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: X X I _ Y such that cI>(i X I) = if; and cl>eo(X) = f. Then 9 = cl>eI(X) has the desired property. 0
3. Co cylinders and Fibrations By dualising the notion of a cylinder on a category we obtain the concept of a co cylinder . By definition a co cylinder on a category C is a cylinder on the dual category coP. Explicitly, we have the following definition.
Definition (3.1). Let C be a category. A co cylinder , P, on C is a functor (co cylinder functor)
( )1: C _
C
together with three natural transformations
co, Cl
: (
)1 _
I de,
s: I de _
( )1
such that cos = CIS = Ide· If we apply ( )1 to an object X of C, we shall write simply Xl, similarly for morphisms.
Examples 1. As in the dual case the basic example of a co cylinder comes from topology. There is a canonical co cylinder W on the category Top which assigns to a topological space X the space X[O,l) of paths in X endowed with the compact open topology (see Hu (1959), tom DieckKampsPuppe (1970) or other textbooks on homotopy theory). Thus an element w of X [0,1) is a path in X, i.e. a continuous map w : [0,1] X. A subbasis for the open sets of X[O,l) is given by the collection of sets M( C, U) where C is an arbitrary compact subset of [0,1], U is an arbitrary open subset of X and M( C, U) is the set of those elements w of X[O,l) such that w( C) ~ U. The natural transformations 12
EO, E} , S are given by the continuous maps EO(X) X[O,}) E} (X) : X[O ,}) seX) : X
t t t
X EO(X)(W) X , E}(X)(W) X[O ,}) , (s(X)(x))(t)
W(O) w(l) x
sending a path w in X to its initial resp. end point, resp. a point x of X to the constant path at that point. 2. For any category C we have the trivial cocylinder Id on C consisting of the identity functor I de : C t C and the corresponding identity transformations. Given a co cylinder P = (( )I,Eo,E},S) on a category C, two morphisms I, 9 : X t Y in C are said to be homotopic, written I ~ g, if there is a morphism ¢ : X t yI in C with EO(Y)¢ = I, E} (Y)¢ = g. We call ¢ a homotopy between I and 9 and write ¢ : I ~ g. As in the development of cylinders, the relation of being homotopic as given by a co cylinder may be neither symmetric nor transitive. Remark. If C is a category which carries both a cylinder and a cocylinder we must distinguish between two notions of homotopy induced by the cylinder, resp. the co cylinder. However, in the basic example where C is the category Top of topological spaces together with the canonical cylinder T, resp . the canonIcal co cylinder W, these notions coincide. This is due to the fact that the cylinder functor ( ) x [0 , 1] on Top is left adjoint to the path space functor ( )[O,}) . This type of question will be dealt with in detail in section 3 of Chapter II. Dual to cofibrations (with respect to a cylinder) one has fibrations (with respect to a co cylinder) defined by a homotopy lifting property. Definition (3.2). Let C be a category with a co cylinder
P = (( )l,Eo,E} , S) . (a) A morphism p : E t B in C has the homotopy lifting property (HLP), with respect to an object Y of C if for any pair of morphisms of C ¢: Y t Bl, I : Y t E such that Eo(B)¢ = pi there 13
is a morphism
E Qo , we define E Q(n ,v,k) by ~ = (n  l(Q)(¢»
for any
(c:, i)
=1=
(v, k) .
By (n ,v,k)(Q)(¢» = , we obtain a natural transformation
(n ,v,k) : Go
t
G(n,v,k).
Definition (5.4). A function). : Q(n,v,k)    t Qn, Q in Cub, is called a filler map if for each 'Y E Q(n,v,k), ).('Y) is a filler of 'Y. If ). is a filler map which makes the diagram
commute , we say that). is compatible with degeneracies. As we stated earlier, we need these filler conditions to be able to apply them to the cubical sets, QI(X, Y) , of higher homotopies between objects X, Y in a category C which has a cylinder I , and thus to be able to specify higher order structure on the cylinder, I.
Definition (5.5). A cylinder I on a category C is said to satisfy the Kan condition E(n,v,k) (0 ~ v ~ 1, 1 ~ k ~ n) if for any objects X, Y of C, the cubical set QI(X, Y) satisfies the Kan condition
E(n, v, k) . We say I satisfies the Kan condition NE(n,v,k) if there is a natural transformation
). : G(n,v,k)QI
t
GnQI,
such that for any X, Y, objects in C,
is a filler map. If, in addition , each )'(X, Y) is compatible with degeneracies, I is 26
said to satisfy the Kan condition DNE(n,lI,k). If for some n, I satisfies E(n, II, k) (resp. NE(n, II, k), resp. DNE( n, II, k)) for all II = 0,1, k = 1"", n, we say that I satisfies the Kan condition E(n) (resp. NE(n), resp. DNE(n)) .
Examples 1. The simple geometric fact that the union of all (n 1 )dimensional faces, except one, of the ndimensional cube [0,1 In is a retract of this cube implies that the canonical cylinder T on the category Top of topological spaces satisfies the Kan condition DNE( n) for all n (see Kamps (1968), (2.24)). 2. For any category C the trivial cylinder Id on C satisfies the Kan condition DNE( n) for all n. The Kan conditions enable one to do a lot of homotopy theory. As a first example of this, we shall look at the homotopy relation, ~, between maps. For the rest of this section let I = (( ) x I, eo, el, a) be a cylinder on a category C.
Proposition (5.6). If I satisfies the Kan condition E(2,1,1), then the homotopy relation is an equivalence relation. Proof. Symmetry: If f, 9 : X + Y in C and ¢> : X x [ + Y is a homotopy between f and g, define a (2,1,1)box, "I, in QI(X, Y) by
"16 = ¢>, "IJ = "If = fa(X). As E(2,1,1) is satisfied, there is a filler)' E C(X
X [2, Y)
of "I. Let
¢>' = ).~ = ).el(X x 1). Then ¢>' is the required homotopy from 9 to f.
27
Transitivity: We are given
cp,,,p: X
f, g, h : X
x I ~ Y,
cp: f
~ Y and homotopies ~ g, "p : 9 ~
h.
We define a (2,1,1)box, " in QI(X, Y) by
,6 = frJ(X),
A6
= cP, ,f = "p.
Let A : X x 12 ~ Y be a filler of , . Then X = Ael (X x I) is the required homotopy.
o Remark. In order to develop what might be called the foundations of (classical) homotopy theory it is sufficient, as we shall see , to assume Kan conditions in low dimensions, 2 and 3. This fact allows one to visualize boxes and fillers as shown above. The reader is advised to do so whenever possible. This will help to create ideas and to understand proofs better. Next we show how a weak form of involution on a cylinder (see (4.1)) can be obtained by means of the Kan condition NE(2) . Propositon (5.7). If I satisfies the Kan condition NE(2,1 ,1) then there is a natural transformation i : ( ) x I ~ ( ) x I with ieo = el and iel = eo.
Proof. Since I satisfies NE(2,1 ,1), there is a natural transformation
A : G(2 ,1,l)QI ~ G 2QI so that for all objects X, Y, A(X, Y) is a filler function . If X is an object of C, then 'x given by
,6 = ,f = eo(X)rJ(X), ,6 = I dx x 1 28
is a (2,1,1)box in QI(X, X x 1). Let
i(X) = (.A(X,X x Ihx)el(X x 1) to get the required natural transformation i : ( ) x I
t (
)
xI.
0
Using NE(2) and E(3) an involution on a cylinder can be approximated even more closely (see Kamps (1968) , (2.8)). We thus see how extra structure in the shape of fillers in the cubical sets QI(X, Y) can be used to obtain a more richly structured homotopy theory. The next question we look at with the help of the filler conditions is that of the properties of the structure maps of a cylinder.
Proposition (5.8). If I satisfies the Kan condition E(2), then eo(X) and el(X) are cofibrations and a(X) is a homotopy equivalence for all objects X of c.
Proof. Suppose we are given maps cp : X x I t Y, j: X x I such that cpeo(X) = feo(X). Using E(2,1 ,1) we obtain a map : X x [2
t
t
Y
Y
such that
eo(X x 1)
= j,
(eo(X) x 1)
= cp,
(el(X) x I)
= f el(X)a(X).
This shows that eo(X) is a cofibration. Similarly, one can prove that el(X) is a cofibration. By definition of a cylinder we have that
a(X)eo(X) = I dx . Using E(2,1,2) we can find a map : X x 12
eo(X x 1)
t
= (eo(X) x 1) = eo(X)a(X) ,
el(X x I)
It follows that
eo(X)a(X)
~
Id x xI
proving that a(X) is a homotopy equivalence.
29
X x I such that
0
= Id x xI ·
Thinking of X x I geometrically, i.e . X x I = X x [0 , 1], the inclusion of the two ends gives us a map , j : X U X 4 X x I, where X U X denotes the topological sum of two copies of X . The geometric significance of this map will become clearer later, but if we are to use it later it will need to have good properties and to these we turn next. We first must set j up in the abstract setting. Let X~XUX~X
(5 .9)
b e a coproduct diagram in C. Let j : X U X with components eo(X) and el(X),
4
X x 1 denote the map
Proposition (5.10). Suppos e that C has coproducts and that the cylinder functor ( ) x 1 preserves these coproducts . Then if I satisfies th e Kan condition E(2 ,1,l) it follows that j is a cofibration. Proof. Let ¢ : (X U X ) X I 4 Y , f : X x 1 4 Y b e maps such that ¢eo( X U X ) = fj. By E(2,1,1), there is a morphism
: X x 12 4 Y such that
eo(X x 1) = f , (el/( X ) x 1) =
¢(~l/ X
1) ,
V
= 0, 1.
As ( ) x I applied to (5. 9) yields a coproduct di agram , we can use uniqueness of induced maps t o conclude th at (j x 1) = ¢. This proves that j is a cofibr ation . We leave t he details to t h e reader. 0 R e m a rk. Again supposing t.h at. ( ) x I preserves coprodu cts, t h e converse of Proposition (5. 10) is also t rue. T h e p roof is left as an exercise. T his means t hat th e Kan condi t ion E(2 ,1,1) can be ch aracterised in terms of a cofibration condition. A similar charact.erisat.ion is possible for t.he ot.her I\.an condit ions in dimension :2 and can also b e giwn fo r Kan condit.ions in higher dimensions . T his shows the extent. t.o which the apparent.ly abst.ract. filler condit.ions have geometric significance. It. also suggests alt.ernative axiom syst.ems for parts of abst.ract homot.opy theory. 30
We now return to the mapping cylinder factorisation to see what additional filler structure yields there. Recall that we had the factorisation
x
f
y
\} f
f :X
Y, introduced in (2.9), and that h : Y t M J denotes one of the structure maps in the definition (2 .8) of the mapping cylinder. We had PJh = Id y . = pJ i J of a morphism
t
Theorem (5.11). Suppose that f has a mapping cylinder, ( ) x I preserves push outs and I satisfies E(2), then (a) hpJ ~ Id M / (b) i J is a cofibration.
Proof. (a) Using E(2) we can find a map A : X x 12
t
M J satis
fying
Aeo(X x 1) = A(eo(X) x 1) = hfcr(X)
= hcr(Y)(f x 1)
and
Ael(X x 1) = 'TrJ. As ( ) x [ preserves pushouts, the diagram
X x [
f
X
eo(X) x I
X
12
jn! xI
Ij
Y x I
X
h
xl
31
MJ x [
is a pushout, and using the commutativity of the square
x f
X
x [
eo(X)
II
we find a map 'ljJ : Mf x [ X
I)
X
X [2
IA
Y x [
'ljJ(7rf
X [
Mf
jfcr(Y)
t
M f so that 'ljJ(jf x I) = itcr(Y).
and
=,\
We have 'ljJoit = itPfjf
and
'ljJo7rf = jfPf7rf,
hence 'ljJo = itPf by the uniqueness clause in the universal property of pushouts . It is straightforward to check that 'ljJl = [d Mf . We now turn to (b). Suppose given 9 : M f   t V, : f '::::!. gin C and p'¢> = pa(E), i.e. the diagram
¢> ExIE'
k
U(E)j
EpB commutes. We call ¢> a homotopy over B and write ¢> : f'::::!. g. B
A morphism over B, 9 : P + p', is a homotopy equivalence over B if there is a morphism over B, g': p' + p, such that
g'g'::::!.Id E and B
gg''::::!.IdE,. B
Such a morphism g' over B is called a homotopy inverse over B of g. Proposition (6.2). If I satisfies the Kan condition DNE(2,1,1) then both ~ and'::::!. are equivalence relations. B
Proof. The proof of Proposition (5.6) can be adapted to the present situation constructing fillers by a natural transformation>. which is compatible with degeneracies according to the definition of DNE(2,1,1) (cf. (5.5)). We leave the details as an exercise. 0
Exercise. Prove that there are relative versions of Lemma (2.3)(b) showing that both ~ and'::::!. are compatible with composition in CA , B
resp. CB. 35
Clearly, a map f : i + i' which is a homotopy equivalence under A is an ordinary homotopy equivalence, viewed as a map
f: X
+
X'.
We are now in a position to state our abstract version of Dold's theorem. This claims that, provided i and i' are cofibrations, and that the cylinder satisfies certain conditions, the converse implication is also true.
Theorem {6.3}. Suppose I is a cylinder on a category C which satisfies DNE(2 ,1,1) and E(3,1,1) and in which ( ) x I preserves weak pushouts. Let
A
/~
X    , X'
f
be a commutative diagram in C where i and i' are cofibrations. Then if f is a homotopy equivalence, f is a homotopy equivalence under A.
We reduce the theorem to the following lemma (under the conditions of the theorem).
Lemma {6.4}. Given a commutative diagram A
/\
XX g where i is a cofibration and g ~ I dx , then there zs a morphism under A, g': i + i, such that g'g ~ Id x . 36
Proof of the reduction. Let I' : X' t X be a homotopy inverse of f, i.e. I'f::= Id x and ff'::= Idx'· Then we have I'i' = I' fi ::= i. By Proposition (2.11) there is a morphism f" : X' t X such that I' ::= f" and f"i' = i. Thus f" is a homotopy inverse of f which is a morphism under A, f" : i' t i. Putting 9 = f" f we get a morphism under A, g: i t i , such that 9 ::= Id x . By Lemma (6.4) there exists a morphism under A, g': i t i, such that g' 9 is a morphism under A, h: i' t i, such that
~ I d x . Then
h=
9' f"
A
hf = g' f" f = g' 9 ::= I dx . Since
fh = fg'f"::= fg'f"ff" = fg'gf"::= ff"::= Idx', we can now apply the same argument to h instead of morphism under A, k: i t i', such that
f and find a
A
kh::=Idx'· Then we have A
A
A
fh::= khfh::= kh::= Idx' hence, by Proposition (6.2), A
fh::=Idxl which proves our theorem.
0
Proof of Lemma (6.4). Let 1> : X x I t X, homotopy and consider 1>' = 1>( i x 1). Then 1>'eo(A) =
1>(i x I)eo(A)
1>: 9
= 1>eo(X)i = gi = i
and as i is a cofibration, there is a homotopy 'lj; : X x I 'lj;(i x 1) = 1>' and 'lj;eo(X) = Id x ·
37
::= Id x , be a
t
X with
We set g' = 7J;el(X), so g'i = i and g' is a map under A. Moreover 7J;( i x I) = ( i x 1). The proof of the lemma will be complete when we have proved
g'g ~ldx. To do this we first consider the (2,1,1 )box "I = ("(6, , "15, "In given by
"If =
,
"15 =
This has a filler p : X
X
7J;(g x 1),
12 
"16
= (69 = ga(X).
X. We set F = l~(p), so
F: X x [  X and F: g'g
~
1dx .
N ext we define
o = (06, , 05, Of, 08, o?) by
03 og
p(i x [2), =
oJ = ia(X)a(X x 1),
or = 0; = (i x 1)(a(A) x I) = 7J;(i x 1)(a(A) x 1).
Thus 0 can be illustrated by the following figure.
38
iu(A)
cj>(i x 1) cj>( i
iu(A)
X
X
cj>( i x 1)
1)
cj>( i
{12 0
cj>(i
813
X
1)
812
iu(X)u(X X I)
1)
cj>(i
X
iu(A)
1)
II
1/;( 9
X
1)(i
X
1)
fL (i
cj>(i
X 12)
F( i
X
X
1)
1)
One verifies easily that 8 is a (3,1,1)box in QI(A,X) . Since I satisfies E(3,1,1), there is a filler G of 8, G : A x 13 t X . Set a = l!(G), a : A x 12 t X . We have
ag
= F(i x 1), aA = al = at = ia(A).
Since i is a cofibration and ( ) x I is assumed to preserve weak pushouts, the diagram
(6.5)
i
AxI
eo(A) x [
A
x II X x[
X
eo(X) x [
X [2
Ii
X
x 12 12
is a weak pushout. Thus there is a map ¢ : X x
[2
t
X so that
¢(i x 12) = a, ¢(eo(X) x 1) = F. 1 2
1
We note that ,1/J : X x [0,1] + Yare homotopies, 1> : f ~ g, 1/J: 9 ~ h, then we have a canonical sum
given by the formula
1>(X,2t), 0$ t $ L (1/J+1>) x,t ( ) = { 1/J(x,2t1), l g is a morphism of 7rYx, then
the track of the reverse homotopy ¢ , is inverse to {¢} . Thus the category 7rY x is an example of the following general notion. Definition (1.1). A groupoid is a small category in which every morphism has an inverse. If X, Y are topological spaces then 7rY x is a groupoid, called the fundamental groupoid of Y under X. Of particular interest is the case where X is a point, in which case 7rYx is abbreviated to 7rY and is called simply the fundamental groupoid of Y. Objects of 7rY are bijective with points of Y and morphisms are bijective with homotopy classes reI end points of paths in Y. In Chapter IV we shall return to the fundamental groupoid in a general situation and construct the fundamental groupoid of a cubical set which satisfies Kan conditions in low dimensions. If 9 is a groupoid and x E Ob(9) is an object of g, then under composition the set of morphisms 9 (x, x) from x to itself is a group, written g(x) , and called the object group, or vertex group, of 9 at
x. Example. If Y is a topological space, then the object group 7rY(Y) of the fundamental groupoid 7rY at an element y of Y is the classical fundamental group 7rl (Y, y) of Y at y. A group G can be regarded as a groupoid with exactly one object, the object group at that object being the group G. Thus the notion of 149
aa groupoid groupoidgeneralises generalisesthat that ofof aa group. group. The The fact fact that that groups groups are are aa special special type type of of groupoid groupoid together together with with the exercise that follows may lead you to believe that groupoids the exercise that follows may lead you to believe that groupoids are are 'everywhere' 'everywhere' in inmathematics. mathematics. We Wewould wouldnot not try tryto todiscourage discourageyou youfrom from view. such aaview. such
Exercise. Exercise. (a) (a)(Equivalence (Equivalence relations) relations) Let Let M M be be aa set set and and
Rr;MxM R C M x M be (Recall the the usual usual notation notation xRy, xRy, ifif be an an equivalence equivalence relation. relation. (Recall (x, (x,y) y) EE R.) R.) Let Let RR be be defined defined by by Ob(R) Ob(R) == M M and and for for each each x,x,yy in in M,, let let M
(x,y) E R (x, y) rt R,
R(x,y) = {{(Y0,x)},
where denotes the the empty empty set. set. Define Define aa composition composition by by where (/)0denotes
(z,y)(y,x) (z,Y)(Y, x) ==(z,x), (z,x), z ) EE R. R. whenever (x, (x,y)(y, y) (y ,z) whenever
• x
(y,x)
(z,y)
•y
)
)
•z
groupoid. Show that that RR isis aa groupoid. Show Note that that each each morphism morphism set set R(x, R(x,y) y) has has at at most most one one element. element. If If Note M xx M, M, that that means means ifif all all elements elements of of M M are are identified identified by by R, R , then then RR ==M we obtain obtain aa groupoid groupoid R, R, where where each each morphism morphism set set R(x R ( x,,y) y) has has exactly exactly we one element. element. Such Such aa groupoid groupoid isis called called aa tree tree groupoid groupoid.. In In particular, particular, one for M = {0), we obtain a 'point groupoid', i.e. a groupoid with exactly for M = {O}, we obtain a 'point groupoid', i.e. a groupoid with exactly one object, object, 0, 0, and and one one morphism, morphism, the the identity identity at at 0; 0; for for M M == {O, {0,1) I} one Z, with two objects 0,l and two nonidentity we obtain a groupoid, we obtain a groupoid, I, with two objects 0,1 and two non identity morphisms morphisms t :
°
t
1,
t
which are are inverse inverse to to each each other. other. which
150
1
:
1
t
°
We shall shall see see that in the theory of groupoids, groupoids, the groupoid IZplays the role of a unit interval. interval. group. Recall that a Gset Gset M M,, is is a set M M (b) (Gs (Gsets) ets) Let G be a group. (b) map, called the operation (action) (action) of G on M, M, together with a map, G x M ~ M,
(g,m)
tt
g. m
such such that
h • (g • m) = (hg) • m ,
1. m = n
for for any g, g, h E G, G, m mEM M.. G K M be defined by Ob(GrxM) O ~ ( G K M== ) M and for for each xx,, y in M, M, let Let GrxM
(GrxM)( x ,y) }. (GKM)(x, y) == {(y,g, {(y,g, xx)) Ig 1 g E G such that y == g. g xx). Define a composition by Define
(z, h, y)(y,g ,x) = (z,hg,x).
•x
(y,g, x ) )
.
(z, h, y)
•
)
y = g. x
z
= h • y = h • (g
• x ) = (hg) • x
Show that GrxM GD<Mis is aa groupoid. groupoid. The The groupoid GrxM G K Mwill will be called the the Show semidirect product p r o d u c t of G and M. M.
'H of grouReturning Returning to to the the general case, case, aa morphism m o r p h i s m ff :: yG t 1t poids is Grpd, of is simply a functor. functor. Thus Thus we we obtain the the category, category, yrpd, groupoids. groupoids. Exercise. Describe the the data needed to to specify specify aa morphism
f :I
t
151
1t.
Show Show such such aa morphism morphism corresponds corresponds to to aa morphism morphism in in the the groupoid, groupoid, H. 3.1.
Exercise. Exercise. Show Show ifif 9, G,Hare 7l are groupoids, groupoids, then then so so also also is is the the product product category 3.1. category 9G xx H. Exercise. Exercise. (i) (i) Write Write down down all all objects objects and and morphisms morphisms in in IZ xx IZ and and draw tree grougroudraw an an arrow arrow diagram diagram illustrating illustrating the the structure. structure. Is Is IZ xx IZ aa tree poid? poid ? (ii) (ii) Let Let L{n} A{n) denote denote the the tree tree groupoid groupoid with with objects objects the the natural natural numbers, 1, ···, . . ,nn (so (so that that L{l} A i l ) == I). Z). Draw Draw arrow arrow diagrams diagrams for for numbers, 0, 0,1,. L{2} , L{3} and L{2} x I. Describe the data needed for a morphism A{2), A(3) and A(2) x Z. Describe the data needed for a morphism from 71. from L{2} A(2) to to H. (iii) Describe (iii) Describe aa morphism morphism from from IZ xx IZ to to H. 3.1. We We define define aa cylinder, cylinder, I,I,on on 9rpd Grpd as as follows. follows.
Definition Definition (1.2). (1.2).
9xI=9xI
eo(9)(x)
= (x,O), el(9)(x) = (x, 1)
(These equations equations describe describe eo(9), eo(G), el(9) el(G) both both on on objects objects and and on on mormor(These of aa phisms ifif we we use use the the general general convention convention that that for for an an object, object, z,z , of phisms groupoid, groupoid, the the identity identity at at zz isis also also denoted denoted by by z.) z.) Finally Finally
0"(9) : 9
X
I ~ 9
the projection projection onto onto the the first first factor. factor. isis the
Exercise. Describe Describe aa morphism morphism from from L{2} A(2) xx IZ to to H 3.1.. Exercise. Such aa morphism morphism isis aa simple simple example example of of aa homotopy homotopy between between Such morphisms of of groupoids. groupoids. We morphisms We next next give give this this notion notion in in general. general. Let Let f, g : G + 3. 1 be morphisms of groupoids and 4 : f g be a homotof, 9 : 9 ~ H be morphisms of groupoids and ¢ : f ~ 9 be a homotopy, ¢: 4 :9G xx IZ ~ +H. 3.1. For For each each object object xx of of 9, G, let let py,
=
(1.3)
¢x = ¢(x,t) E H .
Thus we weobtain obtainaa collection collectionof ofmorphisms morphismsof ofHX, 4,, xx EEOb(9), Ob(G),where where Thus , ¢x, 4, : f (x) + g(x). Then for any a E G(x, y), we have a commutative ¢x : f(x) ~ g(x). Then for any a E 9(x,y), we have a commutative 152
square in 'H
f( x) (1.4)
¢x
g(x) g(a)
f(a) f(y)
¢y
, g(y)
(Note that (1.4) is obtained by applying the functor ¢ to the commutative square
(x,O)
(x, t)
(a,O)
(x,l) (a,l)
(y,O) (y ,t) (y ,l) in 9 x T .) Hence a homotopy ¢ : f ~ 9 between f and 9 gives rise to a natural transformation between f and g. Since a natural transformation between functors of groupoids is automatically a natural equivalence, a homotopy between functors of groupoids determines a natural equivalence between functors of groupoids. We leave it as an exercise to prove that the converse is also true. Exercise. Let f, 9 : 9 + 'H be morphisms of groupoids, let ¢x, x E Ob(9) , be a natural equivalence between f and g. Show that (1.3) uniquely determines a homotopy ¢ between f and g. Thus homotopies of groupoid morphisms correspond to natural equivalences of functors. It follows that a groupoid morphism is a homotopy equivalence if and only if it is a natural equivalence of categories. In the category of groupoids a general 'functor groupoid' construction is available. If g, 'H are groupoids, let 'H g denote the groupoid whose objects are the groupoid morphisms f : 9 + 'H . If f , 9 are objects of 'H g , then the morphisms in 'H g from f to 9 are just the homotopies from f to g, and homotopies are composed in the obvious way, for example 153
by composing squares squares such as as (1.4). (1.4). The composition of homotopies ¢4 : f ~ : 9g ~ N g , 'l/J $J N h will be written 'l/J ?I, + ¢c j :: ff ~ E h. h.
+
Exercise. Exercise. Show that the functor functor groupoid construction 1{i 'HGsatisfies satisfies the 'exponential law' law'.. More precisely, precisely, if (}, G ,H Z , lC K: are groupoids groupoids,, then there is an isomorphism of groupoids
e : lC7l x 9 ~ (lC 9 )7l given by the usual exponential formula
((e(f))(y))(x) = f(x , y). In particular, for for each groupoid (} 4 the functor ( ) x (} G : (}rpd Grpd +r (}rpd Grpd is left adjoint to the functor

( )9 )G: (}rpd Grpd  r (}rpd. Grpd. This means that the category (}rpd Grpd has the structure of what is called a cartesian closed category. If (} to be the groupoid 1, I , we obtain a cocylinder co cylinder functor If we specialise G

