model structure on chain complexes

 +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## 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 complex|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 object]]s in $A$.

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

Similarly, [[cochain complex|cochain complexes]] are identified with [[cosimplicial object]]s

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

At least if $A$ is the category of [[abelian groups]], so that $A^{\Delta^{op}}$ is the category of abelian [[simplicial group]]s it inherits naturally a model category structure from the [[model structure on simplicial sets]], which [[presentable (∞,1)-category|presents]] the [[(∞,1)-category]] of [[∞-groupoid]]s.

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 [[spectrum|spectra]].

This we discuss below in

* _[In non-negative degree](#CochainNonNeg)_

### For unbounded chain complexes

Model structures on unbounded (co)chain complexes can be understood as
presentations of [[spectrum objects]] in model structures of bounded (co)chain complexes.

See

* _[For unbounded chain complexes](#ForUnboundedChainComplexes)_

## Definitions

### In non-negative degree
 {#CochainNonNeg}

Let $C$ be an [[abelian category]].

Recall that by the dual [[Dold-Kan correspondence]] the category $C^\Delta$ of [[cosimplicial object]]s 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](http://arxiv.org/PS_cache/math/pdf/0312/0312531v1.pdf) for more).

#### The standard (Quillen) model structures
  {#StandardQuillenOnBounded}

Let $R$ be a [[ring]] and write $ \mathcal{A} \coloneqq R$[[Mod]] for its [[category]] of [[modules]].

We discuss the

* [Projective structure on chain complexes](#ProjectiveStructureOnChainComplexes)

* [Injective structure on cochain complexes](#InjectiveStructureOnCochainComplexes)

##### Projective structure on chain complexes
 {#ProjectiveStructureOnChainComplexes}

+-- {: .num_theorem #ProjectiveModelStructure}
###### Theorem

There is a [[model category]] structure on the [[category of chain complexes]]
$Ch_{\bullet \geq 0 }(\mathcal{A})$ (in non-negative degree) whose

* [[weak equivalences]] are the [[quasi-isomorphisms]];

* [[fibrations]] are the morphisms that are [[underlying]] [[epimorphisms]] in $R$[[Mod]] in each _[[positive number|positive]]_ degree;

* [[cofibrations]] are degreewise [[monomorphisms]] with degreewise [[projective object|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](#Quillen67)).
See also ([Goerss-Schemmerhorn 06, Theorem 1.5](#GoerssSchemmerhorn06), [Dungan 10, 2.4.2, proof in section 2.5](#Dungan10)).

\begin{prop}\label{ProjectiveModelStructureOnConnectiveChainComplexesIsProper}
  The projective model structure on connective chain complexes is a [[proper model category]].
\end{prop}
(e.g. [Jardine 03, Prop. 1.5](#Jardine03))

\begin{proposition}
\label{ProjectiveModelStructureOnConnectiveChainComplexesIsMonoidal}
{#MonoidalModelStructureOnConnectiveChainComplexes} With respect to the [[tensor product of chain complexes]] this is a [[monoidal model category]].
\end{proposition}
(e.g. [Jardine 03, Prop. 1.5](#Jardine03),  [Schwede & Shipley 2003, p. 312 (26 of 48)](#SchwedeShipley03)).

##### Injective structure on cochain complexes
 {#InjectiveStructureOnCochainComplexes}

Dually

+-- {: .num_theorem #InjectiveModelStructure}
###### Theorem

There is a [[model category]] structure on non-negatively graded [[cochain complexes]]  $Ch^{\bullet \geq 0 }(\mathcal{A})$ whose

* weak equivalences are [[quasi-isomorphisms]];

* cofibrations are the morphisms that are [[monomorphisms]] in $R Mod$ in each _[[positive number|positive]]_ degree;

* fibrations are degreewise [[epimorphisms]] with [[injective object|injective]] [[kernel]],

called the **injective model structure**.

=--

([Dungan 10, Theorem 2.4.5](#Dungan10))

+-- {: .num_remark }
###### Remark

This means that a chain complex $C_\bullet \in Ch_{\bullet}(\mathcal{A})$ is a [[cofibrant object]] in the projective model structure, theorem \ref{ProjectiveModelStructure}, 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 \ref{InjectiveModelStructure}, 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
  {#GeneralResults}

There are _resolution model structures_ on cosimplicial objects in a model category, due to ([DwyerKanStover](#DwyerKanStover)), reviewed in ([Bousfield](#Bousfield))

+-- {: .num_defn}
###### Definition

(...)

=--

+-- {: .num_theorem}
###### Theorem

Let $A$ be an [[abelian category]] and let $\mathcal{G} \in Obj(A)$ be a [[class]] of objects, such that $A$ has enough $\mathcal{G}$-[[injective object]]s.

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](http://arxiv.org/PS_cache/math/pdf/0312/0312531v1.pdf).

If $A$ has enough [[injective object]]s and $\mathcal{G}$ is the class of all of them, this reproduces the
[standard Quillen model structure](#StandardQuillenOnBounded) discussed above:

+-- {: .num_cor}
###### Corollary

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

* weak equivalences are the [[quasi-isomorphism]]s;

* fibrations are the morphisms that are [[epimorphism]]s with injective [[kernel]] in each degree;

* cofibrations are the morphisms which are [[monomorphism]]s 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.

+-- {: .num_cor}
###### Corollary

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

* weak equivalences are cochain [[homotopy equivalence]]s;

* fibrations are the morphisms that are degreewise [[split epimorphism]]s and whose [[kernel]]s are [[injective object]]s;

* cofibration are the morphisms that are in positive degree [[monomorphism]]s.

=--

+-- {: .num_example}
###### Example

If $C = $ [[Vect]] is a category of [[vector spaces]] over some [[field]], we have that every epi/mono splits and that every [[quasi-isomorphism]] is a homotopy equivalence.  Moreover, in this case every chain complex is quasi-isomorphic to its [[homology]] (regarded as a chain complex with zero differentials).

This is the model structure which induces the [[transferred model structure|transferred]] [[model structure on dg-algebras]] over a [[field]] that is used in [[rational homotopy theory]].

=--

#### Projective model structure on connective cochain complexes
  {#CochainNonNegProj}

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](model+structure+on+dg-algebras#Jardine97)).

+-- {: .num_theorem #ProjectiveModelStructureOnConnectiveCochainComplexes}
###### Theorem
**(projective model structure on connective cochain complexs )**

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

* fibrations the degreewise surjections;

* weak equivalences the [[quasi-isomorphism]]s.

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 correspondence|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](#CastiglioniCortinas03).

+-- {: .proof}
###### Proof

We spell out a proof of the model structure below in a sequence of
lemmas. The proof that this is a simplicial model category is
at [[model structure on cosimplicial abelian groups]].

=--

We record a detailed proof of the model structure on $Ch^{\bullet \geq 0}(Ab)$ with fibrations the degreewise surjections, following the appendix of ([Stel 10](#Stel)).

As usual, for $n \in \mathbb{N}$ write $\mathbb{Z}[n]$ for the complex concentrated on the additive group of [[integer]]s 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.

+-- {: .num_lemma #ProjStructGenCofibs}
###### Lemma

For all $n \in \mathbb{N}$ the canonical maps $0 \to \mathbb{Z}[n]$ and $\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]$ are cofibrations, in that they  have the [[left lifting property]] against acyclic fibrations.

=--

+-- {: .proof}
###### Proof

Let $p : A \stackrel{\simeq}{\to} B$ be degreewise surjective and an isomorphism on cohomology.

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

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

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

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

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

Such a lift is equivalently an element $\sigma \in A_{n-1}$ such that

* $d_A \sigma = f_n(1)$

* $p_{n-1}(\sigma) = g_{n-1}(1)$.

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

So then $d_B ( p(z) - g_{n-1}(1) ) = 0$ and again using that $p$ is a quasi-isomorphism this means that there must be a closed $a \in A_{n-1}$ such that $p(a) = p(z)- g_{n-1}(1) + d_B b$ for some $b \in B_{n-2}$. Choose such $a$ and $b$.

Since $p$ is degreewise onto, there is $a'$ with $p(a') = b$. Choosing this the above becomes $p(a) = p(z) - g_{n-1}(1) + p(d_A a')$.

Set then

$$
  \sigma := z - a + d_A a'
  \,.
$$

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

$$
  \begin{aligned}
     d_A \sigma &= d_A z - d_A a + d_A d_A a'
     \\
     & =
     d_A z
     \\
      & = f_n(1)
  \end{aligned}
$$

$$
  \begin{aligned}
     p( \sigma ) &=  p(z) - p(a) + p(d_A a')
     \\
     & = g_{(n-1)}(1)
  \end{aligned}
  \,.
$$

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

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

Such a lift is a choice of element $\sigma \in A_n$ such that

* $d_A \sigma = 0$;

* $p(\sigma) = g_n(1)$;

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

$$
  \sigma := a - d_A a'
  \,.
$$

=--

+-- {: .num_lemma}
###### Lemma

For all $n \in \mathbb{N}$, the morphism $0 \to \mathbb{Z}[n-1,n]$ are acyclic cofibrations, in that they have the [[left lifting property]] again all degreewise surjections.

=--

+-- {: .proof}
###### Proof

For $n = 0$ this is trivial. For $n \geq 1$ a diagram

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

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

=--

+-- {: .num_lemma #ProjStructCharAcyclicFibrations}
###### Lemma

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

=--

+-- {: .proof}
###### Proof

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

So assume $f$ has the RLP. Then from the existence of the lifts

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

one deduces that $f$ is degreewise surjective on closed elements.
In particular this means it is surjective in cohomology.

With that, it follows from the existence of all the lifts

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

for $f$ a lift of the closed element $g_n(1)$ that $f$ is
degreewise surjective on all elements.

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

=--

+-- {: .num_lemma #ProjStrucFactAxiomI}
###### Lemma

Every morphism $f : A \to B$ can be factored as
a morphism with left lifting property against all
fibrations followed by a fibration.

=--

+-- {: .proof}
###### Proof

Apply the [[small object argument]]-reasoning to the
maps in $ J = \{0 \to \mathbb{Z}[n-1,n]\}$.

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

$$
  \array{
    0 &\to& A
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    \coprod_{{n \in \mathbb{n}} \atop {b \in B_n}}
    \mathbb{Z}[n,n+1] &\stackrel{}{\to}&
    B
  }
$$

one obtains a factorization through its [[pushout]]

$$
  \array{
     && A
     \\
       &{}^{\mathllap{j}}\swarrow& \downarrow
     \\
     A \coprod \coprod_{{n \in \mathbb{n}} \atop {b \in B_n}}
     \mathbb{Z}[n,n+1]
     && \downarrow^{\mathrlap{f}}
     \\
     &\searrow_{p}& \downarrow
     \\
     && B
  }
  \,.
$$

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

The map $p$ is manifestly degreewise
onto and hence a fibration.

