E X A M P L E S O F M O D E L

S T R U C T U R E S

damien lejay

April 2017

Abstract. — We give an introduion to the andard model ruures on the

category of unbounded chain complexes by deﬁning ﬁr model ruures on

the category of Abelian groups.

contents

1 Reminder on why we like model categories 1

2 The father of all examples 2

3 Model ruures on Abelian groups 4

3.1 The andard projeive model ruure 5

3.2 The andard injeive model ruure 7

4 The model ruure on chain complexes over a ﬁeld 9

5 Model ruures on chain complexes of Abelian groups 12

5.1 The andard projeive model ruure 12

5.2 The andard injeive model ruure 13

References 14

1 reminder on why we like model categories

The derived category

D(Z)

is by deﬁnition the localisation of the category of

chain complexes where the set of maps we would like to invert is the set of all

quasi-isomorphisms Q:

D(Z) ≃ Q

−1

Ch(Z)

In fa, it is deﬁned as the model of localisation whose set of objes is the

set of all chain complexes (the localisation of a category is only deﬁned up to

equivalences of categories).

Given a category

C

and a set

S

of arrows of

C

, if

C

is known to be a

U

-small

category, then any model of the localisation

S

−1

C

is garanted to be equivalent

to a

U

-small category. Apart from that, essentially nothing is know on the

concrete description of the localisation. If we choose the model which has

the same objes, then the diﬃculty is to underand the set of arrows. In

full generality, the be we know is that it can be described as some huge

‘zig-zags’.

The situation is much better in the following two cases.

— The set S admits a calculus of fraions;

— The category C can be enhanced into a model category.

In the ﬁr case, the morphisms

X → Y

of the localisation may be described

by ‘fraions’ of one of the two types:

X

s

←− Z

f

−→ Y || X

f

−→ Z

s

←− Y

These notes were written for the ‘derived seminar’ I organised at IBS Centre for geometry and

physics in Pohang. Copyright (C) Damien Lejay 2017.

1

with

s ∈ S

, depending on whether

S

admits a right or left calculus of frac-

tions [1].

In the second case, a new model for the localisation may be proposed. The

ruure of model category allows us to isolate the subcategory of ﬁbrant-

coﬁbrant objes. The localisation — called the homotopy category in this

context — is then given by deﬁning maps between ﬁbrant-coﬁbrant objes as

homotopy equivalence classes of morphisms in the category C.

The ﬁr case is less convenient to work with because before arting to

compute the equivalence classes of arrows up to homotopy, one ﬁr has to

replace objes by equivalent ﬁbrant-coﬁbrant objes. Alas, the set

Q

of all

quasi-isomorphisms does not have a calculus of fraions.

Proposition. — The set

Q

of quasi-isomorphisms in

Ch(Z)

does not admit a

calculus of fraions.

Proof. — We will show that

Q

does not satisfy the — left or right — cancellab-

ility condition.

Let

A

be the chain complex

Z

Id

−→ Z

and let

f : A → Z

be the projeion

on the ﬁr term. Then

s : 0 → A

is a quasi-isomorphism such that

f ◦ s = 0

but it is impossible to ﬁnd a quasi-isomorphism

s

′

: Z → B

such that

s

′

◦f = 0

(since f is surjeive, that would mean s

′

= 0).

In the other direion let

i : Z → A

be the injeion to the second term.

Then if

t

is the quasi-isomorphism

A → 0

, we have

t ◦ i = 0

. But since

i

is

injeive, it is impossible to have a quasi-ismorphism

t

′

: B → Z

such that

i ◦ t

′

= 0.

Hence, we are doomed to ﬁnd a model ruure on the category of chain

complexes. The minimum requirement to be able to ﬁnd such ruure with

weak equivalences the set

Q

is that

Q

be able under retras and satisfy the

2 out of 3 property.

Proposition. — The set

Q

of quasi-isomrophisms of

Ch(Z)

is able under retras

and satisﬁes the 2 out of 3 property.

