosimplicial objects in CC is equivalent to the category Ch + (C)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 RR be a ring and write 𝒜R \mathcal{A} \coloneqq RMod 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 0(𝒜)Ch_{\bullet \geq 0 }(\mathcal{A}) (in non-negative degree) whose

called the projective model structure.

The projective model structure on Ch 0Ch_{\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 0(𝒜)Ch^{\bullet \geq 0 }(\mathcal{A}) whose

called the injective model structure.

(Dungan 10, Theorem 2.4.5)

Remark

This means that a chain complex C Ch (𝒜)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 AA be an abelian category and let 𝒢Obj(A)\mathcal{G} \in Obj(A) be a class of objects, such that AA has enough 𝒢\mathcal{G}-injective objects.

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

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

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

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

See Bousfield2003, section 4.4.

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

Corollary

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

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

Corollary

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

Example

If C=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 0(Ab)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 + (Ab)Ab ΔCh^\bullet_+(Ab) \simeq Ab^\Delta by the fact that Ab ΔAb^\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 0(Ab)Ch^{\bullet \geq 0}(Ab) with fibrations the degreewise surjections, following the appendix of (Stel 10).

As usual, for nn \in \mathbb{N} write [n]\mathbb{Z}[n] for the complex concentrated on the additive group of integers in degree nn, and for n1n \geq 1 write [n1,n]\mathbb{Z}[n-1,n] for the cochain complex (00Id0)(0 \to \cdots 0 \to \mathbb{Z} \stackrel{Id}{\to} \mathbb{Z} \to 0 \cdots) with the two copies of \mathbb{Z} in degree n1n-1 and nn.

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

Lemma

For all nn \in \mathbb{N} the canonical maps 0[n]0 \to \mathbb{Z}[n] and [n][n1,n]\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:ABp : A \stackrel{\simeq}{\to} B be degreewise surjective and an isomorphism on cohomology.

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

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

Since p(f 0(1))=0p(f_0(1)) = 0 we have by using that pp is a quasi-iso that f 0(1)=0modimd Af_0(1) = 0 \; mod\; im d_A. But in degree 0 this means that f 0(1)=0f_0(1) = 0. And so the unique possible lift in the above diagram does exist.

Consider now [n][n1,n]\mathbb{Z}[n] \to \mathbb{Z}[n-1,n] for n1n \geq 1. We need to construct a lift in all diagrams of the form

[n] f A σ p [n1,n] g B. \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 σA n1\sigma \in A_{n-1} such that

  • d Aσ=f n(1)d_A \sigma = f_n(1)

  • p n1(σ)=g n1(1)p_{n-1}(\sigma) = g_{n-1}(1).

Since pp is a quasi-isomorphism, and since it takes the closed element f n(1)A nf_n(1) \in A_n to the exact element p n(f n(1))=d Bg n1(1)p_n(f_n(1)) = d_B g_{n-1}(1) it follows that f n(1)f_n(1) itself must be exact in that there is zA n1z \in A_{n-1} with d Az=f n(1)d_A z = f_n(1). Pick such.

So then d B(p(z)g n1(1))=0d_B ( p(z) - g_{n-1}(1) ) = 0 and again using that pp is a quasi-isomorphism this means that there must be a closed aA n1a \in A_{n-1} such that p(a)=p(z)g n1(1)+d Bbp(a) = p(z)- g_{n-1}(1) + d_B b for some bB n2b \in B_{n-2}. Choose such aa and bb.

Since pp is degreewise onto, there is aa' with p(a)=bp(a') = b. Choosing this the above becomes p(a)=p(z)g n1(1)+p(d Aa)p(a) = p(z) - g_{n-1}(1) + p(d_A a').

Set then

σ:=za+d Aa. \sigma := z - a + d_A a' \,.

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

d Aσ =d Azd Aa+d Ad Aa =d Az =f n(1) \begin{aligned} d_A \sigma &= d_A z - d_A a + d_A d_A a' \\ & = d_A z \\ & = f_n(1) \end{aligned}
p(σ) =p(z)p(a)+p(d Aa) =g (n1)(1). \begin{aligned} p( \sigma ) &= p(z) - p(a) + p(d_A a') \\ & = g_{(n-1)}(1) \end{aligned} \,.