=--

+-- {: .num_lemma}
###### Lemma

Every morphism $f : A \to B$ may be factored as a
cofibration followed by an acyclic fibration.

=--

+-- {: .proof}
###### Proof

By a  [lemma above](#ProjStructCharAcyclicFibrations) 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](#ProjStructGenCofibs)
are cofibrations.

The claim then follows again from the [[small object argument]]
apllied to $I$.

=--

+-- {: .num_lemma}
###### Lemma

A morphism $f : A \to B$ that is both a cofibration
(:= LLP against acyclic fibrations ) and a weak equivalence
has the left lifting property against all fibrations.

=--

+-- {: .proof}
###### Proof

By a standard argument, this follows from the
factorization lemma proven [above](#ProjStrucFactAxiomI),
which says that we may find a factorization

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

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

$$
  \array{
    A &\stackrel{j}{\to}& \hat B
    \\
    {}^{\mathllap{f}}\downarrow
    &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{\mathrlap{p}}
    \\
    B &\stackrel{Id}{\to}& B
  }
  \,.
$$

Equivalently redrawing this as

$$
  \array{
    A &\stackrel{Id}{\to}& A &\stackrel{Id}{\to}& A
    \\
    {}^{\mathllap{f}}\downarrow && {}^{\mathllap{p}}
    \downarrow && {}^{\mathllap{i}}\downarrow
    \\
    B &\stackrel{\sigma}{\to}&
    \hat B
    & \stackrel{p}{\to} &
    B
  }
$$

makes manifest that this exhibts $f$ as a retract of $j$
and as such inherits its left lefting properties.

=--

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

### In unbounded degree
  {#OnUnbounded}

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

#### Standard projective model structure on unbounded chain complexes
 {#ProjectiveModelStructureOnUnboundedChainComplexes}

\begin{proposition}
\label{StandardModelStructureOnUnboundedComplexes}
For $R$ a [[commutative ring]] the [[category of unbounded chain complexes]] $Ch_\bullet(R Mod)$ of $R$-[[modules]]  admits the [[structure]] of a

* [[proper model category|proper]]

* [[cofibrantly generated model category|cofibrantly generated]]

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

\begin{tikzcd}[sep=10pt]
      S^{n-1}
      \ar[d, "{ i_n }"]
      \ar[r, phantom, ":\equiv"]
      &
      \big[\;
      \cdots
      \ar[r]
      &
      0
      \ar[r]
      \ar[d]
      &
      0
      \ar[r]
      \ar[d]
      &
      0
      \ar[r]
      \ar[d]
      &
      R
      \ar[r]
      \ar[d, "{\mathrm{id}}"]
      &
      0
      \ar[r]
      \ar[d]
      &
      \cdots
      \;\big]
      \\
      D^n
      \ar[r, phantom, ":\equiv"]
      &
      \big[\;
      \cdots
      \ar[r]
      &
      0
      \ar[r]
      &
      0
      \ar[r]
      &
      \underset{
        \mathclap{
          \raisebox{-3pt}{
            \scalebox{.7}{
              $\mathrm{deg} = n$
            }
          }
        }
      }{
      R
      }
      \ar[r, "{ \mathrm{id} }"]
      &
      R
      \ar[r]
      &
      0
      \ar[r]
      &
      \cdots
      \;\big]
\end{tikzcd}

\begin{tikzcd}[sep=10pt]
      0
      \ar[d, "{ j_n }"]
      \ar[r, phantom, ":\equiv"]
      &
      \big[\;
      \cdots
      \ar[r]
      &
      0
      \ar[r]
      \ar[d]
      &
      0
      \ar[r]
      \ar[d]
      &
      0
      \ar[r]
      \ar[d]
      &
      0
      \ar[r]
      \ar[d]
      &
      0
      \ar[r]
      \ar[d]
      &
      \cdots
      \;\big]
      \\
      D^n
      \ar[r, phantom, ":\equiv"]
      &
      \big[\;
      \cdots
      \ar[r]
      &
      0
      \ar[r]
      &
      0
      \ar[r]
      &
      \underset{
        \mathclap{
          \raisebox{-3pt}{
            \scalebox{.7}{
              $\mathrm{deg} = n$
            }
          }
        }
      }{
      R
      }
      \ar[r, "{\mathrm{id}}"]
      &
      R
      \ar[r]
      &
      0
      \ar[r]
      &
      \cdots
      \;\big]
\end{tikzcd}

* [[combinatorial model category|combinatorial]]

[[model category]] with

* [[weak equivalences]] the [[quasi-isomorphisms]]

* [[fibrations]] the (degreewise) [[epimorphisms]].

* [[cofibrations]] the degreewise [[split monomorphism|split injections]] with cofibrant [[cokernel]].

\end{proposition}
(For partial characterization of the [[cofibrant objects]] see further [below](#BoundedBelowComplexesOfProjectivesAreProjectivelyCofibrant).)
\begin{proof}
Properness and cofibrant generation are discussed in [Hovey, Palmieri & Strickland (1997), remark after theorem 9.3.1](#HoveyPalmieriStrickland97) and [Schwede & Shipley (1998), p. 7](#SchwedeShipley98), see also [Fauk (2006), Thm. 3.2](#Fausk06). The characterization of the cofibrations is in [Hovey (1999), Lem. 2.3.6](#Hovey99) and that of the generating cofibrations are made explicit in [Hovey (1999), Def. 2.3.3](#Hovey99). Cf. also [Muro & Roitzheim (2019), pp. 3](#MuroRoitzheim19).

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

1. $R$[[Mod]] is a [[Grothendieck abelian category]] (by [this example](Grothendieck+category#CateoriesOfModules));

1. when $\mathcal{A}$ is a Grothendieck abelian category then so is $Ch_\bullet(\mathcal{A})$ (by [this example](Grothendieck+category#ChainComplexesInGrothAbCat));

1. all Grothendick abelian categories are locally presentable (by [this example](locally+presentable+category#GrothAbCatsAreLocPresntbl)).

\end{proof}

\begin{remark}\label{NotAllUnboundedComplexesAreProjectivelyCofibrant}
  It is clear that every chain complex in the model structure of Prop. \ref{StandardModelStructureOnUnboundedComplexes} is [[fibrant object|fibrant]]. However, over general rings (not though over [[fields]], see Prop. \ref{UnboundedChainComplexOfVectorSpacesProjectivelyCofibrant} below) *not* every chain complex is [[cofibrant object|cofibrant]], not even those consisting of [[projective modules]] --
a [[counterexample]] is given in [Hovey (1999), Rem. 2.3.7](#Hovey99):

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 [[augmented algebra|augmentation]] by

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

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

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

$$
  \mathcal{A}
  \;\coloneqq\;
  \left(
  \cdots
  \to
  \mathbb{K} \oplus \mathbb{K}x
  \xrightarrow{
    \;\;
    \cdot x
    \;\;
  }
  \mathbb{K} \oplus \mathbb{K}x
  \xrightarrow{
    \;\;
    \cdot x
    \;\;
  }
  \mathbb{K} \oplus \mathbb{K}x
  \to
  \cdots
  \right)
  \,.
$$
Since its [[chain homology]] clearly vanishes in every degree, the morphism it receives out of the [[zero object]] is a [[quasi-isomorphism]] and hence a [[weak equivalence]]
$$
  0 \underset{\in \mathrm{W}}{\longrightarrow} \mathcal{A}
$$
and hence would be an [[acyclic cofibration]] if $\mathcal{A}$ were cofibrant.

But consider then the following [[lifting problem]] with this morphism

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

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

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

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

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

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

On the other hand:

\begin{proposition}\label{BoundedBelowComplexesOfProjectivesAreProjectivelyCofibrant}
**([[bounded-below chain complexes]] of [[projective modules]] are projectively [[cofibrant object|cofibrant]])**
\linebreak
  Every [[bounded-below chain complex]] of [[projective modules]] is [[cofibrant object|cofibrant]] in the model structure of Prop. \ref{StandardModelStructureOnUnboundedComplexes}.
\end{proposition}
([Hovey (1999), Lem. 2.3.6](#Hovey99))

This implies:

\begin{proposition}
\label{UnboundedChainComplexOfVectorSpacesProjectivelyCofibrant}
  For $k$ a [[field]]:

1. every [[object]] in the projective model structure $Ch_\bullet(k Mod)$ (Prop. \ref{StandardModelStructureOnUnboundedComplexes}) is [[cofibrant objects|cofibrant]].

1. the [[cofibrations]] are exactly the [[monomorphisms]].

\end{proposition}
\begin{proof}
 By the assumption that $k$ is a field, every $k$-module (i.e. every $k$-[[vector space]]) is [[projective module|projective]] ([this Prop.](projective+module#ModuleOverAFieldIsProjective)). Therefore Prop. \ref{BoundedBelowComplexesOfProjectivesAreProjectivelyCofibrant} says, in this situation, that every [[bounded-below chain complex]] is cofibrant
Moreover, since every injection of vector spaces splits ([here](Vect#ShortExactSequencesSplit)) the characterization of cofibrations in Prop. \ref{StandardModelStructureOnUnboundedComplexes} says that every [[injection]] into a [[bounded-below chain complex]] of vector spaces is a cofibration (since its [[cokernel]] is clearly itself bounded-below and hence cofibrant by the previous statement) .

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

$$
  cn_0 V_\bullet
  \hookrightarrow
  cn_{-1} V_\bullet
  \hookrightarrow
  cn_{-2} V_\bullet
  \hookrightarrow
  \cdots
  \,.
$$
By the previous paragraph, $cn_0 V_\bullet$ is cofibrant and each morphism in this [[cotower]] is a cofibration. Therefore $cn_0 V_\bullet \hookrightarrow V_\bullet$ is a [[transfinite composition]] of cofibrations, hence a cofibration, and therefore $V_\bullet$ is cofibrant.