Proof. — The 2 out of 3 property is an immediate consequence of the funtori-

ality of the homology funor C 7→ H

∗

(C).

The same is true for the ability under retras, if

g

is a retra of a

given chain map

f

then,

H

∗

(g)

is a retra of

H

∗

(f )

and any retra of an

isomorphism is an isomorphism.

Remark. — More generally, given any funor

F : C → D

, the set of arrows

S

deﬁned by

s ∈ S

if and only if

F(s)

is an isomorphism, will satisfy the 2 out of

3 property and be able under retras.

This means aually that the two condition imposed on a set to be weak

equivalences of a model ruure are faily weak: for any category

C

and set

of arrows

S

, the set of arrows

W

in

C

that become and isomorphism in the

localisation

S

−1

C

is able under retra and satisﬁes the 2 out of 3 property.

And of course we have S

−1

C ≃ W

−1

C.

2 the father of all examples

Before inveigating the possible model ruures on the category of chain

complexes, we shall ﬁr focus on more elementary Abelian categories.

Let

F

be your prefered ﬁeld. Let

Vect

F

denote the category of veor

spaces over

F

. We shall classify all the possible model ruures on

Vect

F

and we shall discover that only two sets of arrows can be taken as set of weak

equivalences for a model ruure.

The key to classify model ruures on

Vect

F

is to ﬁr classify lifting

syems (L, R). We shall see that there are only a handful of them.

2

So let

(L, R)

be a lifting syem i.e the elements of

L

are the maps

l

such

that the following diagram can be ﬁlled up

A C

B D

l

r

for every arrow

r ∈ R

. In the same fashion, the set

R

is required to be deﬁned

as the maps

r

that lift again all maps of

L

. Hence by deﬁnition,

L

contains

all isomorphisms.

Case 1. — L = {iso}. In this case, R = {all} and we have a ﬁr pair

(iso, all)

Case 2. —

L

contains a map

f

that not injeive. Any map

f

retras to the

zero map

Ker f → 0

; in this case, since the kernel of

f

is not zero,

f

retras

to the zero map

π : F → 0.

So by the ability properties of lifting syems, we know that

π

is in

L

.

Computing the set of maps that can be lifted again π, we get

π

l

= {injections}

Let ι be the zero injeion

ι : 0 → F

By computation we have

l

ι = {surjections}.

Hence, we get that

{surjections} =

l

ι ⊂

l

(f

l

) ⊂ L

Case 3. —

L

contains a map

f

that is not surjeive. Then in this case, it is

possible to retra f to ι. By computation,

ι

l

= {surjections}

Since we also have

l

π = {injections},

we have

{injections} =

l

π ⊂

l

(f

l

) ⊂ L

Conclusion. — Since

L

is also able by composition, we have only four possible

choices for L and they correspond to the following lifting pairs

(iso, all) (inj, surj) (surj, inj) (all, iso)

Notice that all those four lifting pairs are also weak faorisation syems.

Indeed, the faorisation for the pair (surj, inj) is the image faorisation

A B

Im(f )

f

and it is the graph faorisation for the pair (inj, surj):

A B

A ⊕ B

f

Γ

π

B

3

Getting back to ﬁnding all model ruures, we are looking for three sets

of maps

(W, C, F)

such that

(W ∩ C, F)

and

(C, W ∩ F)

are weak faoriion

syems and

W

satisﬁes the 2 out of 3 property. Since there are only four

lifting syems, we can deduce that any set

W

has to contain either

iso, inj, surj

or be the set of all arrows.

Suppose

W

contains the set of all injeive maps. Then give any map

f : A → B, since we have the following commutative diagram

0 B

A

f

by the 2 out of 3 property, f is in W and W = {all}.

Suppose now that

W

contains the set of all surjeive maps. Then for any

f : A → B, the following diagram

A 0

B

f

and the 2 out of 3 property means that f is in W and W = {all}.

