related by the Dold-Kan correspondence
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.
At least if is the category of abelian groups, so that 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 -groupoids given by abelian simplicial groups along the Dold-Kan correspondence to 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.
Let be an abelian category.
Recall that by the dual Dold-Kan correspondence the category of cosimplicial objects in is equivalent to the catagory 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).
We discuss the
weak equivalences are quasi-isomorphisms;
called the projective model structure.
weak equivalences are quasi-isomorphisms;
cofibrations are the morphisms that are monomorphisms in in each positive degree;
called the injective model structure.
This means that a chain complex 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.
Then there is a model category structure on non-negatively graded cochain complexes whose
If we take to be the class of all objects of this gives the following structure.
There is a model structure on whose
If 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).
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)
The category of non-negatively graded cochain complexes of abelian groups becomes a model category with
fibrations the degreewise surjections;
weak equivalences the quasi-isomorphisms.
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 by the fact that is a category of cosimplicial objects (see there) in a category with all limits and colimits.
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 with fibrations the degreewise surjections, following the appendix of (Stel).
As usual, for write for the complex concentrated on the additive group of integers in degree , and for write for the cochain complex with the two copies of in degree and .
For let , for convenience.
For all the canonical maps and are cofibrations, in that they have the left lifting property against acyclic fibrations.
Let be degreewise surjective and an isomorphism on cohomology.
First consider . We need to construct lifts
Since we have by using that is a quasi-iso that . But in degree 0 this means that . And so the unique possible lift in the above diagram does exist.
Consider now for . We need to construct a lift in all diagrams of the form
Such a lift is equivalently an element such that
Since is a quasi-isomorphism, and since it takes the closed element to the exact element it follows that itself must be exact in that there is with . Pick such.
So then and again using that is a quasi-isomorphism this means that there must be a closed such that for some . Choose such and .
Since is degreewise onto, there is with . Choosing this the above becomes .
It follows with the above that this satisfies the two conditions on :
Finally consider for all . We need to produce lifts in
Such a lift is a choice of element such that
Since is closed and a surjective quasi-isomorphism, we may find a closed and an such that . Set then
For all , the morphism are acyclic cofibrations, in that they have the left lifting property again all degreewise surjections.
For this is trivial. For a diagram
is equivalently just any element and a lift accordingly just any element with . Such exists because is degreewise surjctive by assumption.
A morphism is an acyclic fibration precisely if it has the right lifting property against and for all .
By the above lemmas, it remains to show only one direction: if has the RLP, then it is an acyclic fibration.
So assume has the RLP. Then from the existence of the lifts
one deduces that 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
for a lift of the closed element that is degreewise surjective on all elements.
Moreover, these lifts say that if is any closed element such that under it becomes exact (), then it must already be exact itself (). Hence is also injective on cohomology and hence by the above is an isomorphism on cohomology.
Every morphism can be factored as a morphism with left lifting property against all fibrations followed by a fibration.
Apply the small object argument-reasoning to the maps in .
Since for a morphism corresponds to an element . From the commuting diagram
one obtains a factorization through its pushout
Since is the pushout of an acyclic cofibration, it is itself an acyclic cofibration. Moreover, since the cohomology of clearly vanishes, it is a quasi-isomorphism.
The map is manifestly degreewise onto and hence a fibration.
Every morphism may be factored as a cofibration followed by an acyclic fibration.
The claim then follows again from the small object argument apllied to .
A morphism that is both a cofibration (:= LLP against acyclic fibrations ) and a weak equivalence has the left lifting property against all fibrations.
By a standard argument, this follows from the factorization lemma proven above, which says that we may find a factorization
with having LLP against all fibrations and being a weak equivalence, and a fibration. Since is assumed to be a weak equivalence, it follows that is an acyclic fibration. By definition of cofibrations as this implies that we have the lift in
Equivalently redrawing this as
makes manifest that this exhibts as a retract of and as such inherits its left lefting properties.
This series of lemmas establishes the claimed model structure on .
There are several approaches to defining model structures on the category of unbounded chain complexes -
model category with
The cofibrations are all in particular degreewise split injections, but not every degreewise split injection is a cofibration.
This is (Rezk-Schwede-Shipley 01, cor 4.6), using the methods discussed at simplicial model category – Simplicial Quillen equivalent models.
Christensen-Hovey construct a family of model category structures on parameterized by a choice of projective class . The cofibrations, fibrations and weak equivalences all depend on the projective class.
is precisely the collection of -epic maps;
is precisely the collection of all objects such that each map in is -epic;
for each object in , there is a morphism in with in .