This proves the first statement. From this the second follows by the characterization of the cofibrations in Prop. \ref{StandardModelStructureOnUnboundedComplexes} and using again that all injections here are split.
\end{proof}
(Alternatively one may argue via the generating cofibration, cf. [MO:a/2457259](https://math.stackexchange.com/a/2457259/58526).)

\begin{proposition}
\label{MonoidalProjectiveModelStructureOnUnboundedChainComplexes}
The [[tensor product of chain complexes]] makes the projective model structure on unbounded chain complexes $Ch_\bullet(R Mod)$ (Prop. \ref{StandardModelStructureOnUnboundedComplexes}) a [[monoidal model category]].
\end{proposition}
For the special case that all [[submodules]] of [[free modules]] are again free (such as over $R = \mathbb{Z}$ the [[integers]], by [this Prop.](free+abelian+group#SubgroupsOfFreeAbelianGroupsAreFree), or for $R = k$ a [[field]] by [this Prop](free+module#EveryModuleOverAFieldIsFree), and in general for $R$ a [[principal ideal domain]], by [this Prop.](free+module#submod)) a short proof is given in [Strickland (2020), Prop. 25](#Strickland20). The general statement is also a special case of [Hovey (2001), Cor. 3.7](#Hovey01) [Fausk (2006), Thm. 6.1](#Fausk06) (which state an even more general result about *[[sheaves]]* of chain complexes).

\begin{proposition}
\label{LocalizedReedyModelStructureOnSimplicialUnboundedChainComplexes}
The  [[category of simplicial objects]] $sCh(R Mod)_\bullet$ in the projective model structure on unbounded chain complexes (from Prop. \ref{BoundedBelowComplexesOfProjectivesAreProjectivelyCofibrant}) carries the structure of a [[combinatorial model category|combinatorial]] [[simplicial model category]] (obtained as a [[Bousfield localization of model categories|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 equivalence|Quillen equivalent]] via :
\[
  \label{QuillenEquivalenceBetweenProjectiveUnboundedChainAndSimpEnhancement}
  const
    \,\colon\,
  Ch_\bullet(R Mod)
  \rightleftarrows
  sCh_\bullet(R Mod)
   \,\colon\,
  ev_0
  \,.
\]
\end{proposition}
\begin{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](#RezkSchwedeShipley01), using methods like those discussed at *[simplicial model category -- Simplicial Quillen equivalent models](simplicial+model+category#SimpEquivMods)*.

Moreover, general facts imply that

1. a Reedy model structure with coefficients in a combinatorial model category is itself combinatorial (see [here](Reedy+model+structure#CombinatorialStructure))

1. the [[left Bousfield localization]] of a combinatorial model category is itself combinatorial (see [here](Bousfield+localization+of+model+categories#Existence)).

\end{proof}

Below, this model structure is recovered as example \ref{CategoricalProjectiveClasses} of the [Christensen-Hovey projective class construction](#InUnboundedDegreeGeneralResults).

\begin{proposition}
\label{OverGroundFieldAllObjectsInSimplicialStructureOnUnboundedComplexesAreCofibrant}
 Over a [[field]] $k$, every object in the model structure on $sCh_\bullet(k Mod)$ (from Prop. \ref{LocalizedReedyModelStructureOnSimplicialUnboundedChainComplexes}) is cofibrant.
\end{proposition}
\begin{proof}
Since [[Bousfield localization of model categories|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](Reedy+model+structure#ReedyModelStructure),
hence (cf. [this Remark](Reedy+model+structure#ReedyCofibrantObjects)) 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](category+of+chain+complexes#AbelianCategoryStructure)), [this Prop.](Reedy+model+structure#LatchingInAbelianCategoryIsDegeneracySubobject) at *[[Reedy model structure]]* says that these are [[monomorphisms]] and hence the claim follows by Prop. \ref{UnboundedChainComplexOfVectorSpacesProjectivelyCofibrant}.
\end{proof}

\begin{proposition}
\label{MonoidalSimplicialEnhancementOfUnboundedChainModelStructure}
  At least over a [[field]] $k$,
  the local model structure on $sCh_\bullet(k)$ from Prop. \ref{LocalizedReedyModelStructureOnSimplicialUnboundedChainComplexes} becomes a [[monoidal model category]] via the $\Delta$-object-wise [[tensor product of chain complexes]], and the [[Quillen equivalence]] (eq:QuillenEquivalenceBetweenProjectiveUnboundedChainAndSimpEnhancement) is a compatibly [[monoidal Quillen adjunction]] with respect to the corresponding monoidal model structure on $Ch_\bullet(k)$ from Prop. \ref{MonoidalProjectiveModelStructureOnUnboundedChainComplexes}, Prop. \ref{MonoidalProjectiveModelStructureOnUnboundedChainComplexes}.
\end{proposition}
This ought to be compatible with the simplicial structure such as to give a [[simplicial monoidal model category]].
\begin{proof}
  First, the plain [[Reedy model structure]] in $sCh_\bullet(k)$ becomes a [[monoidal model category]] under the objectwise [[tensor product of chain complexes]], by [Barwick (2010), Thm. 3.51](monoidal+model+category#Barwick10) (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, $A = \Delta$).

{#SimplicialMonoidalModelStructure} Next, to see that this monoidal model structure passes to the [[Bousfield localization of model categories|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.](homotopy+limit#SimplicialSetIsHomotopyColimitOverItself)) 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](Bousfield+localization+of+model+categories#Barwick10) --- that for $V \,\in\, Ch(k)$ and $\mathscr{W} \,\in\, sCh(k)$ homotopically constant, also their [[internal hom]] $[const(V),\,\mathscr{W}] \,\in\, sCh(k)$ is homotopically constant.

Now, on general grounds, the internal hom in $sCh(k)$ is given by an [[end]] over the internal hom in $Ch(k)$ (to be denoted by the same angular bracket notation), as follows:
$$
  [\mathscr{V},\,\mathscr{W}]
  \;\colon\;
  [s]
  \;\mapsto\;
  \int_{[s'] \in \Delta}
  \big[
    (\Delta[s] \cdot \mathscr{V})_{s'}
    ,\,
    \mathscr{W}_{s'}
  \big]
  \,,
$$
where $(-)\cdot(-)$ denotes the canonical [[tensoring]] of $sCh(k)$ over [[sSet]].

So in the case at hand, where $\mathscr{V} \,=\, const(V)$, we find this to be:
$$
  \begin{array}{l}
    [const(V),\, \mathscr{W}]
    \,\colon\,
    [s] \;\mapsto\;
    \\
    \int_{s'}
    \big[
      \Delta(s',s)
      \cdot V
      ,\,
      \mathscr{W}_{s'}
    \big]
    \\
    \;\simeq\;
    \Big[
      V
      ,\,
    \int_{s'}
      (\mathscr{W}_{s'})^{\Delta(s',s)}
    \Big]
    \\
    \;\simeq\;
    \Big[
      V
      ,\,
      \mathscr{W}_s
    \Big]
    \,,
  \end{array}
$$
where we passed from [[tensoring]] $S\cdot (-)$ to its [[right adjoint]] [[powering]] $(-)^S$, used that an [internal hom preserves limits](hom-functor+preserves+limits#InternalHomFunctor) 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](Introduction+to+Homotopy}Theory#KenBrownLemma)). 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 object|cofibrant]] (as all objects).
\end{proof}

\linebreak

#### Standard injective model structure on unbounded chain complexes
 {#InjectiveModelStructureOnUnboundedChainComplexes}

\begin{proposition}
\label{StandardInjectiveModelStructureOnUnboundedComplexes}
For $R$ a [[commutative ring]] the [[category of unbounded chain complexes]] $Ch_\bullet(R Mod)$ of $R$-[[modules]]  carries the structure of a

* [[cofibrantly generated model category|cofibrantly generated]]

* [[combinatorial model category|combinatorial]]

[[model category]] with

* [[weak equivalences]] the [[quasi-isomorphisms]]

* [[cofibrations]] the (degreewise) [[injections]].

\end{proposition}
This is [Hovey (1999), Thm. 2.3.13](#Hovey99).

#### Christensen-Hovey model structures using projective classes
 {#InUnboundedDegreeGeneralResults}

Let $\mathcal{A}$ be an [[abelian category]] with all [[limit]]s and
[[colimit]]s.

[Christensen-Hovey](#ChristensenHovey) 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.

+-- {: .num_defn #ProjectiveClass}
###### Definition

A **projective class** on $\mathcal{A}$ is a collection $\mathcal{P} \subset ob \mathcal{A}$ of [[object]]s and a collection $\mathcal{E} \subset mor \mathcal{A}$ of [[morphism]]s, 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 $\mathcal{P}$.

=--

+-- {: .num_example #TrivialProjectiveClass}
###### Example

Taking $\mathcal{P} := ob \mathcal{A}$ to be the class of _all_ objects yields a projective class -- called the _trivial projective class_ . The corresponding morphisms are the class $\mathcal{E}$ of all [[split epimorphism]]s in $\mathcal{A}$.

=--

+-- {: .num_example #PullbackProjectiveClass}
###### Example

Let $R$ be a [[ring]] and $\mathcal{A} = $ $R$-[[Mod]] be the category of $R$-[[module]]s. Choosing $\mathcal{P}$ to be the class of all summands of [[direct sums]] of [[finitely presented]] modules yields a projective class.

=--

+-- {: .num_example}
###### Example

Given a pair of [[adjoint functors]]

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

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

* $U^* \mathcal{P} := \{retracts\;of\; F P | P \in \mathcal{P}\}$

=--

+-- {: .num_theorem #ModelStructureOnUnboundedFromProjectiveClass}
###### Definition/Theorem

Given a projective class $\mathcal{P}$ in $\mathcal{A}$ (def. \ref{ProjectiveClass}), 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 [[resolution]]s exist, which is the case in particular if

1. $\mathcal{P}$ is the pullback projective class (def. \ref{PullbackProjectiveClass}) of a trivial projective class (def. \ref{TrivialProjectiveClass}) along a functor $U$ that preserves countable [[direct sums]];

1. (...)

When the structure exists, it is a [[proper model category]].

=--

This is theorem 2.2 in [Christensen-Hovey](#ChristensenHovey).

We shall write $Ch(\mathcal{A})_{\mathcal{P}}$ for this model category structure.

#### Examples
 {#ExamplesOfStructuresOnUnboundedComplexes}

We list some examples for the model structures on chain complexes in unbounded degree discussed [above](#InUnboundedDegreeGeneralResults).

Let $R$ be an associative ring and $\mathcal{A} = R$[[Mod]].

* [Categorical projective class structure](#CategoricalProjectiveClass)

* [Pure projective class structure](#PureProjectiveClass)

##### Categorical projective class structure
 {#CategoricalProjectiveClass}

+-- {: .num_example #CategoricalProjectiveClasses}
###### Example

The **categorical projective class** on $\mathcal{A}$ is the projective class (def. \ref{ProjectiveClass}) with $\mathcal{P}$ the class of [[direct sum]]mands of free modules. The $\mathcal{P}$-model structure on $Ch(\mathcal{A})$ has

* as fibrations the degreewise surjections.

So this reproduces the standard projective model structure from prop. \ref{ProjModelStructureOnUnbounded}.