We can know li all the model ruures on the category Vect

F

:

— (iso, all, all), it is the trivial model ruure;

— (all, iso, all)

, it is also a trivial model ruure but the homotopy cat-

egory in this case is the point;

— (all, all, iso), analogous to the previous one;

— (all, surj, inj);

— (all, inj, surj).

Remark. — The same inveigations can be made for

Set

the category of sets.

However, since it is usually impossible to retra to the empty set, there

are a total of 9 diﬀerent model ruures on

Set

with 3 diﬀerent homotopy

categories.

As we can see, only the la two model ruures are not trivial. But the

very la one is far more useful than the other. We shall call it the andard

model ruure on

Vect

F

. The reason behing this is that both the set of ﬁbrant

objes and coﬁbrant objes are reduced to

0

for the model ruure given

by the triplet

(all, surj, inj)

. Hence, there is no space for tweaking: if we

would like to change some parts of the ruure and keep the ﬁbrations for

examples, then whatever we do, the set of ﬁbrant-coﬁbrant objes is ill

going to be reduced to a point. The same happen if we would like to keep the

coﬁbrations. On the opposite, in the andard model ruure, every obje is

both ﬁbrant and coﬁbrant!

Theorem (Experimental fa). — All useful model categories are based on the

andard model ruure.

3 model structures on abelian groups

The ﬁr main diﬀerence between

Ab

the category of Abelian groups and

Vect

F

is that we ill do not have a complete classiﬁcation of objes in

Ab

[

2

,

3

].

As a consequence, it is impossible to compute all the model ruures on

Ab

. What we shall do inead, is to build model ruures that mimic the

andard model ruure on Vect

F

.

Fa. — The pair (inj, surj) is not a lifting pair in Ab.

4

To see this, have a look to the following commutative square,

0 Z

Z

2

Z

2

mod

2

Id

3.1 The andard projeive model ruure

Idea 1. — Keep the ﬁbration and look for the corresponding left part of the

pair.

Fir let us look at the corresponding coﬁbrant objes.

Definition 3.1. — An abelian group

P

is called projeive if the map

0 → P

satisﬁes the left lifting property again surjeions. That is, we can always ﬁnd a

lift for the following types of diagrams,

X

P Y

Proposition 3.2. — The projeive Abelian groups are the free Abelian groups.

Proof. — If

P

is a free Abelian group then it is projeive as any map with

domain P is only determined by the image of elements of a basis of P.

Suppose P is projeive and let L(P) be the free Abelian group generated

by the elements of

P

. Then because

P

is projeive we get the following

retraion

L(P)

P P

Id

So

P

can be seen as a subgroup of a free Abelian group and since

Z

is a PID,

it means P itself is a free Abelian group.

We can now generalise this setting to describe all the maps that can be

lifted again surjeions.

Proposition. — A map

f : A → B

is inside

l

surj

if and only if it is a (split)

injeion with a free cokernel.

Proof. — If

f

is an injeion with a free cokernel, then it has to be split (thanks

to the lifting property of free modules), so we can write

B ≃ A ⊕ Coker f

and

f

is the identity of

A

. Hence, we can split the problem into a lifting

problem for

Coker f

and

Id

A

. We have already deﬁned the former to be

objes that solve that very problem and isomorphisms can always be lifted.

Suppose now that

f

satisﬁes the left lifting property. Denote by

π

the

projeion

π : B −→ Coker f

To show that the cokernel is free, we will show that it is projeive. Let us be

given any diagram

0 X

Cocker f Y

p

g

5

and we wish to lift

g

. Composing,

g

with

π

gives us a map

B → Y

whose

composition with

f

is equal to zero. Hence we can complete the former

diagram into

A 0 X

B Cokerf Y

f

p

π

g

Using the lifting property of f , we can complete the square in

A X

B Y

0

f

p

g◦π

ϕ

But since

ϕ

composed with

f

is zero, by the universal property of the cokernel,

we get a new map ϕ : Coker f → X such that ϕ = ϕ ◦ π. So we now have

p ◦ ϕ ◦ π = g ◦ π

and since π is an epimorphism we get the following lifting

0 X

Coker f Y

p

g

ϕ

This proves that the cokernel has to be projeive. Since maps with a projeive

cokernel split, we can then rewrite B as the sum

B ≃ Im f ⊕ Coker f

And we can now reri our attention to the case where the cokernel is zero.