Grpd  r (}rpd. Grpd. (( )= l :: (}rpd on (}rpd Grpd giving rise to an adjoint cylinder/cocylinder cylinderjcocylinder pair (1,P) (I,P) on (}rpd Grpd (see section 11.3). 11.3). As in (1.2.5) (1.2 .5) we can define cofibrations of groupoids by a homotopy extension property and, as we are in an adjoint cylinder/cocylinder cylinder j co cylinder situation, fibrations fibrations by a homotopy lifting property (1.3.4, (1.3.4, 11.3.7). II.3.7). However, in the theory of groupoids cofibrations are of minor importance H is a coand we merely state that a morphism of groupoids f :: G (} +  r 'H fibration if if and only if the underlying object map f : Ob(G) Ob(9)  r Ob('H) Ob(H) is injective. (Prove this yourself or see HeathKamps (1976).) A moryourself HeathKamps (1976) .) H is a fibration ifif and only ifif in each phism of (}  r 'H of groupoids p : G diagram of solid arrows


154
x (1.5)
f
.9
eo(X)[ / ; / XxI
¢>
[p 1i
there is a diagonal extending the diagram commutatively. In order to characterise fibrations of groupoids in a different way, we need the following definition.
Definition (1.6). Let 9 be a groupoid, and x E Ob(9). Then the star of 9 at x, written Stgx, is the union of the sets g(x,y) for all y E Ob(9). If f : 9    t 1i is a morphism of groupoids, and x E Ob(9), then Stfx is the restriction of f mapping Stgx
t
St1if(x).
We say f is star injective, star surjective, star bijective according as St fX is injective, surjective, bijective for all x E Ob(9).
Remark. If 1i is a groupoid then we saw earlier that, under evaluation on £ the functors I    t 1i are bijective with morphisms of 1i. Calling a functor I    t 1i a path in 1i, star surjectivity can be interpreted as a path lifting property. The situation can be illustrated by the following picture. x ....   ; t  
9
a
f f(x) ...'))a
= f(a)
Proposition (1. 7). A morphism of groupoids p : 9 fibration if and only if it is star surjective.
t
1i zs a
Sketch of proof Let p be a fibration of groupoids. In order to see that p is star surjective, take X in (1.6) to be the point groupoid o. 155
Then 0 x I is isomorphic to I . Now use that morphisms 0   9 are bijective with objects of 9 and morphisms I   'H are bijective with morphisms of 'H, as mentioned above. Conversely, let p be star surjective. Let (1.5) be a given solid arrow diagram. For each object x of X choose an element Bx in Stgf(x) such that
p(Bx) = ¢(x, t). This is possible, since p is star surjective.
Exercise. Show that
f
and the equation (x , t) =
Bx
determine a morphism : X x I   9 filling in the diagonal of (1.5) commutatively. 0
Exercise. Let G , H be groups, let M be a Gset , N an H  set, (J : G   H a group homomorphism , and K, : M * N a map such that the diagram
GxM·M (JXK,
K,
HxN
. N
commutes. Define
(Jr.x.K, : Gr.x.M by K, : M
*
*
Hr.x.N
N on objects and ((Jr.x.K,)(Y, g, x) = (K,(Y), (J(g), K,(x))
on morphisms. Prove that (a) (Jr.x.K, is a functor , (b) (Jr.x.K, is a fibration of groupoids if (J is an epimorphism of groups. In Chapter IV we shall see how fibrations of groupoids give rise to a type of exact sequence encoding the information contained in the 156
bottom part of the usual exact sequences in homotopy theory such as the Puppe sequence (cf. Puppe (1958)). We now turn to the question of whether the cylinder, resp. cocylinder, on 9rpd is generating.
Proposition (1.8). The cylinder on the category 9rpd of groupoids satisfies DNE(2) and DNE(3).
Corollary. Both the cylinder and the cocylinder on the category of groupoids are generating. Proof of (1.8). We restrict ourselves to sketching the proof for DNE(2,1,1). First we point out that in the category of groupoids, fillers are uniquely determined. Hence E(2,1 ,l) implies DNE(2,1 ,1). Furthermore by adjointness it is sufficient to prove the following . If 'H is a groupoid and f,g,h: I
+
'H
are morphisms such that f(O) = g(O), h(O) unique morphism
9 (1) , there exists a
j +1
+1
i ::; j .
Exercise. Check that you understand why each of these identities occurs. 163
Products and and Function Function Complexes Complexes in in Simp Simp Products As As Simp Simp == Sets ~ e6 ° t is is saa category category ~ ~ ~of of setvalued setvalued diagrams, diagrams, we we can can form form up up products products 'componentwise' 'componentwise',, that that is is dimensionwise, dimensionwise, so so if if X, X ,YY are simplicial simplicial sets, sets, their their product product XX Xx YY isis defined defined by: by: are P
Definition (2.1). (2.1). Definition (X x Y)n 
Xn x Yn,
di(x, y)
(dix, diy),
similarly similarly for for the the degeneracies degeneracies Si(X, y) = (SiX, SiY)·
Given Given this this we we can can take take X X xx .6[1] A[l] to to be be aa cylinder cylinder object in in Simp. Simp. The X with X X xx .6[0] A[0] means means that the the two two end end The natural natural isomorphism of X maps point maps eo : .6[0] + .6[1] el :
.6[0]
+
.6[1]
which are are induced induced by 8S11 and and 8So0 respectively, respectively, more more precisely which eo = .6( , 8D : .6(, [0]) + .6(, [1]) , el
= .6(,8~): .6(,[0]) + .6(,[1]),
=X X xx i,i, i = = 0,1. O , 1 . Similarly define end inclusions, inclusions, ei(X) ei(X) = define
u(X) = X x .6(, (8) : X x .6[1]
+
X x .6[0] ~ X.
This defines defines the the natural cylinder on Simp, Simp, but before examining examining the the This detailed structure of it it,, we we wish to to look look for for aa corresponding corresponding co cocylinder. detailed cylinder. is to to be an an adjoint then we would expect to to have an an interpretation interpretation If it is X I as as aa simplicial simplicial function function complex of maps from from .6[1] A[l] to X. X. It will of Xl search. Suppose Suppose that Y Y and ZZ are are simplicial simplicial sets, sets, pay to widen our search. we want to define define a function function complex ZY ZY which satisfies an adjointness we product,, that is is we want a natural isomorphism relation with product 164
Simp(X, ~ Simp(X y). S i m p ( X , yyZ) Z)E S i m p ( X x Z, Y).
We will turn aside one moment from the direct search to point out a useful fact. Xn is an nsimplex, nsimplex, x fact. If X is a simplicial set and x E X, determines uniquely a simplicial mapping
x:
6[n]t X
so that z(Id[,]) xCI d[n]) == x. (Categorical note: this is just just a special case of the Yoneda lemma cf. MacLane (1971), (1971), III, 111, 2, Nat(6( , [n]), X)) ~ X([nl).) X([n]).) Nat(A(, [nl),X
This useful fact means that we can identify X, Xn with Simp(6[n], X)) S i m p ( A [ n ] ,X for any simplicial set X. In particular for our soughtafter soughtafter function function z, assuming it exists, there will be a natural isomorphism complex, complex, Y YZ, (yZ)n
Simp(6[n], yZ) ~
Simp( Z x 6[n], Y)
so we can now define the simplicial set YZ yZ by
Definition (2.2). yZ YZ
= Simp(Z Y), = S i m p ( Z x 6[]' A[],Y),
i.e. i.e. as above
8i and (Ji with face face and degeneracy maps induced from the Si g i in a fairly yZ . obvious way. We also will use the notation Simp(Z, Simp(Z, Y) for yZ. Notice that (yZ)o (YZ)ois the set of simplicial maps from Z to Y whilst (yz)t (YZ)l is the set of homotopies. Specialising back to our search for a co cylinder taking X' Xl to be xA[l] XL'>[l] in the above sense does the job. cocylinder
Exercise. Write down the structure maps of the co cylinder and cocylinder verify that they do what is required of them. We thus have an adjoint cylinder / co cylinder pair on Simp, cylinder/cocylinder Simp, so we are in the situation of section II.3. 11.3. This cylinder is not generating however, however, as the Kan conditions are not satisfied. To see how bad the situation 165
is, we exhibit a pair of maps which are homotopic in one order, but not in the reverse order. The two maps are eo, el : 6[0] + 6[1]. They are homotopic using the identity function on 6[1] as homotopy, eo ~ el, but as there is no Isimplex going from vertex 1 to vertex 0, el rj:. eo , so homotopy is not symmetric.
Exercise. Give an example of a simplicial set K such that there are three maps Xo , Xl, X2 : 6[0] + K, with Xo ~ Xl, Xl ~ X2 , but Xo rj:. X2. (If you read on a bit after first trying this , it will be obvious, if it is not to start with.) The reason why the Kan conditions are not satisfied is that we have not required them to be satisfied. What do we mean by that?
It sometimes pays to look back at the papers in which ideas were first put forward . This requires an effort , as notation is more complicated and in the original papers the process of simplification of presentation of the ideas has not yet taken place, however the reasons for certain directions that were taken are often clearer even when implicit rather than explicit. In his foundational work on abstract homotopy theory, Kan (1955, 1956) developed cubical sets and filler conditions in them. He also developed the basis for simplicial sets and found that to do a reasonable amount of homotopy theory in this, then new, setting, simplicial analogues of the cubical Kan conditions were a suitable tool. In our treatment of simplicial homotopy theory, the cylinder does not satisfy the cubical Kan conditions because our simplicial sets did not satisfy corresponding simplicial Kan conditions that we now introduce.
Definition (2.3). Let n E IN , 0 :S i :S n and form a simplicial subset A[n, i] by deleting from 6[n]n' the unique nondegenerate nsimplex (corresponding to I d[nJ : [n] + [n] when 6[n] is thought of as 6(, [n])), deleting from 6[n]n1 the element 8~ : [n  1] + [n], and taking the smallest simplicial subset containing the result . This for instance also deletes degenerate images of I d[nJ and 8~. The mental pictures of A[n, i] that one uses are something like the followings for n = 2.
166
/\[2,0]
/\[2,1]
/\[2,2]
In each case the face opposite i is missing as is the top dimensional simplex. Given a simplicial set K, an (n, i)horn, 'Y, in K is a simplicial map 'Y : /\[n, i]
t
K.
(Some authors use the term (n, i)box extending the use in the cubical case.)
Example. Taking n = 2, a (2,0)horn in K consists of two 1simplices, Xl and X2 in K, such that dlX2 = dIXI·
We write 'Y = (, Xl, X2) in agreement with the sort of notation used in Definition (1.5 .3) in the cubical case. Similarly a (2, I)horn, 'Y = (xo, , X2), satisfies dOX2 = dlXo
and a (2, 2)horn, 'Y = (xo, Xl,  ) , satisfies
167
Exercise. Find the conditions on 'Y = (Xo, Xl,' . "  , ' . . ,x n ) where the blank is in the ith position, so that 'Y corresponds to an (n, i)horn in K. Definition (2.4). If 'Y is an (n, i)horn in K, an element called a filler for 'Y if'Y = (xo, Xl, ... , , ... ,x n ) and djx = Xj
0::; j
::; n,
j
X
E Kn is
i= i.
A simplicial set, K, is said to satisfy the Kan condition (n,i) if every (n, i)horn in K has a filler. If K satisfies the Kan condition (n, i) for all 0 ::; i ::; n, then K will be said to satisfy the Kan condition in dimension n. If K satisfies the Kan condition in all dimensions, then K is said to be a Kan complex. Example. If X is a topological space, then Sing(X) is a Kan complex. This follows from the simple geometric fact that the union of all except one of the (n  1)dimensional faces of a standard n simplex is a retract of t::, n. Exercise. (1) Prove the above statement in detail and then extend your proof to show that if we define a simplicial set Top(X, Y)n = Top(X x t::,n, Y) with face and degeneracy maps induced from those of the standard simplices, then Top(X, Y) is a Kan complex. (2) Let C be a small category. We define the nerve of C to be the simplicial set
Ner(C)n = Cat ([n], C) with the 'obvious' face and degeneracy maps. (Here Cat denotes the category of small categories and we are considering the ordered set, [nJ , to be a category in the usual way, thinking of i < j as a morphism from i to j.) Prove (a) For any small category C, any n 2: 1 and any 0 < i < n, N er(C) satisfies the Kan condition (n, i) . 168
(b) N er( C) is a Kan complex if and only if C is a groupoid. (b) Ner(C) (Simplicial (Simplicia1 sets in which the Kan condition (n, (n, i) is satisfied for all n n , i.e. not necessarily for i = = 0 and for i = = n are often often and 0 < i < n, called weak Kan complexes.) Proposition (2.5). If If K K and L L are simplicial sets and L L is a Kan complex) LK is a Kan K an complex. complex, then the function complex) complex, LK complez.
Proof. Suppose we are given an (n (n,, i)horn i)horn 'yY in L", LK , 'Y: !\[n,i] t ~ L". LK. y : A[n,i]
By adjointness this data is equivalent to that contained in the adjoint map map 'Y: K x !\[n , i] ~ L.
From this viewpoint the problem of finding a filler is that of finding a dotted arrow to complete the diagram K x !\[n, i] _ _'Y'......+LL K A[n,i] /' /
I
I
./
/ 
./
./ 'Y
K x .6.[n]
To do this we proceed by induction 'up the skeleta' of K. K . To be more subcomplexes (skjK), = = KT precise, K K is filtered by the sub complexes skj(K) skj(K) where (skjK)r Kr if r 5 ::; j and skjK is the smallest simplicial subset with this property. E KO (skoK), ~ skoK is the constant simplicial set with (skoK)r Ko In dimension 0, skoK for all r. If for If we look at 'Yo skoK x !\[n, i]' L
I
/'
./
/
./
./ 'Yo
skoK x .6.[n]
'Y is defined by a family of (n, horns, one for each Osimplex K.. y (n, i)i)horns, Osimplex in K As L is a Kan complex, complex, each of these has a filler allowing allowing us to fill fill in the dotted arrow. 169
N ow assume that we have a dotted map in the diagram Now fjl
./
skj_1K x I\[n, i]= L
I
,/
,/
/fjl
,/
and look at skjK x I\[n,i] ~ L .
If x E K complex of K Kj, dx denote denote the sub subcomplex K defined by the j , then let ax (j I)dimensional faces of x, ax ~ skj_1K. In the special case where ( j  1)dimensional faces x, dx 2 ~ k j  ~ K . x E .6.[j]j A k l j is is the jsimplex jsimplex corresponding corresponding to Id[j), Idlil, then will write a.6.[j] dAb] for for ax. dx. This corresponds corresponds to the boundary of the the jsimplex jsimplex in topological simplices. simplices. We thus know 17 on ax dx x .6.[n] A[n] as as well as as on {x} {x) xx I\[n,i]. ~ [ ni]., This This gives gives data that corresponds corresponds to a map f : a.6.[j] x
.6.[n] U .6.[j] x I\[n, i]
t
L,
which we need to extend over all of .6.[j] A b ] xx .6.[n]. A[n]. If we list the simplices simplices in .6.[j] A b ] x .6.[n] A[n] on which which we as as yet do do not know 1, 7, then we can fill fill them one L is is a Kan complex, until fy has one by one one using the fact fact that L been extended over all all of the product. The proof is is completed by using 0 induction on jj..
Exercise. The The technique of induction up the the skeleta skeleta is is so so useful Exercise. that we we have have omitted the the details! details! The The details details involve involve an an analysis analysis of the the A[j] x .6.[n] A[n] but not in in i.e. those those in .6.[j] 'missing' simplices, i.e. d A b ] xx .6.[n] A[n] U .6.[j] Ab] xx I\[n, ~ [ ni]. i]., The The aim aim will will be to to develop develop a combinacombinaa.6.[j] mechanism to to list these missing missing simplices simplices in aa systematic systematic way so so torial mechanism that they can be filled filled one one by one. = 2, 2, nn = = 2, ii = = O. 0. The The rsimplices rsimplices in (1) Examine Examine the the case case jj = (1) A[2] xx .6.[2] A[2] can be denoted denoted (ao,al,"',a (ao, al, . . ,a,) x (bo, bl, . ' ,b ,b,)r ) where where ) .6.[2] x (bo,b1," r ao, a1,". ',a  , a,r give give an an rrsimplex of .6.[2]' A [2],similarly for for (bo,· (bo, ··  ,b ,b,)r ).. Thus Thus aO,al,  simplexof (01122). (Loowe have have nondegenerate nondegenerate 4simplices such as as (00112) (00112) xx (01122). we king back at the the definition definition of degeneracies degeneracies in aa product product,, it is is clear king such a simplex is is not degenerate since since the the repetitions repetitions occur in that such 170
different positions in the two factors.)  List all the nondegenerate 4simplices in 6.[2] x 6.[2]. Then find out what faces are common between them and which faces lie in the 3dimensional sub complex 86.[2] x 6.[2] U 6.[2] x 1\[2,0]. Find a way of filling these one by one. (If you do not have a horn to fill at any stage, you may have to construct a lower dimensional face using a 2horn of some type.) (2) Generalise the method of (1) to the general case. (It may help to try a second example in detail.) The method developed above is often used without going into the details, however in certain areas, such as the study of higher order homotopy coherence, such an analysis is essential as only weak Kan complexes are available (cf. CordierPorter (1986».
Corollary (2.6). If K is a Kan complex, then so is the cocylinder KI. 0 Corollary (2.7). If K is a Kan complex and L is in Simp, then homotopy is an equivalence relation on maps from L to K. 0 The proof should be obvious by now.
Remark. We will often say 'K is Kan' instead of 'K is a Kan complex'. We let !Can denote the category of Kan complexes. This category does not inherit a cylinder from the larger category, Simp, since 6.[1] is not an object in !Can so !Can is not closed under ( ) x I as defined on Simp. The same is fortunately not true for the co cylinder by (2.6) above. We take P= (( )LW1,cO,c},s) to be this co cylinder on !Can. Corollary (2.8). The cocylinder P on the category, !Can, of Kan complexes satisfies E( n) for all n . Proof. For each (n, v, k), 0::; v ::; 1, 1::; k ::; n, we can construct a simplicial model for the (n, v, k )box, which we will denote by B(n ,lI,k)' There is an inclusion B(n,lI,k) ~
r, 171
1=6.[1]
and hence a restriction
The (K, L)n Q~ is is given given by QP QP(K, L), == Simp(K, S i m p ( K , LIn), L'"), so so the the set of The functor functor QP (n, ) boxes in ,k)) which is (n,v, v,kk)boxes in Simp( S i m p (K K,,L) L) is is Simp( S i m p (K, K , LB(n,V LB(n,utk)) is isomorphic L) to to Simp(K S i m p ( K xx B(n,v,k), B(n,v,k), L).. Assuming L L is is Kan, Kan, we we can use induction up up the )box. 0 v,kk)box. the skeleton of K K to to give give the the filler filler for for any (n, (n, v, Remark. Remark. With care care the the above above method allows allows one one to to produce fillers fillers that are are compatible with the the degeneracies, degeneracies, however however as as the the fillers fillers for for horns in a Kan complex chosen, it is is difficult to to satisfy satisfy complex have have to to be chosen, naturality conditions conditions on on the the cylinderjcocylinder cylinder/cocylinder pair. pair. This This is is not aa disaster and and by aa slight slight detour detour we we shall shall be be able able to to replace replace reliance on on the DNE(2) condition, by an an alternative alternative argument that will will also also be be the DNE(2) useful useful later later on. The conditions in The initial initial use use of DNEDNEconditions in the the development development of the the theory theory in in Chapter Chapter II was was in in the the proof of Proposition (6.2) (6.2) which which stated stated that A A ~ E
and and ~ !z were were equivalence equivalence relations relations.. For the the other other results in in which which B B
DNE(2) was was aa condition condition,, it it played played that role role by by means means of (6 (6.2), DNE(2) .2) , that is to to say say it it was was the the conclusion conclusion of (6.2) (6.2) that that was was used rather rather than than an an is independent use use of of DNE(2). DNE(2). Further independent Further attention attention to to detail detail reveals reveals the the A
interesting fact fact that that in in all all the the places places in in which which it it is is used used that that ~ !z is is an an interesting equivalence relation, relation, the the only only case case needed needed is is when when the the domain domain or or somesomeequivalence times both both the the domain domain and and the the codomain codomain of of the the morphisms morphisms involved involved times A are cofibrations, cofibrations, i.e. i.e. the the key key result result is is that that DNE(2) DNE(2) implies implies that that ~ IS is an an are equivalence t iif,f , equivalence relation relation on on morphisms, morphisms, ff :: ii +

A
/\
X _ _ _ Xf
f and ifit are are cofibrations. cofibrations. That That isis what what is is used used in in Dold's Dold's where both both ii and where theorem (1.6.3), the identification of trivial cofibrations (I.6.9), the Retheorem (1.6.3), the identification of trivial cofibrations (1.6.9), the Re172
lativity Principle (I.6.13), the pushout condition on trivial cofibrations (I.6.14) and hence in the results on generating cylinders in Chapter II. Analysis of the proof of (6.2) reduces it in our case here to proving that a dotted arrow exists in a diagram of the form B(n,v,k)    , /
Simp(X, Y)
/ Simp(i, Y)
/'
/' /
In     Simp(A, Y) where i is a cofibration in the cases we need. Such a diagram is reminiscent of the HLP and the definition of fibration in a category with A cylinder. This suggests that we might attack ~ being an equivalence relation if we knew Simp( , Y) sends cofibrations to fibrations. So our announced detour is to have a brief look at Kan fibrations.
Kan Fibrations The cylinder/co cylinder pair on Simp is an adjoint pair as in (I.9.5) so by (I.9.7) a map p : E ~ B is a fibration with respect to the cocylinder if and only if it has the homotopy lifting property with respect to the cylinder: in each commutative diagram of solid arrows
there is a diagonal extending the diagram commutatively.
Definition (2.9). A map p : E ~ B in Simp is called a Kan fibration if in any commutative solid arrow diagram
173
J\[n, k]
f
.E
I/~' /Ip .6.[n]
1, 1, 0::; O < kk ::; =< f(x), t > where where < < x, x, t
x E K n, t
E ;6.n
> is is the equivalence equivalence class class determined by (x, (x, t). t).

Exercise. Exercise. Check that If 1 f I,1, thus defined, defined, is is well defined. defined.
Definition (2.17). (2.17). A morphism f : K t LL in Simp Simp is is a weak Definition equivalence if If I1 is a homotopy equivalence equivalence in Top. 'Top. equivalence Remark. There is a notion of weak equivalence in Top 'Top defined Remark. r n ( X ,x) x) of a pointed space (see (see the end of using the homotopy groups 7rn(X, section). For CWcomplexes, CWcomplexes, all weak equivalences are homotopy this section). equivalences so so the above above definition definition could equally well have been given given equivalences equivalence. Also it is is possible (see (see later) later) ( fI being a weak equivalence. in terms of If define homotopy groups 7rr nn (K, ( K ,x) x) for for pointed simplicial simplicia1 sets sets and then to define equivalence if and only if for for all base points xx E EK KO, ff isis a weak equivalence Q, and 0, 7rr nn(f) ( f ) is is an isomorphism: all nn ;::: 0,
1
>
7r r n (f) ( f ) : 7rrn(I{,x) 5 7r.I~,(L, f (x)). n (I 0 and 7ro ~ M. (We have not yet defined 7rn(X) for X, a simplicial Rmodule  but we do not need to do this to describe these constructions! We just treat X as a simplicial set and calculate 7rn(X) with 0 as base point.) (a) Let K(M,O) be the simplicial Rmodule with K(M,O)n = M for all n 2:: 0 and with all di and Si the identity isomorphism on M. (b) Let T(M) be the free Rmodule on the underlying set of M. To indicate whether m E M is to be considered as such or as a generator of T(M) we will write (m) for the second of these. Thus a typical element of T(M) would look like
189
s
t =
L
ri(mi)
i=1
for elements mi E M and ri E R. There is an obvious homomorphism, E(M), from T(M) to M obtained by removing the brackets, thus sending the above t to L rimi in M . Now set To(M) be T(M), TI(M) = T(To(M)) and so on. A typical element of TI (M) might look like
u = rl(r2(m2) + r3(m3))
+ r4(r5(m5) + r6(m6) + r7(m7 )),
but with no limit, of course, on the number of terms involved. There are, as a result, two ways of getting an element of To(M) from u, essentially by removing either the inner or the outer set of brackets. These gIve and
U" = rlr2(m2)
+ rlr3(m3) + r4r5(m5) + r4r6(m6) + r4r7(m7) '
This is quite simple, but later we will need this idea in the chapter on the foundations of homotopy coherence where the possibility of studying the brackettings that result is of first importance. Here we continue setting Tn(M) = T(Tn l (M)) , obtaining n + 1 homomorphisms from Tn (M) to Tn l (M) by removing anyone of the different layers of brackets and evaluating the result. The 'edifice' so far looks very like a simplicial Rmodule with these bracketstripping operators as face maps, but as yet we have no morphisms going up a dimension that could act as degeneracies. Consider (m) a generator of To(M) , then ((m)) is a generator of Tl(M) , hence this defines a morphism So : To(M)    t Tl(M). Removing either pair of brackets gives back (m) , that is dos o = dl So = I d. Similarly from levell to level 2, but now there are two places at which one can insert brackets so one has So, SI : T l (M)    t T2(M) and so on. Thus it looks as if (Tn(M), dj , Si) will be a simplicial R  module. This is the case, but to check that fact 'by hand' is quite tricky. By considering a more general viewpoint, it will be clear what is making this work. 190
RMod + + Sets and a left adjoint We have a forgetful forgetful functor U : RMod F : Sets + 'free' functor F + RMod. These of course come come with a + UF F and counit cE :: F U + + Id. (The unit is 'insert unit '"q :: I d + FU I d. (The generators', the counit is 'remove brackets and evaluate'.) The above construction is:  P Put T = FU, Tn(M) ut T = FU, Tn(M) = = Tn+l(M). Tn+'(M).

