model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
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.
Chain complexes in non-negative degree in an abelian category $A$ are special in that they may be identified via the Dold-Kan correspondence as simplicial objects in $A$.
Similarly, cochain complexes are identified with cosimplicial objects
At least if $A$ is the category of abelian groups, so that $A^{\Delta^{op}}$ is the category of abelian simplicial groups it inherits naturally a model category structure from the model structure on simplicial sets, which presents the (∞,1)-category of ∞-groupoids.
The model structure on chain complexes transports this presentation of the special $\infty$-groupoids given by abelian simplicial groups along the Dold-Kan correspondence to chain complexes.
Analogous statements apply to the category of unbounded chain complexes and the canonical stable (infinity,1)-category Spec of spectra.
This we discuss below in
Model structures on unbounded (co)chain complexes can be understood as presentations of spectrum objects in model structures of bounded (co)chain complexes.
See
Let $C$ be an abelian category.
Recall that by the dual Dold-Kan correspondence the category $C^\Delta$ of cosimplicial objects in $C$ is equivalent to the category $Ch^\bullet_+(C)$ of cochain complexes in non-negative degree. This means that we can transfer results discussed at model structure on cosimplicial objects to cochain complexes (see Bousfield2003, section 4.4 for more).
Let $R$ be a ring and write $\mathcal{A} \coloneqq R$Mod for its category of modules.
We discuss the
There is a model category structure on the category of chain complexes $Ch_{\bullet \geq 0 }(\mathcal{A})$ (in non-negative degree) whose
weak equivalences are the quasi-isomorphisms;
fibrations are the morphisms that are underlying epimorphisms in $R$Mod in each positive degree;
cofibrations are degreewise monomorphisms with degreewise projective cokernel;
called the projective model structure.
The projective model structure on $Ch_{\bullet \geq 0}$ is originally due to (Quillen 67, II.4, pages II.4.11, II.4.12). See also (Goerss-Schemmerhorn 06, Theorem 1.5, Dungan 10, 2.4.2, proof in section 2.5).
The projective model structure on connective chain complexes is a proper model category.
With respect to the tensor product of chain complexes this is a monoidal model category.
Dually
There is a model category structure on non-negatively graded cochain complexes $Ch^{\bullet \geq 0 }(\mathcal{A})$ whose
weak equivalences are quasi-isomorphisms;
cofibrations are the morphisms that are monomorphisms in $R Mod$ in each positive degree;
fibrations are degreewise epimorphisms with injective kernel,
called the injective model structure.
This means that a chain complex $C_\bullet \in Ch_{\bullet}(\mathcal{A})$ is a cofibrant object in the projective model structure, theorem , precisely if it consists of projective modules. Accordingly, a cofibrant resolution in the projective model structure is precisely what in homological algebra is called a projective resolution. Dually for fibrant resolutions in the injective model structure, theorem , and injective resolutions in homological algebra.
This way the traditional definition of derived functor in homological algebra relates to the general construction of derived functors in model category theory. See there for more details. Similar comments apply to the various other model structures below.
There are resolution model structures on cosimplicial objects in a model category, due to (DwyerKanStover), reviewed in (Bousfield)
(…)
Let $A$ be an abelian category and let $\mathcal{G} \in Obj(A)$ be a class of objects, such that $A$ has enough $\mathcal{G}$-injective objects.
Then there is a model category structure on non-negatively graded cochain complexes $Ch^{\bullet \geq 0}(A)_{\mathcal{G}}$ whose
weak equivalences are maps $f : X \to Y$ such that for each $K \in A$ the induced map $A(Y,K) \to A(X,K)$ is a quasi-isomorphism of chain complexes of abelian groups;
$f$ is a cofibration if it is $\mathcal{G}$-monic in positive degree;
$f$ is a fibration if it is degreewise a split epimorphism with degreewise $\mathcal{G}$-injective kernel.
See Bousfield2003, section 4.4.
If $A$ has enough injective objects and $\mathcal{G}$ is the class of all of them, this reproduces the standard Quillen model structure discussed above:
Let $A$ be an abelian category with enough injective objects. Then there is a model category structure on non-negatively graded cochain complexes $Ch^{\geq 0}(A)$ whose
weak equivalences are the quasi-isomorphisms;
fibrations are the morphisms that are epimorphisms with injective kernel in each degree;
cofibrations are the morphisms which are monomorphisms in $A$ in each positive degree.