In this case the following diagram

A A

B B

Id

f f

Id

tells us that we can split A into

A ≃ Ker f ⊕ B

This la diagram will tell us that the kernel of f has to be zero:

Ker f ⊕ B Ker f

B 0

π

Ker f

π

B

Corollary. — The pair made of injeive maps with free cokernels on one side

and surjeive maps on the other side, is a lifting pair.

Proof. — All we need to prove is that any surjeion has the right lifting

property again any injeive map

f : A → B

with free cokernel. But as we

have already said, such a map can be split into the sum of an isomorphism

(which can be lifted) and the zero map

0 → Coker f

which by deﬁnition of

projeiveness can also be lifted.

Theorem. — There exis a model ruure (W, C, F) on Ab where:

— W is the set of all maps;

6

— F is the set of all surjeive maps;

— C is the set of all injeive maps with free cokernel.

This model category is called the andard projeive model ruure.

Proof. — Since

W ∩ F = F

and

W ∩ C = C

, the only thing that is left to

prove is the faoriion property. Let

f : A → B

be any map between Abelian

groups. Let

L(B)

be the free Abelian group with basis the elements of

B

,

the canonical map

π : L(B) → B

is surjeive. We obtain the sought for

faorisation as

A B

A ⊕ L(B)

f

ι

A

f +π

where

ι

A

is the inclusion on the ﬁr faor. The cokernel of

ι

A

is

L(B)

and it

is free.

Remark. — The andard projeive model ruure will exis for every

category of modules over a commutative ring. Ju replace free Abelian

groups by the corresponding projeive modules over the given ring (if the

ring is not a PID).

Remark. — In this model ruure every obje is ﬁbrant but the only coﬁbrant

objes are the free Abelian groups. Many times in the proves, we have used

the fa that for every Abelian group

B

, there exis a surjeion

L(B) ↠ B

with

L(B)

a free Abelian group. This is nothing else than the coﬁbrant replacement

of B.

3.2 The andard injeive model ruure

Idea 2. — Keep the weak equivalences and the coﬁbrations.

In this case let us give a name to the new ﬁbrant objes.

Definition 3.3. — An Abelian group

I

, is injeive if it satisﬁes the right lifting

property again all injeions, i.e if every such diagrams can be ﬁlled,

X I

Y 0

Proposition. — The injeive Abelian groups are the divisible Abelian groups,

that is the Abelian groups

I

such that for every

a ∈ I

and

n ∈ Z

, there exis

b ∈ I

such that a = nb.

Proof. — The divisibility condition for an Abelian group

I

is equivalent to

being able to ﬁll the following diagrams

Z I

Z 0

×n

Hence every injeive Abelian group is divisible. Conversely, given a divisible

Abelian group

I

, a map

f : X → I

and an inclusion

X ⊂ Y

, we will show

that

f

can be extended to

Y

. Let

y

be an element of

Y

that is not in

X

. Then

either

y

is free relative to

X

and

f

may be extended to

X ⊕ Zy

by choosing

any element r ∈ I and letting f (x + ny) = f (x) + n f (y) be the extension.

Either there exis an element

x ∈ X

and a prime number

p ∈ Z

such that

py = x

7

in which case, thanks to the divisibility of

I

, there is an element

r ∈ I

such

that

pr = f (x)

. Then to any element

x + my ∈ X + Zy

, the extended map

is given by

f (x + my) = f (x) + m r

. The extended map is correly deﬁned

since f (py) = pr = f (x).

This extension can be performed on any submodule of

Y

on which

f

has previously been extended. The results then comes from (transﬁnite)

induion.

Definition 3.4. — For a prime number

p

, let

Z

p

∞

be the Abelian group colimit of

the diagram of inclusions Z

p

n

,→ Z

p

n+1

.

