Download NOTE ON COFIBRATION In this overview I assume

NOTE ON COFIBRATION
In this overview I assume, that all the topological spaces are path connected if not
stated otherwise.
Definition 1. Let X, Y be topological spaces. If a map f : X → Y induces isomorphism on the homotopy grups πn (Y ) ∼
= πn (X) for all n ≥ 0, we call it weak homotopy
equivalence (or weak equivalence for short).
Lemma 2. Let f : X → Y, g : Y → Z be maps of topological spaces. Then if two out
of three maps f, g, gf are weak equivalences, the third is a weak equivalence as well.
For a proof it suffices to consider induced group homomorphisms f∗ : π∗ (X) →
π∗ (Y ), g∗ : π∗ (Y ) → π∗ (Z), gf∗ : π∗ (X) → π∗ (Z). If two of these three maps are
isomorphisms, then the reamining one must also be an isomorphism.
The condition of being a weak equivalence ca be also expressed more graphically:
Lemma 3. The map f : X → Y is a weak equivalence iff in the following diagram
where h is homotopy rel S n−1 the upper triangle commutes strictly and the lower triangle commutes up to homotopy for all n > 0:
S n−1
_
i
y
p
h
y
Dn
y
/X
y<
/
q
f
Y
Proof. We can replace the space Y with the mapping cylinder Mf . Obviously they are
homotopy equivalent, so we can restate the statement as follows: The map f : X → Y
is weak homotopy equivalence iff the following diagram commutes for all n > 0 up to
homotopy:
S n−1
_
i
y
Dn
p
h
y
y
q
y
/X
y<
/
f
Mf
The space X is now just a subspace of Mf , so we can consider the long exact
sequence of homotopy groups for pair (Mf , X). We know that a map (Dn , S n−1 , s0 )
represents zero in (Mf , X, x0 ) iff it is homotopic rel S n−1 to a map with image in X.
(the point x0 ∈ X does not play any role here since X is path–connected). It follows,
that πn (Mf , X) = 0 and hence πn (Y ) ∼
= πn (Mf ) ∼
= πn (X).
We wish to define class of maps called cofibrations. To do so, we first have to
definedefine the fibrations.
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NOTE ON COFIBRATION
Definition 4. Serre fibration (fibration) is a map of topological spaces f : X → Y
such that for arbitrary maps p, q, that make the outter square commute, there exists
a lift h such that the following diagram commutes:
p
I n × _ 0
i
w
h
w
In × I
w
w
/X
w;
/
q
f
Y
The fibrations and weak equivalnces then give rise to the cofibrations. We now
define them together with some useful notation.
Definition 5.
(1) Given a following commutative diagram, we say, that the map f : X → Y has
right lifting property (RLP) with respect to the map g (or g has left lifting
property (LLP) with respect to f ) if there exists a lift h : B → Y such that
p = hg, f h = q.
p
A
h
g
~
B
~
~
q
/X
~>
f
/
Y
(2) Trivial fibration is a map which is a weak equivalence and a fibration.
(3) We call a map g : A → B cofibration iff it has LLP with respect to any trivial
fibration f : X → Y .
(4) Trivial cofibration is a map which is both a weak equivalence and a cofibration.
If we now think of category where objects are the maps of topological spaces, what
will be the arrows? Intuitively an arrow between the objects f : X → Y, g : A → B
will be a tuple of maps (i, j) such that the following diagram commutes¿
X
f
/
i
X
g
Y
j
/
Y.
Having that in mind, we can say that a retract of a map f is a map f 0 which is a
retract of f in the beforementioned category.
One of the nice properties of our classes of maps already defined is, that they are
closed under taking retracts. We state this fact in the following lemma.
Lemma 6. Retract of cofibration (fibration, weak equivalence) is a cofibration (fibration, weak equivalence).
Proof. We start by proving that the retract of a cofibration is a cofibration. Let
f : A → B be a cofibration and f 0 : A0 → B 0 retract of f . We need to show that in
the diagram
NOTE ON COFIBRATION
f0
/
a
A0
3
X
p
/
B0
b
Y
where p is a trivial fibration, there exists a lift. If we consider a diagram
f0
/
i
A0
B0
/
j
/
r
A
f0
f
/
s
B
/
a
A0
X
p
/
B0
b
Y
in which the maps (r, s), (i, j) establish the retraction. We have a map h : B → Y
as f is cofibration. The composition hj is then the desired lifting.
The proof for the case of fibrations is essentially the same as for cofibrations.