 Set d TnicTi. dii : Tn(M) Tn(M) + +Tn1(M) TnI(M) equal to TnieTi  Set si Si :: Tn(M) TniF",UT i . Tn(M) + +Tn+l(M) Tn+l(M) equal to TnVqUTi.
Exercise. Exercise. The adjointness relations give that T T is part of a comonad (see Definition (1.4.6) (1.4.6)and dualise). As a consequence one has commutative diagrams T .!!!:....... T2
(a) (a)
m\

\Tm
=F qU where m = F",U
T2_T3 T2 T3 mT mT
and
(b)
(to see how these arise from from the adjointness, consult any book on category theory such as MacLane (1971)). (1971)). Using these facts verify that T(M). = = (Tn(M), (Tn(M),dd;, si) as defined above i , Si) is a simplicial simplicia1 object in RMod.
Remark. Remark. In fact fact your verification should be completely independent of the context, i.e. given categories A, B 23 and an adjoint pair, F :B 23 + + A, U: U : A + + B, 23, with q, as unit and counit, the Fl "', cE as F+ U, U, F: above construction via the comonad T T= = (FU, (FU, m, c) E ) will yield a simpli191
cial object in A. Dualising one gets a monad (U F, fl, "') which yields a cosimplicial object in B. Returning to the simplicial Rmodule T(M)., we note that by any route from Tn(M) down to M via To(M) using any composite of face maps, one obtains the same morphism, which we will denote en. Exercise. Verify the above statement and then prove that the resulting family of morphisms gives a simplicial morphism, e. : T(M).
+
K(M,O).
This finishes our examples for the moment. We next consider the question of a cylinder and a co cylinder on Simp(A). This category is 'simplicially enriched' in a way we will consider in detail later on. In other words given X, Y in Simp(A), there is a simplicial set of morphisms from X to Y in Simp(A). In fact we will see later that this is true for any category A without restriction. Assuming this, we expect , and again it is true, that the resulting Simp(A)(X, Y)o should be the same as Simp(A)(X, Y), and that
S(f}.[l],Simp(A) (X, Y)) = Simp(A)(X, Y)l will be the homotopies between morphisms from X to Y in Simp(A). More generally our experience with !Can suggests that we should examine S(K,Simp(A)(X, Y)) for K a simplicial set (usually a finite simplicial set, but that need not bother us for the moment). We would hope for some object K ®X so that when K = f}.[1J, this would give f}.[l]®X as cylinder. Thus this way of thinking leads one to search for a construction K ®X such that
S(K,Simp(A)(X, Y)) ~ Simp(A)(K®X, Y) but Simp(A)(X, ) gives a functor from Simp(A) to Simp and the above isomorphism makes  ®X into a left adjoint of that functor. Now category theory gives one general constructions of left adjoints using colimits so we can hope to follow through such a construction to try to find a construction of K ®X. However all our fine reasoning has been based on pretty poor foundations using 'if we could prove' arguments, so we will take here a somewhat different route returning 192
to these general arguments later.
Definition (2.18). Let X be an object in Simp(A) and K a finite simplicial set (i.e. each Kn is a finite set). Define K ®X to be the object of Simp(A) given by
(K®X)n = 67 X n. Kn
Although the objects of A need not have 'elements', we will adopt an elementwise description of the face and degeneracy maps of K ®X . We will denote by k ® x an element in the kth summand X n , i.e. for x E X n, k E Kn . With this notation, the face and degeneracy maps gIve di(k ® x) = df(k) ® df(x) si(k ® x) = s{«k) ® sf(x).
Remarks. (i) If one needs to form K ®X where K is not a finite simplicial set then one needs A to be co complete so as to use infinite coproducts. (ii) We note for later use that the analogue of this construction in the nonabelian case works, even when A is Sets. In that latter case, K ®X is nothing more nor less than K x X. Definition (2.19). If X, Yare in Simp(A) , we define Simp(A)(X, Y) by
Simp(A)(X, Y)n = Simp(A) (;6.[n] ®X, Y). Proposition (2.20). Given any finite simplicial set K , there is a natural isomorphism
S(K,Simp(A)(X, Y»
~
Simp(A) (K®X , Y)
Proof. We will give a sketch omitting some details. We have already
seen that K can be written as a colimit
K ~ (U(Kn x ;6.[n]))j,...., = n
j
[n]
Kn x ;6.[n],
see the discussion before Definition (2.15). There we noted that this 193
was bad notation so in fact Kn x ,6,[n] should be replaced by
U,6,[n], the disjoint union of Knmany copies of ,6,[n]. Thus
S(K, Simp(A)(X, Y)) ~ ~
S((U ,6,[n])/ "J, Simp(A)(X, Y))
nS(,6,[n], Simp(A)(X, Y))/ "J nSimp(A)(,6,[n]0X, Y)/ "J, Kn
~
Simp(A)(((U ,6,[n])/ "J)0X, Y)
= Simp(A)(K0X, Y).
0
Remark. The omitted details are to check the compatibility of
"J with the various isomorphisms. Various parts of the proof such as taking U outside to become n are useful consequences of the universal properties of coproducts and products. We give, for fun, the full proof using the end/co end calculus. We hope it will help convince the reader to check up on the foundations of this very useful piece of categorical machinery. First we note that as S(K,L) = NatTrans(K,L), the set of natural transformations between two functors, it has a formulation as an end
S(K, L) ~ f Sets(Kn, Ln). J[nl That being the case we note:
S(K, Simp(A)(X, Y))
AnI Sets(Kn, Simp(A) (,6,[n] 0X, Y)) ~ AnI Q Simp(A) (,6,[n]0X, Y) ~ AnI Simp(A)((~ ,6, [n])0X, Y)
~ Simp(A)((/nl U ,6, [n])0X, Y) Kn
Simp(A)(K0X, Y). We leave it as an exercise to explain why each line follows from the 194
previous one, one, an exercise exercise to be done done with help from from a text that describes describes ends ends and coends (e.g. (e.g. MacLane MacLane (1971))! (1971))!
(i) Using the remark that (K ( K x L)n L), = = Exercise. (i)
ULL,,n, together Kn K"
with the isomorphism isomorphism of (2.20), (2.20), prove the 'enriched' form form of (2.20): (2.20):
~ ( K , S i m p ( d ) ( XY)) Y)) , ~ Simp(A) Simp(A)(K&X, Y) S(K,Simp(A)(X, (Ki8lX, Y) X , Yare Y are in Simp(A) Simp(A) for for an abelian abelian category category A, A, then (ii) If X, (ii) Simp(A)(X,Y) Y) is is a simplicial simplicial abelian abelian group. group. Why? Why ? Simp(A)(X, The result of the last exercise exercise raises raises some some interesting questions. questions. (a) (a) We saw saw the importance earlier earlier of having simplicial simplicial homsets hornsets that are are Kan complexes. complexes. Is Simp(A)(X, Simp(A)(X,Y) Y) a Kan complex? complex ? (b) (b) In the category category Kan, Kan, we took Xl X' =Simp(.6.[1],X), =Simp(A[l], X ) , but except if A have difficulty in doing doing the same same here here as as although A == Ab, we would have Simp(A)(X, Simp(A)(X,Y) Y) may exist, exist, it will not be an object in Simp(A) Simp(A) unless unless A There is is also also the problem of the fact fact that .6.[1] A[l] is is not in A == Ab. There Simp(A), one could take ll.6.[1] Z A [ l ] in its place. place. Simp(A), either, either, but for for A A == Ab, one
examine (b) (b) first first as as ((a) require aa longer study. study. We will examine a) will require We We have have used a 'barred tensor' i8l @ to obtain obtain something something that looks looks a bit like ens or  allows similar definition definition  a cot cotensor allows one one to get like a cylinder. cylinder. A similar a cylinder for for all Simp(A) Simp(A),, A A abelian. abelian.
Definition (2.21). (2.21). Let K K be aa finite finite simplicial simplicial set, set, define define Simp(A)(K, Y) to be the object of Simp(A) determined by the adSimp(A)(K,Y) to object Simp(A) determined adjunction isomorphism isomorphism S(K,Simp(A)(X, (X, Simp(A)(K, Y)). E! Simp(A) Simp(A)(X, =(A)(K, Y)). S(K,Simp(A)(X,Y)) Y)) ~ In other words , Y). words Simp(A) (A)(, ( , Y) Y) is is right adjoint adjoint to to Simp(A)( Simp(A)(, Y).
Proposition (2.22). (2.22). For any finite K, K, and YY in in Simp(A) Simp(A) Simp(A)(K, 0 exists. Simp(A)(K,Y)) Y)) exists. The The proof will will be left left until it can can be proved proved in in aa more more general general setting setting
195
later in this chapter (see Quillen (1967)). Thus the obvious candidate for a co cylinder is Simp(A) (.6[IJ, ). The properties of this co cylinder will depend largely on the answer to question (a) above to which we now turn. Kan Complexes and Simplicial Abelian Groups
Our main aim will be to show that if X is a simplicial abelian group, then its underlying simplicial set is a Kan complex. In fact more is true as we will see: given X and a (n, k)horn 'Y in X, there is an algorithm that gives a filler and as a consequence the filling operation is natural. We will not approach this result formally as yet, since the only way to understand why the result is true is to explore low dimensional examples. The algorithm by itself only tells you the filler  unless you take that algorithm apart. As a first example, we examine a (2 , I)horn in X, so we have:
1\
with al mIssmg. We need an x E X 2 with do x = ao , d2x = a2. As a first attempt we set Xo = soao then do x o = ao as hoped for, but d2Xo = d2s 0 ao = sodlao = sodoa2, which need not be a2. We thus deform Xo to get X l = Xo + y and see what is needed of y. We hope Xl will do, so we need dO XI = ao, d2 XI = a2, hence
°
doy d2y
sOdOa2
+ a2
E
J{ er
do .
This effectively reduces the problem to one where ao = 0, a2 E Set b = SOdOa2 + a2 and try y = SIb. This gives doy
= dOs l b = sodob = 0,
and
d2y
which will do fine and gives Xl
= soao  slsodoa2
+ sla2.
We leave it as an exercise to check this works. 196
= b,
J{ er
do .
As a second example, look at a (2, O)horn: (, aI, a2), dIal = dla2. Take Xo = Sla2; this has dlXo = d2Xo = a2 . Deform Xo to Xl = Y + Xo with Y to satisfy: d2y = 0, dly = al  a2 = b, say and note dlb = o. Set Y = sob to get d2y = 0, dly = al  a2. We leave the (2, 2)horn to the reader. This does suggest a method, but we have not enough complication in dimension 2 to be sure that all the facets of the method are yet visible. We next try a (3, I)horn: (aD, , a2, a3), so dOa3 = d2ao dOa2
= dIaD
d2a3
= d2a2.
Try Xo = soao giving doxo = aD d2Xo = SOd Oa2 d3Xo = SOdOa 3· Going to the other end of the list, we try Xl = Xo + YI to fit aD and a3. The method gives us YI = S2( SOdOa3 + a3). (Note: S2 was used to obtain a3, but So for aD. The degeneracy used depends on whether we are before or after the 'hole' in the list of faces.) The new candidate Xl does not match a2. It gives d2XI = sOdOa2  SOdOa3 + a3, so take X2 = Y2 + Xl· Again Sl of the deficiency in the next face , here d2XI, works, i.e. we set Y2 = slsodoa2  slsodoa3 + sla3 . Finally X2 = slsodoa2  slsodoa3 + Sla3
+ So aD 
s2 s 0dOa3 + S2 a3·
Remark. Although X is assumed to be abelian, we have taken care over the order of terms (e.g. Xl = Xo + Yl, but X2 = Y2 + X l) as the same method works if X is a simplicial group or a simplicial groupoid. Exercises. (a) Try to fill (3,2) , (3,0) and (3,3)horns, then try (4,2) . Having done that and taking note of any difficulties that arise, prove that any (n, k )horn in X has a filler given by a sum of degeneracies . Write out the algorithm that gives the filler. Prove that if'Y is a (n , k)horn in X and f : X + Y is a morphism in Simp(A) , then the filler constructed above, 'Y, say, of'Y is mapped by f to the filler constructed as above for f("() in Y, i.e. the filling operation is natural. Finally go through your proof checking (and if necessary tightening up the proof) 197
that it will apply when X and Yare simply simplicial groups that are not necessarily abelian. (b) Adapt what you did in (a) to prove that a morphism i : X ~ Y of simplicial (abelian) groups has an underlying map that is a Kan fibration if and only if each in : Xn ~ Yn is an epimorphism. We thus have an extremely wellstructured situation. From the naturality of the fillers we can use results from Chapter I with little change. The algorithmic nature of the fillers means that explicit formulae can be given for composed homotopies, for reverse homotopies etc.
Exercises. (i) We have introduced (1.5.4) the condition for fillers in the cubical case to be compatible with degeneracies. Does such a condition make sense here? Are the algorithmic fillers for simplicial abelian groups discussed above going to give us DNE conditions when we use the cylinder or co cylinder to obtain QI(X, Y) or QP(X, Y)? We leave this task as an exercise, but it is not the only way of proceeding.
(ii) This class of examples, Simp(A), with A abelian, is also a class of additive categories, so there is another approach that needs investigation. We can define a cone on X by the pushout
X eo(X) X x I
o
. C(X).
Does this cone have the properties of a cone monad? Does it give a generating cylinder via the X EB C(X) construction? If it does, this gives an approach to deciding what cofibrations and fibrations look like in this setting. Again all the techniques necessary have already been developed. There are however links between Simp(A) and the category of chain complexes in A,Ch(A), that will be investigated shortly. It may help to see how these links interact with the various homotopy structures we have examined earlier. 198
From Simplicial Modules to Chain Complexes The classical passage from spaces to singular homology groups implicitly passed via simplicial abelian groups and from there to chain complexes. At each stage of this process information could be lost and a lot of effort went into analysing the geometric meaning of the 'lost' information. (This is one possible view, for instance, of the .Hurewicz map from homotopy to homology.) No information is lost however in the passage from simplicial abelian groups to chain complexes. As usual we will work with simplicial Rmodules although most of what will be said is equally true of simplicial objects in any abelian category. If X is a simplicial module, then there is a simple way to construct a chain complex starting from X. (If you have not met chain complexes before, a chain complex of modules is a sequence of modules, en, n E 71., with boundary maps or differentials an : en + Cn  I satisfying anIan = 0.) Here we will only be looking at chain complexes that satisfy en = 0 if n < O. Given X we set en = Xn and n
.
an = L:( l)ldi : Xn
Xn  I ·
+
i=O
Exercise. Verify that anIan =
o.
This gives quite a large chain complex and for many purposes, it is better to use the Moore complex of X. One of the main advantages of this latter construction is that it will also work for simplicial groups, simplicial groupoids, simplicial algebras, etc. with little or no extra work although the extra structure available is much more difficult to use than the simple case we will start with here. Given X the Moore complex of X is the complex, (NX, a) where n
(NX)n =
n Ker d
i
i=1
and an = do I NX n.
199
Lemma (2.23). anIan = O.
The Moore complex zs a chain complex, z.e.
Proof. One simply notes that dodo = dod!, and so vanishes on Kerd l ~ NX. 0
Remark. Some authors use different forms of Moore complex. The most common alternative is to consider nI
n Kerdi
i=O
nl
as the object in dimension n and then to take
an = dnInKer di . This i=O
is largely a matter of taste and makes only a minor difference to the theory. Given any chain complex, C = defined to be
(en, an), its
nth
homology group is
Lemma (2.24). If X is a simplicial module,
the nth homotopy group of the simplicial set obtained by forgetting the module structure in X, with base point, O. Proof. The idea is very simple. An element x E Xn is in (N X)n if all of the dix are zero except possibly dox . Thus x E K er an if and only if all its faces are at zero, i.e. if it corresponds to a simplicial map from Sn to X based at zero. Such a map is homotopic to the zero map if it corresponds to a do of something, y say, in dimension (n + 1). We illustrate this in dimension 1 by the following picture which gives the idea of why the result is true.
200
0
0
o
•0 t
0
•
0
x 0 x E KerOl
o
h
or
x
doy = x
The proof has to verify that if h is an arbitrary homotopy h : x ~ 0 then there is another homotopy h' : x ~ 0 of the correct form, i.e. zero everywhere except on the one (n + I)simplex of L.[n] x L.[I] that has x as its doface. (Note that as usual we are abusing notation and are switching from x being an element of Xn to being the representing map, X, of that element.) To prove that such an h' exists one x : L.[n] uses the fillers that were shown to exist in the series of exercises earlier.
o Exercise. The above needs formalising to ensure that there are no gaps. Write out a 'formal' proof. Remark. The above lemma is also true for simplicial groups. The Moore complex of a simplicial module contains all the information on that object. More precisely we have the DoldKan theorem.
Theorem (2.25) (DoldKan). The functor N : Simp(A) 
Ch(A)
is an equivalence of categories. We will again not give a formal proof of this here as the sketch proof given in Curtis (1971) is easily 'filled out' to give such a proof. The idea of the proof is important and so we will spend a little time on that. Given any simplicial module M, and any n, the two morphisms
dn : Mn 
Mn
l
satisfy dns n l = I d, hence any x E Mn can be written 201
(x  SnIdnx)
+ (SnIdnx)
where x  SnIdnx E Ker dn and SnIdnx E SnlMnl. In other words there is a natural isomorphism
Mn
~
Kerd n EB SnIMnl.
The idea is to repeat this with dn 1 and Sn2 and so on. The neatest way of doing this uses a construction that is often useful  the decalage functor, Dec. Given any simplicial object X in a category, C, which need have no special properties, Dec X is the simplicial object given by
(DeeX)n = X n+1 with the di of DeeX being the di+ l of X, and similarly for the degeneraCies. For any X, the last face map gives a map DeeX
t
the final d1 : Xl t Xo together with a homotopy equivalence
X;
Sl :
Xo
t
Xl
form the basis of
DeeX ~ K(Xo,O) where K(Xo,O) is the constant simplicial object with value, Xo. The extra degeneracy is used to build the homotopy from one of the composites to the identity. A version of the argument can be found in Duskin (1975) and in Illusie (1972). We will not need the details here. Returning to our simplicial module M, the natural last face map
d: DeeM
t
M
is an epimorphism split by the last degeneracy map. The kernel of d is K er dn+l in dimension n. We write K = K er d and apply the decomposition to both M n l and K n 1 = Kerd n. This gives
Mn ~ (K er dn l n K er dn) EB sn2(K er dn l ) EBsnl(Ker dn l ) EB Snl Sn2(Mn 2). The natural thing to do is to repeat this process as many times as possible. This gives a decomposition
202
The face and degeneracy maps relative to this decomposition are easily calculated. Thus one feels (and correctly so) that the Moore complex should allow one to reconstruct the simplicial module. The details are sketched out in Curtis (1971).
(c) Simplicial TComplexes Some, but not all, of the arguments above relied on the natural fillers that are there in any simplicial module. These are intimately related to the DoldKan theorem in a subtle way. Of course, simplicial abelian groups model very few homotopy types. The object of study in this section is in some ways an abstraction of the 'simplicial set with canonical filler' structure of simplicial groups. That is not how they arose, but as they also satisfy a DoldKan theorem and given that they model more homotopy types than simplicial abelian groups, they provide a first step towards an algebraic homotopy theory. The remarkable fact is that the canonical filler structure is enough to give them a very rich algebraic structure.
Definition (2.26). A Tcomplex is a simplicial set K together with for each n 2:: 1 a subset Tn ~ Kn of so called thin elements. This structure is to satisfy three simple axioms. (1) Degenerate elements are thin.
(2) Every horn has a unique thin filler. (3) A horn, all of whose faces are thin, has a (unique) thin filler whose last face is also thin. These axioms correspond to the conditions found in the nerve of a groupoid. Recall if C is a small category, N er( C) is the simplicial set given by
Ner(C)n = Cat([n],C). We mentioned earlier in this chapter (just after the definition (2.4) of Kan complexes) that N er( C) was Kan if and only if C was a groupoid. The natural fillers given by composition: for instance in dimension 2,
203
1\ L
fills to give
and and inverse, inverse, again again in in dimension 2, 2, fills to give
a/\b,
~ boa
6'
b b give K is is of rank rank give the the thin elements. elements. In In fact fact if we we introduce introduce the the term K nn to T), Ki to indicate indicate that in in the the Tcomplex (K, (K,T), Ki == Ti Ti if ii 22 n, n , then then K K has has rank 22 if and and only only if K K~ EN N eerr C for for some some groupoid, groupoid, C. C. Exercises. (i) (i) Prove Prove the the above above statement,'K statement,'K is is of rank rank 22 if and and only only if K K~ Z N eerr C C for for some some groupoid, groupoid, C'. C'. Your Your proof should should construct aa groupoid groupoid from from K. K. (ii) complexes. (ii) Investigate Investigate rank 33 TTcomplexes. (iii) (iii) Groupoids Groupoids provide algebraic models models for for all all homotopy Itypes ltypes i.e. i.e. for for all all xx EE X X and and ii > > 1.1.Prove Prove for all all spaces spaces XX which which satisfy satisfy 7ri(X, .rri(X,x) x) == 00 for this this and and then then attempt to to find find aa similar similar result result for for homotopy 2types 2types i.e. i.e. complexes as investigate (as in in (ii)) (ii)) investigate rank 33 TTcomplexes as an an algebraic algebraic structure structure (as and types. (This and see see if they model model all all 22types. (This latter part is is hard  see see Baues Baues (1989), (1989), MacLaneWhitehead MacLaneWhitehead (1950), (1950), Hilton (1953) (1953) for for information that will types (which will help help you you model model 22types (which were were previously called called 3types) 3types) and and Ashley (1978) (1978) for for more more on on simplicial simplicia1Tcomplexes.) Ashley
(iv) Given Given any any Tcomplex L, L, investigate investigate whether there there is is a natural natural (iv) S i m p ( K ,L) L) and and on on K K xx LL for for arbitrary arbitrary K. Ii'. Tcomplex structure structure on on Simp(K, Tcomplex If you you manage manage to to find find such such structures structures,, you you can can then then investigate investigate if the the If resulting co cocylinder is generating. generating. (It (It should should be be clear that the the cylinder cylinder resulting cylinder is K xx .6.[1] A [ l ] is is not aa Tcomplex as as any any Tcomplex Tcomplex is is aa Kan Kan complex complex and and K I( xx .6.[1] A[l]need not be, so so beware of of thinking thinking that this this exercise exercise should should K be 'plain sailing'. In In fact fact the the exact exact structure structure is is an an open open problem 1)!) be will later later meet crossed crossed complexes complexes which which give give aa category category equiequiWe will to that of Tcomplexes via aa version version of the the DoldDoldKan Kan theorem valent to 204
(2.25). It is interesting to see that the category of crossed complexes has generating adjoint cylinder / co cylinder pair, so T complexes should have one as well !
(d) Cubical Sets, Cubical TComplexes etc. The cubical analogues of the simplicial theory is somewhat less developed. Although for intuitive arguments the cubical theory has many advantages , the technicalities with regard to geometric realisations for instance are less evident and in some cases invalid. The problem comes from the very simple structure of the category of cubes. The definition we have given in Chapter I corresponds to a functor from a category, OOP, to Sets. The category, 0, has as objects the natural numbers, 0, 1,···, n, ... and for each m, n a morphism from m to n in 0 is an order preserving map from {O < l}m to {O < l}n satisfying certain rules. These morphisms are generated by the face maps e~ :
{O < 1}nl
+
{O < l}n ,
e
= 0 or 1
if j < i if j = i if j > i and degeneracy maps z~ : n
+ 1 + n if j < i if j ~ i
Note that these morphisms satisfy various equations dual to those given in Chapter I (Definition (5 .1)). The resulting morphisms between 'cubes' can be fairly simply described, but the problem remains as to whether there are enough morphisms in this version of 0 to give a good theory. Explicitly the question of the diagonal
{O < 1}1
+
{O < 1}2
causes problems. Should this be included as a morphism in o? At 205
present in our definition, it is not there. It would give a structure map Xfj : X 2
t
Xl,
for a cubical set, X. How should this be handled? What about an orderreversing operation or involution
rev: 1 I.e.
rev : {O
t
< I}
1,
t
{O < I}
rev(O) = 1, rev(I) = 0 This would, if added in, ensure that homotopy in cubical sets that were based on this extended 0, would always be symmetric. The debate continues. No one has a definitive answer  is there one? If cubical sets are to model all homotopy types in a neat way, the problem of understanding the relevant extended cube category may be equivalent to understanding homotopy theory! That is to say, to understand all the homotopy operations, transformations etc., that must be in any complete algebraic model of homotopy types. This would seem a large job to say the least. One possible extra structure that can be added to cubical sets is that of connections. We will not give a detailed exposition of this here as such would be slightly too technical for this text. The references for what follows are the work of Brown and Higgins on 00 and wgroupoids and the generalised van Kampen theorem, (see bibliography). The basic idea is to include structure related to the max or min maps in the cubical singular complex. Thus for instance, one gets a 'connection' Y
X
r.,a_,,
L a
x
I.e.
Y
x
'j
a
x
x
x
a Icube a gives a 2 cube with two faces equal to a and two 206
constant at the start of a. Such maps act as extra degeneracies; they obey various relations (d. Tonks (1992» and interact nicely with the cubical version of aTcomplex structure (see again the bibliography for the work of Brown and Higgins). The resulting structure is equivalent to that of crossed complexes and hence does not model all homotopy types but does give a rich and fascinating algebraic structure that is simple to define geometrically and which reveals some of the structure of homotopy types. It should be clear from the above that Cub does not have a generating cylinder. To what extent the various strengthened forms have one is still largely open, although as cubical T complexes form a category equivalent to that of crossed complexes, they must have such a structure  but again its description is highly nontrivial. These examples show that even in fairly simple structures there are difficult open problems. We next turn to a more classical situation both arising in topology and in homological algebra, the category of chain complexes. We will also look at more 'esoteric' but related structures such as crossed complexes and higher order analogues of them. 3. Chain Complexes (a) Chain Complexes in an Abelian Category As usual A will be an abelian category, but we will usually assume, for expository reasons , that it is a category of modules over a ring, R, with unit 1. Although we have already met chain complexes in connection with simplicial modules, we repeat their definition here to keep this section reasonably self contained. In fact we will give a slightly more formal treatment of the definition. Definition (3.1). A graded object, e, in A is a sequence C = {Cn : nEll} of objects in A. If elements of objects exist in A, and x E en then we say x has degree n. A morphism of degree r between graded objects, f : e + D say, is a sequence of morphisms in A
207
{fn : Cn + Dn+r : n E Zl}. It is sometimes useful to use special terminology for certain classes of graded objects. For instance, we would say that a graded object C is positive if Cn = 0 if n ::; o. If Cn = 0 only for n < 0, then the term would be nonnegative and if Cn = 0 for n 2: 0 then we say C is negative. It is usual to write cn = C n if C is negative so a superfix indicates a negatively graded object whilst a subfix is an ordinary one. The suspension sC of C is defined by (sC)n = CnI . There is a morphism s : C + sC of degree +1 given by the identity so
deg s(x) = deg(x)
+ 1.
Definition (3.2). A chain complex, (C, d) in A is a graded object, C, in A together with a morphism d : C + C of degree 1 satisfying dd = o. The morphism d is called the differential or boundary morphism of C. A chain map, f : C + D , between chain complexes is a map of degree 0 with df = fd . We denote by Ch(A) the category of chain complexes and of chain maps . Remark. If (C, d) is a chain complex with C a negatively graded object, then it is usual to call (C, d) a co chain complex and write
dn : cn
+
cn+ I
for d_n : C n + C n I. Such objects are the essential building blocks of cohomology: if (C, d) is a chain complex and A is an object of A, then taking
A(C, A)n = A(Cn , A) n with d = A(dn+I , A) , the induced map, we get a 'cochain' complex. Example. Taking A = RMod. Consider the chain complex , 1 , modelled on the cell structure of the unit interval:
10 = Re~ EB Re?, a free Rmodule with generators e~ and e?; a free Rmodule with generator , e l ,.
208
In =
°
if n =I 0,1.
The only nonzero differential will be d : II
d(e l ) = e~
~
10 and is given by
 ego
Definition (3.3). Given graded objects C, D in A, the direct sum C EEl D is given by (C EEl D)n = Cn EEl Dn. If (C, dC), (D, dD) are chain complexes C EEl D is given the differential d(x, y) = (dCx, dDy) . If A = RMod, the tensor product C ® D of two graded objects in A is given by
(C ® D)n =
L:
Ci ®R D j .
i+j=n
If (C, dC) , (D, dD) are chain complexes, then we give C ®D a differential
d(x ® y) = dCx ® y + (_l)deg(x)x ® dDy. (It is usual to drop the suffices on the differentials if no confusion will result.) Remark. Really we need C to consist of right Rmodules and D to consist of a left Rmodules for Ci ®R D j to exist. It will then be merely a chain complex of abelian groups, however if C is two sided as a module as it will be in the most important case then there is no problem and C ® D will again be a graded object or chain complex in A. This is the case in the following example . Example. Suppose D is a chain complex of left Rmodules, then (I ® D)n decomposes as a direct sum of Reg ® D n, Re~ ® Dn and ReI ® Dn l . As for instance Reg ® Dn ~ D n, it is convenient to identify (I ®D)n and DnEElDnEElDnl. In this identification (x, 0, 0) corresponds to eg ® x so d(x, 0, 0) corresponds to eg ® dx, i.e. d(x, 0, 0) = (dx, 0, 0) ; similarly d(O, y, 0) = (0, dy, 0) whilst d(O, 0, z) corresponds to d(e l ® z) = del ® z  el ® dz = e~ ® z  eg ® z  el ® dz. Thus
d(x, y, z) = (dx  z, dy
+ z, dz ).
It is then an easy exercise to check that dod = 0, i.e. that we have a 209
chain complex. This is important as it allows us to define D x I even when A is not a category of modules.
Definition (3.4). If A is an abelian category, and D is an object in Ch(A), then we define D x I by (D x I)n = Dn EEl Dn EEl Dn 1 with differential d(x, y, z) = (dx  z, dy
+ z, dz).
Furthermore let
eo(D): D
+
D x [be defined by eo(D)(x) = (x,O,O)
el(D) : D
+
D x [ be defined by el(D)(y) = (0, y, 0)
and
(1(D) : D x I
+
D be defined by (1(D)(x, y, z) = x
+ y.
Lemma (3.5). (i) D x [ is a chain complex in A. (ii) eo, el and (1 are morphisms in Ch(A) . (iii) (D x I, eo(D), el(D), (1(D)) is a cylinder object in Ch(A). Proof. This is just a matter of checking, so is left as an exercise. 0
Although the elementary cylinder based homotopy theory of chain complexes could now be developed, a quick glance at books on homology theory or homological algebra indicates that this is not how the notion of homotopy of chain maps is usually presented. The more usual presentation of the theory is via the notion of a chain homotopy:
Definition (3.6). Let /,g: C + D be two chain maps. A chain homotopy h from / to 9 is a graded morphism of degree 1 h n : Cn
+
Dn+1
so that
dh
+ hd =
9
f.
(Note the 'abuse' of notation since the two d's are different.)
Proposition (3.7). (a) 1/ H : C x I + D is a homotopy between / and g, then defining h(z) = H(O,O,z) gives a chain homotopy between / and g.
210
(b) II h is a chain homotopy between I and 9 then
H(x, y, z) = I(x) + g(y) + h(z) gives a homotopy, H : C
X
I
~
D between I and g.
0
Exercise. Prove Proposition (3.7) by direct calculation. Remarks. (i) This result is the analogue in this context of the general result on additive homotopies given in Chapter I section 8. We will return to this point shortly and in some detail, but we suggest that the reader investigate to what extent this cylinder is of the same general form as that given in the additive theory. (ii) The development of homology theory would require the introduction of a notion of a weak equivalence. This is defined to be a chain map I : C ~ D such that Hn(f) : Hn(C) ~ Hn(D) is an isomorphism for each n. Here Hn(C) = Ker(d: C n ~ Cnl)/Im(d: Cn +l ~ Cn)
is the nth homology group of C and Hn(f) is the map induced by
f.
If the 'base category' A is a module category, then there is a notion of cofibration based on free modules. The reason for this is that they correspond to chain maps which are built up by attaching chain complexes built freely from 'cells'. More precisely in this notion of cofibration I : C ~ D is a cofibration if (i) it is a monomorphism at each level and (ii) for each n, Dn/ In( Cn) is a free module. This implies that each Dn has a decomposition as C n EB Fa(n) where Fa(n) is a free module on a( n )generators, a( n) usually varying with n. This is less appropriate for chain complexes in general abelian categories, although replacing free modules by projective objects works well. For module categories, the chain complexes of free modules do form a category of cofibrant objects with a structure based on these notions of (homology) weak equivalence and cofibration. Any homotopy equivalence of chain complexes (in the sense of the cylinder structure) is a weak equivalence. The connection between the two notions of cofibrations is more subtle and will only be described when we have an alternative description of the cylinder structure and in particular an explicit description of 211
the cylinder based cofibrations. Here we merely note the existence of the other theory and refer the reader to books on homological algebra for its application and to Hartshorne (1966) for the theory of derived categories which shows another aspect of that homology based theory. We thus have a cylinder and want to know its structure, that is for instance, which of the Kan filler conditions does it satisfy. We will not answer this explicitly, again leaving a detailed investigation as an extended exercise. To help with this however we will look at condition E(2 ,I,I). In this case the geometric picture is (0, 1)
B(2,I,I) =
(a, 1)
(1,1)
(a,O)
(1,0)
,
(O,a)
(0,0)
where a is the Icell from 0 to 1. Thus if we have a map f defined on 'B(2, I, 1) 0 D' with values in E, we need to extend it over [2 0 D and hence have to find values corresponding to the summands (a, a) 0 D n  2 and (I, a) 0 D n  1• We define
1: /2 0
D
+
E
by
1(k 0
d) =
l(k 0 d)
f (k 0 d) if k E B (2, 1, 1) ,
= 0 if k = (a, a), dE D n  2
and
1((1, a) 0
d) = f ((0, a) 0 d)
+ f ((a , 1) 0
d)  f ((a, 0) 0 d).
Now extend linearly over the whole of /2 0 D. We leave you to check that this works when you take into account all the different differentials  this is not trivial as working out a notation that suits you and helps you keep track of the nine different factors of (12 0 D)n is quite tricky. 212
In fact, when you have worked through the details, it becomes clear that this cylinder satisfies all DNEconditions in all directions and dimensions. In each case you send the top dimensional cell to zero and the remaining face to a sum of the values on the other faces. Exercise. Prove that on Ch(A), the cylinder defined above satisfies all DNEconditions. Hence the cylinder is generating  and a lot more. We suggested above that, as Ch(A) was an abelian category and the functor  X I was additive, there might be a connection between this structure and the theory outlined in Chapter I, section 8. A moment's examination shows that D x I is not of the form Dffi J (D) for any functor Jon Ch(A), since if (x , y, z ) E (D x I)n, d(O, 0, z) is (z , z, dz) and so D x I does not split simply like that. There is however a second cylinder structure that fits exactly into the relative injective homotopy theory of Chapter II, section 4. This structure was first studied by Kleisli (1962) . Other useful sources are Simson and Tyc (1974), and Kamps (1978 c). Let A be an abelian category, then there is a category Qr(A) of graded Aobjects. (This category is isomorphic to the category A Z of functors from the set, ?l., of integers to A, where as usual a set is considered as having a trivial category structure in which the only morphisms are the identities.) The forgetful functor
U : Ch(A)
+
Qr(A)
which forgets the differential, has a right adjoint
R : Qr(A)
+
Ch(A)
given by
R(X)n = Xn ffi X n 1 with differential d(x, y)
= (y,O).
Exercises. (i) Check that R is right adjoint to U. Explicitly prove that if D is any chain complex and X is any graded object in A then there is a natural bijection 213
Ch(A)(D, R(X))
~
Qr(A)(U(D), X)
(ii) Prove that U also has a left adjoint. Thus taking C = Ch(A), V = Qr(A), we get an example of the situation described in the subsection on relativeinjective type comonads. Following that, we set J = RU to get a cone monad on Ch(A). We make this structure explicit as follows.
Definition (3.8). If D is a chain complex,
 J(D) is the chain complex with J(D)n = Dn E9 Dn 1 ,  j(D): D
~
 J.L(D) : J2(D)
d(x, y) = (y,O)
J(D) is given by j(D)(x) = (x,dx) ~
J(D) is given by J.L(D)(x, y, z, t) = (x, z).
The cylinder (D x I, eo(D), el(D), O"(D)) is specified by
 (D x 1) is the chain complex with D x [ = D E9 J(D) , so (D x I)n = Dn E9 Dn E9 Dn 1 d(x, y, z) = (dx, z, 0)  the structure maps eo(D) , el(D) and O"(D) are
eo( D)( x) = (x, 0, 0) e 1 ( D ) (x) = (x, x, dx)
O"(D)(x, y, z) = x. From the general theory of abstract additive homotopy (sections 1.8 and II.4) we know this cylinder is generating. This raises the obvious question as to the relationship between this relative injective homotopy theory and that defined geometrically or via the tensor with an algebraic model of 6[1]. Just for the moment we will denote the earlier form of cylinder by D ® I , using D x [ for this relative injective cylinder; the
214
structure maps symbols eo, el, a will be used with both.
Proposition (3.9). There is a natural isomorphism between  0 I and 
X
I compatible with the structure maps.
Proof. Suppose (a, b, c) E (D
I)n, then define
X
:C(K®X, Y) ~Simp(K, C(X , Y)) given as as the the adjoint adjoint of of the the composite composite an isomorphism, isomorphism, where where ¢>4 isis given isis an _
KxC(K®X,Y) 
Olx ld
C(X, K ®X)xC(K ®X,Y) ~C(X,Y).
cotensor of of KK by by YY exists exists ifif there there isis an an object object C(K, C(IC,Y) Dually the the cotensor Dually Y) and and aa morphism morphism (3: K
~C(C(K,Y),Y)
238
inducing in similar fashion fashion an isomorphism isomorphism
C(X, C(K, Y)) C(X, C(K, Y)) ~ E Simp(K,C(X, Simp(K,C(X,Y)). Y)). KOX, the required isomorphism Remark. For the tensor K®X, 1,4>, '\6, (64)1) .
258
to obtain
>'5 We o2 = We set set y,5 = yi ,8 == A.>.. Similarly Similarly we we apply apply E(2,1,2) E(2,1,2) to to obtain obtain aa 2cube 2 cube pf.L with boundary boundary with
y; = p We set y? ,f = ,f f.L and obtain
canonical At see that that the the degenerate degenerate 2cube 2cube (l(d¢>l isis aa canonical At this this stage stage we we see choice choice for for ,d. y:. Thus Thus we we obtain obtain aa (3,1,1)box (3,l ,I)box
259
"/ = ((J(J¢l, ,)..,f.L,)..,f.L).
We apply E(3,1,1) to obtain a filler of ,,/, 2cube with boundary
r
E Q3' Then a =
rt
is a
as desired. That completes the proof. Note that the construction of a can also be illustrated by the following figure.
f.L
f.Lt f.Lt
(J(J¢l
)..
f.L
)..2 0 )..2 0
)..
o The following lemma approximates the 'interchange of coordinates' in a cubical set.
Lemma (1.5) (Interchange Lemma). Let Q be a cubical set which satisfies E(2) and E(3). Let a be any 2cube of Q, let Da = (¢,¢','I/J,'ljJ') denote its boundary. Then there exists a 2cube f3 E Q2 with boundary Df3 = ('I/J,'I/J',¢,¢') .
260
Proof. By Lemma (1.4) (1.4) there there exist A, X A' E Q2 Q2 such such that Proof. DA == ((J7/Jo,7/J,(J7/Jo,7/J) (C,1$o, $ 7 A', A', (14), c;4, O!) a)
to obtain obtain a filler filler of "y, to desired boundary. boundary. desired 0
rI? E
Q3. Then Then f3,6' = = rl I?; isis aa 2cube 2cube with the the Q3.
Exercise. Visualise Visualise the the proof of Lemma (1.5). (1.5). Exercise. you should should be be able able to to do do the the following following exerexerAfter these preparations you CIse. cise.
Exercise. (a) (a) Prove Prove Proposition Proposition (1.2) (1.2).. It will will help help to to look look back at Exercise. the sort sort of arguments arguments that were were given given in in Chapter Chapter II and and as as we we did did there there the attempt to to think think the the proof out out geometrically geometrically before before attempting aa more more formal approach. approach. formal (b) (b) Now Now do do the the same same for for Proposition (1.3). (1.3). Remember to to check check that the the composition composition is is well well defined. defined. To To help help increase increase your your awareness awareness of of exactly exactly what ingredients are are needed for for these these results, results, we we suggest suggest that you v, k), k), E(3, E(3, v, v, k) k) which which are are really used you list those those Kan Kan conditions conditions E(2, E(2,v, in in your your proofs. proofs. The The following following lemma lemma provides provides aa basic basic tool tool for for computation in in aa funfundamental groupoid. groupoid. damental
Lemma Lemma (1.6). (1.6). Let Q Q be be a cubical cubical set which which satisfies E(2) E(2) and E(3). E(3). Let O!a be be a 2cube of Q Q with with boundary DO! = (4),4>', 7/J, 7/J').
261
Then in IIQ we have the equation
{1jJ'}
+ {¢>} = {¢>'} + {1jJ} .
Proof. First we choose CT E Q2 such that
DCT = (¢>, CTi, (J¢>o, 1jJ'). Thus CTt represents {1jJ'}
+ {¢>}.
Then we choose r,7r E Q2 such that
Dr = (1jJ, (J1jJo, (J1jJo, rf) and D7r = (¢>',7ri, r f,(J¢>D· Finally, by Lemma (1.4) we can choose p E Q2 such that D p = (1jJ', (J1jJ~, 1jJ', (J1jJi).
Let A E Q3 be a filler of the (3,1,1 )box , = (0', , CT, 7r, r, p).
Then
At shows that 7rt is congruent to CTI, hence {7rn = {1jJ'}
+ {¢>}.
On the other hand by Lemma (1.5) we have
{7rn + {rn = {¢>'}, furthermore
{rn
+ {1jJ} =
01/10'
hence
{7rn = {¢>'} + {1jJ} which completes the proof.
0
The construction of the fundamental groupoid is functorial in an obvious sense.
Exercise. Let Cub' denote the full subcategory of Cub whose objects are those cubical sets which satisfy E(2) and E(3). Let f : Q > Q' be a morphism of Cub', i.e. Q, Q' are cubical sets which satisfy E(2) and E(3) and f = (in : Qn t Q~ : n E IN) is a cubical map. 262
Show that the formulae
I*(a) = lo(a)
for a E Qo
I*{¢} = {h(¢)}
for ¢ E Qt
determine a morphism of groupoids III
= 1* : IIQ + IIQ'.
Moreover, II is a functor, II : Cub'
+
9rpd.
We now apply the construction of the fundamental groupoid to an abstract homotopy theory induced by a cylinder
1=« ) x I,eo,e},CT) on a category, C. Recall that for any objects X, Y of C we have an induced cubical set, QI(X, Y), with ncubes given by
(see section 1.5). For the rest of this section we assume that I satisfies the Kan conditions E(2) and E(3), i.e. for any objects X, Y of C the cubical set QI(X, Y) satisfies E(2) and E(3).
Definition (1. 7). Let X, Y be objects of C. Then the fundamental groupoid IIQI(X, Y) of the cubical set QI(X, Y), for which we will simply write II(X, Y) , will be called the fundamental groupoid of Y under X. Exercise. Show that the objects and morphisms of the fundamental groupoid II(X, Y) admit the following explicit description. The objects of II(X, Y) are the morphisms of C, I: X + Y, from X to Y. If I, 9 : X + Yare objects of II(X, Y), then the morphisms in II( X, Y), {¢}: I : = } g, from I to 9 are the equivalence classes (which have been called tracks (see the definition after Proposition (1.2») of 263
homotopies ¢ : j ~ 9 from j to 9 with respect to the following relation, == . If ¢, ¢' : j ~ 9 are homotopies then ¢ == ¢' if and only if there is a 'homotopy of homotopies'
q, : X
X [2 t
Y
such that
q,eo(X x I) = ¢, q,el(X x I) = ¢' q,(eo(X) x I) = jrJ(X), q,(el(X) x I) = gcr(X). The identity (zero element) at j, Of : j
===> j,
is the track {cf} of the constant homotopy, cf, at j, where
cf = jcr(X) : j
~
f.
Exercise. (a) Show that in the category Top of topological spaces the fundamental groupoid II(X, Y) of Y under X constructed via the canonical cylinder T on Top coincides with the fundamental groupoid 7rYx of Y under X as defined in section III.I. In particular, the notion of a track introduced here in the abstract case is consistent with that used in Chapter III. (b) Show that for the cylinder on the category 9rpd of groupoids defined in section III.1 the fundamental groupoid II(X, Y) of Y under X coincides with the function groupoid Y x. (HINT: Prove that for groupoids the congruence relation == is equality.) (c) Dualise the construction of the fundamental groupoid to a category with a co cylinder. What happens in an adjoint cylinder/cocylinder pair situation? From a categorical point of view our assumption on the cylinder I means that the functor Ql : cop x C t Cub into the category of cubical sets restricts to the full subcategory Cub' of Cub defined above whose objects are those cubical sets which satisfy E(2) and E(3). Composition with the fundamental groupoid functor
264