Remark. — Since every cyclic group Z

p

n

can be represented as the quotient

Z

p

n

≃ Z ⟨1/p

n

⟩/Z

the group Z

p

∞

may be seen as

Z

p

∞

≃ Z[1/p]/Z.

where

⟨.⟩

denotes group generation and

[.]

denotes ring generation. It can also

be described as

Z

p

∞

≃ Q/pZ.

Proposition (Classiﬁcation of injeive Abelian groups). — Every injeive

Abelian group is a sum of the groups Q and Z

p

∞

for every prime p.

Proof. — Fir

Q

and

Z

p

∞

are injeive because they are divisible. This is

obvious for the ﬁr one and a consequence of Bezout theorem for the second

one.

Let

I

be any injeive Abelian group, we will show that

I

splits as the sum

of its injeive subgroups ‘divisibly generated’ by one element. Let

x

be an

element of

I

. If

x

is not a torsion element, since

I

is divisible, it deﬁnes an

injeion

ι

x

: Q → I

and as

Q

is injeive,

I

splits into a sum

I ≃ Q ⊕ J

where

J

is also divisible (every quotient of a divisible group is divisible).

Hence we can now suppose that

I

is equal to its torsion subgroup. For

a given prime number

p

, let

x

be an element of

p

-torsion of

I

. Since

I

is

divisible, for every k ≥ 1, there exis an element x

k

∈ I such that

x = p

k−1

x

k

The subgroup of

I

generated by the

x

k

for

k ≥ 1

is isomorphic to

Z

p

∞

. The

map sends the generators

p

−k

to

x

k

and all we need to show is that every

linear equation

N

X

k=1

a

k

x

k

= 0

with all

a

k

integer numbers not divisible by

p

, image of a element

P

N

k=1

a

k

/p

k

in

Z

p

∞

, is in fa impossible. For every

k

, by conruion

p

k

x

k

= 0

. It follows

that

p

N−1

a

N

x

N

= 0

. But since

p

N

x

N

= 0

it means that

p

divides

a

N

which is

impossible unless a

N

= 0. This shows that the built map

Z

p

∞

−→ I

is injeive. The end of the proof consis in a usual (transﬁnite) induion.

Remark. — By the classiﬁcation of divisible Abelian groups, we see that they

are exaly the Abelian groups where the ruure of

(Z, +)

-module can be

extended to a ruure of (Q, +)-module.

Proposition (Fibrant replacement). — The induion funor

Q ⊗

(Z,+)

: Ab −→ Mod

(Q,+)

8

is left adjoint to the forgetful funor and is such that the unit of the adjunion is

a monomorphism i.e for every Abelian group A, the canonical map

A ,→ Q ⊗

(Z,+)

A

is an injeion.

In particular, we get the following replacements:

Z ,→ Q ; Z

p

n

,→ Q/p

n

Z ≃ Z

p

∞

Proof. — The adjunion is a andard result in the theory of modules over

monoids. The only thing we have to prove is that the unit is a monomorphism.

Let

a

be any element of

A

, then its image by the unit is

(0, a)

which is equal

to 0 if and only if a is equal to zero in A.

Now that we have dealt with the necessary prerequisites, we can describe

the new model ruure.

Theorem. — There exis a model ruure on the category of Abelian groups

where

— any map is a weak equivalence;

— a map is a coﬁbration if and only if it is injeive;

—

a map is a ﬁbration if and only if it is a (split) surjeion with divisible

kernel.

This model ruure is called the andard injeive model ruure.

The proof of this theorem is a complete mirror of the proof of the exience

of the projeive model ruure (once you know how to build a ﬁbrant

replacement). Indeed, it is the projeive model ruure on the category

Ab

op

.

Remark. — As we see, the key fa about the category of Abelian groups is

that we are always able to ﬁnd, for every group

A

, a surjeive map to

A

with

free domain and an injeive map from

A

to a divisible group. We say that

the category of Abelian groups has both enough projeives and injeives.