If we take $\mathcal{G}$ to be the class of all objects of $A$ this gives the following structure.
There is a model structure on $Ch^{\bullet\geq 0}(A)_{tot}$ whose
weak equivalences are cochain homotopy equivalences;
fibrations are the morphisms that are degreewise split epimorphisms and whose kernels are injective objects;
cofibration are the morphisms that are in positive degree monomorphisms.
If $C =$ Vect is a category of vector spaces over some field, we have that every epi/mono splits and that every quasi-isomorphism is a homotopy equivalence. Moreover, in this case every chain complex is quasi-isomorphic to its homology (regarded as a chain complex with zero differentials).
This is the model structure which induces the transferred model structure on dg-algebras over a field that is used in rational homotopy theory.
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).
(projective model structure on connective cochain complexs )
The category $Ch^{\bullet \geq 0}(Ab)$ of non-negatively graded cochain complexes of abelian groups becomes a model category with
fibrations the degreewise surjections;
weak equivalences the quasi-isomorphisms.
Moreover this is a simplicial model category-structure with respect to the canonical structure of an sSet-enriched category induced from the dual Dold-Kan equivalence $Ch^\bullet_+(Ab) \simeq Ab^\Delta$ by the fact that $Ab^\Delta$ is a category of cosimplicial objects (see there) in a category with all limits and colimits.
The first part of this theorem is claimed, without proof, in Castiglioni-Cortinas 03, Def. 4.7.
We spell out a proof of the model structure below in a sequence of lemmas. The proof that this is a simplicial model category is at model structure on cosimplicial abelian groups.
We record a detailed proof of the model structure on $Ch^{\bullet \geq 0}(Ab)$ with fibrations the degreewise surjections, following the appendix of (Stel 10).
As usual, for $n \in \mathbb{N}$ write $\mathbb{Z}[n]$ for the complex concentrated on the additive group of integers in degree $n$, and for $n \geq 1$ write $\mathbb{Z}[n-1,n]$ for the cochain complex $(0 \to \cdots 0 \to \mathbb{Z} \stackrel{Id}{\to} \mathbb{Z} \to 0 \cdots)$ with the two copies of $\mathbb{Z}$ in degree $n-1$ and $n$.
For $n = 0$ let $\mathbb{Z}[-1,0] = 0$, for convenience.
For all $n \in \mathbb{N}$ the canonical maps $0 \to \mathbb{Z}[n]$ and $\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]$ are cofibrations, in that they have the left lifting property against acyclic fibrations.
Let $p : A \stackrel{\simeq}{\to} B$ be degreewise surjective and an isomorphism on cohomology.
First consider $\mathbb{Z}[0]\to \mathbb{Z}[-1,0] = 0$. We need to construct lifts
Since $p(f_0(1)) = 0$ we have by using that $p$ is a quasi-iso that $f_0(1) = 0 \; mod\; im d_A$. But in degree 0 this means that $f_0(1) = 0$. And so the unique possible lift in the above diagram does exist.
Consider now $\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]$ for $n \geq 1$. We need to construct a lift in all diagrams of the form
Such a lift is equivalently an element $\sigma \in A_{n-1}$ such that
$d_A \sigma = f_n(1)$
$p_{n-1}(\sigma) = g_{n-1}(1)$.
Since $p$ is a quasi-isomorphism, and since it takes the closed element $f_n(1) \in A_n$ to the exact element $p_n(f_n(1)) = d_B g_{n-1}(1)$ it follows that $f_n(1)$ itself must be exact in that there is $z \in A_{n-1}$ with $d_A z = f_n(1)$. Pick such.
So then $d_B ( p(z) - g_{n-1}(1) ) = 0$ and again using that $p$ is a quasi-isomorphism this means that there must be a closed $a \in A_{n-1}$ such that $p(a) = p(z)- g_{n-1}(1) + d_B b$ for some $b \in B_{n-2}$. Choose such $a$ and $b$.
Since $p$ is degreewise onto, there is $a'$ with $p(a') = b$. Choosing this the above becomes $p(a) = p(z) - g_{n-1}(1) + p(d_A a')$.