II :: Cub'
t
Qrpd Grpd
yields a functor cop Cop
xx C ,  t Qrpd. Grpd.
X + Y is a morphism of C and Z Z is an object of In particular, if ff : X t Y C we have induced morphisms of groupoids


f* : II(Z, X) f* X )   t II(Z, II(Z, Y) Y) and f* f * :: II(Y, II(Y, Z) Z )   t II(X, II(X, Z) Z) defined defined in an obvious obvious way.
Give a detailed description of f, f *, more more precisely Exercise. Give f* and f*, show that f, f * are are determined by the following following formulae: formulae: f* and f* show
where Z where 9g :: Z

t

f*(g) = fg, f*(k) = kf,
X X and k :: Y Y
t
Z Z are are morphisms of C, C,
f*{¢} = {f¢} : fg => fh f*{1/!} = {1/!(f x

In :kf => if
where ¢4 : Z x II + X and and 1/! $ : YY x II   t Z Z are are homotopies, homotopies, ¢4 : 9g ~ N h, where t X $:k ~ E l. 1. resp. 1/! resp. are now in a position to describe describe how the cylinder, cylinder, II,, on C gives gives We are rise to a 2dimensional 2dimensional structure on C. C.
0: The constituents in dimension 0, 0, called Ocells, 0cells, are are Dimension 0: the the objects X, X , etc. etc. of C. Dimension 1: 1: The The constituents constituents in in dimension 1, 1, called Icells, 1cells, are are the X   t Y, Y, etc. etc. of C. C. the morphisms morphisms ff :: X Dimension 2: 2: The The constituents constituents in dimension 2, 2, called 2cells, 2cells, are are X, Y) (i.e. the F :: ff => g, g, etc. etc. in II( II(X,Y) (i.e. F F == {¢} (4) for for a homotopy the tracks F ¢4 :: ff ~ N g) g),, where where X, X, Yare Y are any objects of C.

*
We We will will use the the following following pictures to to illustrate these three types types of cells. cells.
265
x
Ocell
X  f  y
Icell
f
r .JJF  \Y
X
~
2cell
h
Another useful notation for a 2cell which we have borrowed from Grandis (1991) is
We have a composition of Icells according to the composition of morphisms in the category C :
X~Z.
XLy~Z
Then we have a composition of 2cells according to the composition in the fundamental groupoid IT(X, Y) :
f
f
r\
0!'\ 9 ,Y
X
X .JJG+F Y
~
~
h
h
If F:f~g:X+Y, G:g~h:X+Y
are tracks in IT(X, Y), then we will call G+F:f~h:X+Y,
the vertical composition of F and G. Using the vertical composition of 2cells we can define another composition of 2cells which will be called horizontal composition. The symbolic picture is 266
f
kf
k
r .JJ.F  \Yr .JJ.K  \Z "J "J 9 1
r.JJ.KoF  \Z "J Ig
X
X
Definition (1.8). If F :f
===}
9:X
t
Y and K : k
===}
1: Y
t
Z
are 2cells the horizontal composition of F and K which we will denote KoF is given by
KoF = g*K + k*F. Exercise. Show that the definition of KoF makes sense and gives
KoF: kf
===}
Ig : X
t
Z.
The horizontal composition KoF can be calculated in a different way.
Lemma (1.9). For the horizontal composition KoF of F and K as in Definition (1.8) we have
KoF = g*K + k*F = I*F + f*K. Proof. Let F be represented by : X
X
I
t
Y, : f
~ g,
X
1
t
Z, 1/;: k
~
let K be represented by
1/; : Y thus F
= {},
K
I,
= {1/;}. Consider the composite a = 1/;(
X
1) : X
X
12
t
Z.
Then we have Da = (k,I,1/;(f x 1),1/;(g x 1)).
Thus by Lemma (1.6) we obtain
KoF = g*K+k*F=g*{1/;}+k*{} = {1/;(gx1)}+{k} 267
{Iif>} 
+ {1/;(f
I)}
x
+ f* {1/;} =
I*{ if>}
I*F + f* K.
0
Vertical and horizontal composition are interrelated by the socalled interchange law. Proposition (1.10){Interchange Law). Let
F:J=?g:XtY, G:g=?h:XtY
K :k
=?
I :Y
t
Z, L: I =? m : Y
t
Z
be 2 celis. Then we have
(L + K)o(G
+ F) =
(LoG)
J
+ (KoF).
k
rv\0!\ 9 Y I Z
X
~~ h
m
Proof. By Definition (1.8) and Lemma (1.9) we have
(L
+ K)o(G + F)
h*(L + K)
+ k*(G + F) h* L + h· K + k.G + k.F
=
+ I*G + g* K + k.F (LoG) + (KoF). 0 h* L
Exercise. (a) Prove that horizontal composition is associative when defined. (b) If J : X   t Y and 9 : Y   t Z are morphisms of C, prove that OgOOf
(c) If F : J =? 9 : X
t
=
Ogf·
Y is a 2 cell, prove that
268
FoO Idx = F = OIdyoF.
In the following exercise some of the algebraic properties described above are rephrased in categorical terms.
Exercise. (a) Show that the objects of C and the 2cells form a category under horizontal composition with identities the 2cells of the form OIdx' (b) Let X, Y, Z be any objects of C. Show that composition in C and horizontal composition give rise to a functor (composition functor) J.l : II(X, Y) x II(Y, Z)   II(X, Z)
from the product category II(X, Y) x II(Y, Z) to II(X, Z). The structure on C given by Ocells, Icells, 2cells, vertical and horizontal composition and the algebraic properties described has been called a 2category in the literature. Moreover, since every 2cell is invertible with respect to vertical composition the terminology groupoid enriched category applies. A detailed treatment of enrichment of categories can be found in Kelly (1983). Enough of the general theory for our purposes will be handled in this section and later in the slightly different context of simplicially enriched categories in Chapter V. Thus we can summarise our results as follows .
Theorem (1.11). If I is a cylinder on a category, C, which satisfies E(2) and E(3), then I induces on C the structure of a groupoid enriched category. 0 Remark. The following combinatorial property of the induced functor QI : cop x C   Cub
has been crucial for the proof of Lemma (1.9), and hence of the Interchange Law (1.10). For any objects X, Y, Z ofC and any 1> E QI(X, Y)l, 'Ij; E QI(Y, Z)l such that
269
there exists a E QI(X, Zh such that Da = (7/Jo.¢,'l/JI·¢,¢o7/J,¢i7/J) .
This property has been set as an axiom (Kamps (1973 a) , Axiom (A) ; see also Grandis (1991)) in abstract homotopy theory based on the notion of a general (semi)cubical homotopy system
Q : cop x C  t Cub (cf. Remark after (II.3.12)). The following exercise shows that the fundamental groupoid is well behaved with respect to homotopies.
Exercise. (a) Let f, 9 : X  t Y be morphisms of C. Then if f ~ g , for any object Z of C there is a natural equivalence of functors between the induced morphisms of fundamental groupoids,
(b) If f : X  t Y is a homotopy equivalence, then for any object Z of C, the induced functor f* : II(Z, X)
t
II(Z, Y)
is a natural equivalence of groupoids. (c) 'Dualise' (a) and (b).
Remark. For a 'Review of the elements of 2categories' we refer the reader to Kelly and Street (1974) , see also Gabriel and Zisman (1967). Homotopy theory in groupoid enriched categories goes back to Gabriel Zisman (1967). Category theoretic aspects of homotopy theory in groupoid enriched categories have been considered by Fantham and Moore (1983). The fundamental groupoid of a cubical set has been applied to abstract homotopy theory by Kamps (1972 b , 1973 a, 1978 b) . The 2dimensional categorical foundations of homotopy theory have been investigated in detail by Grandis (1991 ,1994). Another 2dimensional categorical setting for homotopy theory (special double groupoid with special connection) has been developed by Brown and Spencer (1976) as a suitable algebraic tool to handle a 2dimensional 270
van Kampen theorem. In the remainder of this section we show how to define and manipulate homotopy commutative squares in the groupoid enriched setting of Theorem (1.11). First we introduce a modification of the notion of a homotopy commutative square as defined in (11.2.15) . According to (11.2.15) a homotopy commutative square in C X  !  X I
fl)
y     , yl g
consists of a square in C (the outer diagram) together with a specified homotopy
¢: gp
~
plf.
If we replace the homotopy ¢ by its track,
H = {¢} : gp => pi!, we obtain the notion of a track homotopy commutative square. Definition (1.12). A track homotopy commutative square in C
x P
!
II)
XI
Ip'
y     , yl g
is a square in C together with a specified track
H: gp => pi! in the fundamental groupoid II(X, y/). 271
In particular, a commutative square in C
X __I__. X'
pI y   .. . . Y' 9
(i.e. gp = p' f) can and will be considered as the track homotopy commutative square
X _=1_. X' p
O~
y   ..... y' 9
with the track
Ogp
of the constant homotopy, cgp = gpa(X) : gp
~
gpo
Note that a track homotopy commutative square contains less information than a homotopy commutative square with a specified homotopy. Next we define a vertical and a horizontal composition of track homotopy commutative squares.
Definition (1.13). (a) The vertical composition of track homotopy commutative squares in C
X
1
pH) Y
f;! Z
X'
Ip' 9
Y'
Ii h 272
Z'
is defined to be the track homotopy commutative square
x
f
X'
qpl LJ lq'p' zz' h with
L=
q~H
+p*K,
where p* : II(Y, Z') + II(X, Z') , q~ : II(X, Y') + II(X, Z') are the induced morphisms of fundamental groupoids. (b) The horizontal composition of track homotopy commutative squares
X
f!! Y
f
f'
X'
p'l ~ 9
Y'
X"
lp" g'
Y"
is defined to be the track homotopy commutative square
f'f X" X ''
f~! YY" g'g with K = f*H'
+ g:H
where g: : II(X, Y') + II(X, Y"), f* : II(X', Y") the induced morphisms of fundamental groupoids. Example. If a commutative square 273
+
II(X, Y") are
I'
X'
X"
p'l
III'
Y'
Y"
g'
is attached from the right to a track homotopy commutative square
I
X
X'
f!) Y
Ip' Y'
9
we obtain
X
I
f!) Y
f'
. X'
X"
ip' 9
(Proof. /*Ogl11'
Ip"
Y'
g'
f'1 . X"
X
Y"
=
f:~ Y
g'g
Ip" Y"
+ g~H = Oglplf+ g:H = g:H.)
Exercise. Try and extract the algebraic properties from the composition of track homotopy commutative squares. In particular, show that there is an interchange law for track homotopy commutative squares in the situation
.

