Download The homotopy category is a homotopy category. Arne Str¢m

The homotopy category is a homotopy category.
By
Arne Str¢m
In
[4] Quillen defines the concept of a category of
(a model category for short).
models for a homotopy theory
A model category is a category
K together with three
distinguished classes of morphisms in
C
("cofibrations"), and
K:
F
("fibrations"),
W ("weak equivalences").
These
MO-M 5 of (4].
classes are required to satisfy axioms
A
closed model category is a model category satisfying the extra
axiom
M 6 (see
[4]
for the statement of the axioms
To each model category
HoK
M0- M6 ).
K one can associate a category
called the homotopy category of
K.
is obtained by turning the morphisms in
Essentially,
W into
HoK
isorn~rphisrns.
It is shown in [4] that the category of topological
spaces is a closed model category if one puts
fibrations}
and
F
= {Serre
W = {weak homotopy equivalences},
and takes
C to be the class of all maps having a certain lifting property.
From an aesthetical point of view, however, it
would be
nicer to work with ordinary (Hurewicz) fibrations, cofibrations
and homotopy equivalences.
The corresponding homotopy category
- 2 -
would then be the ordinary homotopy category of topological
spaces, i.e. the objects would be all topological spaces and
the morphisms would be all homotopy classes of continuous maps.
In the first section of this paper we prove that this is
indeed feasible, and in the last section we consider the case
of spaces with base points.
\
- 3 -
1.
The model category structure of To2.
Let
Top
be the category of topological spaces and
continuous maps.
By fibrations (cofibrations) we shall mean
maps having the homotopy lifting (extension) property with
respect to all spaces.
F = {fibrations},
Let
C
= {closed
cofibrations}, and
W = {homotopy equivalences}.
If
i: A
+
X and
shall say that
i
respect to
and that
(RLP)
p,
p: E
B are morphisms in
+
has the left lifting property
with respect to
p
Top, we
(LLP)
with
has the right lifting property
i, if every commutative square of the
form
X
in
Top
admits a diagonal
Proposition 1.
F, C and W.
(a)
(b)
(c)
(d)
---->
X
+
B
E.
The following relations hold between
p E F <=> p has the RLP with respect to all i E Q flW.
i E C <=> i has the LLP with respect to all p E F n W.
pE FnW <=> p has the RLP with respect to all i€ C.
1€ C ():W <=> i has the LLP with respect to all p E F.
Proof. (a) follows from [61, Theorem 8, and the definition
of fibrations.
=>in (b) follows from (6], Theorem 9.
To prove
<=
we
first note that if i has the LLP with respect to all pEFnW, then
i is a cofibration. It remains to show that i is closed.
We may
- 4 assume that
i
is an inclusion,
=A
E
and define
I
X
p: E
~
u
X
X by
X
(0 ,1)
i: Ac X.
c X
X
I'
p(x,t) = x.
p
Let
is a homotopy
equivalence, and it is also a fibration, for given
and
G: Y
lifting
I
x
X with
+
G: Y
x
I
+
E
g: Y
~
E
G = pg, we can construct a suitable
0
by letting
G(y,t) = (G(y,t), t+ (1-t)prig(y)),
where
pri: E
+
I
is the projection map.
(This construction
works for any inclusion map.)
Hence
has the
i
j(a) = (a,O).
by
j : A + E
LLP
there is a map
f: X
A = f- 1 pr! 1 (O)
is closed in
+ E
and
i'
Then
and consequently
pj = lxi,
f =pi= p'i',
Now define
It follows that
j •
x.
[6],
Theorems 8 and 9, and (b).
Every continuous map
Proposition 2.
i
p.
extending
(c) and (d) follow from
factored as
with respect to
where
p
f: X
and
are closed cofibrations, and
p'
i
~
Y can be
are fibrations,
and
p'
are
homotopy equivalences.
f = pi:
Proof.
5.27) that
f
It is well knO\'ln (see for instance r1),
can be factored
f = nj,
X as a strong deformation retract of
fibration.
=
(x,O),
retract of
W, and
j: XC W imbeds
n: W + Y is a
As in the proof of Proposition 1 (b) let
E = X x I u W x ( 0, 1)
i(x)
where
E
and define
7r'(w,t) = w.
and
Then
i (X) = pri-1 ( 0 ) •
i: X
+
E,
TI': E ~ W by
i(X) is a strong deformation
It follows that
1
is a
- 6 -
MO
and
M5
are obviously true and
above, \'Thile
2.
M6
M2
is just Proposition 2
follows from Proposition 1, r112
and
M5.
Some lemmas.
The following lemmas will be useful in the next section.
Lemma 4.
If
i: A c X is a cofibration and
i#: Aye XY
compact space, then the map
Y is a
induced by
i
is also
a cof'ibration (with respect to the compact-open topology).
Proof:
functions
H, ~ are as in (6], Lemma 4, then corresponding
If
H:
xY
X
H(f,t)(y)
l(f)
I
+
xY
xY
¢:
and
I
+
are given by
= H(f(y),t),
= sup
¢f(y).
y€ y
Lemma 5.
i
and
ij
Proof.