=--

##### Pure projective class structure
 {#PureProjectiveClass}

+-- {: .num_example }
###### Example

The **pure projective class** on $\mathcal{A}$ has as $\mathcal{P}$ summands of sums of finitely presented modules. Fibrations in the corresponding model structure are the maps that are degreewise those epimorphisms that appear in $\mathcal{P}$-exact sequences.

=--

#### Gillespie's approach using cotorsion pairs

[Hovey](#HoveyOverview) has shown that, roughly speaking, [[model structures]] on [[abelian categories]] correspond to cotorsion pairs.  See [[abelian model structure]].

[Gillespie](#HoveyOverview) 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]].

+-- {: .num_theorem}
###### Theorem
Let $\mathcal{A}$ be a [[Grothendieck abelian category]].  Suppose $(\mathcal{D}, \mathcal{E})$ is a hereditary [[cotorsion pair]] that is cogenerated by a set, such that $\mathcal{D}$ is a [[Kaplansky class]] on $\mathcal{A}$ and $\mathcal{A}$ has enough $\mathcal{D}$-objects.

Then there is an [[abelian model structure]] on the category of complexes $Ch(\mathcal{A})$ such that the trivial objects are the acyclic complexes.
=--

Gillespie uses this result to get a **[[monoidal model category|monoidal]]** [[model structure]] on $Ch(Qcoh(X))$, the category of complexes of [[quasi-coherent sheaves]] on a [[quasi-compact scheme|quasi-compact]] [[separated scheme|separated]] [[scheme]] $X$.  This gives a better understanding of the [[derived category of quasi-coherent sheaves]] $D(Qcoh(X))$, and in particular gives immediately the [[derived functor]] $\cdot \otimes^{\mathbf{L}} \cdot$ (which is usually a problem due to sheaves not having enough projectives).

#### Cisinski-Deglise approach using descent structures

A third approach is due to [Cisinski-Deglise](#CisinskiDeglise).

Let $\mathcal{A}$ be a [[Grothendieck abelian category]].  We will define a notion of **descent structures** on $\mathcal{A}$.

+-- {: .num_defn}
###### Definition
For each object $E$ of $\mathcal{A}$ and integer $n \in \mathbf{Z}$, we define the complexes $S^n E$ and $D^n E$ as follows: let $(S^n E)^n = E$ in degree $n$ and 0 elsewhere; and let $(D^n E)^n = (D^n E)^{n+1} = E$ and 0 elsewhere.  There are canonical morphisms $S^{n+1} E \hookrightarrow D^n E$.
=--

+-- {: .num_defn}
###### Definition
Let $\mathcal{G}$ be an essentially small set of objects of $\mathcal{A}$.  A morphism in $Ch(\mathcal{A})$ is called a **$\mathcal{G}$-cofibration** if it is contained in the smallest class of morphisms in $Ch(\mathcal{A})$ that is closed under [[pushouts]], [[transfinite compositions]] and [[retracts]], generated by the inclusions $S^{n+1} E \to D^n E$, for any integer $n$ and any $E \in \mathcal{G}$.  A complex $C$ in $Ch(\mathcal{A})$ is called **$\mathcal{G}$-cofibrant** if the morphism $0 \to C$ is a $\mathcal{G}$-cofibration.
=--

+-- {: .num_defn}
###### Definition
A [[chain complex]] $C$ in $Ch(\mathcal{A})$ is called **$\mathcal{G}$-local** if for all $E \in \mathcal{G}$ and $n \in \mathbf{Z}$, the canonical morphism
$$ Hom_{\mathbf{K}(\mathcal{A})}(E[n], C) \to Hom_{\mathbf{D}(\mathcal{A})}(E[n], C) $$
is an isomorphism.  Here $\mathbf{K}(\mathcal{A})$ and $\mathbf{D}(\mathcal{A})$ denote the [[homotopy category of complexes]] and the [[derived category]] of $\mathcal{A}$, respectively.
=--

+-- {: .num_defn}
###### Definition
Let $\mathcal{H}$ be a small family of complexes in $Ch(\mathcal{A})$.  An complex $C$ in $Ch(\mathcal{A})$ is called **$\mathcal{H}$-flasque** if for all $n \in \mathbf{Z}$ and $H \in \mathcal{H}$,
$$ Hom_{\mathbf{K}(\mathcal{A})}(H, C[n]) = 0. $$
=--

Finally we define:

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

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

+-- {: .num_theorem}
###### Theorem
Let $(\mathcal{G},\mathcal{H})$ be a descent structure on the [[Grothendieck abelian category]] $\mathcal{A}$.  There is a [[proper model category|proper]] [[cellular model category|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](#CisinskiDeglise) for the notion of a **weakly flat** descent structure.

+-- {: .num_theorem}
###### Theorem
Suppose $(\mathcal{G}, \mathcal{H})$ is a _weakly flat_ descent structure on $\mathcal{A}$.  Then the $\mathcal{G}$-model structure is further [[monoidal model category|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
 {#Properness}

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

\begin{prop}\label{PushoutAlongDegreewiseInjectionsPreservesQuasiIosmorphism}
**([[pushout]] along degreewise [[injections]] presrves [[quasi-isomorphism]])** \linebreak
  Let
  $$
   \array{
    A &\overset{\; i \;}{\longrightarrow}& B
    \\
    {}^{\mathllap{f}}
    \big\downarrow
    &{}^{{}_{(po)}}&
    \big\downarrow {}^{\mathrlap{g}}
    \\
    C &\underset{\; j \;}{\longrightarrow}& D
   }
  $$
  be a [[pushout]] [[commuting diagram|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]].

[[formal duality|Dually]], the [[pullback]] of a quasi-isomorphism along a degreewise [[surjection]] is again a quasi-isomorphism.
\end{prop}
(e.g. [Strickland 2020, Prop. 24](#Strickland20))
\begin{proof}
  The [[pushout]] of chain complexes is degreewise a pushout in the underlying [[abelian category]].
  Since pushout in abelian categories preserves monomorphisms ([this Prop.](abelian+category#PullbackPreservesEpimorphisms)) it follows that also $j_n$ is a monomorphism. Finally, the [[pasting law]] implies that the induced morphism of [[cokernels]] of $i$ and $j$ is an [[isomorphism]]. In summary, this means that we have a [[commuting diagram]] of chain complexes as follows:
  $$
   \array{
    A
    &\overset{\;\;\; i \;\;\;}{\hookrightarrow}&
    B
    &\longrightarrow&
    cok(i)
    \\
    {}^{\mathllap{f}}
    \big\downarrow
    &{}^{{}_{(po)}}&
    \big\downarrow {}^{\mathrlap{g}}
    &&
    \big\downarrow {}^{\mathrlap{\simeq}}
    \\
    C
    &\underset{\;\;\; j \;\;\;}{\hookrightarrow}&
    D
    &\longrightarrow&
    cok(j)
    \,.
   }
  $$

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

$$
  \array{
    \cdots
    &\to&
    H_{n+1}
    \big(
      cok(i)
    \big)
    &
    \xrightarrow{\; \delta \;}
    &
    H_n(A)
    &
    \xrightarrow{\; i_\ast\;}
    &
    H_n(B)
    &
    \xrightarrow{\;\;\;}
    &
    H_n\big(cok(i)\big)
    &
    \xrightarrow{\;\delta\;}
    &
    H_{n-1}(A)
    &
    \to
    &
    \cdots
    \\
    &&
    {}^{\mathllap{}}
    \big\downarrow
    {}^{\mathrlap{\simeq}}
    &&
    {}^{\mathllap{f_\ast}}
    \big\downarrow
    {}^{\mathrlap{\simeq}}
    &&
    {}^{\mathllap{g_\ast}}
    \big\downarrow
    &&
    \big\downarrow
    {}^{\mathrlap{\simeq}}
    &&
    {}^{\mathllap{f_\ast}}
    \big\downarrow
    {}^{\mathrlap{\simeq}}
    \\
    \cdots
    &\to&
    H_{n+1}
    \big(
      cok(j)
    \big)
    &
    \xrightarrow{\; \delta \;}
    &
    H_n(C)
    &
    \xrightarrow{\; j_\ast\;}
    &
    H_n(D)
    &
    \xrightarrow{\;\;\;}
    &
    H_n\big(cok(j)\big)
    &
    \xrightarrow{\;\delta\;}
    &
    H_{n-1}(C)
    &
    \to
    &
    \cdots
  }
$$
From this the [[five lemma]] implies that $g_\ast$ is an isomorphism on [[chain homology]], hence that $g$ is a quasi-isomorphism.
\end{proof}

### Left/right exact functors and Quillen adjunctions {#ExactFuncsAndQuillenFuncs}

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](#GeneralResults) where fibrations are the degreewise [[split monomorphism]]s with [[injective object|injective]] [[kernel]]s.

+-- {: .num_prop}
###### Proposition

If

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

is a pair of [[adjoint functor]]s where $L$ preserves [[monomorphism]]s, then

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

is a [[Quillen adjunction]].

=--

+-- {: .proof}
###### Proof

Every functor preserves split epimorphism. Being a [[right adjoint]] in particular $R$ is a [[left exact functor]] and hence preserves kernels. Using the characterization of [[injective object]]s as those $I$ for which $Hom(-,I)$ sends monomorphisms to epimorphisms, we have that $R$ preserves injectives because $L$ preserves monomorphisms, by the adjunction isomorphism.

Hence $L$ preserves all cofibrations and $R$ all fibrations.

=--

### Cofibrant generation

+-- {: .num_prop}
###### Proposition

The injective model structure on $Ch^{\geq 0}(R Mod)$ (from theorem \ref{InjectiveModelStructure}) is a [[cofibrantly generated model category]].

=--

This appears for instance as [Hovey, theorem 2.3.13](#Hovey), 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](#ChristensenHovey).

### Inclusion into simplicial objects

Let $\mathcal{A} = $ [[Ab]] be the category of [[abelian group]]s. The [[Dold-Kan correspondence]] provides a [[Quillen equivalence]]

$$
  (N \dashv \Gamma) : Ch_\bullet^+ \stackrel{\overset{N}{\leftarrow}}{\underset{\Gamma}{\to}}
  sAb
$$

between the projective model structure on connective chain complexes and the [[model structure on simplicial T-algebras|model structure on simplicial abelian groups]]. This in turns sits as a [[transferred model structure]] along the [[forgetful functor]] over the [[model structure on simplicial sets]]

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

The combined Quillen adjunction

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

prolongs to a Quillen adjunction on the projective [[model structure on simplicial presheaves]] on any [[site]] $C$ , which we denote by the same symbols

$$
  (N F  \dashv U \Gamma) : [C,Ch_\bullet]_{proj} \stackrel{\leftarrow}{\to} [C, sSet]_{proj}
  \,.
$$

With due care this descends to the [[local model structure on simplicial presheaves]] which [[presentable (infinity,1)-category|presents]] the [[(∞,1)-sheaf (∞,1)-topos]] on $C$. Then the above Quillen adjunction serves to embed [[abelian sheaf cohomology]] on $C$ into the larger context of [[nonabelian cohomology]] on $C$. See [[cohomology]] for more on this.