.

.
I
•
. . •I 274
Describe the role played by track homotopy commutative squares of the form
x
f
X'
O!J
resp.
y==y
XX'
f
(Here we have chosen the suggestive notation
X=X for the identity morphism I dx : X    t X.)) You should end up with what has often been called a double category in the literature (for a definition see again Kelly and Street (1974)). As an application we give a diagramatic proof of a lemma of Vogt (1972) on homotopy equivalences in our abstract setting.
Proposition (1.14) (Vogt's Lemma (1973». Let f : X be a morphism of C which is a homotopy equivalence. Let 9 : Y be any homotopy inverse of f and let
t t
Y X
¢;:gf~Idx
be any homotopy. Then there exists a homotopy 1jJ : fg
~
Id y
such that f ¢; == 1jJ(J x 1)
and ¢;(g x I)
== g1jJ
(cf. Definition (1.1) and the exercise after (1.7)). Proof. The assertion can be reformulated as follows. Let f : X    t Y be a homotopy equivalence, let 9 : Y homotopy inverse of f and let
H: gf
===}
275
Id x
t
X be any
be any track. Then there exists a track
K: fg
~
Id y
such that
f*H
= f*K
and g*H
= g*K.
The method of proof is based on the interchange law for track homotopy commutative squares which allows one to compute a certain 'composite of four track homotopy commutative squares ' in two different ways. We apply this principle in an naive way to situations which are slightly more complicated. For a thorough treatment of 'pasting problems' in this type of context we refer to Dawson and Pare (1993) . We now construct the track K : By hypothesis there is a track
L: fg
~
Id y
.
The trick is to choose
K = L+ f*g*H  (fg)*L. Thus K is the track in the following composite rectangle.
y===y===y
fj
9]
X==X
f]~ I y
9
I;j
X
I f f
y
I y
jt
y===y===y We now compute the track T in the composite diagram
276
x
f
9
X
\\
X
f
9\ Ij
9\
f
x
f
Y
X
f
y
1\ ~ y
y
y
X
(1.15)
X
y
9
r~ It ,y y
\
y
in two different ways. Using horizontal composition we obtain
But composing differently we obtain
which equals the trivial track Of, hence
We note that the latter argument might be expressed in the form:
'In diagram (1.15) we first cancel H with H, then L with L to obtain the trivial track Of .' Next, by making various internal cancellations we see that the tracks in the following two rectangles coincide.
277
g[=g[~T, 1 glg LJ[ ~ ~ IILJ [ * T f (! gl LJ I I xx I gl IJ iI 9 1 If (! 11~ II~ Ig II fl I x I y
yYX 9 ·x
y
Y
Y
9
gl
xX =
x
I. yY
lg
yYx 99
1 
x
x
9 II~ 11~
y= Y
y Yx
9
9
xXY= f
x
;
Y
19Ig II~
x
x X
x
y
.y
9 1 7
i; x
y
II~ II
Ig
Y Y
y Y
9
y YX
9 9
x
However, by cancelling HH in the central square ()f of the left diagram it,, we find that the track in with H in the square immediately below it the left diagram equals
g*H + + g*K. g*H g*K. Since the tracks in the right hand square cancel to the trivial track Ogfg Ogf,, , we have proved that
g*H g*H == g*K g*K as as required. required . 0
Exercise. Check all the details in the preceding proof. Remark. f , gg,, H, Remark. A quadrupel ((I, H, K) K) as in the conclusion of Vogt's lemma will be called a strong homotopy equivalence. equivalence. Such strong homotopy equivalences seem first to have been considered by Lashof (1970). (1970). Vogt's lemma shows that any homotopy equivalence fI is part of a strong homotopy equivalence. Spencer (1977) (1977) has considered Vogt's lemma in an abstract setting. The diagramatic approach presented here is due 1995). to Hardie and Kamps (1989, 1995). 2. Track Homotopy Categories and Dold's Theorem
Let C be a category with a cylinder
1 = (( ) x I,eo,e1,CJ) 278
which satisfies the Kan conditions DNE(2) and E(3). We assume that C has pushouts and ( ) x J preserves pushouts. As exhibited in section 1.6 classical homotopy theory under a fixed object, A, of C is homotopy theory in the category, CA , of objects under A. Thus it is concerned with morphisms under A, f: i t i', i.e. commutative triangles in C of the form
A (2.1 )
('\
X   ....., X'
f
and homotopies respecting this structure (see (1.6.1)). The corresponding homotopy relation, homotopic under A, denoted a quotient category,
~ , gives rise to
the homotopy category of C under A. Thus a morphism
[f]A : i
t
i'
of h( CA ) is the equivalence class with respect to ~ (homotopy class under A) of a morphism under A, f: i t i' (see Remark (1.6.15)). Furthermore [f] A : i t i' is an isomorphism in h( CA) if and only if f : i t i' is a homotopy equivalence under A. In these terms Dold's theorem (1.6.3) can be reformulated as follows.
Theorem (2.2). If i and i' are cofibrations, then [f]A : i t i' is an isomorphism of h( CA ) if and only if f : X t X' is a homotopy equivalence. This means that for certain objects, i, i' of CA , namely the cofibrations, the isomorphisms i t i' in h( CA ) can be characterised by means of ordinary homotopy equivalences. In this section we will give a separate proof of this using a different approach to homotopy theory under A based on track homotopy commutative squares of the form 279
A==A
. If)
(2 .3)
z
i'
XX'
/
This will give rise to a track homotopy category under A, 1{A, with the same objects as h(C A ) such that isomorphisms between arbitrary objects can be characterised by means of homotopy equivalences (see Theorem (2 .5) below). As an application we give a proof of Dold's theorem separating arguments involving homotopy equivalences from arguments using cofibrations in a transparent way. Definition (2.4). The objects of the track homotopy category under A, 1{A, are the objects under A, i.e. the morphisms i : A ~ X of C with domain A. If i : A ~ X, i' : A ~ X' are objects under A, then the set 1{A(i, i') of morphisms i ~ i' of 1{A is obtained from the set of track homotopy commutative squares of the form (2.3) by factoring out by the equivalence relation
A A
. If) X
. If)
z
A
z
i'
/
o
A
'"
X
/
i'
X'
FJ
X' X
/'
X'
where F : /' ===> / is a track in II(X, X') and the diagram on the right is the vertical composite of the two squares, i.e. we have
A
A
iIf) X
/
A
i~
i' o
A
X'
X 280
f'
i'
. X'
if there exists a track F : f'
f in II(X, XI) = H +i*F
===}
HI
such that
(cf. Definition (1.13)(a)). The morphism i denoted by
t
i ' in JtA represented by diagram (2.3) will be
{f, H} : i
t
i'.
Composition in JtA is induced by horizontal composition of track homotopy commutative squares (cf. Definition (1.13)(b)) . The identity
is represented by the commutative square
A
A
i
X
X
Exercise. Check all the details to ensure that JtA is a category. We now propose to characterise the isomorphisms of Jt A .
Theorem (2.5). Let {f, H} E JtA( i, i/) be represented by the track homotopy commutative square (2.3). Then {j, H} is an isomorphism ofJtA if and only if f is a homotopy equivalence. Proof. Let {j , H} E JtA(i,i/) be an isomorphism and let {g, K} E JtA(i' , i),
represented by the track homotopy commutative square
281
A
A
KJ
i'
XI
i
9
X 0
be inverse to {j, H}. Then we have the relations
A==A==A . H~ .1 z dI z
A==A
X_o X'
X
X==X
A
A
KJ
f
9
0
resp.
A
A
., KJ z·IIJ
i'
z
X'
9
oX
o
f
'"
X'
i'
X'
A i'
X'
From the definition of the relation", in (2.4) we infer that there are tracks F:ldx~gf
F' : Id x ' ~ fg.
resp.
Writing F = {cp}, F' = {cp'} where cp : I dx
~
cp' : Id x ' ~ fg
gf,
are homotopies (cf. the exercise after (1.7)) , we see that 9 is a homotopy inverse of f. Hence f is a homotopy equivalence. Conversely, assume f is a homotopy equivalence. Then by Vogt's lemma (1.14) we can choose g, F, F' in such a way that (j , g,F, F ' )
is a strong homotopy equivalence (see the remark after (1.14)) , l.e. 9 : X' + X is a homotopy inverse of f and 282
F: gl => Id x , F': Ig => Id x ' are tracks such that
= f* F'
I*F
and g*F
= g*F' .
Then the composite track homotopy commutative square
A
A
If)
A
Ii i'
X ==X
IF)X'===X'
g'
X
represents an element {g,K} E 1{A(i', i). We claim that this element is inverse to {I, H} in 1{A . This can be seen by composing diagrams. We compute {g,KH/,H} :
A
A
IIJ i'
l
X
I
If)
A=A
[i .
[i
X=X
=
A=A
A
i[
i[
[i
X=X
X
f[~
,X'=X'X 9
283
f[~ I
XX'X
I
9
x
x
A==A
I
I~
=
xL.x,~x
il
Ii
X
X
F)
X ===== X
Whilst computing {j, H}{g, K},
A=A=A=A
If)
Ii il ~ II F;)
A=A=A=A
If) "'
X=X
1
X=X
rv
i'
11 F;)
II
II X'=X'~X~X'
X'=X'XX' 9
Ii il ~
I
I
IIFJ k I
X'=X'=X'=X' we need to use that F cancels with F' since I.F = cancelling H with H we obtain
This completes the proof.
A
A
t
r:
0
284
f* F' " Then
after
Corollary (2.6). (a) If f, f' : A t X are objects under A such that f ~ f', then f and f' are isomorphic as objects of 'HA. (b) Every object f in 'HA is isomorphic to a cofibration.
Proof. (a) Choose a homotopy cp : f
~
f',
consider the diagram
A==A fII)
f'
X==X where H is the track {cp} of cp and apply Theorem (2.5). (b) In the mapping cylinder factorisation of f : A t X,
A~Mf~X (cf. (1.2.9), (1.5.11)), if is a cofibration and Pf is a homotopy equivalence. Now consider the diagram
A
A
MfX Pf
and apply Theorem (2.5).
0
Exercise. (a) Construct a track homotopy category, 'HB, over a fixed object B of C starting from track homotopy commutative squares of the form
E p
f Hft. d/
B
E' p'
B
285
(i) in a category, C, with a suitable co cylinder (dualise Definition
(2.4)); (ii) in a category, C, with a suitable cylinder; (iii) in a category, C, with a suitable adjoint cylinder/co cylinder pair. Which properties of the cylinder / co cylinder do you need? (b) Characterise the isomorphisms in
1{ B.
Exercise. Let C be a category suitably structured by a cylinder resp. co cylinder. (a) Construct a track homotopy category, 1{PC, by introducing an equivalence relation, "', in the set of track homotopy commutative squares as follows:
x
f'
X'
FJ I
X
p
II)
y 9
, X'
X
p'
p
Y'
Y
I
II) 9
X' p'
Y'
GJ Y ___ Y' g'
where F : I that means
::::::::} I', X
p Y
G: g' ::::::::} 9 are tracks in II(X, X') resp . II(Y, yl) ,
I
II) 9
, X'
p'
X
'"
y'
p Y
where 286
f'
H;J g'
X' p' Y'
H' =p~F+H +p*G.
The objects of ,}fPC are the pairs, p,p' etc., in the sense of EckmannHilton, i.e. the morphisms of the category, C. The category HPC is called the category of homotopy pairs. (b) Characterise the isomorphisms in HPC. (For the definition and investigation of HPC in the topological case we refer to Hardie (1982) and HardieJansen (1982)).
Remark. In the topological case, the utility of track homotopy, especially the homotopy pair category, has been revealed by Hardie; in particular, track homotopy has proved to be an appropriate tool to handle Toda brackets (Hardie (1982), HardieJansen (1982, 1983, 1984), Hardie (1991, 1993), HardieKampsMarcum (1991), HardieKamps (1993), HardieMahowald (1993)). Various track categories have been investigated by HardieKamps (1987 a,b, 1989, 1992). In each case the isomorphisms can be characterised by means of homotopy equivalences. Marcum (1990) has a general treatment of homotopy equivalences in 2categories. A categorical approach to matrix Toda brackets has been given by HardieKampsMarcum (1995). Next we investigate the relation between classical homotopy theory under A and track homotopy under A. Let i : A + X, i': A + X' be objects under A. By Ii, i/JA we will denote the set of morphisms IfJA, from i to i' in the classical homotopy category, h(C A ), under A. We have a canonical map
e : Ii, i'JA + HA( i, i') induced by the assignment which sends a commutative triangle in C
A
1\
XX'
f
287
to the track homotopy commutative square
A
A== A
i
i'
0:)
X
X    X'
f
f
A i'
X'
Thus we have
e[f]A = {f, Oi'}. This definition induces a functor
e :h(C A) ~ 1{A from the classical homotopy category under A to the track homotopy category under A which is the identity on objects. Exercise. Check the details.
Proposition (2.7). If i is a cofibration, then for any object under A, i': A ~ X' , the map
e : Ii, i']A ~ 1{A(i, i') is a bijection.
Proof. Let {f, H} E 1{A(i, i') be represented by the track homotopy commutative square (2.3). Choose a homotopy
}
= {II>( i x I)} = {'P} = H,
it follows that
8[f/]A = {j, H} by definition of the relation", in (2.4). This proves that 8 is surjective. To see that 8 is injective, let [f]A, [f']A E [i, i/]A and suppose
8[f]A = 8[f']A. Then by Definition (2.4) there exists a track H : f ====> f' in II(X, X') such that
i*H = Oi'. In order to obtain [f]A = [f/]A, we apply the following lemma which will be of independent use (cf. section 3 of this chapter). 0
Lemma (2.8). Let f, f' : i  + i' be morphisms under A, where i : A  + X is a cofibration and i' : A  + X' is arbitrary. Suppose H : f ====> f' is a track in II(X, X') such that i*H = Oil . Then f is .
homotopzc under A to f', f
A
~
f'·
Sketch of proof. Choose a homotopy 0:' : X x I  + X', 0:': f ~ f', such that H = {O:'}. Then by assumption we have a homotopy of homotopies
,\ : A x
[2  +
X'
with boundary
D,\
= (i'a(A), i'a(A), O:'(i
x I), i'a(A».
i'a(A)
i'U(A:D':U(A) z
0:'( i x
I)
z
Using Remark (1.6.6) we can 'extend' ,\ to a homotopy of homotopies 289
x
X /2 ~
X' connecting f and
f'
via homotopies under A.
0
Exercise. Complete the details of the preceding proof. We are now in a position to give the promised proof of Dold's theorem (2.2) separating arguments involving homotopy equivalences (Theorem (2.5)) from those using cofibrations (Proposition (2.7)). We will make use of the following characterisation of isomorphisms in an arbitrary category. Exercise. Prove that a morphism k : K ~ L of an arbitrary category IC is an isomorphism if and only if the set maps
f*:
IC(K,K) ~ IC(K,L) and
induced by composition with
f*:
IC(L,K) ~ IC(L,L)
f, are bijective.
Proof of Theorem (2.2). The converse implication being clear, we assume that f is a homotopy equivalence. By the preceding exercise, we have to show that the functions
(2.9)
f* : [i, i]A
(2.10)
f* : [i', i]A
~ ~
[i, i/]A [i', i/]A,
induced by composition with [f]A in h(CA), are bijective. In order to prove the bijectivity of the first map we consider the following commutative diagram of sets
[i, i]A _=f*~_ [i, i/]A (2.11)
ej 1{A(i, i)
je fo
1{A(i, i/)
where the arrow fo is induced by composition with {J, Oil} in 1{A . Since f is a homotopy equivalence, by Theorem (2.5) the map fo is a bijection. Since i is a cofibration, by Proposition (2.7) the vertical arrows in (2.11) are bijections. Hence the upper arrow in (2 .11), i.e. the map f* of (2 .9) is bijective. Similarly, using that i ' is a cofibration, we can prove that 290
the map f* in (2.10) is a bijection.
0
Remark. In the topological case the approach to Dold type theorems via track homotopy is due to HardieKamps (1989,1992); HardieKampsKieboom (to appear) have a careful analysis of the cubical structure involved. We conclude this section with an exercise which shows that the track homotopy category under A, 'H,A, can be interpreted as a category of fractions. Let e : CA + 'H,A denote the functor which is the identity on objects and maps a morphism f : i + i ' under A to the morphism {f,Od of 'H,A. Then by Theorem (2.5) e factors through the category of fractions CA(~l) where ~ is the class of those morphisms under A, f : i + ii, as in (2.1) such that the underlying morphism f : X + X' of C is an ordinary homotopy equivalence, thus we have a commutative diagram
where fJ is the universal functor (see (II.2 .5)). Exercise. Show that, under suitable conditions, the functor r is an isomorphism of categories. HINT: Analyse the proof of HardieKampsPorter (1991) where the dual situation, objects over B, has been handled for topological spaces. We suggest that you should work with an adjoint cylinder / co cylinder pair and that some of the arguments of section II.2 will need to be brought in. We leave the detailed analysis and the formulation of the final result to you. 3. Exact Sequences
Exact sequences of, say, abelian groups are an important tool in 291
homological algebra allowing one to exchange information between various groups and homomorphisms. For instance, if X is a topological space and A is a subspace of X, then we have a long exact sequence ... t
Hn+1(A)
t
Hn+1(X)
t
Hn+1(X, A)
t
Hn(A)
t
Hn(X)
t . • . ,
the homology sequence of the pair (X, A) (d. Dold (1972), 111.3). Similarly, if X is a topological space, A is a subspace of X and Xo E A, we have a long exact sequence ... t
t
... t
7rn+1(A,xo)
7rn(A, xo)
7r1(A,xo)
t
t
t
7rn+1(X,xo)
7rn(X, xo)
7r1(X,XO)
t
t
t
7rn+1(X,A,xo)
7rn(X, A, xo)
7r1(X,A,xo)
t
t .•.
7ro(A,xo)
t
7ro(X,xo),
the homotopy sequence of the triplet (X, A, xo) (d. for instance, Hu (1959), IV.7) . In this case, however, the groups 7r1(A, xo), 7r1(X, xo) are not necessarily abelian and the last three terms are merely sets with a distinguished base point. In this section we shall see that the special features of this type of exact sequence can be modelled by an exact sequence associat~d to a fibration of groupoids. Furthermore, we shall show how exact sequences of this type arise in abstract homotopy theory. (a) Groupoid Exact Sequences We recall the definition of an exact sequence of abelian groups.
Definition (3.1). (a) A pair (a,(3) of composable homomorphisms of abelian groups
(3.2) is called exact, if Ima = Ker (3
where 1m a is the image a( A) of a and K er (3 denotes the kernel of (3, i.e. the inverse image (31(0) of the zero element of C . (b) A (finite or infinite) sequence of homomorphisms of abelian groups of the form 292
(3.3)
... +
An+l
Ct'n+l t
An
an t
an_l
A n 1  t An2
+ ...
is called exact, if each segment (an+b an) when defined is exact. As a first application the property of a homomorphism of abelian groups being surjective, injective, resp. bijective can be expressed in terms of exactness.
Exercise (3.4). Let a : A + B be a homomorphism of abelian groups, let 0 denote the trivial group. Prove that
(a) is exact if and only if a is surjective.
(b)
O+A~B
is exact if and only if a is injective.
(c)
O+A~B+O
is exact if and only if a is an isomorphism. Definition (3 .1) literally applies to non necessarily abelian groups to give rise to the notion of an exact sequence of groups. (Notational convention: If composition in a group is written multiplicatively the neutral element of a group and the trivial group will be denoted by 1.) Surjectivity, injectivity, resp. bijectivity of a group homomorphism can be characterised as in the preceding exercise (3.4). If (3.2) is an exact sequence of groups then 1m a = K er {J is a normal subgroup of B . Exactness can also be defined for pointed sets.
Definition (3 .5). (a) A pointed set is a pair (M,m) where Mis a set and m E M is an element of M . The element m is called the base point of (M, m) . The base point of a pointed set is often denoted by *. If no confusion can arise we omit the base point and simply write M for (M, m) resp . (M, *). A pointed set consisting only of the base point *, will also be denoted by *. A group may also be viewed as a pointed set, the base point being the neutral element of the group. 293
(b) A pointed map 0:' : (M, m) + (N, n) of pointed sets is a set map 0:' : M + N such that 0:'( m) = n. Composition of maps gives rise to a category Sets*, the category of pointed sets. Note that each pointed set of the form * is a zero object of Sets*. (c) A pair (0:', (3) of composable maps of pointed sets
L~MLN is called exact, if 1m 0:' = K er (3, where 1m 0:' is the image 0:'( L) of 0:' and Ker (3 denotes the kernel of (3, i.e. the inverse image (31(*) of the base point of N. The definition of an exact sequence of pointed sets of the form (3.3) is obvious. (d) A map 0:' : M + N of pointed sets is called weakly injective, if
Clearly, if 0:' : M
+
N is a pointed map, the sequence M~N+*
is exact if and only if (the underlying map of) 0:' is surjective whereas the sequence *+M~N
is exact if and only if 0:' is weakly injective. Since a map that is weakly injective is not necessarily injective (consider the pointed map 0:': ({0 , 1,2},0)
+
({0,1} , 0)
where 0:'(0) = 0, 0:'(1) = 0:'(2) = 1,) a characterisation of injectivity as in Exercise (3.4)(b) is no longer valid for pointed sets, however, when studying the exact sequence associated to a fibration of groupoids we shall encounter a situation where weak injectivity does imply injectivity. In order to be able to exhibit that groupoid exact sequence, we need
294
some preparation.
Definition (3.6). If 9 is a groupoid, x, y E Ob (9). Then x belongs to the same component as y (written x = y), if 9(x, y) is nonempty.
=is an equivalence relation on Ob (9). The set of equivalence classes Ob (9)/ =is called the set of comExercise. Prove that
ponents of 9, denoted 'lro9. If x E Ob (9) we write [x]g or simply [x] for the component of x in 9. Exercise. Let that the formula
f :9
~
H be a morphism of groupoids. Prove
f*([x]g) = [f(x)]?i for x E Ob (9) gives rise to a functor
'lro : 9rpd ~ Sets. Definition (3.7). Let p : 9 ~ H be a morphism of groupoids, y E Ob (H) a fixed object of H. Then the fibre F = pl(y) of p over y is defined as the subgroupoid of 9 determined by
Ob(F)
= pl(y) = {x
E
Ob(9) \p(x) = y}
~
Ob(9)
F(x, x') = {a E 9(x,x') \p(a) = ly} for x, x' E Ob (F) . Suppose p : 9 ~ H is a fibration of groupoids, x E Ob (9) is a fixed object of 9, y = p(x) and F = pl(y) is the fibre over y. Then we can define an operation
H(y) x 'lroF
~
'lroF
of the object group, H(y), of H at y on the set of components, 'lroF, of the fibre over y. Let h E H(y), i.e. h : y ~ y is a morphism of H, let Xo E Ob (F), i.e. Xo E Ob (9) such that p(xo) = y. Since p is a fibration of groupoids there exists a morphism 9 : Xo ~ Xl of 9 such that p(g) = h. Then Xl E Ob (F .) Now define 295
h • [xolF = [xI1F. In order to show that • is well defined, let Xo, x~ E Ob (F) such that [xolF = [xolF. Then there is an element a E F(xo, xo), i.e. a E y(xo, xo) and p(a) = ly . Let 9 : Xo + Xl, g' : Xo + Xl be morphisms of y such that p(g) = h = p(g') . Then g'ag l : Xl + Xl is a morphism of y such that
p(g'ag l ) = p(g')p(a)p(gtl = hlyh l = ly, i.e. g'ag l E F(XI ,xD, hence
[xI1F = [xI1F. Exercise. Prove that • satisfies the properties of an operation: (h'h) • a
= h' • (h • a),
ly. a
= a
for any h, h' E H(y), a E '!roF. Now define a connecting map
8: H(y)
+
'!roF
by
8(h) = h • [xlF and consider the sequence
(3.8)
1
+
F(x) ~ y(x) ~ H(y) ~ '!roF ~ '!rOY ~ '!roH
where i : F + y is the inclusion of the fibre . The first four terms of (3.8) are groups, and the last three terms will be considered as sets with base point
[xlF E '!roF , [xlg E '!rOY , resp. [yl'H E '!roH . Then 8, i* and p* are pointed maps. We are now in a position to state the main result of this section.
Theorem (3.9). The sequence (3.8) of groups and pointed sets is 296
exact. Further: (a) If h, k E 11. (y), then o( h) = o( k) if and only if there is agE 9 (x) such that p(g) = k l h. (b) If a,{3 E 'TroF, then i*(a) = i*({3) if and only if there is an hE 11.(y) such that h • a = (3. The sequence (3.8) will be called the exact sequence of a fibration of groupoids.
Proof. We restrict ourselves to consider the segment
11.(y) ~ 'TroF ~ 'Tro9 .
(3.10)
First we show that (b) implies exactness of (3.10). Let h E 11.(y). Then by (b)
hence 1m 0 ~ K er i*. Conversely, let a E 'TroF be such that
i*(a) = [x]g =i*([X]F). Then by (b) there is an h E 11. (y) such that
o(h)
= h • [X]F = a,
hence Keri* ~ Imo. We now prove (b). Let XO,XI E Ob(F). Suppose
i*([xo]F)
= i*([XI]F), i.e.
[xo]g
= [XI]g.
Then 9(xo, Xl) is nonempty. Let 9 E 9(xo, Xl). Then h = p(g) E 11.(y) and
h • [XO]F = [XI]F. Conversely, if h E 11.(y) and h • [XO]F = [XI]F, then h = p(g) for some 9 E 9(xo, Xl), hence [xo]g = [XI]g. D
Exercise. Complete the proof of Theorem (3.9). (If necessary, refer to R. Brown (1970).) Exercise. Consider the exact sequence (3.8) of a fibration of groupoids. 297
a:
(a) Prove that the connecting map 1{(y) + 'TroT is injective if and only if it is weakly injective. (b) Prove that i* : 'TroT + 'TroY is injective if and only if 1{ (y) operates trivially on 'TroT (i.e. h • a = a for any h E 1{(y), a E 'TroT); in particular i* : 'TroT + 'TroY is injective if 1{(y) is the trivial group.
Remark. From Theorem (3.9)(a) we infer that the connecting map
a : 1{(y) + 'TroT in the exact sequence of a fibration of groupoids (3.8) induces an injection
1{(y) / py(x)
+
'TroT
where 1{ (y) / py (x) denotes the set of left cosets of py (x) in 1{ (y). Varying the object x E Ob (T) in a suitable way one can show that 'TroT is a disjoint union of cosets of the form 1{ (y) / py (x) . (For the details we refer to R. Brown (1970) .) Thus in the set of components of a fibre of a fibration of groupoids, a grouptheoretical structure can be restored to a certain extent retaining a basis of group theoretical arguments. For a detailed discussion of this topic with concrete applications to group theory and homotopy theory we refer to HiltonRoitberg (1981). A unifying categorical approach to exactness in algebraic topology has been given by Grandis (1992 b) . We conclude this subsection with an exercise showing that the exact sequence of a fibration of groupoids gives rise to an exact orbit sequence for Gsets (see also HeathKamps (1982)).