Set then
It follows with the above that this satisfies the two conditions on $\sigma$:
Finally consider $0 \to \mathbb{Z}[n]$ for all $n$. We need to produce lifts in
Such a lift is a choice of element $\sigma \in A_n$ such that
$d_A \sigma = 0$;
$p(\sigma) = g_n(1)$;
Since $g_n(1)$ is closed and $p$ a surjective quasi-isomorphism, we may find a closed $a \in A_n$ and an $a' \in A_{n-1}$ such that $p (a) = g_{n}(1) + d_B(p(a'))$. Set then
For all $n \in \mathbb{N}$, the morphism $0 \to \mathbb{Z}[n-1,n]$ are acyclic cofibrations, in that they have the left lifting property again all degreewise surjections.
For $n = 0$ this is trivial. For $n \geq 1$ a diagram
is equivalently just any element $g_{n-1}(1) \in B$ and a lift $\sigma$ accordingly just any element $\sigma \in A$ with $p(\sigma) = g_{n-1}(1)$. Such exists because $p$ is degreewise surjctive by assumption.
A morphism $f : A \to B$ is an acyclic fibration precisely if it has the right lifting property against $0 \to \mathbb{Z}[n]$ and $\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]$ for all $n$.
By the above lemmas, it remains to show only one direction: if $f$ has the RLP, then it is an acyclic fibration.
So assume $f$ has the RLP. Then from the existence of the lifts
one deduces that $f$ is degreewise surjective on closed elements. In particular this means it is surjective in cohomology.
With that, it follows from the existence of all the lifts
for $f$ a lift of the closed element $g_n(1)$ that $f$ is degreewise surjective on all elements.
Moreover, these lifts say that if $f_n(1)$ is any closed element such that under $p$ it becomes exact ($d_B g_{n-1}(1) = p(f_n(1))$), then it must already be exact itself ($d_A \sigma_{n-1}(1) = f_n(1)$). Hence $f$ is also injective on cohomology and hence by the above is an isomorphism on cohomology.
Every morphism $f : A \to B$ 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 $J = \{0 \to \mathbb{Z}[n-1,n]\}$.
Since for $n \in \mathbb{N}$ a morphism $\mathbb{Z}[n,n+1]\to B$ corresponds to an element $b \in B_n$. From the commuting diagram
one obtains a factorization through its pushout
Since $j$ is the pushout of an acyclic cofibration, it is itself an acyclic cofibration. Moreover, since the cohomology of $\coprod_{{n \in \mathbb{n}} \atop {b \in B_n}} \mathbb{Z}[n,n+1]$ clearly vanishes, it is a quasi-isomorphism.
The map $p$ is manifestly degreewise onto and hence a fibration.
Every morphism $f : A \to B$ may be factored as a cofibration followed by an acyclic fibration.
By a lemma above acyclic fibrations are precisely the maps with the right lifting property against morphisms in $I = \{0 \to \mathbb{Z}[n], \mathbb{Z}[n]\to \mathbb{Z}[n-1,n]\}$, which by the first lemma above are cofibrations.
The claim then follows again from the small object argument apllied to $I$.
A morphism $f : A \to B$ that is both a cofibration (:= LLP against acyclic fibrations ) and a weak equivalence has the left lifting property against all fibrations.
By a standard argument, this follows from the factorization lemma proven above, which says that we may find a factorization
with $j$ having LLP against all fibrations and being a weak equivalence, and $p$ a fibration. Since $f$ is assumed to be a weak equivalence, it follows that $p$ is an acyclic fibration. By definition of cofibrations as $LLP(Fib \cap W)$ this implies that we have the lift in
Equivalently redrawing this as
makes manifest that this exhibts $f$ as a retract of $j$ and as such inherits its left lefting properties.
This series of lemmas establishes the claimed model structure on $Ch^\bullet_+(Ab)$.
There are several approaches to defining model structures on the category of unbounded chain complexes $Ch(\mathcal{A})$ -
For $R$ a commutative ring the category of unbounded chain complexes $Ch_\bullet(R Mod)$ of $R$-modules admits the structure of a
with generating (acyclic) cofibrations being, for $n \in \mathbb{Z}$:
model category with
fibrations the (degreewise) epimorphisms.
cofibrations the degreewise split injections with cofibrant cokernel.
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:
$R$Mod is a Grothendieck abelian category (by this example);
when $\mathcal{A}$ is a Grothendieck abelian category then so is $Ch_\bullet(\mathcal{A})$ (by this example);
all Grothendick abelian categories are locally presentable (by this example).