### Cofibrations

We discuss cofibrations in the [model structures on unbounded complexes](#OnUnbounded).

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

+-- {: .num_prop}
###### Proposition

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

1. in each degree $n \in \mathbb{Z}$ the object $C_n$ is relatively projective in $\mathcal{A}$;

1. every morphism from $C$ into a weakly  contractible complex in $Ch(\mathcal{A})_{\mathcal{P}}$ is [[chain homotopy|chain homotopic]] to the [[zero morphism]].

=--

This appears as ([ChristensenHovey, lemma 2.4](#ChristensenHovey)).

+-- {: .num_prop}
###### Proposition

A morphism $f : A \to B$ in $Ch(\mathcal{A})_{\mathcal{P}}$ is a cofibration precisely if it is degreewise

1. a [[split monomorphism]];

1. with cofibrant [[cokernel]].

=--

This appears as ([ChristensenHovey, prop. 2.5](#ChristensenHovey)).

### Relation to module spectra

For $R$ any ring, there is the [[Eilenberg-MacLane spectrum]] $H R$. This is an [[algebra spectrum]], hence there is a notion of $H R$-[[module spectra]]. These are [[Quillen equivalence|Quillen equivalent]] to chain complexes of $R$-modules. See [[module spectrum]] for details.

## Related concepts

* [[model structure on equivariant chain complexes]]

* [[model structure on chain complexes of super vector spaces]]

* [[model structure on dg-modules]]

* [[model structure on dg-algebras]]

* [[stable model category]]

* [[(∞,1)-category of chain complexes]]

## References
 {#References}

### For bounded chain complexes

The projective model structure on connective chain complexes (Theorem \ref{ProjectiveModelStructure}) is due to

* {#Quillen67} [[Daniel Quillen]], Section II.4 item 5 in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

see also:

* {#DwyerSpalinski95} [[William Dwyer]], [[Jan Spalinski]], Section 7 of: *[[Homotopy theories and model categories]]* ([[DwyerSpalinski_HomotopyTheories.pdf:file]])

  in: [[Ioan Mackenzie James|I. M. James]], *[[Handbook of Algebraic Topology]]*, North Holland 1995 ([ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2), [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

* {#GoerssSchemmerhorn06} [[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](http://arxiv.org/abs/math.AT/0609537), [doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

The projective model structure on connective *co*chain complexes (Theorem \ref{ProjectiveModelStructureOnConnectiveCochainComplexes}) is claimed, without proof, in:

* {#Hess06} [[Kathryn Hess]], p. 6 of _Rational homotopy theory: a brief introduction_, contribution to _[Summer School on Interactions between Homotopy Theory and Algebra](https://jdc.math.uwo.ca/summerschool/)_, University of Chicago, July 26-August 6, 2004, Chicago ([arXiv:math.AT/0604626](http://arxiv.org/abs/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](http://dx.doi.org/10.1090/conm/436))

* {#CastiglioniCortinas03} 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](http://arxiv.org/abs/math/0306289), [doi:10.1016/j.jpaa.2003.11.009](https://doi.org/10.1016/j.jpaa.2003.11.009))

Of course the description of model categories of chain complexes as ([[presentable (infinity,1)-category|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-isomorphism]]s) 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](http://arxiv.org/abs/math/0102087), [pdf](http://arxiv.org/PS_cache/math/pdf/0102/0102087v1.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:

* {#Hovey99KTh} [[Mark Hovey]], *Model category structures on chain complexes of sheaves* (1999) [[K-theory:0366](https://faculty.math.illinois.edu/K-theory/0366), [pdf](https://faculty.math.illinois.edu/K-theory/0366/sheaves.pdf), [[Hovey-SheavesOfChainComplexesPreprint.pdf:file]]]

* {#Hovey01} [[Mark Hovey]], _Model category structures on chain complexes of sheaves_, Trans. Amer. Math. Soc. **353** 6 (2001) [[ams:S0002-9947-01-02721-0](https://www.ams.org/journals/tran/2001-353-06/S0002-9947-01-02721-0), [jstor:221954](https://www.jstor.org/stable/221954)]

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

* {#Dungan10} [[Gregory Dungan]], _Review of model categories_, 2010 ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.696.123&rep=rep1&type=pdf), [[DunganModelCategories.pdf:file]])

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

* [[Herman Stel]], _[[schreiber:master thesis Stel|∞-Stacks and their Function Algebras -- with Applications to ∞-Lie Theory]]_ (2010)
{#Stel}

The resolution model structures on cofibrant objects go back to

* [[William Dwyer]], [[Dan Kan]], C. Stover, _An $E_2$ model category structure for pointed simplicial spaces_, J. Pure and Applied Algebra 90 (1993) 137–152
{#DwyerKanStover}

and are reviewed in

* {#Bousfield} [[Aldrich Bousfield]], _Cosimplicial resolutions and homotopy spectral sequences in model categories_ Geometry and Topology, volume 7 (2003)

### For unbounded chain complexes
  {#ForUnboundedChainComplexes}

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

* [[Nicolas Spaltenstein]], *Resolutions of unbounded complexes*, Compositio Mathematica **65** 2 (1988) 121-154 [[numdam:CM_1988__65_2_121_0](http://www.numdam.org/item?id=CM_1988__65_2_121_0), [eudml:89885](https://eudml.org/doc/89885)]

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

* {#Hinich97} [[Vladimir Hinich]],  *Homological algebra of homotopy algebras*, Communications in Algebra **25** 10   (1997) 3291-3323 [[arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), [doi:10.1080/00927879708826055](https://doi.org/10.1080/00927879708826055), Erratum: [arXiv:math/0309453](http://arxiv.org/abs/math/0309453)]

and maybe independently due to:

* {#HoveyPalmieriStrickland97} [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]],  after theorem 9.3.1 in: *Axiomatic stable homotopy theory*, Memoirs Amer. Math. Soc. **610** (1997) [[ISBN:978-1-4704-0195-5](https://bookstore.ams.org/memo-128-610), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf)]

(where the relevant insights are credited to [Weibel (1994)](https://ncatlab.org/nlab/show/homological+algebra#Weibel94))

and shown to be [[cofibrantly generated model category|cofibrantly generated]] in:

* {#SchwedeShipley98} [[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](https://arxiv.org/abs/math/9801082), [doi:10.1112/S002461150001220X](https://doi.org/10.1112/S002461150001220X)]

* {#Hovey99} [[Mark Hovey]], Thm 2.3.1 in: *[[Model Categories]]*, Mathematical Surveys and Monographs, **63** AMS (1999) [[ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover)]

and shown to be ([[cofibrantly generated model category|cofibrantly generated]] and in addition) [[proper model category|proper]] and [[monoidal model category|monoidal]] in:

* {#Strickland20} [[Neil Strickland]], *The model structure for chain complexes* [[arXiv:2001.08955](https://arxiv.org/abs/2001.08955)]

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

* {#SchwedeShipley03} [[Stefan Schwede]], [[Brooke Shipley]], _Equivalences of monoidal model categories_, Algebr. Geom. Topol. **3** (2003) 287-334 [[arXiv:math.AT/0209342](http://arxiv.org/abs/math.AT/0209342), [euclid:euclid.agt/1513882376](https://projecteuclid.org/euclid.agt/1513882376)]

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

* {#RezkSchwedeShipley01} [[Charles Rezk]], [[Stefan Schwede]], [[Brooke Shipley]], Cor. 4.6 in: *Simplicial structures on model categories and functors*, American Journal of Mathematics **123** 3 (2001) 551-575  [[arXiv:0101162](http://arxiv.org/abs/math/0101162), [jstor:25099071](https://www.jstor.org/stable/25099071)]

Generalization to [[presheaves]] of chain complexes:

* {#Fausk06} [[Halvard Fausk]], *T-model structures on chain complexes of presheaves* [[arXiv:math/0612414](https://arxiv.org/abs/math/0612414)]

The article

* {#ChristensenHovey} [[Dan Christensen]], [[Mark Hovey]], _Quillen model structures for relative homological algebra_, Math. Proc. Cambridge Philos. Soc. **133** 2 (2002) 261-293 [[arXiv:math/0011216](https://arxiv.org/abs/math/0011216), [doi:10.1017/S0305004102006126](https://doi.org/10.1017/S0305004102006126)]

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

* [[Dan Christensen]], _Derived categories and projective classes_ , (2005) ([hopf archive](http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen/derived))

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

* {#HoveyOverview} [[Mark Hovey]], _Cotorsion pairs and model categories_, 2006 ([pdf](http://homepages.math.uic.edu/~bshipley/hovey.pdf))

Some generalizations and simplifications of the original approach are discussed in

* {#Gillespie} James Gillespie, _Kaplansky classes and derived categories_, 2007, [pdf](http://phobos.ramapo.edu/~jgillesp/updated%20qcsheaf.pdf)

Generalization to [[bicomplexes]]:

* {#MuroRoitzheim19} [[Fernando Muro]], [[Constanze Roitzheim]], *Homotopy Theory of Bicomplexes*, Journal of Pure and Applied Algebra **223** 5 (2019) 1913-1939 [[arXiv:1802.07610](https://arxiv.org/abs/1802.07610), [doi:10.1016/j.jpaa.2018.08.007](https://doi.org/10.1016/j.jpaa.2018.08.007)]

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

* [[Denis-Charles Cisinski]], [[Frédéric Déglise]], *Local and stable homological algebra in Grothendieck abelian categories*, Homology, Homotopy and Applications **11** 1 (2009) 219–260 [[arXiv:0712.3296](http://arxiv.org/abs/0712.3296), [hha:1251832567](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-11/issue-1/Local-and-stable-homological-algebra-in-Grothendieck-abelian-categories/hha/1251832567.full)]

* {#CisinskiDeglise09} [[Denis-Charles Cisinski]], [[Frédéric Déglise]], *Triangulated categories of mixed motives*, Springer Monographs in Mathematics, Springer (2019) [[arXiv:0912.2110](http://arxiv.org/abs/0912.2110), [doi:10.1007/978-3-030-33242-6](https://doi.org/10.1007/978-3-030-33242-6)]

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

* {#Jardine03} [[J. F. Jardine]], _Presheaves of chain complexes_, K-theory 30.4 (2003): 365-420 ([pdf](https://pdfs.semanticscholar.org/360c/fd4eb76da956e0c8dd897377f6d6f342e44e.pdf), [[JardinePresheavesOfChainComplexes.pdf:file]])

A model structure on noncommutative [[dg-algebra]]s whose proof strategy is useful also for cochain complexes is in

* [[J. F. Jardine]], _[[JardineModelDG.pdf:file]]_, Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications **17**, AMS (1997) 55-58.