4 the model structure on chain complexes over a field

Let us move on to the category

Ch(F)

of unbounded chain complexes over

your cheriched ﬁeld

F

. This time we will build a model ruure with non-

trivial weak equivalences, namely the set Q of quasi-isomorphisms.

But ﬁr we may warn the reader of the following fa.

Proposition. — Any chain complex in Ch(F) retras to its homology.

Proof. — Let

... C

1

C

0

C

−1

...

2

1

0

−1

be any chain complex over

F

. Let us denote by

B

n

the image of the map

n+1

in

C

n

and let

Z

n

be the kernel of

n

inside

C

n

. Finally let

H

n

be the quotient

Z

n

/B

n

. Since on a ﬁeld, every submodule is a dire summand, we get that

for every n,

C

n

≃ B

n

⊕ H

n

⊕ B

n−1

.

Hence it is very raighforward to build a retraion map

... C

1

C

0

C

−1

...

... H

1

H

0

H

−1

...

2

1

π

1

0

π

0

−1

π

−1

0 0 0 0

9

with raighforward seion

... C

1

C

0

C

−1

...

... H

1

H

0

H

−1

...

2

1

0

−1

0 0

ι

1

0

ι

0

0

ι

−1

Remark. — Although every chain complex retras to its homology, it is not

true that every map of chain complexes f retras to H(f ).

As a consequence, we have the following description of the derived cat-

egory of chain complexes over F.

Theorem. — The homotopy category

D(F)

is equivalent to the category of graded

veor spaces Mod

Z

F

.

Proof. — Thanks to the previous proposition, the homology funor

H : Ch(F) → Vect

Z

F

is left adjoint to the inclusion of graded veor spaces inside chain complexes

Vect

Z

F

Ch(F)

H

with π being the unit of the adjunion and ι being the counit.

Moreover, a chain map is an isomorphism precisely if its image by

H

is

an equivalence of graded veor spaces. Thus, the derived category is even a

reﬂeive localisation of Ch(F).

As a consequence, we do not need any model ruure on the category of

chain complexes to be able to underand its homotopy category. Nonethe-

less, underanding the andard model ruure on the category of chain

complexes over a ﬁeld will be an important ep toward the model ruures

we are going to build on the category of chain complexes of Abelian groups.

Theorem. — There exis a model ruure on the category of chain complexes

over F where

— the weak equivalences are the quasi-isomorphism;

— the ﬁbrations are the epimorphism;

— the coﬁbrations are the monomorphisms.

This model rure is called the andard model ruure.

Proof. — In view of the total symmetry of the given ruure, we shall

only prove that the pair made of trivial coﬁbrations and ﬁbrations is a weak

faorisation syem. The same proof applied to

Ch(F)

op

will prove that

coﬁbrations and trivial ﬁbrations also form a weak faorisation syem.

Let us ﬁr setup some notations. We will call

ι

n

the following trivial

coﬁbration

... 0 0 0 ...

... F[n + 1] F[n] 0 ...

Id

F

A map of complexes

f

has the right lifting property again

ι

n

if and only if

f

n+1

is surjeive, hence

{ι

n

}

l

n∈Z

= {ﬁbrations}

10

Let

π

n

denote the surjeion

π

n

: F[n] −→ 0

. A map

f

has the left lifting

property again

π

n

if and only if

H

n

(f )

is injeive. Finally let us denote by

p

n

the following ﬁbration

... F[n + 1] F[n] 0 ...

... F[n + 1] 0 0 ...

Id

F

Id

F

A map of chain complexes

f

has the left lifting property again

p

n

if and

only if H

n+1

(f ) is surjeive and f

n

is injeive. As a consequence

l

{π

n

, p

n

}

n∈Z

= {trivial coﬁbrations}

Laly we need to show that it is aually possible to lift any of our deﬁned

trivial coﬁbrations again a ﬁbration. Let

f : X → Y

be a trivial coﬁbration

and g : R → S be a ﬁbration and let be given a commutative square

X R

Y S

r

f

∼

g

s

We will conru a map

ϕ : Y → R

to ﬁll up this diagram. A andard rategy

would be to art lifting

ϕ

0

and then induively build the other

ϕ

n

but since

we are dealing with unbounded chain complexes there isn’t really a right

place to art and we will see that being able to lift is in fa some kind of a

global property of a complex. Let us adopt another rategy.

