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model structure on chain complexes

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

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.

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 complexs 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.

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.

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 catagory 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.

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 }(A) whose

called the injective model structure.

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

Remark

This means that a chain complex C Ch (𝒜)C_\bullet \in Ch_{\bullet}(\mathcal{A}) is a cofibrant object in the projective model structure, theorem 1, 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 2, 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 𝒢\mathcal{G}-injective kernel.

See Bousfield2003, section 4.4.

If AA has enough injective objects and 𝒢\mathcal{G} is the clss 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.

With fibrations being surjections in all degrees

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

Theorem

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

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.

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).

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

model category with

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

See (Hovey-Palmieri-Strickland 97, remark after theorem 9.3.1, Schwede-Shipley 98, p. 7).

Proposition

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

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

Below this model structure is recovered as one example of the Christensen-Hovey projective class construction, as example 5.

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

Given a projective class 𝒫\mathcal{P} in 𝒜\mathcal{A}, 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 of a trivial projective class along a functor UU that preserves countable direct sums;

  2. blah-blah

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 example for the model structure on chain complexes is 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 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. 1.

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

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) is a cofibrantly generated model category.

This appears for instance as Hovey, theorem 2.3.13.

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

Inclusion into simplicial objects

Let 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. 2 and write Ch(𝒜) 𝒫Ch(\mathcal{A})_{\mathcal{P}} for the corresponding model structure on unbounded chain complexes, theorem 5.

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.

Examples

On dg-modules

References

For bounded chain complexes

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

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

Of course the description of model categories of chain complexes as (presentations of) special cases of (stable) (,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 in

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

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

  • Gregory Dungan, Model categories (pdf)

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

The resolution model structures on cofibrant objects go back to

  • William Dwyer, Dan Kan, C. Stover, An E 2E_2 model category structure for pointed simplicial spaces, J. Pure and Applied Algebra 90 (1993) 137–152

and are reviewed in

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

A general textbook account is in chapter 2 of

For unbounded chain complexes

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

Spaltenstein wrote a famous paper

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

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

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

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

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

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

The article

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

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

Finally a third approach to the unbounded case is discussed in

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

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

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

Revised on September 11, 2013 13:02:52 by Urs Schreiber (77.251.114.72)