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

0 A σ p [n] g B. \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 σA n\sigma \in A_n such that

  • d Aσ=0d_A \sigma = 0;

  • p(σ)=g n(1)p(\sigma) = g_n(1);

Since g n(1)g_n(1) is closed and pp a surjective quasi-isomorphism, we may find a closed aA na \in A_n and an aA n1a' \in A_{n-1} such that p(a)=g n(1)+d B(p(a))p (a) = g_{n}(1) + d_B(p(a')). Set then

σ:=ad Aa. \sigma := a - d_A a' \,.
Lemma

For all nn \in \mathbb{N}, the morphism 0[n1,n]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=0n = 0 this is trivial. For n1n \geq 1 a diagram

0 A p [n1,n] g B \array{ 0 &\to& A \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \mathbb{Z}[n-1,n] &\stackrel{g}{\to}& B }

is equivalently just any element g n1(1)Bg_{n-1}(1) \in B and a lift σ\sigma accordingly just any element σA\sigma \in A with p(σ)=g n1(1)p(\sigma) = g_{n-1}(1). Such exists because pp is degreewise surjctive by assumption.

Lemma

A morphism f:ABf : A \to B is an acyclic fibration precisely if it has the right lifting property against 0[n]0 \to \mathbb{Z}[n] and [n][n1,n]\mathbb{Z}[n] \to \mathbb{Z}[n-1,n] for all nn.

Proof

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

So assume ff has the RLP. Then from the existence of the lifts

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

one deduces that ff 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

[n] f A σ [n1,n] g B \array{ \mathbb{Z}[n] &\stackrel{f}{\to}& A \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow \\ \mathbb{Z}[n-1,n] &\stackrel{g}{\to}& B }

for ff a lift of the closed element g n(1)g_n(1) that ff is degreewise surjective on all elements.

Moreover, these lifts say that if f n(1)f_n(1) is any closed element such that under pp it becomes exact (d Bg n1(1)=p(f n(1))d_B g_{n-1}(1) = p(f_n(1))), then it must already be exact itself (d Aσ n1(1)=f n(1)d_A \sigma_{n-1}(1) = f_n(1)). Hence ff is also injective on cohomology and hence by the above is an isomorphism on cohomology.

Lemma

Every morphism f:ABf : 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[n1,n]} J = \{0 \to \mathbb{Z}[n-1,n]\}.

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

0 A f n𝕟bB n[n,n+1] B \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

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

The map pp is manifestly degreewise onto and hence a fibration.

Lemma

Every morphism f:ABf : 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[n],[n][n1,n]}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 II.

Lemma

A morphism f:ABf : 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

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

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

A j B^ f σ p B Id B. \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

A Id A Id A f p i B σ B^ p B \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 ff as a retract of jj and as such inherits its left lefting properties.

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

In unbounded degree

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

Standard projective model structure on unbounded chain complexes

Proposition

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

model category with

(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. RRMod is a Grothendieck abelian category (by this example);

  2. when 𝒜\mathcal{A} is a Grothendieck abelian category then so is Ch (𝒜)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 𝕂\mathbb{K} any field, let R𝕂𝕂xR \coloneqq \mathbb{K} \oplus \mathbb{K} \cdot x be its ring of dual numbers, i.e. with x 2=0x^2 = 0.

Denote its augmentation by

ϵ:𝕂𝕂x 𝕂 a+bx a \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 𝕂\mathbb{K} as an RR-module.

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

𝒜(𝕂𝕂xx𝕂𝕂xx𝕂𝕂x). \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

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

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

But consider then the following lifting problem with this morphism

(00R00) ϵ 𝒜 ϵ (00𝕂00),. \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 𝒜\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 RR in degree 0, in order to make (in degree 0), this diagram of RR-modules commute:

R id ϵ R ϵ 𝕂 \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

0 𝕂𝕂x id × id 𝕂𝕂x x 𝕂𝕂x. \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 𝒜\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 kk a field:

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

  2. the cofibrations are exactly the monomorphisms.

Proof

By the assumption that kk is a field, every kk-module (i.e. every kk-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 V_\bullet is the colimit of its stages of lower connective covers:

cn 0V cn 1V cn 2V . cn_0 V_\bullet \hookrightarrow cn_{-1} V_\bullet \hookrightarrow cn_{-2} V_\bullet \hookrightarrow \cdots \,.

By the previous paragraph, cn 0V cn_0 V_\bullet is cofibrant and each morphism in this cotower is a cofibration. Therefore cn 0V V cn_0 V_\bullet \hookrightarrow V_\bullet is a transfinite composition of cofibrations, hence a cofibration, and therefore V 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 (RMod)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=R = \mathbb{Z} the integers, by this Prop., or for R=kR = k a field by this Prop, and in general for RR 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 (which state an even more general result about sheaves of chain complexes).