Exercise. Let G be a group, MaGset. Then for any a E M
orbc(a) = G  a = {g • a I9 E G} ~ M is called the orbit of a. Let
M/G={GalaEM} denote the orbit set. If a E M, then the subgroup of G
G( a)
= {g E Gig • a = a}
is called the stability (isotropy) group of a. Now let G, H be groups, let M be a Gset, Nan H  set, (J : G 298
+
H
a group epimorphism, and K. : M 
N a map such that the diagram
GxM·....· M
~ x K.l
!K.
HxN
•. N
commutes. Then the induced map of semidirect products ~rxK.
: GrxM 
H rxN
is a fibration of groupoids (see section IIL1). Let a E M. Define
F
= K.1(a),
K
= Ker~.
Prove that (i) the operation of G on M restricts to an operation of K on F, (ii) the fibre ~rxK. over K.a is KrxF, (iii) the exact sequence of the fibration of groupoids ~rxK. can be interpreted as an exact sequence of stability groups and pointed orbit sets (exact orbit sequence) 1  K(a) 
G(a) 
H(K.a) 
FIK 
MIG 
NIH
with base points orbK(a) E FI K, orbG(a) E MIG, resp. orbH(K.a).
(b) Exact Cofibration Sequences In this subsection we investigate exact sequences in abstract homotopy theory resulting from fibrations between fundamental groupoids . Let C be a category with a cylinder,
1=« )x I,eo,el,a), which satisfies DNE(2) and E(3). We assume that C has pushouts and ( ) x I preserves pushouts. The following proposition tells us that cofibrations in C give rise to fibrations of groupoids.
Proposition (3.11). If a morphism of C, i : A X, is a cofibration, then, for any object, Z , of C, the induced morphism of 299
fundamental groupoids i* : II(X, Z) 
II(A, Z)
(cf. section 1) is a fibration of groupoids. Proof. We show that i* is star surjective (cf. Definition (IIL1.6), Proposition (IIL1.7)). Let f be an object of II(X, Z) and H : i*f
==> j
be a morphism of II(A , Z), i.e. I,j are morphisms of C, I: X j :A Z , and H = { j then S[2J(i, j) is empty. If i = j, then because of the simplicity of the structure of the ordered set S[2](i, i) is isomorphic to L[O], as there are no non trivial strings in [2J, that will compose to give you identity elements. Dimension 0:
S[2J(0, 1)0 = {(OI)}, similarly for
S[2J(1,2)0 = {(12)}. However
319
S[2](0, 2)0
= {(02), (01, 12)}.
Dimension 1:
S[2](0,lh and S[2](1,2)1 consist only of degenerate copies of the single Osimplex. S[2](02)1 consists of ((01,12)) plus degenerate copies of (02) and (01,12) . These are ((02)) and ((01),(12)) respectively. Note that
d1 ((01, 12)) = (02) do( (01,12)) = (01,12), so ((01,12)) is a nondegenerate Isimplex joining (02) to (01,12). In higher dimensions all simplices are degenerate. Thus S[2](i,j) is either empty, isomorphic to 6[0] or in one case, isomorphic to 6[1].
Exercise. (i) Show that S[3](i, j) is either empty, 6[0], 6[1] or 6[1] x 6[1]' this latter occurring when (i,j) = (0,3) . (ii) Investigate S[41. At this point it is worth noting the connection between these S[n] and the homotopy coherence structure we found in low dimensions the same diagrams occur except that here the (n  1)cubes are triangulated as 6[1]nl in the usual way. This allows one to realise this data as 'gluing' instructions for higher order simplices
F(i) x 6[j  i]
+
F(j)
in Top(F(i), F(j)) so as to fit together to get higher homotopies
F(i) x 6[ljii
+
F(j).
In other words, to be made precise shortly, a homotopy coherent diagram of type [n] in Top corresponds exactly to a 'simplicial functor '
S[n]+ Top. The meaning to be attached to this phrase 'simplicial functor' is more exactly that of a simplicially enriched functor or Sfunctor, again an idea taken from enriched category theory (cf. Kelly (1983)) . For the notation and terminology which will be used in the following 320
definition we refer the reader to Definition (III.4.1). In particular, Scategories will be denoted by bold letters C etc., with the corresponding C for the underlying category. Definition (2.1). An Sfunctor (or simplicially enriched functor) T from an Scategory A to an Scategory B consists of  a function Ob(A) ~ Ob(B)  for each pair of objects, A, A' of A, a simplicial map
TA,A' : A(A, A')
+
B(TA, TA')
such that if A, A', A" are in A, the diagrams
A(A,A')
TA ,A'
X
X
A(A',A") __c_o_m_p__ A(A,A")
TA' ,A"I
ITA ,A"
B(T A, TA') x B(T A', TAli)
comp
B(T A, TAli)
and
IdA/A(A,A)
.6[0]
/
,~
TA,A B(TA,TA)
commute. For us the point of this is the following theorem of Cordier (1982) . Theorem (2.2). Let A be a small category, then the data for a homotopy coherent diagram of type A in Top (cf. Definition (1.1)) is equivalent to that given by an S functor
321
SeA)
+
Top.
0
We will not give the proof. Although the idea is clear, the proof is quite complex. From our point of view this result means that if we define a homotopy coherent diagram of type A in an arbitrary Scategory B to be an S  functor F :S(A) + B , then in the case of B = Top, we retrieve Vogt's notion. To get the value of such an F :S(A) + B on a particular simplex, er, in say S(A)(A, A')nl we can use the fact that
S(A)(A, A')nl = F(A)nl(A , A') = Fn(A)(A , A') to realise er as an ordered set erA = (iI , " ', 1m) of composable morphisms in A, starting at A and ending at A', together with some bracketting of erA. The corresponding simplex erA E Ner(A)m, considered as a functor [m] + A yields an induced Sfunctor
S(erA) : S[m] +S(A) and hence a simplicial map
S(erA) : S[m](O, m) +S(A)(A , A'). There will be an (n  I) simplex a' E S[m](O, m)nl given by the bracketting, this time of (OI,I2, " ',m 1 m), that relates er to erA and S(erA)(a') = er. This means that F(er) E B(F A, F A')nl can be read off from the composite F S( erA) evaluated at a'. Finally we note that S[m](O, m) ~ 6[I]ml, to give a description of F(er) as one of the (n I)simplices making up an (m  I)cube within B(F A, F A') , the other simplices coming from the other brackettings of erA . 0
This motivates the following definition due to Cordier (1982).
Definition (2.3). Let B be an arbitrary S  category. We denote by Nerh .c. (B) , and call it the homotopy coherent nerve of B , that simplicial class having as its collection of nsimplices
N erh.c.(B)n = S  Cat (S[n],B). We have indicated how to prove:
Theorem (2.4) (Cordier (1982)). A homotopy coherent dia322
gram of type A in an Scategory B corresponds precisely to a simplicial map Ner(A)
t
Nerh .c.(B).
0
For a detailed proof see the original paper. Our aim in this section is to get the reader to understand Vogt's theorem (Vogt (1973)), its generalisation by Cordier and Porter (1984), together with one or two related results which fit into the same setting. These results are in part proved by arguments based on the sort of filling arguments that we have used repeatedly in this book. Some of those results need aspects of the theory of homotopy coherent ends or co ends and as we have not assumed knowledge of these techniques, those results will of necessity be left without proof here. The analysis of homotopy coherent diagrams in B can be thus reduced to the study of N erh.c.(B) which is independent of the diagram scheme, here denoted A. We want to study homotopy coherent diagrams of given type, but this implies that we must find how to compare them. If we consider the simple case A = [1], a homotopy coherent diagram is nothing but a morphism in B. Clearly to consider natural transformations as the basis for comparison of two such morphisms would lead to a morphism being a commutative square
F(O) G(O)
F(Ol)[
[G(Ol)
F(l)
G(l)
which seems much too strong a notion, rather we want a homotopy coherent square to represent the appropriate notion of homotopy coherent (h.c.) morphism.
Definition (2.5) (Vogt (1973) and Cordier (1982». If F,G are h.c. diagrams of type A in an Scategory B, then a h.c. morphism f : F t G is a h.c. diagram of type A x [1] in B restricting on A x {O} to F and on A x {I} to G. 323
This means that we have
f :S(A x [1])
+
B
or alternatively
f : Ner(A) x 6[lJ
+
Nerh.c.(B),
restricting to F and G on the ends. Homotopy coherent morphisms need not compose and when they do, they do not do it uniquely! To see why note that if f : Fo + F1 , and f' : Fl + F2 are h.c. morphisms then all that f and f' give us is a map
Ner(A) x /\[2,lJ
+
Nerh .c. (B)
whilst we need to fill /\[2, 1J and then restrict to the d1face of the result, however if Nerh.c.(B) is not a (weak) Kan complex then we would have a problem. Before turning to an analysis of the filler conditions satisfied by Nerh .c. (B), we will examine in a series of exercises the particular case when B is associated to a 9rpdenriched category B as in Chapter IV. This not only provides some idea on how to attack the general case, but provides important insight into the relations between the ideas used in Chapter IV and those here. Exercises. (i) In III.2(a) the nerve of a groupoid was shown (by you!) to be a Kan complex and in III. 2( c) an exercise asked you to show that if C was a groupoid, N er C was a rank 2 T complex. Reexamining the ideas there, prove: if C is a groupoid then any (n, i)horn for n ~ 2 fills uniquely. (ii) Prove that if C and D are categories then
Ner(C x D)
~
NerC x NerD .
(In fact we used this just now when we wrote Ner(A x [1]) in the form Ner(A) x 6[lJ .) (iii) Suppose B is a 9rpdenriched category, so each B(B, B') is a groupoid and the composition
324
B(B, B') x B(B', B")
t
B(B, B")
is a morphism of groupoids. Attempt to define an Scategory, B, on the same underlying context by setting
B(B,B') = NerB(B,B'). See if this gives an Scategory by checking the axioms (III. 4.1). (iv) Suppose B is a 9rpdenriched category, and
F :S(A)
t
B
is a h.c. diagram of type A in the corresponding Scategory B. Examine the cases A = [3], and A = [4] to find if the fact that each B(B, B') has unique fillers in dimensions 2: 2 can be interpreted in terms of the h.c. structure. What does this imply about the maps
FA,A' :S(A)(A, A')
t
B(F A, FA'),
in a general situation? Would it be possible to simplify the theory of h.c. diagrams in those cases where the receiving category B was assumed to have a specified 9rpdenriched structure? (v) If B is the Scategory corresponding to the groupoid enriched category B, investigate the Kan filler conditions on Nerh.c.(B) . One of the main points about those Scategory structures that come, as in these exercises, from 9rpdenriched settings is that the Scategory so formed is locally Kan. Recall (III. 4) that this means that each B(B, B') is a Kan complex.
Proposition (2.6). lIB is a locally Kan Scategory then
Nerh.c.(B) satisfies the weak K an condition. Although we will omit the detailed proof, which can be found in Cordier and Porter (1986), the idea is somewhat similar to many proofs we have already seen, once the encoding of the problems is done. For this we need some notation: we write I for 6[1] to shorten expressions. The monoid structure on I is given by
325
m:]2
m(O,O) = 0, m(O, 1) = m(l,O) = m(l, 1) = 1
t],
 proj : I
t
.6[OJ is the unique map
 di denote the face maps in N er (A) . Finally given
f : KI
B(X, Y)
t
g: K2
and
t
B(Y, Z)
two simplicial maps, we will write
gf : KI
X
K2
t
B(X, Z)
for the composite y
KI
X
K2 f x ~ B(X, Y)x B(Y, Z) cx,z, B(X, Z).
Now the data encoded by some
F :S(A)
t
B
is equivalent to:  to each object A of A, F assigns an object F(A) of Bj  to each composable string a = (/0, · .. , fn) E An+I(A, B) or alternatively in Ner(A)n+l, F assigns a simplicial map
F(a): In
t
B(FA,FB)
such that (i) if fo (ii) if
/i
= Id,
F(a)
= ] d, for
= F(doa)(proj
°
n. The cellular approximation theorem says any map is homotopic to a cellular one, cf. for instance Fritsch and Piccinini (1990), Theorem 2.4.11). ~ is the class of all expansions, that is finite composites of elementary expansions as recalled in the previous section. The category of fractions C(~1) is in this case equivalent to the homotopy category, since the end inclusions into a cylinder are expansions. (This result is essentially Corollary (3.5) of the next section, so we will not give more details here.) Remark. The simple morphism classes of X certainly form a class, but as we need to consider them as a set, we will assume that this is the case. This set will be denoted A(X) for X an object of C, thus A(X)
= {< 0:' >: 0:' a morphism of C(~1), domain of 0:' = X}.
The set A(X) has a distinguished element namely subset E(X) = {
E E(X) and f3 E < 0: >, then f3 is also an isomorphism in C(~l), also < I dx > E E(X).
Definition (2.2). Let C, ~ be as above with Q : C t C(~l) then this defines a categorical simple homotopy theory if the following two axioms are satisfied: (8 1) If I
:X
t
Y and s : X
~,
Z are in C, then the pushout
X
_=I_y
91
Iw w
Z exists and if s E
t
I'
so is s'.
(8 2) If I,g are morphisms in C such that Q(f) = Q(g) then there exist simple morphisms s, t E ~ such that sl = tg . We shall describe the development of this theory basing our description on Eckmann (1970) as 8iebenmann (1970) does not give as much detail. From now on we shall assume (8 1) and (8 2) hold for the remainder of this section.
Lemma (2.3). (i) Any morphism 0: in C(~l) can be written in the lorm 0: = Q(stIQ(f), with s E ~. (ii) Any simple isomorphism can be written in thelormQ(st1Q(t), with s,t E~. Proof. The argument for (i) parallels that used to prove Corollary (II.2.17). For convenience we include the details. Any morphism 0: in C(~l) can be written as a composite
with, of course, s}, ..• ,Sn E ~ . The situation Q(h)Q(S2)1 corresponds to a diagram of the form
Z~X..D...Y, 347
and hence gives a pushout (by (S 1))
II X....:....:....y
Is,
S21
Zw I{
with
s~
E I:. Thus we have Q(Sl)lQ( s;t1Q(fDQ(h) ... Q(fn)
C¥
=
Q(S;slt1Q(f{h)··· Q(fn)
with S~Sl E I: , since I: is closed under composition. So , by this means, we can shorten any word representing c¥ until it has the form Q(s)lQ(f) thus proving (i) . Now assume c¥ is a simple isomorphism, then in the initial representation
we may assume each Ii is also in I:, by the definition of a simple isomorphism. Following through the reduction steps we used to prove (i), at each stage (S 1) implies that in the pushout
X i _....::.I..:....i_ y; "+1
I
Y;+l
If
I
S:+1
II
W
is in I: as well as Si+l· This implies that in the final representation as Q(s)lQ(f) we have I is in I: as well. 0
of
c¥
c¥
Corollary (2.4). Given any simple morphism class < c¥ > with E C(I: 1 ), there is a morphism I in C such that < c¥ > = < Q(f) > . Proof. By Lemma (2 .3),
c¥
< c¥ > = < Q(f) > as claimed.
= Q(s)lQ(f) so Q(s)c¥ = Q(f) and 0
348
To simplify notation, in future we will usually write < of < Q(f) > .
1>
instead
Lemma (2.5). Given I,g, two morphisms in C having the same domain, then < 1 > = < 9 > il and only il there exist s, tEL; such that sl = tg. Proof. If < 1 > = < 9 >, we have Q(f) = ,Q(g) for some simple isomorphism ,. We know, = Q(sot 1Q(to) for some so, to E L;, hence Q(sof) = Q(tOg).
We now use axiom (8 2) to show the existence of elements such that
Sl, t1
E L;
as required, again since ~ is closed under composition. The converse is clear. 0 We next turn to the structure of A first proving that it defines a functor from C to Sets*, the category of pointed sets.
Lemma (2.6). Let 1 : X   Z, h: X   Y, morphisms in C. In any pushout square X =g=
hi Y
II : Z
  Zl be
w
Ih' g'
T
writing g' = h* (g), then
Proof. This merely states that the composite of two pushout squares is a pushout square:
349
x
hi Y
I p.o.
h*(J)
·Z
Ih' W
it p .o.
h:(Jd
ZI
X
I
hi Y
WI
which is a routine exercise in category theory.
Id p.o.
h*(Jd)
. ZI
I
WI
0
Exercise. Check that you can prove that a composite of pushout squares is itself a pushout square. (We will need this several more times in this section.) Given a morphism h : X given by
t
Yin C, we denote by A(h) the function
A(h) : A(X)
t
A(Y)
A(h)< I >=< h*(J) >. Remark. There is a technicality here that is important when it comes to the possible removal of the 'all isomorphisms are simple' rule. Any pushout is only defined up to isomorphism, so the above definition only makes sense if the natural isomorphisms between pushouts are simple. In more detail if Wand W' are two push outs of h along I as above and h*(J) and h*(J)' are the corresponding morphisms with domain Y, then there is an isomorphism a : W   t W' with h*(J)' = ah*(J) . We will have < h*(J) > = < h*(J)' > if a is simple, not otherwise. As a is an isomorphism, this poses no problem because we have required all isomorphisms to be simple. A refined definition of 'simple morphism' might be attempted where the natural isomorphisms arising from nonuniqueness of such universal constructions as pushout , would be defined to be simple, but in general, isomorphisms would have to be shown to be simple. The apparent alternative of choosing a pushout etc. for each 'corner' does not get around the difficulty as it all too easily leads to coherence problems as the composite of two chosen pushout squares need no longer be the chosen pushout square for the composite corner. The role of the 'isomorphisms are simple' rule is crucial in the next few pages. 350
Lemma (2.7). Jf< 9 >=< f >, then < h*(g) >=< h*(f) >, i.e. A( h) is well defined.
Proof. Suppose < 9 > = < f >, then by Lemma (2.5), there exist morphisms s, t E ~ such that tg = sf. By Lemma (2.6) then we get h~(t)h*(g)
= h~(s)h*(f)
where h' , hl/ are given by the pushouts
f
X
z lh'
hI Y
h*(f)
·W
and 9
X h Y but as s,t
E~,
h~(s)
Zl
lhl/ h*(g)
and hl/(t) E
~
W' and hence
< h*(g) > = < h*(f) > as required.
0
We also note that A(h) < Ix pointed sets.
> = < Iy >, so A(h) is a morphism of
Lemma (2.8). The assignment of A(X) to X and of A(h) to h yields a functor
A : C    t Sets*. It remains to check that A preserves composition. If h : X    t Y, j : Y    t Z then for any f : X    t W we have that the outer square of the diagram
Proof.
351
f
x
hi
I I
p .o.
Y
ilz
w u
h*(f) p.o.
j*h*(f)
V
is a pushout as the individual squares are pushouts, that is
(jh)*(f) = j*h*(f) . It then follows easily that A(j)A(h) required. 0
< f > = A(jh) < f > , as
The following proposition is the first real indication that something interesting is happening. We still do not know if A is nontrivial and such a fact would be particular to examples, but we do know from the next result that it is a 'homotopy invariant ' in some sense. Proposition (2.9). For any
E 1::,
8
8 :
X
+
Y , the function
A( s) is a bijection. Proof. If s E 1::, then in the pushout diagram
f Z X
I
s
Y we have
Is' W
l'
s' E 1::. Hence < 1's > = < 8'f > = < f > . Now define cp : A(Y )
+
A(X)
by
cp( < 9
»
=
then 352
< g8 >
cpA( s ) < I > = cp < f' > = < f's > = < I > so cpA(s) is the identity mapping on A(X) . For the composition A( s)cp < 9 >, we have a composite pushout square s 9 X    Y "..... Zl
sl
If
If'
Y   _· W t
so A(s)cp< 9 > = < g't > . Now g'ts
g'
T
= t"gs so
Q(g't) = Q(t")Q(g) and < g't > = < 9 > as required and A(s) is a bijection.
0
Remark. The whole point of A (and later on E) is that there may be other isomorphisms in C(~l), not just simple ones, but if h is any morphism in C with Q( h) an isomorphism in C(~l), then the above proposition together with Lemma (2.12) shows that A(h) is an isomorphism. The pointed sets A(X) have a very naturally defined abelian monoid structure, which makes the subsets E(X) into an abelian group. The basic idea is that if I : X ~ Y, g: X ~ Z are in C, (we can think of them as 'inclusions') then we can glue Y and Z together along the copies of X, forming a new space YU Z with a new map from X to it.
x
This is, of course, just the pushout of I and 9 and so we already have an established notation for this. More formally we consider I : X ~ Y, g : X ~ Z giving classes < I > and < 9 > E A(X).
Definition (2.10). The sum of < I > and < 9 > is defined to be
< I > + < 9 > = < I' 9 > = < g'l > where < f' > = A(g)
and < g' > = AU) < 9 > .
This addition is clearly commutative. Moreover the formal categorical properties of push outs yield, without any problem, the following : 353
Proposition (2.11). The addition, +, defines onA(X), the structure of an abelian monoid in which < Ix > is the neutral element. If h : X t X', then A( h) satisfies A(h)«
f >
+ +A(h)< 9 >
and hence A with this structure gives a functor from C to the category of abelian monoids. Proof. As the proof that + is well defined and associative is fairly easy, we will leave it as an exercise, limiting our attention to giving the key ideas for the second part. In the pushout
X_..::....f_y
k
gj z we have < f > + < 9 > = < pushout squares X
f'
f' 9 > . We form
9
hj X'
W
Z
jh' z, h*(g)
f'
two new diagrams using
W
jh" h~(J')
W'
and X
f
hj
y
g'
j
h
. Y'
X' h*(J)
h*(gl)
354
W
W'
This gives us a new square h*(f)
X' h,(g) ]
Z'
Y' ]);,(g')
h~(f')
W'
which not only is commutative but is a pushout. As this uses a routine type of argument, we will omit the proof of this fact, leaving it as an exercise. Accepting this, we get A(h)( < f
>+
h~(f')h*(g)
>
by Lemma (2.6) < h* (f) > + < h* (g) > A(h)< f > +A(h) . 0
Earlier we introduced the subset E(X) of A(X) defined by those < a > for which a was an isomorphism in C(~l) or equivalently by those < f > such that Q(f) is an isomorphism in C(~l). The natural result to expect is that E would define an abelian groupvalued functor on C, and to this end we now start to work. We first investigate more fully those f such that < f > is in E(X).
Lemma (2.12). Given f in C, Q(f) has a left inverse if and only if there is an 1 in C with 1f E~. Proof. If Q(f) has a left inverse, a = Q(s)lQ(g) with, of course, s E ~, then Q(s)lQ(gf) = 1, so Q(gf) = Q(s). By axiom (S 2) there are morphisms r, t in ~ such that rgf = ts E ~ so taking 1 = rg gives the required result. Conversely if there exists an 1 in C such that 1f = s E ~ then Q(s)lQ(1) is left inverse to Q(f). 0
Exercise. Complete the proof of the result mentioned earlier: If h is such that Q( h) is an isomorphism, then A( h) is an isomorphism of abelian monoidii. Of course, the previous lemma applies in particular if Q(f) is an isomorphism, in which case Q(1) is also an isomorphism. 355
Proposition (2.13). If f : X
Y, h: X   t X' in C, and Q(f) is an isomorphism, then Q(h*(f)) is an isomorphism. t
Proof. By the previous Lemma (2.12), there is a morphism 1 with s E ~, but then h~(1)h*(f) = h*( s) (in the notation of Lemma (2.6) and h*(s) E ~ by axiom (8 1)). Again using Lemma (2 .12), but in the opposite direction we find Q( h*(f)) has a left inverse namely
1f =
Q(h*(s))lQ(h~(1)). Rewriting we get
Q(h~(1))Q(h*(f))Q(h*(s)tl = Id,
so Q(h~(1)) has a right inverse. We know however that Q(1) is an isomorphism as Q(f) is one, so we can rerun the above argument with f replaced by 1, h replaced by h', to get that Q(h~(1)) has a left inverse, but then Q(h~(f)) must be an isomorphism and so we can conclude that Q(h*(f)) is an isomorphism.
o Corollary (2.14). If < f >, < 9 > E E(X), then
< f > + < 9 > E E(X) . Proof. We have
< f > + < 9 > = < f'g > where f' = g*(f) . As Q(f) is an isomorphism, so is Q(g*(f)) and hence so is Q(f' g) as required. 0 Corollary (2.15). If h : X
t
= Q(f')
X' and < f > E E(X), then
A(h) < f > E E(X').
0
We leave this proof to the reader as it is so similar to the above. We next check that E(X) is an abelian group, and as we already have enough to imply it is an abelian monoid it only remains to check for inverses . We will use the remark after Proposition (2 .9), namely that if h : X   t X' is such that Q( h) is an isomorphism then A( h) is a bijection.
356
Proposition (2.16). If < f > E E(X), there is a < 9 > E E(X) such that
< f > + < 9 > = < Idx > . Thus E(X) is an abelian group, and E defines functors from C and Ab, of abelian groups.
C(~l) to the category,
Proof. If < f > E E(X), then by definition in the category C(~l) the morphism Q(J) is an isomorphism and there is some 1 such that 1f = s E ~ by (2.12). Suppose / : X + Y and 1 : Y + Z. As A(J) is a bijection, there is a morphism 9 : X + W, say, with A(J) < 9 > = < f*(g) > = < 1 > . This fits into a pushout square
/ Y X
g\ W
\/.(g) /'
z'
where Q(J') is an isomorphism, since Q(J) is (compare (2.13)) . We also have that Q(1) is an isomorphism and as < /*(g) > = < 1 >, so must be Q(J*(g)) but then Q(g) = Q(J't1Q(J*(g))Q(J) must also be an isomorphism, i.e. < 9 > E E(X). We finally note that the pushout square gives
+
= < /*(g)/ > = < 1/ > = < s > = I dx
as required. The other statements of the proposition are now easy consequences 0 of this. Thus with just two conditions on the class, ~, one has a rich structure for comparing those morphisms inverted by Q : C + C(~l) and those constructible from the morphisms of ~ and their formal inverses. To illustrate the basic properties of A and E, we will look at the derivation of a basic formula.