Fir we may split Y as a sum

Y ≃ X ⊕ Coker f

and since

f

is a quasi-equivalence, its cokernel is aually quasi-isomorphic to

zero. As it is always possible to lift an isomorphism, we can suppose that

X

is

zero and that

Y

is quasi-isomorphic to zero. Hence what we shall prove is that

trivial coﬁbrant objes lift againts ﬁbrations. We wish to lift the following

diagram

0 R

Y S

∼

g

t

Since

g

n

is a surjeion for every

n

, it is possible to ﬁnd a map

ϕ

n

: Y

n

→ R

n

lifting s

n

. This gives us a map ϕ of graded veor spaces but it may fail to be

a chain map. Let ψ be deﬁned as the failure of ϕ to be a chain map:

ψ

n

=

n+1

◦ ϕ

n+1

− ϕ

n

◦

n+1

By conruion

ψ

is a chain map from

Y

to

R[1]

and since

ϕ

is a lift again

g

,

the image of ψ is contained in (Ker g)[1],

(Ker g)[1]

Y R[1]

g[1]

ψ

Now thanks to the exaness of the complex

Y

, it is aually homotopic to zero

(since exa complexes are split over a ﬁeld). Hence, we can ﬁnd a homotopy

h between ψ and 0:

ψ

n

=

n+1

◦ h

n+1

− h

n

◦

n+1

We now deﬁne a new map

τ = ϕ − h

11

By conruion

τ

is a chain map from

Y

to

R

. By conruion also, since

h

has its values in Ker g, we have

g ◦ τ = t

And the diagram is ﬁlled.

We now approch the very ﬁnal ep: showing that any map can be faored

into a trivial coﬁbration followed by a ﬁbration. Let

f : X → Y

be any map of

chain complexes. Let also

e

Y

be the chain complex quasi-isomorphic to zero

such that

e

Y

0

= Y

,

e

Y

−1

= Y

and the only non-zero map is the identity of

Y

. Let

π :

e

Y → Y

be the ﬁbration deﬁned by projeion on the

0

faor. Then we can

faor the map f as follows,

X Y

X ⊕

e

Y

f

ι

∼

f ⊕π

where ι is the inclusion on the ﬁr faor.

5 model structures on chain complexes of abelian groups

5.1 The andard projeive model ruure

The situation for the andard projeive model ruure on chain complexes

of modules over a ring is aually a bit worse than for the projeive model

ruure on the category of modules or on chain complexes over a ﬁeld.

Indeed, it is not true for example that for any ring, a chain complex of free

modules is going to have the left lifting property again all epimorphisms.

Here is a famous counter example: the ground ring is

Z

4

and the following

chain complex is only made of free modules,

. . . Z

4

Z

4

Z

4

. . .

×2 ×2 ×2 ×2

It is an exa complex but as it does not split, it is not homotopic to zero! Hence

it can’t be a coﬁbrant obje and as a consequence, ﬁnding free resolutions is

no longer enough to build coﬁbrant replacement... On a PID, the situation is

aually much better.

Proposition. — Any exa chain complex of free Abelian groups is homotopic to

zero.

Proof. — Let

C

be an exa chain complex of free Abelian groups. For every

n ∈ Z

,

B

n

⊂ C

n

is a subgroup of a free Abelian groups, hence it is also free.

Moreover, since

C

is exa, the quotient

C

n

/B

n

is isomorphic to the image

B

n−1

which is free (as a subgroup of a free Abelian group) so that

C

n

can

aually be split

C

n

≃ B

n

⊕ B

n−1

And split exa chain complexes are homotopic to zero.

Thanks to this proposition we can now deﬁne the andard model ruure

on the category of chain complexes of Abelian groups.