Proposition

The category of simplicial objects sCh(RMod) 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:Ch (RMod)sCh (RMod):ev 0. 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 kk, every object in the model structure on sCh (kMod)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 sCh (kMod)V_\bullet^\bullet \in sCh_\bullet(k Mod) is Reedy cofibrant, hence (cf. this Remark) that the comparison morphisms from the latching objects L rV rV rL_r V^r_\bullet \to V^r_\bullet are monomorphisms for all rr \in \mathbb{R}. But since Ch (kMod)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 kk, the local model structure on sCh (k)sCh_\bullet(k) from Prop. becomes a monoidal model category via the Δ\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 (k)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 (k)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 “M(A)\mathbf{M}(A)” there does refer to the Reedy model structure on presheaves, Func(A op,M)Func(A^{op}, \mathbf{M}) (cf. p. 265), which means that the condition that A A^{\leftarrow} consists of epimorphisms is satisfied for our case where, under this notational convention, A=ΔA = \Delta).

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

Observing that every object in sCh(k)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)sCh(k) are the homotopically constant simplicial objects, it is sufficient to check — by Barwick (2010), Prop. 4.47 — that for VCh(k)V \,\in\, Ch(k) and 𝒲sCh(k)\mathscr{W} \,\in\, sCh(k) homotopically constant, also their internal hom [const(V),𝒲]sCh(k)[const(V),\,\mathscr{W}] \,\in\, sCh(k) is homotopically constant.

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

[𝒱,𝒲]:[s] [s]Δ[(Δ[s]𝒱) s,𝒲 s], [\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)sCh(k) over sSet.

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

[const(V),𝒲]:[s] s[Δ(s,s)V,𝒲 s] [V, s(𝒲 s) Δ(s,s)] [V,𝒲 s], \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()S\cdot (-) to its right adjoint powering () S(-)^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,𝒲 kCh(k)V,\,\mathscr{W}_{k'} \,\in\,Ch(k) are cofibrant and fibrant (as all objects of Ch(k)Ch(k), by the above discussion), the functor [V,]:Ch(k)Ch(k)[V,-] \,\colon\, Ch(k) \to Ch(k) is a right Quillen functor and as such preserves weak equivalences between the fibrant objects 𝒲 \mathscr{W}_{\bullet} (by Ken Brown’s lemma). This means that if the simplicial chain complex 𝒲 \mathscr{W}_\bullet is homotopically constant then so is the simplicial chain complex [const(V),𝒲]:[s][V,𝒲 s] [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 (k)Ch_\bullet(k) is a monoidal Quillen adjunction, but this is immediate since constconst 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 RR a commutative ring the category of unbounded chain complexes Ch (RMod)Ch_\bullet(R Mod) of RR-modules carries the structure of a

model category with

This is Hovey (1999), Thm. 2.3.13.

Christensen-Hovey model structures using projective classes

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

Christensen-Hovey construct a family of model category structures on Ch(𝒜)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 𝒜\mathcal{A} is a collection 𝒫ob𝒜\mathcal{P} \subset ob \mathcal{A} of objects and a collection mor𝒜\mathcal{E} \subset mor \mathcal{A} of morphisms, such that

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

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

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

Example

Taking 𝒫:=ob𝒜\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 𝒜\mathcal{A}.

Example

Let RR be a ring and 𝒜=\mathcal{A} = RR-Mod be the category of RR-modules. Choosing 𝒫\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

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

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

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

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

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

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

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

  1. 𝒫\mathcal{P} is the pullback projective class (def. ) of a trivial projective class (def. ) along a functor UU 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(𝒜) 𝒫Ch(\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 RR be an associative ring and 𝒜=R\mathcal{A} = RMod.

Categorical projective class structure
Example

The categorical projective class on 𝒜\mathcal{A} is the projective class (def. ) with 𝒫\mathcal{P} the class of direct summands of free modules. The 𝒫\mathcal{P}-model structure on Ch(𝒜)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 𝒜\mathcal{A} has as 𝒫\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 𝒫\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 𝒜\mathcal{A} is a Grothendieck abelian category, then a cotorsion pair induces an abelian model structure on the category of (unbounded) complexes Ch(𝒜)Ch(\mathcal{A}), where the weak equivalences are quasi-isomorphisms.

Theorem

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

Then there is an abelian model structure on the category of complexes Ch(𝒜)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))Ch(Qcoh(X)), the category of complexes of quasi-coherent sheaves on a quasi-compact separated scheme XX. This gives a better understanding of the derived category of quasi-coherent sheaves D(Qcoh(X))D(Qcoh(X)), and in particular gives immediately the derived functor L\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 𝒜\mathcal{A} be a Grothendieck abelian category. We will define a notion of descent structures on 𝒜\mathcal{A}.