It is clear that every chain complex in the model structure of Prop. is fibrant. However, over general rings (not though over fields, see Prop. below) not every chain complex is cofibrant, not even those consisting of projective modules – a counterexample is given in Hovey (1999), Rem. 2.3.7:
For $\mathbb{K}$ any field, let $R \coloneqq \mathbb{K} \oplus \mathbb{K} \cdot x$ be its ring of dual numbers, i.e. with $x^2 = 0$.
Denote its augmentation by
Via this ring homomorphism we regard $\mathbb{K}$ as an $R$-module.
Now in the category $Ch_\bullet(R Mod)$, consider the following unbounded chain complex:
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
and hence would be an acyclic cofibration if $\mathcal{A}$ were cofibrant.
But consider then the following lifting problem with this morphism
Since the morphism on the right is clearly degreewise surjective and hence a fibration in the model structure, cofibrancy of $\mathcal{A}$ would imply that a lift in this diagram exists. But to be even a lift of the underlying graded modules this lift would have to be the identity morphism on $R$ in degree 0, in order to make (in degree 0), this diagram of $R$-modules commute:
But that underlying lift fails to be a chain map in degrees (-1,0), where the following diagram does not commute
It follows that the lift does not exist, hence that we have found an object $\mathcal{A}$ in the model structure from Prop. which is not cofibrant.
On the other hand:
(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. .
This implies:
For $k$ a field:
every object in the projective model structure $Ch_\bullet(k Mod)$ (Prop. ) is cofibrant.
the cofibrations are exactly the monomorphisms.
By the assumption that $k$ is a field, every $k$-module (i.e. every $k$-vector space) is projective (this Prop.). Therefore Prop. says, in this situation, that every bounded-below chain complex is cofibrant Moreover, since every injection of vector spaces splits (here) the characterization of cofibrations in Prop. says that every injection into a bounded-below chain complex of vector spaces is a cofibration (since its cokernel is clearly itself bounded-below and hence cofibrant by the previous statement) .
Now every chain complex $V_\bullet$ is the colimit of its stages of lower connective covers:
By the previous paragraph, $cn_0 V_\bullet$ is cofibrant and each morphism in this cotower is a cofibration. Therefore $cn_0 V_\bullet \hookrightarrow V_\bullet$ is a transfinite composition of cofibrations, hence a cofibration, and therefore $V_\bullet$ is cofibrant.
This proves the first statement. From this the second follows by the characterization of the cofibrations in Prop. and using again that all injections here are split.
The tensor product of chain complexes makes the projective model structure on unbounded chain complexes $Ch_\bullet(R Mod)$ (Prop. ) a monoidal model category.
The category of simplicial objects $sCh(R Mod)_\bullet$ in the projective model structure on unbounded chain complexes (from Prop. ) carries the structure of a combinatorial simplicial model category (obtained as a left Bousfield localization of the Reedy model structure), whose weak equivalences are the maps that are quasi-isomorphisms under the total chain complex functor, and such that the underlying model category is Quillen equivalent via :
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
a Reedy model structure with coefficients in a combinatorial model category is itself combinatorial (see here)
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.
Over a field $k$, every object in the model structure on $sCh_\bullet(k Mod)$ (from Prop. ) is cofibrant.
Since left Bousfield localization does not change the class of cofibrations, we need to show that every object $V_\bullet^\bullet \in sCh_\bullet(k Mod)$ is Reedy cofibrant, hence (cf. this Remark) that the comparison morphisms from the latching objects $L_r V^r_\bullet \to V^r_\bullet$ are monomorphisms for all $r \in \mathbb{R}$. But since $Ch_\bullet(k Mod)$ is an abelian category (cf. here), this Prop. at Reedy model structure says that these are monomorphisms and hence the claim follows by Prop. .
At least over a field $k$, the local model structure on $sCh_\bullet(k)$ from Prop. becomes a monoidal model category via the $\Delta$-object-wise tensor product of chain complexes, and the Quillen equivalence (1) is a compatibly monoidal Quillen adjunction with respect to the corresponding monoidal model structure on $Ch_\bullet(k)$ from Prop. , Prop. .