If
j: B +A
i: A+ X are maps such that
are cofibrations, then
We can assume that
There exists a halo
retraction
and
r: U
+
U around
A.
Since
is also a cofibration.
j
i
A in
and
j
are inclusion maps.
X together with a
U is also a halo around
B c U is a cofibration ([2], Satz 2, Korollar).
Now consider a commutative diagram
(1)
_.....;;..f_> y
B
in
X,
- 7 where
~ 0 (w)
= w(O).
The diagram
B
(\
(2)
u
is also commutative, and since
G: U
admits a diagonal
It follows that
+
Y1 •
B c: U is a cofibration, ( 2)
GIA
is then a diagonal in
(1).
is a cofibration.
j
Recall that a well-pointed space is a space
*
with a base point
~
X together
X such that the inclusion maP {*} c X
is a closed cofibration.
Lemma 6.
Consider a pullback diagram
f'
E' - - - > E
lp
1
f
B' - - - > B
where
p
is a fibration.
pointed and that
f
and
Suppose that
p
E, B, B'
are well-
respect base points.
Then
(with the obvious base point) is also well-pointed.
the fiber of
Proof.
p
Consider the sequence
j
*
i
f'
F C E'-> E
is the base point of
base point of
In particular,
is well-pointed.
pq.c
where
E'
B'.
By
[6],
E'
and
F
is the fiber over the
Theorem 12,
i
and
f'i are
8 -
=
cofibrations.
fibration;
ij
The pointed
Let
E
is well-pointed,
hence, by Lemma 5 above,
follows that
3.
Since
Top *
Top
All base points will be denoted by
Top *
are defined exactly as
except that all maps and homotopies are required to
From now on homotopies, fibrations, etc.
will be referred to as free homotopies, fibrations, etc.
It is clear that if a map
i: A-+ X
in
Top *
cofibration (that is, when considered as a map in
is a cofibration in
Top *
It
be the category of pointed spaces and continuous
respect the base points.
in
is a cofibration.
case~
Fibrations and cofibrations in
Top,
is a co-
is a cofibration.
base point preserving maps.
in
j
(f'i)j
Top * •
is a free
Top), then it
On the other hand, a fibration in
is also a free fibration.
Just as in the free case one can prove that all cofibrations
in
Top * are imbeddings.
closed, an inclusion
only if
Also, if the base point of
A c X in
(X x 0 u A x I) I* xI
deformation retract) of
Top *
X
is
is a cofibration if and
is a retract (and hence a strong
X x I/* x I
The arguments are similar to those in
(A
need not be closed).
[5]
and
[6).
We shall need the following result, analogous to (6],
Lemma 4.
Proposition 7.
Let
i: AC X be an inclusion map in
and suppose that there exists a continuous function
w:
X -+ I
~
Top· ,
- 9 1jJ -1( 0)
such that
= {*}.
Then
i
is a cofibration if and only
if there exist
a continuous function
and a homotopy
H: X
H(x,t)
whenever
A
(f.
If such
that
cj>
Proof.
Since
i
with
H0
AC
= ~X
cj> - l (0),
and
H exist they can be chosen in such a way
x EX.
Suppose first that
~
X
X
0 u A
p: X x 0 u A x I
p(x,t)
I
X rel A such that
+
for all
K = { (X' t)
and define
+
Min(t,l)J(x)) > cj>(x).
and
cj>(x) < ljJ(x)
I
x
cj>: X
+ K
i
X
is a cofibration.
II
Let
t < l)J(x)}
by
= (x,Min(t,l)J(x))).
is a cofibration,
p extends to
r: X x I
+
K.
cj>
and
H can then be defined by
cj>(x)
H(x,t)
= sup
tEI
(Min(t,l)J(x)) - prir(x,t))
= prxr(x,t).
It is clear that
Conversely,
cp
if
cj>
and
and
H have the desired properties.
H are given, we can define a
retraction
r: X
x
I/*
x
I --> (X
x
0 U A
x I)/~ x
by
. (H(x,t) ,0),
r(x,t)
tljJ(x) =< cj>(x)
=
(H(x,t),t- cj> (x)/ljJ(x)),
tljJ(x) > cj>(x).
I
- 10 -
Top w
· Our main interest will be in the full subcategory
lie
of well-pointed spaces, rather than the whole category
Top •
It is an easy consequence of the product theorem for cofibrations ( [6] , Theorem 6)
Top *
that the mapping cylinder in
of
a map between well-pointed spaces is well-pointed. Consequently,
Top w is a cofibration in
a map in
Top w
if and only if it
Top * •
is a cofibration in
Dually, it follows from Lemmas 4 and 6 above that the
mapping track of a map in
in
Topw
Top
is a fibration in
w
is well-pointed, and so a map
Topw
if and only if it is a
Top * •
fibration in
A necessary and sufficient condition for a pointed space
to be well-pointed is that there exist
and a homotopy
and
(P,~)
P(x,t)
=*
t >
~(x)
P: X x I
([6],
a well-pointing couple for
Lemma 8.
exi~s
when
If
A
c. X
a
~:
X
I
with
X with
+
Lemma 4).