Definition

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

Definition

Let 𝒢\mathcal{G} be an essentially small set of objects of 𝒜\mathcal{A}. A morphism in Ch(𝒜)Ch(\mathcal{A}) is called a 𝒢\mathcal{G}-cofibration if it is contained in the smallest class of morphisms in Ch(𝒜)Ch(\mathcal{A}) that is closed under pushouts, transfinite compositions and retracts, generated by the inclusions S n+1ED nES^{n+1} E \to D^n E, for any integer nn and any E𝒢E \in \mathcal{G}. A complex CC in Ch(𝒜)Ch(\mathcal{A}) is called 𝒢\mathcal{G}-cofibrant if the morphism 0C0 \to C is a 𝒢\mathcal{G}-cofibration.

Definition

A chain complex CC in Ch(𝒜)Ch(\mathcal{A}) is called 𝒢\mathcal{G}-local if for all E𝒢E \in \mathcal{G} and nZn \in \mathbf{Z}, the canonical morphism

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

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

Definition

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

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

Finally we define:

Definition

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

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

Theorem

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

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

We call this the 𝒢\mathcal{G}-model structure on Ch(𝒜)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 (𝒢,)(\mathcal{G}, \mathcal{H}) is a weakly flat descent structure on 𝒜\mathcal{A}. Then the 𝒢\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

A i B f (po) g C j D \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

Then also gg 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 nj_n is a monomorphism. Finally, the pasting law implies that the induced morphism of cokernels of ii and jj is an isomorphism. In summary, this means that we have a commuting diagram of chain complexes as follows:

A i B cok(i) f (po) g C j D cok(j). \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:

H n+1(cok(i)) δ H n(A) i * H n(B) H n(cok(i)) δ H n1(A) f * g * f * H n+1(cok(j)) δ H n(C) j * H n(D) H n(cok(j)) δ H n1(C) \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 *g_\ast is an isomorphism on chain homology, hence that gg is a quasi-isomorphism.

Left/right exact functors and Quillen adjunctions

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

Proposition

If

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

is a pair of adjoint functors where LL preserves monomorphisms, then

(Ch +(L)Ch (R):Ch +(𝒜)Ch (R)Ch +(L)Ch +() (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 RR is a left exact functor and hence preserves kernels. Using the characterization of injective objects as those II for which Hom(,I)Hom(-,I) sends monomorphisms to epimorphisms, we have that RR preserves injectives because LL preserves monomorphisms, by the adjunction isomorphism.

Hence LL preserves all cofibrations and RR all fibrations.

Cofibrant generation

Proposition

The injective model structure on Ch 0(RMod)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 𝒜=\mathcal{A} = Ab be the category of abelian groups. The Dold-Kan correspondence provides a Quillen equivalence

(NΓ):Ch +ΓNsAb (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

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

The combined Quillen adjunction

(NFUΓ):Ch 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 CC , which we denote by the same symbols

(NFUΓ):[C,Ch ] proj[C,sSet] proj. (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 CC. Then the above Quillen adjunction serves to embed abelian sheaf cohomology on CC into the larger context of nonabelian cohomology on CC. See cohomology for more on this.

Cofibrations

We discuss cofibrations in the model structures on unbounded complexes.

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

Proposition

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

  1. in each degree nn \in \mathbb{Z} the object C nC_n is relatively projective in 𝒜\mathcal{A};

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

This appears as (ChristensenHovey, lemma 2.4).

Proposition

A morphism f:ABf : A \to B in Ch(𝒜) 𝒫Ch(\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 RR any ring, there is the Eilenberg-MacLane spectrum HRH R. This is an algebra spectrum, hence there is a notion of HRH R-module spectra. These are Quillen equivalent to chain complexes of RR-modules. See module spectrum for details.

References

For bounded chain complexes

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

see also:

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

Of course the description of model categories of chain complexes as (presentations of) special cases of (stable) (,1)(\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 (and shown to be a monoidal model category) in:

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

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 2E_2 model category structure for pointed simplicial spaces, J. Pure and Applied Algebra 90 (1993) 137–152

and are reviewed in

  • Aldrich 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):

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

and maybe independently due to:

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

and shown to be cofibrantly generated in:

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

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

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

Generalization to presheaves of chain complexes:

The article

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

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

Generalization to bicomplexes:

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

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

Last revised on May 9, 2023 at 13:01:05. See the history of this page for a list of all contributions to it.