357
Proposition (2.17). Suppos e 1 : X morphisms in C, then in A(Y) ,
A(I)( < gl
»
t
= A(I) < 1 >
Y , and 9 : Y
t
Z are
+
Proof. Consider the diagram
gl
f
x~z I [r If" p.o.
Y
p.o.
·V
W
~ I*(gl)
in which the two inner squares, and thus their composite, are pushouts . Of course f' = 1*(1) , so
< 1*(1) > + < 9 > = < f" 9 > = < ul*(I) > but
from the outer square.
0
3. Generation of Simple Homotopy Equivalences At the end of section one, we posed a problem which is of great importance to this theory, namely how is one to generate 'simple homotopy equivalences' if one does not have objects that can play the role of cells. The Eckmann Siebenmann theory presupposes the initial data includes a class of 'simple equivalences', but does not suggest how such a class might be built in the 'abstract' . A search through the literature, however, reveals a geometrically based theory, roughly parallel to the EckmannSiebenmann theory. (This can be found for 358
instance in the book of Cohen (1973) or the paper by Eckmann and Maumary (1970).) In this geometric version of the theory, much of the development is based on a mapping cylinder calculus, replacing each cellular map f : X ~ Y between finite (or locally finite) CWcomplexes by the corresponding cofibration if : X ~ Mf, and then constructing A(X) and E(X) using these cofibrations. A key point earlyon in this geometric development is that the inclusion of X into the cylinder X x I as one of the ends, is a simple cofibration. As the view of abstract homotopy theory taken here is based on a cylinder functor, this observation is particularly attractive and suggests that these 'endinclusions' could be used as a generating class of 'simple expansions'. It then remains to decide what the generating rules should be. The EckmannSiebenmann pushout rule (S 1) suggests one possible such rule, but the requirement that simple equivalences be homotopy equivalences restrains the way in which this must be presented. The difficulty is that in the EckmannSiebenmann model for classical simple homotopy theory, C is the category of finite CWcomplexes and cellular inclusions (which are cofibrations) and the simple maps in E are simple cofibrations, so trivial cofibrations, and we know trivial cofibrations are preserved by pushouts. General homotopy equivalences need not be preserved by general pushouts, so a naive pushout construction in our generating rules would cause problems.
Definition (3.1). Given a generating cylinder I on C, let E be the class of morphisms in C generated by the following rules: (SE 0) All isomorphisms are in E. (SE 1) For any object X in C, the end inclusion
eo(X) : X
~
X x I
is in E. (SE 2) If morphisms s, tin C are such that st is defined, then if any two of s, t and st are in E, so is the third. (SE 3) Simple Pushout Principle (S.P.P.)
359
Given a pushout A
_=1_.
B
ic l l
i'
·D
I'
with i a cofibration in L:, then i' is in L:. The morphisms in L: will be called simple equivalences or simple morphisms for short.
Remarks on the Generating Rules. (a) The sense in which the simple equivalences are 'generated' by the above rules does need a bit of explanation. The use of such a recursive definition is standard in many areas of algebra. Here the first two rules give us some basic elements of L:. Forming up composites of these and using the cancellation that is implicit in (SE 2) yields more, then the S.P.P. generates still more, that are fed back into (SE 2) and then closed up once again with the S.P.P. and so on, taking the union of all these classes. As usual, one can 'construct' L: in another way. First check that the family of subclasses of morphisms of C satisfying (SE 0) to (SE 3) is nonempty. This is clear since the class of all morphisms satisfies these rules. Now take the intersection of the family, and check it satisfies the rules. It must be L: , as it is clearly the smallest such class. (b) In the classical geometric case , a simple equivalence is any morphism homotopic to a composite of elementary expansions and contractions. By the cellular approximation theorem (see Fritsch and Piccinini (1990), Theorem 2.4.11), any map I between CWcomplexes is homotopic to a cellular map, f', say, but of course, if I was a homeomorphism, there was no implication that f' was one as well. Because of this , for a long time it was not known if all homeomorphisms were simple. This was proved, however, by Chapman in 1974 (d. Cohen (1973), Appendix), but the theorem is very complex and uses a lot of deep geometric topology. Thus although most of the isomorphisms that we use will be provably 360
simple by other methods, we will assume that all isomorphisms are simple as an axiom. This axiom is really only needed when identifying the simple equivalences for additive simple homotopy theory. To aid the reader to keep track of the isomorphisms used, we have labelled as simple isomorphisms those provably simple from the axioms with (SE 0) replaced by a weaker condition: "All identities are in ~" . (c) The composite and cancellation rule (SE 2) is slightly 'nonconstructive' at least when cancellation is involved, however it is unlikely that such a rule can be omitted and still allow for a functor E from C to the category of abelian groups. The sort of formulae given in (2.17) indicate that if 9 and gl are simple (i.e. have their corresponding < >class trivial) and if 1 gives an isomorphism in C(~l), then 1 must also be simple. (d) From the Simple Pushout Principle one can derive various more powerful 'Principles' which will be used frequently. The first one is the alternative version of S.P.P.: If i is a cofibration and 1 is simple, then
f'
is simple.
A more complex one is a Simple Relativity Principle (S.R.P.) which states: If in the diagram
x'y' i and i' are cofibrations and all square faces are pushouts, then s E implies t E ~.
~
Finally the most general form is a Simple Gluing Principle (S.G.P.) which states:
361
Let
So Ao        =     Bo
~ Al     I~    Bl SI
f
f' A2
S2
= B2
~ A      ~ . B s
be a commutative cube where the left and righthand squares are pushouts such that i and if are cofibrations. Then if so, SI, S2 are in E, so zs s .
The proofs that S.P.P. implies both of S.R.P. and S.G.P. is essentially the same as that of the Relativity Principle and Gluing Principle in a Cofibration Category sketched in Chapter II. This will be set as exercises in the next section. We will use both the S.R.P. and the S.G.P. frequently as they get around the problem of presenting essentially the same argument time and time again. So far however we only know two types of simple equivalences, identities and morphisms of the form eo(X). Lemma (3.2). Let C, I, E be as above, then for any object, X, the natural morphisms
and a(X) : X x I
t
X
are in E. Proof. Recall that a(X)el(X) = Id x = a(X)eo(X), now use 0 (SE 0), (SE 1) and (SE 2). 362
Before we proceed further we will check that in the only well known case of a simple homotopy theory, 'our' simple equivalences are the same as the 'classical' ones.
Theorem (3.3). For C = CWfin, the class of simple equivalences defined above coincides with the class defined by elementary expansions and contractions. Proof. In CWfin, the class ~classical generated from the elementary
expansions and contractions satisfies the SEaxioms, the only minor difficulty occurs when checking the Simple Pushout Principle but this follows from the known "Sum formula" given, for instance, in Siebenmann (1970) for the 'proper' case. (We will derive a similar formula later on in the abstract case.) As ~ is generated by these rules, it is the minimum class of morphisms satisfying the SEaxioms so ~ ~ ~classical . To prove the converse, it suffices to prove that the usual elementary expansions can be generated by this cylinder method. We consider the class of inclusions
(In
X
{O}) u (ar
X
I)
+
r
X
1=
r+l.
By induction n, it is clear that these expansions are simple in the cylinder sense by using (SE 1) and (SE 2), but then using pushouts, they generate all elementary expansions and so ~classical ~ ~.
0
It would be silly to talk of simple homotopy theory if simple equivalences were not homotopy equivalences. We mentioned this point earlier on, the following theorem completes its clarification:
Theorem (3.4). Let I be a generating cylinder on the category C. Then all simple equivalences are homotopy equivalences. Proof. Rephrasing the statement, the class, ~h.e., of homotopy equi
valences satisfies the SEaxioms, so as ~ is the class generated by the axioms and the initial generators (Le. the identities and the Oend inclusions) are homotopy equivalences, it follows that ~ ~ ~h . e . as required. 0
363
Corollary (3.5). The category of fractions C(L;l) is isomorphic to that, Ho( C) , obtained by inverting the class, L;h.e., of homotopy equivalences. Proof. As L;
~
L;h.e., the quotient functor
C ~ Ho(C) factors through C(L;l), however any homotopic pair of morphisms f, 9 : X ~ Y in C is already identified in C(L;l) since, if H : f ~ g, f = Heo(X) , 9 = Hel(X) , as eo(X) and el(X) are in L; , they become isomorphisms in C(L;l). However cr(X) is the inverse of both of them, so they become equal. But then f and 9 must also be identified. Finally this implies all homotopy equivalences are already isomorphisms 0 in C(L;l) , i.e. Ho(C) ~ C(L;l). 4. The Mapping Cylinder Calculus Now that we have a class of simple equivalences to work with, it might be thought that the only thing that would be needed would be to verify the axioms, (S 1) and (S 2), of section 2. This is not the case however since although (S 1) seems to be the Simple Pushout Principle it is not, as in (S 1) no mention is made of the morphisms being cofibrations. In fact as we noted in se~tion 2, Eckmann and Siebenmann model their theory not on the category of finite CW complexes , and all (cellular) maps, but on that of finite CWcomplexes and all cellular inclusions. Rethinking our approach this suggests working with a category C cojib having the same objects as C but with only cofibrations as the morphisms , then taking L; to be the class generated as in (3.1) but taking care with cancellation, i.e. if a, bEL; , then ab E L;, but if a and ab E L; , do not allow oneself to conclude bEL; unless you know b is a cofibration and similarly if band ab E L;. Even this causes problems however, (S 1) is now easily verified, but (S 2) seems hard to check in this generality. In fact we know of no proof even in the classical case , that does not rely on constructions involving structure not available in the abstract case. This suggests the following 'exercise' to which we have no solution:
364
Exercise. Investigate what extra structure on the cylinder I is needed to provide a system in which (S 2) can be verified. Does this 'failure' of (S 2) in the general abstract case, mean that the reader should shut the book at this point and put it back on the shelf? The answer is clearly negative. At the same time, the early 1970's, as Eckmann and Siebenmann published their approach, various authors, Eckmann and Maumary (1970), Bolthausen (1969), Stocker (1970) and Cohen (1973) , published geometric approaches that avoided the use of the category of CWinclusions by working with mapping cylinders. This theory does adapt well to an abstract setting and has the advantage of being geometrically, almost visually, pleasing. This aids one in building up an intuition of the structure, an intuition which is slightly less evident in the abstract setting used by Eckmann and Siebenmann, as some of that structure is compressed there. The basic method of attack in this 'geometric' approach uses mapping cylinders and hence is ideally suited to a cylinder functor approach to homotopy theory. For convenience we first recall the mapping cylinder construction given earlier (1.2 .8) . Given f : X ~ Y in C, where as always in this section C has a generating cylinder, I, then we form the pushout
X  f   Y
[il
eo(Xl [
XxlMf 7rf
The object Mf is the mapping cylinder of gives a morphism p f : M f ~ Y such that Pf7rf
= fa(X)
and Pf]f
f.
The pushout property
= I dy
and forming the composite if = 7rfe}(X), we have f = Pfif· As we are assuming that I is generating then we also have that if is a cofibration and hpf ~ Id Mj (cf. 1.5.11 and 1.8.1).
365
Proposition (4.1). Assuming that I is generating, then for any morphism f : X + Y in C, f = p fi f where if is a cofibration and p f is simple. If f is simple, so is if· Proof. For the first part we only need to check that Pf is simple as
the rest is part of the results we have just recalled. The Simple Pushout Principle gives that as eo(X) is simple, so is iI, but as PfiI = Id y , the cancellation rule, (SE 2) , gives Pf is simple. Another use of cancellation shows that if f is simple, so is if. 0
Exercise. (i) Show that if I is a generating cylinder, then the classes: cofibrations = cofibrations with respect to I weak equivalences = ~ U isomorphisms is a category of cofibrant objects in the sense of Definition (II.1.4). (ii) Adapting the arguments outlined in Chapter II, prove that (a) the Simple Pushout Principle (S.P.P.) implies its alternative version, (b) the S.P.P. implies the S.G.P., (c) the S.G.P. implies the S.R.P., (d) the S.R.P. implies the S.P.P. (Refer to the last section for the meaning of S.G.P. and S.R.P.) These exercises have the advantage of shortening many proofs in what follows. From our earlier development of the EckmannSiebenmann theory we know approximately in which direction to go. In fact the construction of A(X) there does not depend on the axioms (S 1) and (S 2) . Those are only used when proving A is a functor and that A(X) has a natural monoid structure. The description of A(X) given in section 2 is not adapted to this new setting so for the moment we will use that old version as a starting point only. Our plan is thus to redefine A(X) for this new context and then to attempt to verify functoriality etc. using the generating rules for simple equivalences directly, taking as a model the approach of Eckmann and Maumary. We will see that this works quite well. 366
Consider the class of all morphisms with domain X in C, and define a relation on it by f : X t Y is related to f' : X t Y' if there is a simple equivalence s : Y t Y' such that sf is homotopic to f', i.e. the diagram
Y _ _ _ Y' s
is homotopy commutative. As it stands this relation does not look to be symmetric although we will show later that it is. The relation is clearly reflexive and transitive, so we replace it by the equivalence relation it generates. The equivalence classes are known as simple morphism classes and we will denote by < f > , the simple morphism class containing f. We thus redefine A(X) in this setting to be A(X) = {< f
>: f in C, domain of f
= X} .
Exercises. (i) Show that if f ~ f', then < f > = < (ii) Prove that if / ~ 9 and f E ~, then 9 E ~.
f' > .
(iii) Suppose we have sf ~ f' with s a simple equivalence. Then by Theorem (3.4), s is a homotopy equivalence. Let t be a homotopy inverse of s. Prove that t is in ~ as well and that tf' ~ f so the relation defined above was indeed symmetric. (iv) If f, 9 are two morphisms in C having the same domain and s, t E ~ are such that sf = tg, show that < / > = < 9 > in the new meaning of the symbols, (cf.(2.5)) . (v) Show using (2.4) and (3.5), that the two definitions of A(X) do, in fact, give the same answer, i.e. define 'isomorphic' pointed sets. (You may need to prove refinements of one or two results to make the obvious ideas work.)
Lemma (4.2). With the assumption 0/(4.1), we have for any f:XtY = . 367
0
The proof is immediate given (4.1). This lemma enables us to define sums etc. of simple morphism classes using constructions analogous to those in section 2 but with f replaced by if. Of course this will mean a lot of checking of 'well definition' of operations and much of this section is about that alone. To aid in keeping track of the constructions we will use a mapping cylinder calculus and a corresponding diagram calculus. Suppose f : X    t Y, and g : X    t Z are in C, then we denote by Mf,g, the double mapping cylinder given by the pushout If
X
. Mf
I
19
qj
Mg
and write if,g : X
t
. Mf,g
qg
Mf,g for the diagonal, if,g = qfif = qgig.
Remark on Schematic Diagrams. This double mapping cylinder construction is different from that used in 1.8 , the reason is that the form used there has no suitable morphism of X into it . In the form used here there are two copies of X x I, joined along a copy of X, to that are glued the copies of Y and Z. The diagram for M f would be X
•
MJ
Y
•
and that for Mf,g
•z
Mg
x•
MJ
•y
These diagrams correspond to all of X , Y and Z being points but they provide a visual schema which we shall use several times to plan out proofs. The triple mapping cylinder of f, g and h : X    t W is given by the pushout
368
with i f,g ,h : X
+ Mf,g,h
being the diagonal. The diagram is W
·""Mh
yx
• y "" • "Mj
z• It is immediate that Mf,g,h could be defined by combining Mg,h with M f or Mf,h with Mg. The various versions are naturally isomorphic and will be considered to be equal. (Really we should note that such natural isomorphisms are in fact simple.)
Corollary (4.3). Both if,g and if,g,h are cofibrations.
0
Notation. We have to redevelop the base on which to prove the functoriality of A. This will be in terms of an action and so given the situation above we define gf : Y + Mf,g
to be the co fibration qflJ . We note that gf f
= qflJPfif
so < 2.f ,g > = < 9 ff
~ qfi f
= if,g : X
+ Mf,g
>.
We next need to see how the mapping cylinder construction behaves with respect to homotopies. If f, 9 : X + Y and f ~ g, then
< f > = < 9 > so < if > =
,
but as we are wanting to check that our definitions work well and have 369
no ambiguities, this argument is not sufficient. In fact we need to construct explicit simple comparisons between if and ig so as to combine them later on with double mapping cylinders, etc. The results we will prove play the role of the EckmannSiebenmann axiom (S 2) in the development .
Proposition (4.4). Let f : X + Y be a morphism in C and F : X x I + Y be a homotopy with Feo(X) = f. Then there is a simple equivalence c : Mf U
Xx{o}
X xI
+
MF.
This simple equivalence is relative to X and Y in the sense that the diagrams
XXxI eo(X) and
commute, where i : M f
+
Mf U
Xx{o}
X x I is the canonical morphism
Proof. Consider the pushout diagram eo(X U X) XUX     , (XUX)xI
eo + el\
\.
X x 1J   , M eo+e\ 370
.
in which all morphisms are cofibrations. By Proposition (1.2.10) the fact that eo + el is a cofibration implies that the natural map from Meo +e1 to X x I x I is split:
Here 57r = (eo(X) + el(X)) x I, 5j = eo(X X 1) and r5 = Id. We note that as j and eo(X x 1) are simple, both 5 and r are simple equivalences. We can construct M eo +e1 in another way, as the colimit of the diagram
X
eo(X)
X xl
e1(x)1 X eo(X). X x I
.~
eo(x)1
X x I         . M eo +e1 If we add to the bottom of this diagram, F : X x I ~ Y, we get Me o+e IF U Y, which by calculating the colimit in a different order is iden
tifiable as M f U X xl, whilst MF is given by the pushout of eo(Xx1) X x {o} along F, i.e. we have a pushout XxI=F_y
eo(X x
I)I
IjF
Xxlxl' MF 7rF The splitting of
5
thus induces a splitting of Mf
U
Xx{o}
r'
XxI='=;MF C
371
and using the S.R.P., we obtain that c is simple. The statements about c (and similar facts about r') being relative to X and Yare now verified 0 by diagram chasing. Exercise. The above result (4.4) can be proved as a special case of a more general result: If f : X    t Y is a morphism and i : Xo cofibration, then the naturally induced morphism Mfi U X
t
Xo
t
X is a simple
Mf
is simple. Prove this using the Simple Gluing Principle. The previous result follows taking F for f and eo(X) for i. Corollary (4.5). Let f : X
t
Y be a morphism in C and
F:XxltY be a homotopy with Feo(X) = f, then the endinclusion eo (X) : X
t
X x I
induces a simple equivalence s: Mf
t
MF
relative to X and Y. This has as homotopy inverse a morphism, 7r :
MF
t
Mf
given by the composition of
r' : M F
t
Mf
U
X x {O}
X x I
and the morphism from M f U X x I to M f that is induced by a(X) : X x I
t
X,
(i. e. that contracts the' whisker' X x I). Proof. To prove that s : M f strong S.G.P. in the cube
t
MF is a simple equivalence, use the
372
X
f
eo(X)
X x I ~eo(X) ~eo(X x 1) ~ eo(X) x I ~ XxI XxlxI =~,
I
F
y
~
   =  + ,
Y
~
M f     s     ' MF
first noting that composition cancellation (i.e. (SE 2)) gives that eo(X) x I is simple. The checking that this induced map has the homotopy inverse stated is routine. The stated map is easily shown to be a retraction onto M f and the homotopies required can be read off from 0 earlier sections.
Proposition (4.6). Given a category C with a generating cylinder, I, if f, 9 : X   t Yare homotopic morphisms in C then there is an object M together with simple equivalences, s : Mf
t
M, t: Mg
t
M
such that sif = tig and sit = tjg. Proof. Let F : X x I Consider the diagram Mf U
Xx{o}
t
Y be the homotopy joining f and g.
X x I+ M f
i;Y
Ys
c
XxI
~ MF
"'>
X I1(X)
i~~M
in which the square faces are pushouts. Similarly we have a diagram replacing Mf by Mg and gluing along X x {I}, giving a morphism 373
t : Mg t M such that iF = tig. (It is important here to note that the bottom square is the same in both diagrams.) By the Simple Relativity Principle sand t are simple (and if we need that fact, both are split by explicitly described morphisms induced from the splitting of c and the analogous one for the diagram for Mg). This gives us s, t satisfying sif = ti g. Now consider slJ and tjg . All of the constructions are relative to Y, so a diagram chase proves
as required.
0
This also shows that if
f c::: 9 and 9 is simple, so is f.
Corollary (4.7). If F : f c::: f' : X are simple morphisms
t
s' : Mf,g t N, t' : M!"g
Y and 9 : X
t
t
N
relative to X, Y and Z. Therefore
< gf > = < gl' > and < fY > = < j'g > . Proof. We have by definition gf
= qflJ , fY = qgjg where
Zf
XMf
ig
I
I
qj
Now consider the diagram
Y
qf M f ', Mf,g
y
s
s'
X    : Mg
~
Zg
~
M·N q' 374
Z, there
and a similar diagram with I replaced by I' and s by t, inducing  t N. The S.R.P. implies that both s' and t' are simple equivalences, which are relative to X, Y and Z and then
t' : Mj' ,g
< Ig > = < qgjg > = < s'qgjg > = < ijg > and from the similar diagram we get
< I'g > = < ijg > as well. We also note s'qf
< gf as required.
= q's and t'qj' = q't, but slt = tjj' so > = < qflt > = < qrij' > = < gl' >
0
Looking back at section 2, the obvious definition of AU) for  t Y would be
I :X
AU) < 9 > = < gf > . Corollary (4.7) thus interprets as saying that homotopic morphisms I, I' give the same induced mapping, AU) = AU') and also goes some way to proving that if < 9 > = < g' > then < gf > = < g'f > and hence that AU) defined as above is well defined. Clearly we have if 9 c:::: g' then this is the case but we still have to check what happens if we replace 9 by some sg with s E I:. How are gf and (sg)f related? As a step in this direction we look at the relation between working with I and with i f . We have that M il is given by the pushout: zf
X (4.8)
eo(X)
I
, Mf s
X x [
'M' I
but as X x I is the mapping cylinder of I dx , this means M il ~ M Jdx ,!
375
This isomorphism is simple as it is compatible with the inclusion of Y into both sides and those inclusions are both simple. The pushout (4 .8) also gives a simple cofibration
M,
t
MIdx'[
which by abuse of notation we will also denote by s.
Proposition (4.9). For any 9 : X valence
s' : M"g
t
t
Z, there is a simple equz
MiJ ,g.
Proof. We leave the details as an exercise in the use of the Simple 0 Relativity Principle. Similarly one can prove 'multiple' mapping cylinder versions.
Proposition (4.10). For any j : X    t Y, 9 : X    t Z and h : X    t W, there is a simple equivalence between M"g,h and MiJ ,g,h relative to X, Y , Z and W . 0 This result, which came out as a biproduct of the verification that our tentative definition of AU) < 9 > is well defined, is a good step on the path to checking that AU) preserves sums, i.e. is a monoid homomorphism. This is a bit premature  we have not yet defined sums in A(X) in this general context, but the following is clearly needed.
Theorem (4.11) . Given j,g,h as above, there is a simple equivalence between MgJ,hI and M"g ,h relative to Y , Z and W. Proof. The two objects to be linked have schematic representation:
376
. . •
w
x
Y
w
and
• Y
x
~•
/x
. . •
z
•
• Y
z•
Y
The plan of how to link these up is to break the process into simple 'moves' as indicated below:
x ...
w
Y
W
Y
1/;1
I .YI ... z x
..x
x ...
W
Y
(1)
Xj»r (i) . x.I ... x
I I X..Y