[[!redirects model structures on chain complexes]]

[[!redirects model structure on cochain complexes]]
[[!redirects model structures on cochain complexes]]

[[!redirects model structure on connective chain complexes]]
[[!redirects model structures on connective chain complexes]]

[[!redirects model structure on unbounded chain complexes]]
[[!redirects model structures on unbounded chain complexes]]

[[!redirects projective model structure on chain complexes]]
[[!redirects projective model structures on chain complexes]]

[[!redirects injective model structure on chain complexes]]
[[!redirects injective model structures on chain complexes]]

[[!redirects injective model structure on connective cochain complexes]]

[[!redirects projective model structure on unbounded chain complexes]]
[[!redirects projective model structures on unbounded chain complexes]]

[[!redirects injective model structure on unbounded chain complexes]]
[[!redirects injective model structures on unbounded chain complexes]]

[[!redirects model category of chain complexes]]
[[!redirects model categories of chain complexes]]


model structure on chain complexes of super vector spaces

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[category]] of [[chain complexes of super vector spaces]] over a [[field]] of [[characteristic zero]] carries a _projective_ [[model category]] [[structure]] whose [[weak equivalences]] are the underlying [[quasi-isomorphisms]] and whose [[fibrations]] are the degreewise [[surjections]] (all either in unbounded degree, in non-negative degree or in non-positive degree).

The model structure is hence the direct generalization of the projective [[model structure on chain complexes]] of plain [[vector spaces]], to which it reduces on the objects concentrated in even super-degree.

## Related concepts

* [[model structure on differential graded-commutative superalgebras]]

* [[super L-infinity algebra]]

## References

A unified treatmeant generalizing to arbitary super [[Fermat theories]] is in

* {#CarchediRoytenberg12} [[David Carchedi]], [[Dmitry Roytenberg]],  section 6.1 of  of _Homological Algebra for Superalgebras of Differentiable Functions_ ([arXiv:1212.3745](https://arxiv.org/abs/1212.3745))

[[!redirects model structures on chain complexes of super vector spaces]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[model category]] structures on [[Eilenberg-Moore categories]] of [[coalgebras over a comonad]].

## Related concepts

* [[model stucture on dg-coalgebras]], [[model structure on dg-comodules]]

* [[model structure on monoids in a monoidal model category]]

* [[model structure on algebras over an operad]]

* [[model structure on coalgebras over a comonad]]

* [[model structure on simplicial T-algebras]]

## References

Sufficient conditions for the exstence of a model structure on coalgebras over a comonad are stated as theorem 5.8 in

* {#HessShipley14} [[Kathryn Hess]], [[Brooke Shipley]], _The homotopy theory of coalgebras over a comonad_,  Proc. London Math. Soc. (2014) 108 (2): 484-516.([arXiv:1205.3979](https://arxiv.org/abs/1205.3979))

This refers to the concept of "Postnikov presentation" of model categories due to appendix section 5 of

* [[Kathryn Hess]], _Homotopic Hopf-Galois extensions: foundations and examples_, New topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geom. Topol. Monogr., vol. 16, Geom. Topol. Publ., Coventry, 2009, pp. 79–132. MR 2544387 (2010j:55010) ([arXiv:0902.3393](https://arxiv.org/abs/0902.3393))

[[!redirects model structures on coalgebras over a comonad]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[classical model structure on topological spaces]] $Top_{Qu}$ restricts to [[compactly generated topological spaces]] and further to [[weak Hausdorff spaces]] among these, to yield [[Quillen equivalence|Quillen equivalent]] model structures $k Top_{Qu}$  and $h k Top_{Qu} $there. Since these model structures on k-spaces are Cartesian monoidal closed as model categories (Prop. \ref{CartesianMonoidalModelClosureOfkTop} below) while $Top_{QU}$ is not, they provide a more [[convenient category of topological spaces|convenient]] foundations for much of [[homotopy theory]] in terms of [[model categories]].

## Statement

Recall (from [here](compactly+generated+topological+space#SequencesOfCoReflections)) the sequence of [[adjoint functors]]

\[
  \label{SequenceOfAdjunctions}
  h k Top
  \underoverset
    {\underset{}{\hookrightarrow}}
    {\overset{ h }{\longleftarrow}}
    {\;\;\;\; \bot \;\;\;\; }
  k Top
  \underoverset
    {\underset{k}{\longleftarrow}}
    {\overset{}{\hookrightarrow}}
    {\;\;\;\; \bot \;\;\;\;}
  Top
\]

exhibiting the [[coreflective subcategory]] inside all of [[Top]] of [[compactly generated topological spaces]] and further the [[reflective subcategory]] of [[weak Hausdorff spaces]] among these.

### On k-Spaces
 {#OnkSpaces}

\begin{proposition}\label{ClassicalModelStructureOnCompactlyGeneratedSpaces}
**(classical model structure on compactly generated topological spaces)**
\linebreak
  The [[classical model structure on topological spaces]] $Top_{Qu}$ restricts along $k Top \xhookrightarrow{\;} Top$ (eq:SequenceOfAdjunctions) to a [[cofibrantly generated model category]] structure $k Top_{Qu}$ on [[compactly generated topological spaces]], and the [[coreflective subcategory|coreflection]] becomes a [[Quillen equivalence]]:

$$
  k Top
  \underoverset
    {\underset{k}{\longleftarrow}}
    {\overset{}{\hookrightarrow}}
    {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;}
  Top
$$
\end{proposition}
(e.g. [Hovey 1999, Thm. 2.4.23](#Hovey99))
A **proof** is spelled out with [this Thm.](classical+model+structure+on+topological+spaces#ModelStructureOnTopcg) at *[[classical model structure on topological spaces]]* (which is [this Thm.](Introduction+to+Homotopy+Theory#ClassicalModelStructureOnCompactlyGeneratedTopologicalSpaces) at *[[Introduction to Homotopy Theory]]*).

\begin{prop}\label{CartesianMonoidalModelClosureOfkTop}
  The model structure on compactly generated topological spaces from Prop. \ref{ClassicalModelStructureOnCompactlyGeneratedSpaces} is a [[Cartesian monoidal category|cartesian]] [[monoidal model category]];

\end{prop}
(e.g. [Hovey 1999, Prop. 4.2.11](#Hovey99))
A **proof** is given with [this Prop.](classical+model+structure+on+topological+spaces#PushoutProductInTopCGSendsCofCofToCof) at *[[classical model structure on topological spaces]]* (which is [this Prop.](Introduction+to+Homotopy+Theory#PushoutProductInTopCGSendsCofCofToCof) at *[[Introduction to Homotopy Theory]]*).

### On weakly Hausdorff k-Spaces

Similarly:

\begin{proposition}\label{ClassicalModelStructureOnCompactlyGeneratedSpaces}
**(classical model structure on compactly generated weak Hausdorff spaces)**
\linebreak
  The [[model structure on compactly generated topological spaces]] $k Top_{Qu}$ from Prop. \ref{ClassicalModelStructureOnCompactlyGeneratedSpaces} restricts along $h k top \xhookrightarrow{\;} k Top $ (eq:SequenceOfAdjunctions) to a [[model category]] structure on [[weak Hausdorff space|weakly Hausdorff]] k-spaces $h k Top_{Qu}$, and the [[reflective subcategory|reflection]] is a [[Quillen equivalence]]:

$$
  h k Top_{Qu}
  \underoverset
    {\underset{}{\hookrightarrow}}
    {\overset{ h }{\longleftarrow}}
    {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\; }
  k Top_{Qu}
$$

\end{proposition}
(e.g. [Hovey 1999, Thm. 2.4.25](#Hovey99))

Again, this is a [[Cartesian monoidal category|Cartesian]] [[monoidal model category]] (e.g. [Hovey 1999, Thm. 2.4.23](#Hovey99)).

### On Delta-generated spaces
 {#ModelStructureOnDeltaGeneratedTopologicalSpaces}

In the same way as [above](#OnkSpaces) (e.g. [Gaucher 2007, p. 7](#Gaucher07), [Haraguchi 2013](Delta-generated+topological+space#Haraguchi13)):

the model structure on [[compactly generated topological spaces]] restricts further along the inclusion of [[Delta-generated topological spaces]] $D Top \hookrightarrow k Top$, to give a [[Quillen equivalence|Quillen equivalent]] [[model structure on Delta-generated topological spaces]]:

$$
  Top_{Qu}
  \underoverset
    { \underset{ k }{\longrightarrow} }
    { {\hookleftarrow} }
    { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; }
  k Top_{Qu}
  \underoverset
    { \underset{ D }{\longrightarrow} }
    { {\hookleftarrow} }
    { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; }
  D Top_{Qu}
  \,.
$$

## Related concepts

* [[model structure on topological spaces]]

  * [[classical model structure on topological spaces]] ([[classical model structure on pointed topological spaces|on pointed spaces]])

    * [[model structure on compactly-generated topological spaces]]

    * [[model structure on Delta-generated topological spaces]]

  * [[Strøm model structure]]

* [[model structure on diffeological spaces]]

## References

Textbook accounts:

* {#Hovey99}  [[Mark Hovey]], *[[Model Categories]]*, Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))

Lecture notes:

* [[J. F. Jardine]], *Serre fibrations and the model structure for CGWH* ([pdf](https://www.uwo.ca/math/faculty/jardine/courses/homth/homth-lecture002a.pdf))

See also:

* {#Gaucher07} [[Philippe Gaucher]], p. 7 of: *Homotopical interpretation of globular complex by multipointed d-space*, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 ([arXiv:0710.3553](https://arxiv.org/abs/0710.3553))

* {#SterlingKapulkin23} [[Sterling Ebel]], [[Chris Kapulkin]], Thm. 4.9 in: *Synthetic approach to the Quillen model structure on topological spaces* [[arXiv:2310.14235](https://arxiv.org/abs/2310.14235)]

[[!redirects model structure on compactly-generated topological spaces]]

[[!redirects model structure on compactly generated weak Hausdorff spaces]]
[[!redirects model structure on compactly-generated weak Hausdorff spaces]]

[[!redirects model structure on Delta-generated topological spaces]]
[[!redirects model structure on Delta generated topological spaces]]


model structure on cosimplicial abelian groups

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#Contents#
* table of contents
{:toc}

## Idea

Let $Ab^{\Delta}$ be the [[category of cosimplicial objects]] in the category [[Ab]] of [[abelian group]]s -- the category of _[[cosimplicial abelian group|cosimplicial abelian groups]]_ .