First, the plain Reedy model structure in $sCh_\bullet(k)$ becomes a monoidal model category under the objectwise tensor product of chain complexes, by Barwick (2010), Thm. 3.51 (beware that the notation “$\mathbf{M}(A)$” there does refer to the Reedy model structure on presheaves, $Func(A^{op}, \mathbf{M})$ (cf. p. 265), which means that the condition that $A^{\leftarrow}$ consists of epimorphisms is satisfied for our case where, under this notational convention, $A = \Delta$).
Next, to see that this monoidal model structure passes to the left Bousfield localization of $sCh_\bullet(k)$ at the realization equivalences:
Observing that every object in $sCh(k)$ is a simplicial homotopy colimit of simplicially constant objects (by an argument as in this Prop.) and recalling that the local objects in $sCh(k)$ are the homotopically constant simplicial objects, it is sufficient to check — by Barwick (2010), Prop. 4.47 — that for $V \,\in\, Ch(k)$ and $\mathscr{W} \,\in\, sCh(k)$ homotopically constant, also their internal hom $[const(V),\,\mathscr{W}] \,\in\, sCh(k)$ is homotopically constant.
Now, on general grounds, the internal hom in $sCh(k)$ is given by an end over the internal hom in $Ch(k)$ (to be denoted by the same angular bracket notation), as follows:
where $(-)\cdot(-)$ denotes the canonical tensoring of $sCh(k)$ over sSet.
So in the case at hand, where $\mathscr{V} \,=\, const(V)$, we find this to be:
where we passed from tensoring $S\cdot (-)$ to its right adjoint powering $(-)^S$, used that an internal hom preserves limits and then the enriched Yoneda lemma in its end-form (discussed at co-Yoneda lemma).
But since the objects $V,\,\mathscr{W}_{k'} \,\in\,Ch(k)$ are cofibrant and fibrant (as all objects of $Ch(k)$, by the above discussion), the functor $[V,-] \,\colon\, Ch(k) \to Ch(k)$ is a right Quillen functor and as such preserves weak equivalences between the fibrant objects $\mathscr{W}_{\bullet}$ (by Ken Brown’s lemma). This means that if the simplicial chain complex $\mathscr{W}_\bullet$ is homotopically constant then so is the simplicial chain complex $[const(V),\mathscr{W}] \,\colon\, [s] \,\mapsto\, [V,\, \mathscr{W}_s]$, which was to be shown.
It remains to observe that the Quillen equivalence to $Ch_\bullet(k)$ is a monoidal Quillen adjunction, but this is immediate since $const$ is by aconstruction a strong monoidal functor and the tensor unit is cofibrant (as all objects).
For $R$ a commutative ring the category of unbounded chain complexes $Ch_\bullet(R Mod)$ of $R$-modules carries the structure of a
model category with
cofibrations the (degreewise) injections.
Let $\mathcal{A}$ be an abelian category with all limits and colimits.
Christensen-Hovey construct a family of model category structures on $Ch(\mathcal{A})$ parameterized by a choice of projective class . The cofibrations, fibrations and weak equivalences all depend on the projective class.
A projective class on $\mathcal{A}$ is a collection $\mathcal{P} \subset ob \mathcal{A}$ of objects and a collection $\mathcal{E} \subset mor \mathcal{A}$ of morphisms, such that
$\mathcal{E}$ is precisely the collection of $\mathcal{P}$-epic maps;
$\mathcal{P}$ is precisely the collection of all objects $P$ such that each map in $\mathcal{E}$ is $P$-epic;
for each object $X$ in $\mathcal{A}$, there is a morphism $P \to X$ in $\mathcal{E}$ with $P$ in $\mathcal{P}$.
Taking $\mathcal{P} := ob \mathcal{A}$ to be the class of all objects yields a projective class – called the trivial projective class . The corresponding morphisms are the class $\mathcal{E}$ of all split epimorphisms in $\mathcal{A}$.
Let $R$ be a ring and $\mathcal{A} =$ $R$-Mod be the category of $R$-modules. Choosing $\mathcal{P}$ to be the class of all summands of direct sums of finitely presented modules yields a projective class.
Given a pair of adjoint functors
between abelian categories and given $(\mathcal{P}, \mathcal{E})$ a projective class in $\mathcal{B}$ then its pullback projective class $(U * \mathcal{P}, U^* \mathcal{E})$ along $U$ on $\mathcal{A}$ is defined by
Given a projective class $\mathcal{P}$ in $\mathcal{A}$ (def. ), call a morphism $f \in Ch(\mathcal{A})$
a fibration if $\mathcal{A}(P,f)$ is a surjection in Ab for all $P \in \mathcal{P}$;
a weak equivalence if $\mathcal{A}(P,f)$ is a quasi-isomorphism in $Ch(Ab)$ for all $P \in \mathcal{P}$.