Taking to be the class of all objects yields a projective class – called the trivial projective class . The corresponding morphisms are the class of all split epimorphisms in .
Given a pair of adjoint functors
between abelian categories and given a projective class in then its pullback projective class along on is defined by
Given a projective class in , call a morphism
a fibration if is a surjection in Ab for all ;
a weak equivalence if is a quasi-isomorphism in for all .
When the structure exists, it is a proper model category.
This is theorem 2.2 in Christensen-Hovey.
We shall write for this model category structure.
We list some example for the model structure on chain complexes is unbounded degree discussed above.
Let be an associative ring and Mod.
So this reproduces the standard projective model structure from prop. 1.
The pure projective class on has as summands of sums of finitely presented modules. Fibrations in the corresponding model structure are the maps that are degreewise those epimorphisms that appear in -exact sequences.
Gillespie shows that if is a Grothendieck abelian category, then a cotorsion pair induces an abelian model structure on the category of (unbounded) complexes , where the weak equivalences are quasi-isomorphisms.
Then there is an abelian model structure on the category of complexes such that the trivial objects are the acyclic complexes.
Gillespie uses this result to get a monoidal model structure on , the category of complexes of quasi-coherent sheaves on a quasi-compact separated scheme . This gives a better understanding of the derived category of quasi-coherent sheaves , and in particular gives immediately the derived functor (which is usually a problem due to sheaves not having enough projectives).
A third approach is due to Cisinski-Deglise.
Let be a Grothendieck abelian category. We will define a notion of descent structures on .
For each object of and integer , we define the complexes and as follows: let in degree and 0 elsewhere; and let and 0 elsewhere. There are canonical morphisms .
Let be an essentially small set of objects of . A morphism in is called a -cofibration if it is contained in the smallest class of morphisms in that is closed under pushouts, transfinite compositions and retracts, generated by the inclusions , for any integer and any . A complex in is called -cofibrant if the morphism is a -cofibration.
A chain complex in is called -local if for all and , the canonical morphism
Let be a small family of complexes in . An complex in is called -flasque if for all and ,
Finally we define:
A descent structure on is a pair , where is an essentially small set of generators of , and is an essentially small set of -cofibrant acyclic complexes such that any -flasque complex is -local.
Now one defines a model structure associated to any such descent structure.
Let be a descent structure on the Grothendieck abelian category . There is a proper cellular model structure on the category , where the weak equivalences are quasi-isomorphisms of complexes, and cofibrations are -cofibrations.
Also, a complex in is fibrant if and only if it is -flasque or equivalently -local.
Suppose is a weakly flat descent structure on . Then the -model structure is further monoidal.
Let and be abelian categories. Let the categories of chain complexes and be equipped with the model structure described above where fibrations are the degreewise split monomorphisms with injective kernels.
is a Quillen adjunction.
Every functor preserves split epimorphism. Being a right adjoint in particular is a left exact functor and hence preserves kernels. Using the characterization of injective objects as those for which sends monomorphisms to epimorphisms, we have that preserves injectives because preserves monomorphisms, by the adjunction isomorphism.
Hence preserves all cofibrations and all fibrations.
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.
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
The combined Quillen adjunction
With due care this descends to the local model structure on simplicial presheaves which presents the (∞,1)-sheaf (∞,1)-topos on . Then the above Quillen adjunction serves to embed abelian sheaf cohomology on into the larger context of nonabelian cohomology on . See cohomology for more on this.
We discuss cofibrations in the model structures on unbounded complexes.
An object is cofibrant precisely if
This appears as (ChristensenHovey, lemma 2.4).
A morphism in is a cofibration precisely if it is degreewise
This appears as (ChristensenHovey, prop. 2.5).
For any ring, there is the Eilenberg-MacLane spectrum . This is an algebra spectrum, hence there is a notion of -module spectra. These are Quillen equivalent to chain complexes of -modules. See module spectrum for details.
An original source for the standard model structure on with having enough injectives is
Of course the description of model categories of chain complexes as (presentations of) special cases of (stable) -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
it seems that the injective model structure on chain complexes has been made fully explicit in print only in proposition 3.13 of
(at least according to the remark below that).
The projective model structure is discussed after that 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
and are reviewed in
A general textbook account is in chapter 2 of
Work specifically on model structures on unbounded complexes includes the following.
Spaltenstein wrote a famous paper
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
discusses model structures on unbounded chain complexes with generalized notions of epimorphisms induced from “projective classes”.
Some generalizations and simplifications of the original approach are discussed in
Finally a third approach to the unbounded case is discussed 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