P0
= ~X
Let us 0all
X.
Top
is a cofibration in
(P,~)
a well-pointing couple
+
X
for
w
there
X such that
P(Axi)cA.
Proofo
for
X and
Let
(PX,~X)
and
A, respectively.
(PA'~A)
Choose
be well-pointing couples
~:
X
+
I
and
H: X x I
satisfying the conditions of Proposition 7 with respect to
and define
a: X -
{!k}
+I
A c X is a cofibration,
with
P0
= ~x·
Also,
~A
by
PA
a(x)
= 1-
~(x)/~X(x).
extends to a homotopy
P:
+
~X'
Since
X x I
can be extended to a continuous
+
X
X
- 11 -
w: X
+
I
by putting
r(x)lj!
AH(x,l) + ¢(x)'
vHx) =
cp(x) = l/Jx(x).
(!x(x),
(Recall that
H(x,l)
=
¢(x) < wx(x).)
For,
{*}.
l/JAH(x,l) = ¢(x) =
we should have
=*
A when
rl co)
We then have
H(x,l)
€
and
X E.
A.
However,
o.
if
X '¢
Wx(x)
= 0,
'¢
:k
and w(x) =
o,
But this would imply
it is clear that
for all
and therefore
X
x'
E.
A,
contradicting the assumption that
*•
The required couple
(P,$)
t/$(x)),
P(x,t)
t < l/J(x),
=
Px(P(x,l) ,t- l[i(x)),
l/J(x)
is now given by
=
t
~
iV(x),
Min ( 1 , l/J ( x) + wxP ( x, 1) ) •
We shall use this lemma to prove
Proposition 9.
A map
i: A + X
in
Top w
is a cofibration
if and only if it is a free cofibration.
Proof.
Only "only if" needs proof.
Suppose, then, that
- 12 -
AC.X
i:
that
i
Topw•
is a cofibration in
is an inclusion.)
Let
(For simplicity we assume
(P,w)
as described in Lemma 8, and then let
be a well-pointing couple
~
and
H
be as in Pro-
position 7.
Define
H': X xI+ X and
P ( H (X, t ) ,
H'(x,t)
~
H'
9(
X)
and
X+ I
~':
Min [ t , ~ (X) / W(X)] ) ,
by
X -#
*,
X :
*,
=
=
~ (X ) -
~~
l/J ( X) + Sup WH ( X, t ) •
t 4L I
then satisfy the conditions of
it follows that
i
r6J,
Lemma
4,
and
is a free cofibration.
The dual statement is also true:
Proposition 10.
A map
p: E
+
B
in
Top w
is a fibration
if and only if it is a free fibration.
Proof.
It follows from [5] , Theorem 4 that if
p
is a free
fibration, then it has the pointed homotopy lifting property with
respect to all well-pointed spaces.
It is also true that a map in
Topw
is a homotopy equivalence
if and only if it is a free homotopy equivalence ([1], 2.18)
- 13 -
Theorem 11.
The category
Topw , with the classes of
(pointed) fibrations, closed cofibrations, and homotppy equivalences, satisfies the axioms
This follows from Theorem 3 and Propositions 9 and
Proof.
10.
Ml - M6.
It is not hard to show that the constructions
in the proof
of Proposition 2, when performed on well-pointed spaces, yield
well-pointed spaces.
One could hardly expect
MO
to hold in
Top w,
but
does have sums, finite products, pullbacks of fibrations, pushouts
of cofibrations, smash products, suspensions, loop spaces, etc.,
and this goes a long way.
A simple consequence of the product theorem for cofibrations
and Proposition 9 is the following "smash product theorem".
Proposition 12.
in
Top
w
If
AC X
Bc Y
and
are cofibrations
and at least one of them is closed, then
XI\ B UA/\. YC XI\ Y
is also a cofibration in
Topw•
(X 1\ B
x/\Y.)
be given the subspace topology induced by
Remark.
Results analogous to 10, 11
in the lar~ category
for which
Top 0
A 1\ Y should here
and
and
12
above hold
consisting of all pointed spaces
there exist functions
~:
X
~
I
with
but the proofs become a bit more complicated.
~ -1( 0)
X
= { *} ,
- 14 -
References.
l.
T.Tom Dieck, K. H. Kamps, D. Puppe,
Homotopietheorie,
Lecture Notes in Mathematics 157,
Berlin-Heidelberg-New York, 1970.
2.
A. Dold,
Die Homotopieerweiterungseigenschaft
(= HEP) is eine lokale Eigenschaft,
Invent. Math. ~ (1968), .185-189.
3.
I. M. Hall,
The generalized Whitney sum, Quart.J.
Math. Oxford Ser. (2) 16 (1965),360-384.
4.
D. G. Quillen,
Homotopical Algebra, Lecture Notes in
Mathematics 43, Berlin-HeidelbergNew York, 1967.
5.
A. Str¢m,
Note on cofibrations, Math.Scand. 19
(1966), 11·14.
6.
A. Str¢m,
Note on cofibrations II, Math. Scand.
22 (l9v8), 130-142.