Then this constitutes a model category structure precisely if cofibrant resolutions exist, which is the case in particular if
$\mathcal{P}$ is the pullback projective class (def. ) of a trivial projective class (def. ) along a functor $U$ that preserves countable direct sums;
(…)
When the structure exists, it is a proper model category.
This is theorem 2.2 in Christensen-Hovey.
We shall write $Ch(\mathcal{A})_{\mathcal{P}}$ for this model category structure.
We list some examples for the model structures on chain complexes in unbounded degree discussed above.
Let $R$ be an associative ring and $\mathcal{A} = R$Mod.
The categorical projective class on $\mathcal{A}$ is the projective class (def. ) with $\mathcal{P}$ the class of direct summands of free modules. The $\mathcal{P}$-model structure on $Ch(\mathcal{A})$ has
So this reproduces the standard projective model structure from prop. .
The pure projective class on $\mathcal{A}$ has as $\mathcal{P}$ summands of sums of finitely presented modules. Fibrations in the corresponding model structure are the maps that are degreewise those epimorphisms that appear in $\mathcal{P}$-exact sequences.
Hovey has shown that, roughly speaking, model structures on abelian categories correspond to cotorsion pairs. See abelian model structure.
Gillespie shows that if $\mathcal{A}$ is a Grothendieck abelian category, then a cotorsion pair induces an abelian model structure on the category of (unbounded) complexes $Ch(\mathcal{A})$, where the weak equivalences are quasi-isomorphisms.
Let $\mathcal{A}$ be a Grothendieck abelian category. Suppose $(\mathcal{D}, \mathcal{E})$ is a hereditary cotorsion pair that is cogenerated by a set, such that $\mathcal{D}$ is a Kaplansky class? on $\mathcal{A}$ and $\mathcal{A}$ has enough $\mathcal{D}$-objects.
Then there is an abelian model structure on the category of complexes $Ch(\mathcal{A})$ such that the trivial objects are the acyclic complexes.
Gillespie uses this result to get a monoidal model structure on $Ch(Qcoh(X))$, the category of complexes of quasi-coherent sheaves on a quasi-compact separated scheme $X$. This gives a better understanding of the derived category of quasi-coherent sheaves $D(Qcoh(X))$, and in particular gives immediately the derived functor $\cdot \otimes^{\mathbf{L}} \cdot$ (which is usually a problem due to sheaves not having enough projectives).
A third approach is due to Cisinski-Deglise.
Let $\mathcal{A}$ be a Grothendieck abelian category. We will define a notion of descent structures on $\mathcal{A}$.
For each object $E$ of $\mathcal{A}$ and integer $n \in \mathbf{Z}$, we define the complexes $S^n E$ and $D^n E$ as follows: let $(S^n E)^n = E$ in degree $n$ and 0 elsewhere; and let $(D^n E)^n = (D^n E)^{n+1} = E$ and 0 elsewhere. There are canonical morphisms $S^{n+1} E \hookrightarrow D^n E$.
Let $\mathcal{G}$ be an essentially small set of objects of $\mathcal{A}$. A morphism in $Ch(\mathcal{A})$ is called a $\mathcal{G}$-cofibration if it is contained in the smallest class of morphisms in $Ch(\mathcal{A})$ that is closed under pushouts, transfinite compositions and retracts, generated by the inclusions $S^{n+1} E \to D^n E$, for any integer $n$ and any $E \in \mathcal{G}$. A complex $C$ in $Ch(\mathcal{A})$ is called $\mathcal{G}$-cofibrant if the morphism $0 \to C$ is a $\mathcal{G}$-cofibration.
A chain complex $C$ in $Ch(\mathcal{A})$ is called $\mathcal{G}$-local if for all $E \in \mathcal{G}$ and $n \in \mathbf{Z}$, the canonical morphism
is an isomorphism. Here $\mathbf{K}(\mathcal{A})$ and $\mathbf{D}(\mathcal{A})$ denote the homotopy category of complexes? and the derived category of $\mathcal{A}$, respectively.