This entry discusses structures of [[model categories]] on $Ab^\Delta$.

By the dual [[Dold-Kan correspondence]] there is an [[equivalence of categories]] $Ab^\Delta \stackrel{\overset{\Xi}{\leftarrow}}{\underset{N}{\to}} Ch^\bullet_+(Ab)$ with the [[category of cochain complexes]] in non-negative degree. Since [[Ab]] is an [[abelian category]], we have by general results various model structures on cochain complexes. Via the Dold-Kan equivalence, all of these induce model structures on $Ab^\Delta$.

## Properties

### Simplicial enrichment of the projective structure {#SimplicialEnrichmentOfProjective}

Since [[Ab]] has all [[limit]]s and [[colimit]]s, the [[category of cosimplicial objects]] (as described there) $Ab^\Delta$ inherits canonically the structure of an [[sSet]]-[[enriched category]] which is [[power]]ed and [[copower]]ed.

Write $Ab^\Delta_{proj}$ for the model structure that is induced by the dual [[Dold-Kan correspondence]] $Ab^\Delta \simeq Ch^\bullet_+(Ab)$ from the [[model structure on cochain complexes]] whose fibrations are the _degreewise surjections_ (and weak equivalences the usual [[quasi-isomorphism]]s). This is described in detail here.
So this induces the model structure $Ab^\Delta_{proj}$ whose fibrations are also the degreewise surjections in $Ab$ (using that the [[Moore complex|normalized cochain complex]]-functor preserves surjections.)

+-- {: .un_prop}
###### Proposition

The canonical $sSet$-enrichement of $Ab^\Delta$ is compatible with the model category structure $Ab^\Delta_{proj}$  in that the combination gives $Ab^\Delta$ the structure of a [[simplicial model category]].

=--

+-- {: .proof}
###### Proof

We need check the [[pushout-product axiom]] of an [[enriched model category]] of the standard [[model structure on simplicial sets]] $sSet_{Quillen}$

So we need to show that for $i : C \to C'$ any cofibration in $sSet_{Quillen}$ and $j : X \to Y$ a fibration of cosimplcial abelian groups (degreewise surjection) the morphism

$$
  k : X^{C'} \to X^C \times_{Y^C} Y^{C'}
$$

induced by the [[power]]ing $(-)^{(-)} : Ab^\Delta \times sSet \to Ab^\Delta$ is a fibration, which is acyclic if $i$ or $j$ is.

That $k$ is a fibration is easily checked. To see acyclicity we first notice the following

**Lemma.** If $i : C \to C'$ is a weak equivalence then for every cosimplicial abelian group $A$ we have $A^i$ is a weak equivalence.

To see this observe that $A^C$ is the diagonal of an evident [[bisimplicial object|bisimplicial abelian group]] and that $A^i$ is then in one argument a degreewise [[quasi-isomorphism]]. Since forming total complexes preserves degreewise equivalences, the lemma follows.

To continue the main proof, notice that we have a [[short exact sequence]]

$$
  0 \to X^C \times_{Y^C} Y^{C'} \to X^C \oplus Y^{C'} \stackrel{f}{\to}
  Y^C \to 0
$$

with $f : (x,y) \mapsto j^C(x) - Y^i(y)$. This induces a [[fiber sequence|long exact sequence in cohomology]]

$$
  \cdots \to
  H^p(X^C \times_{Y^C} Y^{C'})
  \to
  H^p(X^C) \oplus H^i(Y^{C'})
  \to
  H^p(Y^C)
  \to
  H^{p+1}(X^C \times_{Y^C} Y^{C'})
  \to
  \cdots
  \,.
$$

If $i$ is a weak equivalence, then by the above lemma we have that

$$
  ker(H^p(X^C) \oplus H^p(Y^{C'}) \to H^p(Y^C))
  \simeq
  H^p(X^C)
  \,.
$$

Inspection of the [[connecting homomorphism]] then shows that $H^p(Y^C) \to H^{p+1}(X^C \times_{Y^C} Y^{C'})$ is the 0-map. In total this implies that we have an [[isomorphism]]

$$
  H^p(X^C \times_{Y^C} Y^{C'}) \stackrel{\simeq}{\to}
  H^p(X^C)
$$

for all $p$, and hence that

$$
  X^C \times_{Y^C} Y^{C'} \to  X^C
$$

is a weak equivalence. Since by the above lemma also $X^i : X^{C'} \to X^C$ is a weak equivalence, it follows by [[category with weak equivalences|2-out-of-3]] that the morphism $k$ is indeed a weak equivalence if $i$ is.

An analogous argument shows that $k$ is a weak equivalence if $j$ is.

This argument is essentially that on page 41 of ([Toën](#Toen))

=--

## Related concepts

* [[model structure on simplicial abelian groups]]

## References

The [[model structure on cosimplicial algebras]] is discussed in detail in

* {#Toen} [[Bertrand Toën]], _Champs affines_, Selecta Math. new series **12** (2006), no. 1, 39-135 ([arXiv:math/0012219](https://arxiv.org/abs/math/0012219), [doi:10.1007/s00029-006-0019-z](https://doi.org/10.1007/s00029-006-0019-z))

The above proof that $Ab^\Delta_{proj}$ is a simplicial model category mimics the proof on page 41 there. Indeed, the claim is that the model structure on cosimplicial algebras is the [[transferred model structure]] induced by the above from the evident forgetful functor.


model structure on cosimplicial algebras > history

< [[model structure on cosimplicial algebras]]


model structure on cosimplicial objects

## References

* A. Bousfield, _Cosimplicial resolutions and homotopy spectral sequences in model categories_ ([arXiv:math.AT/0312531](http://arxiv.org/abs/math.AT/0312531))


model structure on cosimplicial rings

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#Contents#
* table of contents
{:toc}

## Idea

A [[model category]] structure on [[cosimplicial objects]] in unital, [[commutative algebras]] over some [[field]] $k$.

Under the [[monoidal Dold-Kan correspondence]] this is [[Quillen equivalence|Quillen equivalent]] to the [[model structure on dg-algebras|model structure on commutative non-negative cochain dg-algebras]].

## The model structure

Let $k$ be a [[field]] of [[characteristic zero]].

+-- {: .num_defn #CosimplicalCommutativeAlgebras}
###### Definition

Write $cAlg_k^\Delta$ for the [[category]] of [[cosimplicial objects]] in the [[category]] of unital, [[commutative algebras]] over $k$.

=--

+-- {: .num_remark #ComparisonFunctors}
###### Remark

Sending $k$-[[associative algebras|algebras]] to their underlying $k$-[[modules]] yields a [[forgetful functor]]

$$
  U \colon cAlg_k^\Delta \longrightarrow k Mod^\Delta
$$

from cosimplicial $k$-allgebras (def. \ref{CosimplicalCommutativeAlgebras}) to [[cosimplicial objects]] in $k$-[[vector spaces]].

Moreover, the [[Dold-Kan correspondence]] provides the [[normalized cochain complex]] functor

$$
  N \colon k Mod_k^\Delta \to Ch^{\geq 0}(k)
$$

from [[cosimplicial objects|cosimplicial]] $k$-[[vector spaces]] to [[cochain complexes]] (i.e. with [[differential]] of degree +1) in non-negative degrees.

=--

+-- {: .num_prop}
###### Proposition

Say that morphism $f \colon A \to B$ in $cAlg_k^{\Delta}$ (def. \ref{CosimplicalCommutativeAlgebras}) is

1. a _weak equivalence_ if its image $N(U(f)) \colon N(U(A)) \to N(U(B))$ under the comparison functors from remark \ref{ComparisonFunctors} is a [[quasi-isomorphism]] in $Ch^{\geq 0}(k)$;

1.a _fibration_ if $f$ is an [[epimorphism]] (i.e. degreewise a [[surjection]]).

Then

1. this defines a [[model category]] structure, to be called the _projective model structure_ on comsimplicial commutative $k$-algebras. $(cAlg_k^\Delta)_{poj}$.

1. this is a [[cofibrantly generated model category]]

1. and a [[simplicial model category]].

=--

e.g. [Toën 00, theorem 2.1.2](#Toen00)

+-- {: .proof}
###### Proof

The first two statements follow by observing that $(cAlg_k^{\Delta})_{proj} is$the [[transferred model structure]] along the [[forgetful functor]] $U \circ N$ from remark \ref{ComparisonFunctors} of the [[projective model structure on chain complexes]], by [this prop.](transferred+model+structure#SufficientConditions).

The third statement is the content of prop. \ref{SimplicialModelStructure} below.

=--

## Properties

### Simplicial model category structure

There is also the structure of an [[sSet]]-[[enriched category]] on $cAlg_k^\Delta$ (def. \ref{cAlgDelta})

+-- {: .num_defn #cAlgDelta}
###### Definition

For $X$ a [[simplicial set]] and $A \in Alg_k^{\Delta}$ let $A^X  \in Alg_k^\Delta$ be the corresponding $A$-valued [[cochains on simplicial sets]]

$$
  A^X \;\colon\; [n]  \mapsto (A_n)^{X_n} = \underset{X_n}{\prod} A_n
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

If we write $C(X) \coloneqq Hom_{Set}(X_\bullet,k)$ for the cosimplicial algebra of [[cochains on simplicial sets]] then for $X$ degreewise finite this may be written as

$$
  A^X = A \otimes C(X)
$$

where the tensor product is the degreewise tensor product of $k$-algebras.

=--

See also [Castiglioni-Cortinas 03, p. 10] (#CastiglioniCortinas03).

+-- {: .num_defn}
###### Definition

For $A,B \in Alg_k^\Delta$ define the [[sSet]]-[[hom-object]] $Alg_k^\Delta(A,B)$ by

$$
  Alg_k^\Delta(A,B) \coloneqq Hom_{sSet}(A, B^{\Delta[\bullet]})
  = Hom_{sSet}(A, B \otimes C(\Delta[\bullet]))
  \in sSet
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

For $B \in Alg_k$ regarded as a constant cosimplicial object under the canonical embedding $Alg_k \hookrightarrow Alg_k^\Delta$ we have

$$
  Alg_k^\Delta(A, B^{\Delta[n]})
  =
  Alg_k^\Delta(A, B \otimes C(\Delta[n]))
  \simeq
  Alg_k(A_n,B)
  \,.
$$

=--

+-- {: .proof}
###### Proof

Let $f : A \to B \otimes C(\Delta[n])$ be a morphism of cosimplicial algebras and write

$$
  f_n : A_n \to B
$$

for the component of $f$ in degree $n$ with values in the copy $B = B \otimes k$ of functions $k$ on the unique non-degenerate $n$-[[simplex]] of $\Delta[n]$. The $n+1$ coface maps $C(\Delta[n])_n \leftarrow C(\Delta[n])_{n-1}$ obtained as the pullback of the $(n+1)$ face inclusions $\Delta[n-1] \to \Delta[n]$  restrict on the non-degenerate $(n-1)$-cells to the $n+1$ projections $k \leftarrow k^{n+1} : p_i$.