Theorem. — There exis a model ruure on the category

Ch(Z)

of chain

complexes of Abelian groups where:

— the weak equivalences are the quasi-isomorphisms;

— the ﬁbrations are the epimorphisms;

—

the coﬁbrations are the split monomorphisms with cokernel made of free

Abelian groups.

12

This model ruure is called the andard projeive model ruure.

Remark. — In the case of the category of chain complexes over a non PID ring,

the andard model ruure ill exisits but the coﬁbrant objes are a bit

diﬀerent: there are the complexes

P

made of projeive modules such that

every map

P −→ K

with

K

quasi-isomorphic to zero, is homotopic to the zero map. We shall call

such a chain complex dg-projeive.

The proof of the exience of the projeive model ruure is essentially

the same as the one on chain complexes over a ﬁeld; the previous proposition

guaranties that chain complexes of free Abelian group are coﬁbrant.

In order to prove the two faorisations some extra work is needed and in

particular, the following propoion mu be used.

Proposition. — Given a chain complex of Abelian groups

C

there exis a chain

complex of free Abelian groups L and a trivial ﬁbration

L C

∼

Proof. — We are going to give the reciepe to build a dg-projeive resolution

(the fa that the ring is a PID is not relevant here). Fir truncate on the right

the chain complex and then build induively a projeive resolution. This is

always possible since there are enough projeives. We then have

P

≥0

C

≥0

∼

Then funorially complete the projeive resolution into a projeive res-

olution

P

≥n

→ C

≥n

for

n ≤ 0

. I.e for every negative

n

we want to have the

following diagram

P

≥n

P

≥n−1

C

≥n

C

≥n−1

∼

∼

Then take the ﬁltered colimit indexed by

ω

0

to get a dg-projeive resolu-

tion. This happens because both weak equivalences and ﬁbrations are able

under ﬁltered colimits. As such, any ﬁltered colimit of dg-projeive chain

complexes is again dg-projeive and every ﬁltered colimit of resolutions is a

resolution. Laly, every right bounded chain complex of projeive modules is

dg-projeive

5.2 The andard injeive model ruure

For injeive modules, the same problems appear as with the projeive mod-

ules. Namely, some chain complexes of injeive modules may not be ﬁbrant

in the andard model ruure. As before, the situation on a PID is much

better.

Proposition. — Any exa chain complex of divisible Abelian groups is homotopic

to zero.

Proof. — The proof boils down to the following fa: every quotient of a

divisible group is again divisible. This means that given any complex of

divisible groups

C

, for every

n

the boundaries subgroup

B

n

⊂ C

n

is divisible

and then

C

n

may be split into

C

n

≃ B

n

⊕ B

n−1

if

C

were exa. This means

that C is homotopic to zero.

Theorem. — There exis a model ruure on the category of chain complexes of

Abelian groups where:

— the weak equivalences are the quasi-isomorphisms;

13

— the coﬁbrations are the monomorphisms;

—

the ﬁbrations are the epimorphisms with kernel made of divisible Abelian

groups.

This model ruure is called the andard injeive model ruure.

In the same way that we reﬁne the deﬁnition of coﬁbrant objes, we may

deﬁne a dg-injeive chain complex as a complex

I

made of injeive modules

such that for every map

K −→ I

with K quasi-isomorphic to zero, is homotopy equivalent to zero.

But contrarily to the projeive case, we need to use one more assumption

to be able to build a dg-injeive resolution for every chain complex: we need

to use the fa that ﬁltered colimits in the Abelian category we are working

in are exa i.e that they preserve the monomorphisms. Apart from that the

proves are totally similar.

references

[1] Gabriel P. and Zisman M., Calculus of fraions and homotopy theory,

vol. 35. Springer Berlin Heidelberg, 1967.

[2] Fuchs L., Inﬁnite Abelian groups. Vol. I. Academic Press, 1970.

[3] Fuchs L., Inﬁnite Abelian groups. Vol. II. Academic Press, 1973.

14