Let $\mathcal{H}$ be a small family of complexes in $Ch(\mathcal{A})$. An complex $C$ in $Ch(\mathcal{A})$ is called $\mathcal{H}$-flasque if for all $n \in \mathbf{Z}$ and $H \in \mathcal{H}$,
Finally we define:
A descent structure on $\mathcal{A}$ is a pair $(\mathcal{G},\mathcal{H})$, where $\mathcal{G}$ is an essentially small set of generators of $\mathcal{A}$, and $\mathcal{H}$ is an essentially small set of $\mathcal{G}$-cofibrant acyclic complexes such that any $\mathcal{H}$-flasque complex is $\mathcal{G}$-local.
Now one defines a model structure associated to any such descent structure.
Let $(\mathcal{G},\mathcal{H})$ be a descent structure on the Grothendieck abelian category $\mathcal{A}$. There is a proper cellular model structure on the category $Ch(\mathcal{A})$, where the weak equivalences are quasi-isomorphisms of complexes, and cofibrations are $\mathcal{G}$-cofibrations.
Also, a complex $C$ in $Ch(\mathcal{A})$ is fibrant if and only if it is $\mathcal{H}$-flasque or equivalently $\mathcal{G}$-local.
We call this the $\mathcal{G}$-model structure on $Ch(\mathcal{A})$. As in Gillespie’s approach we can sometimes get a monoidal model structure. We refer to Cisinski-Deglise for the notion of a weakly flat descent structure.
Suppose $(\mathcal{G}, \mathcal{H})$ is a weakly flat descent structure on $\mathcal{A}$. Then the $\mathcal{G}$-model structure is further monoidal.
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.
Model categories of chain complexes tend to be proper model categories.
(pushout along degreewise injections presrves quasi-isomorphism)
Let
be a pushout square of chain maps between (unbounded) chain complexes, such that
$i$ is a degreewise injection;
$f$ is a quasi-isomorphism.
Then also $g$ is a quasi-isomorphism.
Dually, the pullback of a quasi-isomorphism along a degreewise surjection is again a quasi-isomorphism.
The pushout of chain complexes is degreewise a pushout in the underlying abelian category. Since pushout in abelian categories preserves monomorphisms (this Prop.) it follows that also $j_n$ is a monomorphism. Finally, the pasting law implies that the induced morphism of cokernels of $i$ and $j$ is an isomorphism. In summary, this means that we have a commuting diagram of chain complexes as follows:
This implies a morphism of the corresponding long exact sequences in chain homology of the form:
From this the five lemma implies that $g_\ast$ is an isomorphism on chain homology, hence that $g$ is a quasi-isomorphism.
Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let the categories of chain complexes $Ch_\bullet^+(\mathcal{A})$ and $Ch_\bullet^+(\mathcal{B})$ be equipped with the model structure described above where fibrations are the degreewise split monomorphisms with injective kernels.
If
is a pair of adjoint functors where $L$ preserves monomorphisms, then
is a Quillen adjunction.
Every functor preserves split epimorphism. Being a right adjoint in particular $R$ is a left exact functor and hence preserves kernels. Using the characterization of injective objects as those $I$ for which $Hom(-,I)$ sends monomorphisms to epimorphisms, we have that $R$ preserves injectives because $L$ preserves monomorphisms, by the adjunction isomorphism.
Hence $L$ preserves all cofibrations and $R$ all fibrations.
The injective model structure on $Ch^{\geq 0}(R Mod)$ (from theorem ) is a cofibrantly generated model category.
This appears for instance as Hovey, theorem 2.3.13, where it is stated for unbounded (in both directions) chain complexes.
For results on model structures on chain complexes that are provably not cofibrantly generated see section 5.4 of Christensen, Hovey.
Let $\mathcal{A} =$ Ab be the category of abelian groups. The Dold-Kan correspondence provides a Quillen equivalence
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
prolongs to a Quillen adjunction on the projective model structure on simplicial presheaves on any site $C$ , which we denote by the same symbols
With due care this descends to the local model structure on simplicial presheaves which presents the (∞,1)-sheaf (∞,1)-topos on $C$. Then the above Quillen adjunction serves to embed abelian sheaf cohomology on $C$ into the larger context of nonabelian cohomology on $C$. See cohomology for more on this.
We discuss cofibrations in the model structures on unbounded complexes.