Accordingly, from the naturality squares for $f$

$$
  \array{
    A_n &\stackrel{f_n}{\to}& B
    \\
    \uparrow^{\mathrlap{\delta_i}} && \uparrow^{\mathrlap{p_i}}
    \\
    A_{n-1} &\stackrel{f_{n-1}}{\to}& B^{n+1}
  }
$$

the bottom horizontal morphism is fixed to have components

$f_{n-1} = (f_n \circ \delta_0, \cdots, f_n \circ \delta_n)$

in the functions on the non-degenerate simplices.

By analogous reasoning this fixes all the components of $f$ in all lower degrees with values in the functions on degenerate simplices.

=--

The above [[sSet]]-[[enriched category theory|enrichment]] makes $cAlg_k^\Delta$ into a [[simplicially enriched category]] which is [[copower|tensored]] and [[power|cotensored]] over $sSet$.

And this is compatible with the model category structure:

+-- {: .num_prop #SimplicialModelStructure}
###### Proposition

With the definitions as above, $(cAlg_k^\Delta)_{proj}$ is a [[simplicial model category]].

=--

[Toën 00, theorem 2.1.2](#Toen00)

### Relation to the model structure on cochain dgc-algebras

Under the [[monoidal Dold-Kan correspondence]] this is related to the [[model structure on dg-algebras|model structure on commutative non-negative cochain dg-algebras]].

## References

Details are in

* {#Toen00} [[Bertrand Toën]], section 2.1 of _Affine stacks (Champs affines)_ ([arXiv:math/0012219](http://arxiv.org/abs/math/0012219)) .

See also

* [[Paul Goerss]], [[Rick Jardine]], _[[Simplicial homotopy theory]]_

The generalization to arbitrary cosimplicial rings is proposition 9.2 of

* {#CastiglioniCortinas03} J.L. Castiglioni G. Cortiñas, _Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence_ ([arXiv:math/0306289](http://arxiv.org/abs/math/0306289))

There also aspects of relation to the [[model structure on dg-algebras]] is discussed. (See [[monoidal Dold-Kan correspondence]] for more on this).

[[!redirects model structure on cosimplicial algebras]]


model structure on cosimplicial simplicial sets

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#Contents#
* table of contents
{:toc}

## Idea

For $\Delta$ the [[simplex category]] the [[functor category]] $sSet^{\Delta}$ is that of [[cosimplicial object]]s in [[simplicial set]]s: [[cosimplicial simplicial set]]s.

There are various standard [[model category]] structures on this category. The [[Reedy model structure]] is discussed in ([BousfieldKan](#BousfieldKan)), the injective structure is discussed in ([Jardine](#Jardine)).

## Properties

+-- {: .num_remark}
###### Remark

The [[totalization]] of a cosimplicial simplicial set $X^\bullet$ coincides with the [[sSet]]-enriched [[hom-object]]

$$
  Tot(X_\bullet) = sSet^{\Delta}(\Delta, X_\bullet)
  \,,
$$

where $\Delta : [k] \mapsto \Delta[k]$ is the canonical cosimplicial simplicial set given by the [[simplex]]-assignment.

Since $\Delta$ is cofibrant in the [[Reedy model structure]] it follows that totalization of Reedy-fibrant cosimplicial simplicial sets preserves weak equivalences. The following lists situations in which totalization respects weak equivalences even without this assumption.

=--

+-- {: .num_remark}
###### Remark

Totalization is closely related to [[descent objects]]. If
$A$ is a [[simplicial presheaf]] and $Y \to X$ is a [[hypercover]], then the descent object is the [[sSet]]-[[hom object|enriched hom]]

$$
  Desc(Y,X) = sPsh(Y,A)
  \,.
$$

If we decompose

$$
  Y = \int^{[n] \in \Delta} \Delta[n] \cdot Y_n
$$

into its cells by a [[coend]], where now each $Y_n$ is a Set-valued [[presheaf]] (see [[co-Yoneda lemma]]), then this is

$$
  \begin{aligned}
    Desc(Y,A)
    &=
    sPSh(\int^{n \in \Delta} \Delta[n] \cdot Y_n, A)
    \\
    & =
    \int_{n \in \Delta} sPSh(\Delta[n] \cdot Y_n , A)
    \\
    & = \int_{n \in \Delta} sSet(\Delta[n], sPSh(Y_n,A))
   \\
    &= Tot( sPSh(Y_\bullet, A))
  \end{aligned}
  \,,
$$

where the equality signs are [[isomorphisms]] of [[simplicial sets]], the outside integral sign denotes the [[end]], and in the integrand we are using that [[simplicial presheaves]] are [[simplicially enriched category|simplicially enriched]] and [[tensoring|tensored]] over simplicial sets.

So a standard class of examples of cosimplicial simplicial sets to keep in mind are those obtained by evaluating a [[simplicial presheaf]] degreewise on the components of a [[hypercover]]. Its totalization then is the corresponding [[descent object]].

=--

+-- {: .num_prop}
###### Proposition

For $G^\bullet \to H^\bullet$ a morphism of [[cosimplicial object|cosimplicial]] [[groupoid]]s which is degreewise an [[equivalence of categories|equivalence]], also the induced morphism of [[totalization]]s

$$
  Tot(G^\bullet) \to Tot(H^\bullet)
$$

is a weak equivalence (of [[simplicial set]]s).

=--

This is ([Jardine, corollary 12](#Jardine)).

+-- {: .num_defn #RestrictedTotalization}
###### Definition

Let $\Delta_+ \hookrightarrow \Delta$ be the [[subcategory]] of the [[simplex category]] on the co-face maps. Write $rTot$ for the corresponding [[totalization]], called the **restricted totalization**.

=--

+-- {: .num_prop}
###### Proposition

For $G^\bullet \to H^\bullet$ a degreewise weak equivalence of [[strict 2-category|strict]] [[2-groupoid]]s, the resulting morphism of connected components of [restricted totalizations](#RestrictedTotalization)

$$
  rTot(G^\bullet) \to rTot(H^\bullet)
$$

is a [[weak equivalence]].

=--

This is ([Prezma, theorem 6.1](#Prezma)).

Retsricted to $\pi_0$ this statement appeared as ([Yekutieli, theorem 2.4](#Yekutieli)). Notice  that it is indeed necessary to use the restricted totalization instead of the ordinary totalization here.

## References

The [[Reedy model structure]] on $sSet^{\Delta}$ is discussed in  Chapter X of

* [[Aldridge Bousfield]] and [[Dan Kan]], _Homotopy limits, completions and localizations_ Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.
 {#BousfieldKan}

The [[model structure on functors|injective model structure]] is discussed in

* [[Rick Jardine]], _Cosimplicial spaces and cocycles_ ([pdf](http://www.math.uwo.ca/~jardine/papers/preprints/cosimp5.pdf))
 {#Jardine}

Totalization of cosimplicial strict [[2-groupoids]] is considered in

* [[Amnon Yekutieli]], _Combinatorial Descent Data for Gerbes_ ([arXiv:1109.1919](http://arxiv.org/abs/1109.1919))
 {#Yekutieli}

and

* [[Matan Prezma]], _Descent data of cosimplicial 2-groupoids_ ([arXiv:1112.3072](http://arxiv.org/abs/1112.3072))
 {#Prezma}

[[!redirects model structure on cosimplicial spaces]]


model structure on crossed complexes

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#Contents#
* table of contents
{:toc}

## Idea

A [[model category structure]] on the category of [[crossed complexes]], due to [Brown & Golasiński 1989](#BrownGolasiński89).

The key facts used here are that the category of crossed complexes is monoidal closed, and that there is a good interval object, essentially the groupoid $\mathbf I$ extended to be a crossed complex.

Notice that by other work of Brown and Higgins, crossed complexes are equivalent to [[strict omega-groupoid|strict globular omega-groupoids]], and also to strict cubical omega-groupoids with connections. In fact the monoidal closed structure on crossed complexes can be deduced from that on strict cubical omega-groupoids.

## Related entries

* [[model structure]]

* [[crossed complex]]

## References

The original article:

* {#BrownGolasiński89} [[Ronald Brown]], [[Marek Golasiński]]: _A model structure for the homotopy theory of crossed complexes_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, **30** 1 (1989) 61-82 [[numdam:CTGDC_1989__30_1_61_0](http://www.numdam.org/item/?id=CTGDC_1989__30_1_61_0)]

See also

* Ronald Brown, [[Philip J. Higgins]], and [[Rafael Sivera]], _[[Nonabelian algebraic topology]]: filtered spaces, crossed complexes, cubical homotopy groupoids_, EMS Tracts in Mathematics Vol 15, (2011).

* [[Dimitri Ara ]] and [[Francois Metayer|François Métayer]],  The Brown-Golasiński model structure on strict ∞-groupoids revisited. Homology Homotopy Appl. 13 (2011), no. 1, 121–142.

* [[Georges Maltsiniotis]] La catégorie cubique avec connexions est une catégorie test stricte. Homology, Homotopy Appl. (11)  (2) (2009) 309 -- 326.


model structure on cubical presheaves

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=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[site]], there is a [[model category]] structure on the category of [[cubical objects]] in the [[presheaf topos]] over that site, hence on the category of [[cubical set]]-valued [[presheaves]], such that at least the corresponding [[homotopy category]] is [[equivalence of categories|equivalent]] to that of the corresponding local [[model structure on simplicial presheaves]] ([Jardine 03, theorem 71](#Jardine03)) (the latter being known to [[locally presentable (infinity,1)-category|present]] the [[(infinity,1)-topos]] over the given site).

## Related concepts

* [[model structure on simplicial presheaves]]

* [[model structure on simplicial sheaves]]

* [[model structure on presheaves of spectra]]

* [[model structure for (2,1)-sheaves]]

## References

* {#Jardine03} [[Rick Jardine]], _Categorical homotopy theory_, 2003 ([web](http://www.math.uiuc.edu/K-theory/0669/))