It remains us to show that a retract f 0 of a weak equivalence f is again a weak
equivalence. We use the functoriality of the maps, so we have the diagram in the
category of groups:
i∗
πi A 0
f 0∗
πi B 0
j∗
/
/
πi A
r∗
/
f 0∗
f∗
s∗
πi B
πi A 0
/
πi B 0 .
In the diagram above, both squares commute, f∗ , ri∗ , sj∗ are all isomorphisms and
it follows that f 0 ∗ is an isomorphism. Hence f 0 is a weak homotopy equivalence.
We now want to show the following: Any map f : X → Y can be written as a
composition of trivial cofibration and fibration. We first work with the diagram:
tn,a,b Dn × 0
/
a
X
f
i
tn,a,b Dn × I
/
b
Y
and we consider the pushout
tn,a,b Dn × 0
/
a
/
tn,a,b Dn × I
b
X
i1
X1 A
f
AA
AA
p1 AAA
-
Y
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NOTE ON COFIBRATION
The space X1 is just the space X with the cells Dn glued on it. Hence, X is
homotopy equivalent to X1 . Overmore, from the universal property of pushout, the
map i1 has LLP with respect to all fibrations.
In a similar way again by glueing, we now build i2 : X1 → X2 , . . . , in : Xn−1 → X n
etc. Finally we take colimit of the sequence:
X → X1 → X2 → . . . → X∞ = colimXn
Again, there is a map i∞ : X → X∞ and a mapping p∞ : X∞ → Y , such that
f = p∞ i∞ . Using the so–called small object argument, we see, that i∞ is a weak
equivalence:
Let us take a map s : S k → X. The image of S k in X∞ by the induced map
i∞ s : S k → X∞ can intersect only finite number of cells, so there must exist and
n ∈ N such that the image s(S k ) lies in Xn . As in is a homotopy equivalence, we get
that i∞ is a weak homotopy equivalence.
Also i∞ has LLP with respect to all fibrations (follows from the definition of fibration). Hence i∞ has LLP with respect to all trivial fibrations and i∞ is a (trivial)
cofibration.
Following the small object argument again, the map p∞ is clearly a fibration.
We have shown, that any map f : X → Y can be written as a composition of trivial
cofibration and fibration.
In a similar fashion, we can also prove, that the map f can be written (or decomposed) as a composition of cofibration and trivial fibration. To show that, we use the
same techniques - colimits of pushouts and small object argument. The key difference
is in using the pushouts in the diagram
/
a
tn,a,b S n−1
X
f
i
/
tn,a,b Dn
b
Y
the rest of the proof is left to the reader.
Having proven that any arrow can be decomposed canonically, we can charecterize
the cofibrations more geometerically.
Lemma 7. Suppose f : X → Y is a trivial cofibration. Then it is a retract of a
relative CW complex inclusion.
Proof. We use the previous facts above and factorize f : X → Y as a composition pi :
X → X∞ → Y , where i is a trivial cofibration having the LLP with respect to all
fibrations and p is a fibration
X
i
/
X∞
p
f
/Y
Y
As f, i are weak equivalnces, the map p is a trivial fibration, so there exists a lift
r : Y → X∞ . So we see, that f is a retraction of i.
id
NOTE ON COFIBRATION
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We can see the topological spaces as a motivation for a more general notion
Definition 8. A model category is a category C with three distinguished classes of
arrows:
(1) weak equivalences
(2) fibrations
(3) cofibrations
Map which is both weak equivalence is called trivial fibration and map which is both
weak equivalence and cofibration is called trivial cofibration.
We also require the following axioms to hold:
MC1 Finite limits and colimits exist in C
MC2 Let f, g be maps in category C. Then if two out of three maps f, g, f g are
weak equivalences, the third must also be a weak equivalence.
MC3 If f is a retract of g and g is a fibration (weak equivalence, cofibration), then
f is a fibration (weak equivalence, cofibration).
MC4 Given any commutative diagram as in ...a lift exists there in following cases (1)
p is a fibration and i a trivial cofibration, and (2) p is a trivial fibration and i
a cofibration.
MC5 Any map f can be factored in two ways : (1) f = pi, where p is a fibration
and i a trivial cofibration, and (2) f = pi, where p is a trivial fibration and i
a cofibration.
To make it clear that the category of topological spaces C where fibrations are
serre fibrations and weak equivalences are weak homotopy equivalences satisfies all
the axioms written above( so it is a model category), we must chceck the ramaining
two facts that were not proven already:
MC1 : T op has all finite limits and colimits (which we can see from the existence of
products and equalizers and their duals as well).
MC4 The existence of the second lift is clear as this is the way we originally defined
cofibrations. The second statement is obtained by the fact stated in lemma 7,
because any trivial cofibration is retract of relative CW inclusion and so it has
LLP with respect to all fibrations.