Let $\mathcal{P}$ be a given projective class on an abelian category $\mathcal{A}$, def. and write $Ch(\mathcal{A})_{\mathcal{P}}$ for the corresponding model structure on unbounded chain complexes, theorem .
An object $C \in Ch(\mathcal{A})_{\mathcal{P}}$ is cofibrant precisely if
in each degree $n \in \mathbb{Z}$ the object $C_n$ is relatively projective in $\mathcal{A}$;
every morphism from $C$ into a weakly contractible complex in $Ch(\mathcal{A})_{\mathcal{P}}$ is chain homotopic to the zero morphism.
This appears as (ChristensenHovey, lemma 2.4).
A morphism $f : A \to B$ in $Ch(\mathcal{A})_{\mathcal{P}}$ is a cofibration precisely if it is degreewise
with cofibrant cokernel.
This appears as (ChristensenHovey, prop. 2.5).
For $R$ any ring, there is the Eilenberg-MacLane spectrum $H R$. This is an algebra spectrum, hence there is a notion of $H R$-module spectra. These are Quillen equivalent to chain complexes of $R$-modules. See module spectrum for details.
The projective model structure on connective chain complexes (Theorem ) is due to
see also:
William Dwyer, Jan Spalinski, Section 7 of: Homotopy theories and model categories (pdf)
in: I. M. James, Handbook of Algebraic Topology, North Holland 1995 (ISBN:9780080532981, doi:10.1016/B978-0-444-81779-2.X5000-7)
Paul Goerss, Kirsten Schemmerhorn, Theorem 1.5 in: Model categories and simplicial methods, Notes from lectures given at the University of Chicago, August 2004, in: Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007(arXiv:math.AT/0609537, doi:10.1090/conm/436)
The projective model structure on connective cochain complexes (Theorem ) is claimed, without proof, in:
Kathryn Hess, p. 6 of Rational homotopy theory: a brief introduction, contribution to Summer School on Interactions between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago (arXiv:math.AT/0604626), chapter in Luchezar Lavramov, Dan Christensen, William Dwyer, Michael Mandell, Brooke Shipley (eds.), Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007 (doi:10.1090/conm/436)
J. L. Castiglioni, G. Cortiñas, Def. 4.7 of: Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence, J. Pure Appl. Algebra 191 (2004), no. 1-2, 119–142, (arXiv:math.KT/0306289, doi:10.1016/j.jpaa.2003.11.009)
Of course the description of model categories of chain complexes as (presentations of) special cases of (stable) $(\infty,1)$-categories is exactly opposite to the historical development of these ideas.
While the homotopical treatment of weak equivalences of chain complexes (quasi-isomorphisms) in homological algebra is at the beginning of all studies of higher categories and a “folk theorem” ever since
it seems that the injective model structure on chain complexes has been made fully explicit in print only in proposition 3.13 (according to remark 3.14 there) of
The projective model structure is discussed after that (and shown to be a monoidal model category) in:
Mark Hovey, Model category structures on chain complexes of sheaves (1999) [K-theory:0366, pdf, pdf]
Mark Hovey, Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353 6 (2001) [ams:S0002-9947-01-02721-0, jstor:221954]
An explicit proof of the injective model structure with monos in positive degree is spelled out in
An explicit proof of the model structure on cochain complexes of abelian groups 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
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:
Stefan Schwede, Brooke Shipley, p. 7 of: Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 2 (2000) 491-511 [arXiv:math/9801082, doi:10.1112/S002461150001220X]
Mark Hovey, Thm 2.3.1 in: Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books]
and shown to be (cofibrantly generated and in addition) proper and monoidal in:
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
Generalization to bicomplexes:
Finally a third approach to the unbounded case is discussed (in a context of mixed motives) in:
Denis-Charles Cisinski, Frédéric Déglise, Local and stable homological algebra in Grothendieck abelian categories, Homology, Homotopy and Applications 11 1 (2009) 219–260 [arXiv:0712.3296, hha:1251832567]
Denis-Charles Cisinski, Frédéric Déglise, Triangulated categories of mixed motives, Springer Monographs in Mathematics, Springer (2019) [arXiv:0912.2110, doi:10.1007/978-3-030-33242-6]
A discussion of the homotopy theory of presheaves of unbounded chain complex is in
A model structure on noncommutative dg-algebras whose proof strategy is useful also for cochain complexes is in
Last revised on April 16, 2024 at 14:38:38. See the history of this page for a list of all contributions to it.