Z
Y
•
Y
Z
I
•Y
w
•

(3) Z•
~·  · Y / x
although in fact (1) has to be modified and (2) and (3) will be done together. Stage (1) in its simplest form reduces to
..
x
..
x
Y
1/;1 » OY
I
.Y
Xo
I II ..
..I
x
Y
x
Y
Y
T
S
which is made up of two compatible moves, one on the top half diagram, the other on the bottom. The first object S is built up from two copies of Mf U Y x I , glued along Y whilst the two halves of T are both M f x } , glued along a copy of Mf. The homotopy, f x I: X x 1 Y x I joins eo(Y)f and e1(Y)f, but Meo(Y)f ~ M f U Y x I, from the composite pushout
377
X  f  .y
eo(X)]
x
eo(Y)
Yx[
]
]iI . Mf
x I
and the isomorphism is simple, as is easily checked by 'cancellation'. Using this we find by Corollary (4.5), there is a simple equivalence Meo(Y)f ~ M f U Y x [ ~ M f x !
relative to X and Y xl. The inclusion that we want is given by the cube
Y
Jf =. Mf
~Y Mf U
Mf U Y x [ 
x [
sl~
~ Mf x!
s =M f x!
~ S     ~ T t
with left and right faces pushouts. As () : Y ~ Mf U Y x [ is a cofibration, the Simple Gluing Principle would give us that t was a simple equivalence as needed, if, that is, we knew that the naturally defined map cp : M f
t
Mf x!
was a cofibration. However although we have an explicit description of cp, and we know that it is simple and a strong deformation retract, these do not suffice to prove the cofibration property. We thus 'make it into a cofibration' replacing it by i", : M f ~ M", and composing s with the simple morphism j", : M f x ! ~ M", to define s' : M f U Y x I ~ M", . The square involving s' commutes up to 378
homotopy but since B : Y + M f UY x I is a cofibration, using (1.2.11) s' can be deformed to give a new morphism s : M f U Y x I + Mip making the square commute. We form a new cube (which we will abstain from drawing) which is as above except Mip replaces M fx !, s replaces s and the new pushout of the right hand side is T with t : S + T being the induced morphism. Now we can invoke the S.G.P. to claim that t is a simple equivalence which is relative to Y and to the crucial copies of X to which Mg and Mh are to be attached. We next attach Mg and Mh to S and to T getting objects Sg,h and Tg ,h and, by a new use of the S.G.P. a simple equivalence
tg,h : Sg,h
+
Tg,h'
Recall that T g,h is obtained from two copies of Mip glued along iip. The morphism iip is a simple cofibration so we can find a right inverse r for iip so that r is simple. Applying this to both copies of Mip that together make up T we get a simple equivalence r U r from T to M f . We need to see what happens to the attaching copies of X (for Mg and Mh) in this process, but as r is essentially the identity on if it is clear, that the attaching points become amalgamated, as required. eee
!!!I
e
I
ifni
e
._e_e
'"
ee
/
o
Exercise. (i) In the special situation, that
1=(( ) x I,eo,e},a) is a generating cylinder on C in which ( ) x I preserves pushouts, show that, under suitable filler conditions, the map 'P : Mf
+
Mfx!
is a cofibration. (ii) Investigate the situation in the additive case.
We next prepare for the study of composite morphisms mentioned earlier. 379
Theorem (4.12). Let f : X ~ Y, g: Y ~ Y' be given morphisms in C. Forming the pushout in the diagram
then there is a simple equivalence Mgf~M
relative to X and Y'. Proof. Since we have a standing assumption in this section that C has a generating cylinder, we have that there is a homotopy
h: Id Mg
~
jgpg.
Composing h : Mg x I ~ Mg with (igf) x [ : X gives a homotopy, denoted , where i : X ~ Y was a morphism with domain X and where i and 9 : X ~ Z are simply equivalent if there is a morphism s E ~ such that s : Y ~ Z and sic::: g. There is another description of this equivalence relation that is sometimes useful to have at our disposal. It is as follows. We say i , g, as above, are simply equivalent if there is a sequence {Ii : X ~ Yi}o::;i::;n for some n, with io = i, in = 9 and for each k, 0 ~ k ~ n, either there is a simple equivalence, Sk : Y k ~ Yk+l, so that sdk = ik+l or there is a simple equivalence, Sk : Yk+l ~ Yk, so that skik+l = ik.
Proposition (4.13). The two notions of simple equivalence coincide. Proof. Clearly the second type of simple equivalence is at least as strong as the first one, so assume given i ,9 and s : Y ~ Z , s E ~ , such that si c::: g. Then by Proposition (4.6), there is an object M and simple equivalences
relative to X. By Theorem (4.12), we have an object pushout
382
M given
by the
(relative to X) and as s is simple, PI is simple, and a simple equivalence MSf
t
M.
SO we have Y
/!!
relative to X.
Mf ~
M~
Msf
t
M ~ Mg ~ Z
0
As in section 2, we will require that A(X) is always a set and not a proper class. The proof of that in any particular case will depend on the particular structure of that situation and not on generally applicable arguments.
Theorem (4.14). The construction A defines an abelian monoid valued homotopy functor on C. Proof. We will break the proof up into lemmas. To start with we suppose f : X t Y, < 9 > E A(X) with 9 : X t Z, and as suggested earlier we define
AU) < 9 > = f* < 9 > = < gf > where gf was introduced earlier (after Corollary (4.3)).
Lemma (4.15). If < 9 > = < g' >, then < gf > = < g'f >, so f* is well defined. Proof. By Proposition (4.13), it is sufficient to assume g' = sg for sEE. By Theorem (4.12), there is a simple equivalence Msg t M where
383
and as
8
is simple, so is is and hence Pl . We thus have
Now pushout along if : X   7 M f giving qf : M f  7 Mf,g, qj : Mf   7 Mf,sg and say q : M f   7 M' , and a diagram under Mf
Mf
Now compose with It : Y
7
Mf to get
< gf >=< qflt >=< 7J.lt >=< qjlt >=< (8g)f > as required.
0
We note that instead of using (4.13) we could have used Corollary
(4.7) with the other form of simple equivalence. As noted after Corollary (4.7), that result together with the above definition of A(f) shows: Lemma (4.16). If f
~
l' : X
7
Y, AU) = AU').
0
The next lemma will examine the functoriality of A.
Lemma (4.17) (i) If f = Id x , AU) = IdA(x) . (ii) If f : X   7 Y, 1': Y   7 Z, AU' f) = AU')AU). Proof. (i) If f = Id x , then 9 : X   7 W, in the pushout,
It = eo(X),
384
if
=
el(X) and for
we have ql : X x I + MIdx,g ~ Mg U (X x I) is the inclusion, i 2, into the second part of Mg U (X x I), hence
gId X = i2eO(X) ~ i2el (X) = qgig, however qg is simple so < g1d x > = < ig > = < 9 > as required. (ii) Suppose f, f' as specified and let 9 : X + W, then
gl'l : Z
+
M!'!,g
represents A(f'1) < 9 >, whilst
(gl)1' : Z
+
Mf' ,gJ
represents A(f')A(f) < 9 > . We therefore will need to compare Mf'I,g with M!',gJ relative to Z. Schematically Mf' ,gJ is represented by the diagram Y xI
eeeee W
X
Y
Y
Z
since MgJ is given by the composite pushout: JI Y ~ MI
eo(Y)
I
I
ql ==MI,g
IjgJ
YxI
and is thus represented by Y xI
eeee W X Y Y
An application of the Simple Relatively Principle will allow us to 'crush' Y x I using the structure morphism a(Y), and giving as a result a simple equivalence, relative to Z from MgJ to the object represented by eeee W
X
Y
Z
but the object given by the two right hand parts of this diagram, MI 385
and MI" is what was called M in Theorem (4.12) and there, with the necessary changes in notation, it was shown that M was equivalent relative to X and Z, to the mapping cylinder MI' f of the composite. Thus we have simple equivalences relative to Z, between MI' ,gf and MI'f,g i.e.
A(J' f) < 9 > = A(J')A(J) < 9 > as required.
0
So far we have shown that A is a homotopy functor from C to Sets . In fact each A(X) has a distinguished element , namely < Id x >, and we leave it as an Exercise to prove that if / : X .. Y,
A(J)< Id x
>=< Id y > .
Lemma (4.18). 1/ / : X .. Y is a homotopy equivalence, then A(J) is an isomorphism. Proof. This is immediate given Lemma (4.16).
0
We next turn to the monoid structure on A(X). Of course this will be given by +=
where in the pushout square
the morphism if,g is the diagonal if,g = qfif
= qgi g.
Lemma (4.19). The addition is well defined. Proof. We first note that as if,g = ig,J, + is commutative so we do not need to bother about order. Again we use Proposition (4.13) to reduce the question of welldefinition to when 9 is replaced by sg for s E L;. Suppose 9 : X .. Z and s : Z .. Z' . By Theorem (4.12) , 386
there is an object M given by the pushout
and a simple equivalence between Msg and M relative to X. As PI is simple , this gives , as in the proof of Lemma (4.15) , a diagram Mg
PI

M '
Msg
~I / X
Now use pushout along if to conclude that Mf,g and Mf,sg are linked by a chain of simple equivalences relative to X and hence that < i f,g > = < i f,sg > as required. 0
Lemma (4.20).
< 9 > + < Id x > = < 9 > . Proof. I7(X).
This should be clear, using the collapse of X x I VIa 0
Lemma (4.21). The addition is associative.
Proof. Let 9 : X
Y, h: X + Z, k: X + W, then to compare Mig,h,k and the triple mapping cylinder Mg ,h,k, we use Proposition (4.10). The triple diagonal, ig,h,k, then satisfies +
(< 9 > + < h » + < k > = < ig,h,k > but is independent of the order of formation of the sum, which comple0 tes the proof. Finally we have to verify that AU) is a monoid homomorphism.
387
Lemma (4.22). If f : X AU) ( < 9 >
t
+ + AU)
.
Proof. We note that AU)« where (ig ,h)f : Y
t
9
> + < h » = < (ig ,h)! >
Mig ,d, whilst
AU) < 9 > +AU) < h > =
< igf,hf >,
but that these two are the same is given by Theorem (4.11). This completes the proof of Theorem (4.14).
0
0
We now turn to E and start with a useful result which describes on A(X) in a slightly different way:
Lemma (4.23). If f : X
t
Y and 9 : X
t
+
Z, then
< f > + < 9 > = < 9f > where 9 : Y
t
W is any representative of AU) < 9
> in A(Y).
Proof. When we first introducedg! (after Corollary (4.3)) , we noted that g! = q!j! and
so
< f > + < 9 > = < g! f > but AU) < 9 > = < g! > and if 9 E < g! > then there is a simple s with 9 ~ sg!, hence 9f ~ sg! f, and we can conclude
< f > + < 9 > = < 9f > as stated.
0
Proposition (4.24). If f : X    t Y is a homotopy equivalence, then there is a 9 : X    t Z such that < f > + < 9 > = < I dx > . Proof. Let f' : Y    t X be homotopy inverse to f, then since AU) : A(X)    t A(Y) is a bijection, there is some < 9 > E A(X) with 388
AU) < 9
> = < f' >, but then
as required.
+ < 9 > = < l' f > = < I dx >
0
Remark. It is of note that we only really use that f : X ~ Y has a left homotopy inverse f', i.e. that f' f ~ I dx , since then AU')AU) is the identity on A(X), i.e. AU) is a split monomorphism, and we can take < 9 > = AU') < f' > . (Compare section 2.) Because of this A(X) may have invertible elements other than those inE(X). Recall that E(X) = {< 9 > E A(X) : 9 is a homotopy equivalence}. By the above Proposition (4.24), we find that each element of E(X) has an inverse within A(X). In fact, as we expect E(X) is an abelian group and E is a functor from C to the category of abelian groups.
Lemma (4.25). For any AU) < 9 > E E(Y).
f :X
~ Y, and
< 9 > E E(X),
Proof. As 9 is a homotopy equivalence, so is ig and hence ig is a trivial cofibration. It follows, that, since I is assumed to be generating, in the pushout square
ZI
XMI
i,j
jqJ
ql is also a trivial cofibration. As j 1 is a homotopy equivalence, we have that gl = qllt, is one as well, i.e. AU) < 9 > = < gl > E E(Y) as required. 0 Lemma (4.26). (i) If < 9
>, < h > E E(X), < 9 > + < h > E E(X).
(ii) If
< 9 > E E(X), there is an element < h > E E(X) with < 9 > + < h > = < I dx > . 389
Proof. (i) As 9 and h are homotopy equivalences, ig and ih are trivial cofibrations, hence so is ig,h, i.e. < ig,h > is in E(X) as required. (ii) We saw (Lemma (4.21») that each element in E( X) has an inverse in A(X). The inverse was found by using a homotopy inverse g' for 9 and then  < 9 > = A(g') < g' > . However g' is also a homotopy 0 equivalence, so A(g') < g' > E E(X) as required. If f : X    t Y then we will write E(f) for the restriction of A(f) : A(X)    t A(Y) to the submonoid E(X). With this notation we summarise these results in the theorem:
Theorem (4.27). The submonoids E(X) of the corresponding A(X) are abelian groups and together with the restrictions E(f) of the induced homomorphisms, define a homotopy functor,
E: C    t Ab, to the category of abelian groups.
0
Our next aim is to see what general tools we have that may aid calculation of elements of these groups and in some cases allow calculation of E(X) itself. Explicit calculations in the general case are made more complicated to start with by the necessity of replacing morphisms by the corresponding mapping cylinder inclusions. In fact, this is not always necessary, but special case analysis would have got in the way of the presentation and would have obscured certain symmetries in the constructions which are themselves useful. We examine one of these 'special cases' in more detail for later use.
Proposition (4.28). Let f : X    t Y be a cofibration and 9 : X    t Zany morphisms in C, then if X  f   Y
i,j
ji,
MgW
390
is a pushout,
AU) < 9 > = < 2g > . Proof. We factor factor f as as PIi p iI with PI pf simple, simple, and form form the diagram Proof.
X
~Mg
~
2g
~
Y
'W 2g
As both f and iiIf are are cofibrations, cofibrations, the Simple Simple Relatively Relatively Principle Principle apapplies pf is is simple. simple. Hence Hence plies showing showing that PI
AU) < 9 > = < qti I >  < PIqIit > as as required. required.
0
Corollary Corollary (4.29). (4.29). If I f f :: X and
Y and 9g :: X + +Z Z are are cofibrations cofibrations
+ dY
X_..:....f_y
gj Z
k w
is is aa pushout, then then
AU) < 9 > = < g' > . Proof. Proof. Applying Applying the the same same argument argument again, again, but this this time time to to gg,, does does 0 the the job.
391
Corollary (4.30). If f : X and
+
Y and 9 : X
+
Z are cofibrations
X_:.f__. Y
k
g\ Z
f'
·W
is a pushout, then
+ < 9 > = < g'f >
Proof. Lemma (4.23) gives that
< f > + < 9 > = < 9f > where < 9
> = AU) < 9 >, so the result follows from Corollary (4.29).
o These results imply that for cofibrations, the methods developed in section 2 still apply. This allows special results to be proved simply, for instance: Proposition (4.31). Iff: X then
+
Y, g: Y
+
Z are cofibrations,
AU) < 9 f > = AU) < f > + < 9 > . Proof. Both the statement and the proof are those of Proposition (2 .17), except that there no mention is made of cofibrations. 0
This formula enables us to prove the first significant result of a calculatory nature about A and E. The Simple Gluing Principle specifies roughly, that if a morphism is induced in a cube with pushouts in the opposite faces, and if its 'components' are simple then it itself is simple. The Gluing Theorem (1.7.1) says that if its components are homotopy equivalences then so is it . The Sum Theorem shows that in special cases, if its components are trivial cofibrations and it is also, there is a simple formula giving its simple equivalence class in terms of those of its components. We first need to set up some notation. 392
We have a cube
(4.32) 91 Y2   
h
,
X2
~ Y     ~ X I
with both the X and the Y faces pushouts and all morphisms cofibrations. We will write 90 for 91901 = 92902. The morphisms Ii determine classes < Ii > E A(Y;) and for our first result these will not be assumed to be in E(Y;). The formula will look at < I >, building it up from the A(9i) < Ii > . We therefore need to form, for instance, the pushout
Y1  II   X1 91
1
I
YYUX 1
II
and from the commutativity of the front square in the cube (4.32) we get
Y ~X=Y il) YUX 1 ~X for a unique hI' We note that if II is a trivial cofibration so is Similarly we can factor h : 1'2 + X 2 as:
i1'
Y2 .J::...." X 2 = Y2 i2) Y2 U Xo ~ X 2 . For our argument to work we will need to be in a situation where these induced morphisms are cofibrations.
393
Theorem (4.33). If in the cube (4.32) the left and right faces are push outs, all morphisms are cofibrations as is the induced morphism, h2 : Y2 U Xo + X 2 , then A(f) < f
> = A(f)(A(gI) < it > + A(g2) < h >  A(go) < fa ».
Before proving this we note: Theorem (4.34) (Sum Theorem). Given the conditions of (4.33) but in addition assume fa, it, h are trivial cofibrations, then
< f > = E(gI) < it > + E(g2) < h >  E(go) < fa > . Proof of (4.34) given (4.33). The Gluing Theorem (1.7.1) shows f is also a trivial cofibration, therefore A(f) is an isomorphism. As all < Ii > are in the corresponding E(Xi ), the result follows from (4.33).0 Proof of (4.33). First we note a lemma:
Lemma (4.35). In the square
we have hI is a cofibration and
< hI > = A(92) < h2 > . Proof. The square arises in the following diagram
394
x o  
~ yo, Yl 901
The front face face of the cube is a pushout by assumption. assumption. The right hand face face is one by construction, construction, and hence the composite composite of the two is a pushout. This composite composite is also the composite composite of the left hand and back faces, faces, but the left hand face face is a pushout by construction, and thus so is the back face. face. Finally we have the back rectangle rectangle is a pushout by assumption, so we can conclude that the square square in the lemma is one. one. As h2 assumed to be a cofibration, is2 a cofibration, ?j cofibration, we get hz is assurhed cofibration, and 92 hI is a cofibration hl cofibration and
Return to the main proof. We have hI hl is a cofibration cofibration and f = = hdI. hlf;, so by (4.31) (4.31)
A(id < f > = A(jl) < il > + < hI > . f 91 gl are cofibrations, cofibrations, so by (4.29) (4.29),, The morphisms II,
=
A(91) < II >
395
so hence
A(f) < f > = A(h1)A(il) < f > = A(f)(A(91) < II »
+ A(h192) < h2 >,
so we have to calculate A(h 192) < h2 > . However have (by (4.31))
h
= hd2' so we also
or
This, of course, implies that
A(h192) < h2 > = A(h192i2) < h >  A(h92i2) < i2 > but h192i2 = f92 from the diagram in (4.35), so
A(h 192) < h2 > = A(f)A(92) < h >  A(f)A(92) < i2 > . Finally i2 was given by the pushout
so < i2 > = A(902) < fo > and hence A(92) < i2 > = A(90) < fo > as required. 0 As a second application of Proposition (4.31), we will prove that the formula of that result remains valid even if f and 9 are not cofibrations , provided they are homotopy equivalences.
Proposition (4.36). If f : X topy equivalences, then
>
Y and 9 : Y
>
E(f) < 9f > = E(f) < f > + < 9 > . 396
Z are homo
Proof. The morphisms 9 : Y + Z and 9 f a( X) : X X [ + Z define a morphism H : M f + Z satisfying Hlt = 9 and Hi f = gf. We form the mapping cylinder MH and obtain a commutative diagram:
Z in which if induces k and It induces j . By the two bottom triangles, j and k are simple. All the morphisms, with the possible exception of j and k, are cofibrations. Using (4.32) A(iH)(A(i f ) < if >
+ < iH »
A(iHi f ) < if> +A(iH)< iH >
also A(iHif) = A(iHif) A(iHif) < i gf >
The right hand square similarly gives us A(iHlt) < iHlt
>
A(iHlt) < It > + A(iH) < iH > = A(iH)< iH > =
and A(iHjf) < iHlt
>
=
A(iHlt) < jig> A(iHlt) < ig >.
Putting these together A(iH)A(if) < igf
>= A(iHi f ) < if> +A(iHlt) < ig >. 397
Now iH is a homotopy equivalence, hence A(i H) is an isomorphism, but this means A( if) < igf
> = A(i f) < if> + A (it ) < ig > .
Operating on this with A(p f) gives the result, since etc. 0
=
As a final theme in the calculations related to E(X), we look at the abstract version of some results of Cockcroft and Moss (cf. Cohen (1973) , § 24). To state some of the results we will need some terminology. If A is an abelian group and G is a group, a right action of G on A is a function act : A x G
+
A
act(a,g) = a • g
such that (a. g) • g' = a • gg' and a • 1 = a, if 1 EGis the identity element. In this context a crossed homomorphism from G to A is a function t:G+A
such that t(gg') = t(g) • g'
+ t(g').
Now let X be an object of C, then we will denote by 1t(X) the group of homotopy classes of self homotopy equivalences of X. There is a naturally defined right group action of 1t(X) on E(X) defined as follows . If
•
. Lemma (4.37). With this action there is a crossed homomorphism TO :
1t(X)
+
398
E(X).
Proof. We have to define TO 70 and prove
To[gf] = TO([g]) 70[gfl = ro([gl) • [f] [fl + To[1] ro[fl but taking To[1] rO[f] =< < f >, (4.36).. = >, this is just just (4.36)
0
We will use TO TO to define a new right action of E ( X ) on the set E (X) H(X) E(X) (i.e. (i.e. forgetting forgetting the group structure): for H(X) E(X) for [g] [g] E E ( X ) and < E E (X)
* [g] == < f > •* [g] To[g]. >*[91 [gl + +.ro[gl. Sx, of simple homotopy types of objects, objects, Y, Y, Now consider the set, set, Sx, X.. homotopy equivalent to X Theorem (4.38) E(X) (4.38) (cf. (cf. Cockcroft Cockcroft and Moss) Moss).. The The orbits of of E (X) *action ofH(X) of X ( X ) are in oneone oneone correspondence with Sx. under the *action
E(X) Proof. Suppose < f >, >, < !' f' > E E ( X ) and that for some some g ] €EXH(X), ( X ) , wwee hhave a v e < f > ** 'yY = f l >>,, t then hen 'yY ==[ [g] = = < f > * 'Y = =
< f > • 'Y + TO ('Y) 1 E(gt < f > + < 9 > < fg > .
= < !' f ' > if there is a simple equivalence s and a square, Thus < f > * 'yY =
'
f X 'Y'
g[
s
X, Y
f
commuting up to homotopy. Thus if < < f > and < !' f 1 > are in the same orbit, their codomains Y e. Y and Y' have the same homotopy typ type. Conversely, Conversely, suppose given any s : Y Y + Y' Y',, where f :: X + +Y Y and!, + Y' are homotopy equivalences, and f ' :: X + equivalences, then if we write s for j for a homotopy inverse for for sand s and f for a homotopy inverse for f ,, defining 9 js!, X. g= =f sf ' gives gives a self homotopy equivalence equivalence of X.

399
Thus the oneone correspondence is set up by assigning to the orbit + Y. 0
< f > • 'H(X) , the simple homotopy type of Y where f : X Corollary (4.39). The mapping
'H(X) + E(X) is surjective if and only if the homotopy type of X coincides with its simple homotopy type. 70 :
Proof. If 70 is surjective, then as < I dx > * l' = 70(')'), there is only 0 one orbit of E(X) under the *action and conversely. Theorem (4.40). Given an object X of C and Y , a homotopically
equivalent object. Define Ex(Y) = {< f > If : X   Y is a homotopy equivalence} . Then if Y and Y' are both homotopically equivalent to X , the following are equivalent: (a) Ex(Y) n Ex(Y') "I 0 (b) Y and Y' have the same simple homotopy type. (c) Ex(Y) = Ex(Y') . 0 We leave the proof as an exercise. This gives a partition of E(X) such that the sets Ex(Y) are exactly the orbits of E(X) under the *:action. Now let vx = #(Sx), the cardinality of Sx,
Eo(X) = {< f >: E(f) = IdE(x)} ~ E(X) and set
'H(X, Y) = Fixing
f :X
U : X  Y If
+
is a homotopy equivalence}.
Y, the correspondence
'H(X)   'H(X , Y) g I   t fg defines a bijection. (This is part of the structure groupoid of homotopy classes of homotopy equivalences within C.) Since Ex(Y) is a quotient of 'H(X, Y) we get
#(Ex(Y)) ~ #('H(X, Y)) = #('H(X)) . 400
We We also also have have by by Theorem Theorem (4.40) (4.40)
#(E(X)) #(E(X)) ::;I lIx#(Ex(Y)) vx#(Ex(Y)) ::;I lIx#(H(X)). vx#(x(X)). Theorem Theorem (4.41). (4.41). In I n the the above above notation, notation,

lIx#(Eo(X)) #(E(X)) ::;I lIx#(H(X)). vx#(WX)). vx#(Eo(X)) ::;I #(E(X))
Proof. Proof. Suppose Suppose 90 go :: XX 7 +YY is is aa homotopy homotopy equivalence, equivalence, then then ifif
!f ::XX 7 XX isis such , such that that EE Eo(X) Eo(X), < 90! > = + < 90> E Ex(Y)
so
++Eo(X) Eo(X) ~cEx(Y) Ex(Y) and and #(Ex(Y)) #(Ex(Y)) ~2 #(Eo(X)). #(Eo(X)). We We conclude conclude
lIx#(Eo(X)) vx#(Eo(X)) ::;I#(E(X)). #(E(X)).
0
ITIfE(X) E (X)isisinfinite infinitebut butH(X) x ( X )isisfinite, finite,then thenclearly clearlylIx vx must mustbe beinfinite. infinite.
401
VII Injective Simple Homotopy Theories
The simple homotopy theories given by nonadditive cylinders  that is the topological, simplicial and groupoid examples  seem to be more complicated than those given by cylinders of injective type. For the injective and relative injective cylinders, one can readily give a complete description of the simple equivalences and can analyse E(X) in some cases. Because of this we will look at these injective style theories in some detail. 1. Simple Equivalences in Injective Simple Homotopy Theory
We suppose that we are given an additive monad, C = (C,j, JL), on an abelian category, A , and that we have
x
x I = X EB C(X)
etc., giving a cylinder structure as with the three types we dealt with in I.8 and II.4. Considering the formation of the mapping cylinder M f of a map f : X + Y in this setting, we have M f ~ Y EB C(X)
with j f the inclusion of Y. Thus it is feasible to expect that simple equivalences can be built up from inclusions as direct summands with some C(X) as complementary summand, isomorphisms and the corresponding projections. This guess is not far off the mark as we shall see. Its main fault lies in the following fact. Suppose the inclusion X~XEBJ{
402
is a trivial cofibration and
K EB C(M)
~
C(L)
then the obvious composite of inclusions
X ~ X EBK ~ X EBC(L) equals the inclusion til
X + X EB C(L).
Thus, since £" is 'probably simple' and £1 is also, £ must also be 'probably simple'. As in Milnor's paper (1966) where stably free projectives are considered, we therefore shall have to consider 'stably cofree injectives'. We recall that in the homotopy theory given by C = (C,j, J.L) the trivial cofibrations have the form X + X EB K
where K is such that j (K) is a split monomorphism. To avoid repeating this phrase we will call such a K a Cinjective object of A or relative Cinjective of A if it seems useful to emphasise that this is relative to the monad, C. We shall say that K is co/ree if there is a family {La : a E A} of objects and an isomorphism K ~
II
C(La) = EB C(La).
aEA
aEA
(Note: this is a wider and different use of cofree than that occurring in Hilton and 8tammbach (1971).)
Remark. If A is a finite set then K ~ C(EBLa), so cofree means isomorphic to some C(L). The other case with A arbitrary is only used so as to be able to verify axiom (8 4) in those cases in which it is not trivially true. We say K is stably co/ree if there are two cofree objects C 1 and C 2 and an isomorphism,
403
We next define a class of morphisms in A, which we will denote by S . S is the smallest class of morphisms satisfying the following: (1) For any object X and any stably cofree K the inclusion incx : X 
X EB K
is in S. (2) For any X, K as in (1) the projection prx : XEBKX is in S . (3) All isomorphisms are in S. (4) The composite of any two morphisms in S is again in S. Remarks. Since each morphism of the generating classes  inclusions, projections and isomorphisms  is a homotopy equivalence, it is clear that all morphisms in S must be homotopy equivalences. It will often be convenient in what follows to use a matrix notation (as found in Bass (1968) for instance). Suppose we have two direct sums Al EB A2 and BI EB B2 and a map () between them. Using the inclusions and projections of these sums we can define ()i, j : Ai Bj for i, j = 1, 2 by ()i j
= prBj() incA;.
It is then natural to think of () as being represented by a matrix () = (()11
()21 )
()I2
()22
For instance if A2 = B2 then for a: Al 
aEB I d
A2
=
.
BI we get
(ao Id0) A2
as alternative notations .
Proposition {1.1}. Any map in S can be written as a composite X
incx +
X EB K
~
+
Y EB L
pry +
Y
with K, L stably co/ree. Conversely any such composite is in S. 404
Proof. The final statement is obvious. For the first part, it suffices to prove that the composite of two such maps is again of the same form . Suppose we have a composite incx
X + X EEl with
J{,
() J{ +
=
pry
incy
111
prz
Y EEl L + Y + Y EEl M + Z EEl N + Z
=
L, M and N all stably cofree, 9 = prz Winey,
f
=pryOinex.
If we represent 0 by the matrix
o=
(00 00 11
21 ) 22
12
then
f = pry 0 inex
implies that 011
= f.
Similarly
Now consider the possible composite ~
incx
X + X EEl
J{
prz
EEl M + Z EEl N EEl L + Z .
(Note the abuse of notation in the use of inex and prz.) Both and N EEl L are stably cofree and if we can find q> such that
(i)
q> is an isomorphism
and
(ii)
gf = prz q> inex
then the proof will be complete. Let TL ,M : L EEl M   M EEl L be the twist isomorphism, so TL ,M=
0 ( Id L
then form
405
Id M 0
)
'
J{
EEl M
This is clearly an isomorphism and it is easily checked to have g gff in representation, as required. the top left corner of its matrix representation, 0
Proposition (1.2). (1.2). Suppose ff = = incx :: X S, then K must be stably cofree. cofibration in S,
+ t
X EB $ K is a trivial
Proof. ff has a factorisation as in (1.1) (1.1) X
incx t
X EB L
0 ~
X EB K EB M
prX$K t
X EB K.
Composition with pprx X EB K + X shows B is the identity on rx : X $K +X shows that 8 X . By well known results on direct sums (cf. (cf. Bass (1968), (1968), the factor X. Lemma 3.3, 3.3, p.18), p.18), this implies that
KEBM
~
L.
M and L are stably cofree, P, Q with M M EB P M cofree, hence there exist cofree P, $P and L L EB K EB P) EB $ Q cofree. cofree. It follows follows that K $ ((M M EB $ P) $ Q is cofree cofree and that K is thus stably cofree. 0 K cofree.

The proof proof of the following following proposition is left as an exercise. exercise.
Proposition (1.3). (1.3). IIff f :: X + Y Y is any retract in S, S , then Y E $ K with K K stably cofree and under this isomorphism, isomorphism, f is Y ~ X EB identifiable as incx. 0 Proposition (1.4). If (1.4). If Y~X~Z
is a pair of morphisms with v a retract in i n SS then, in i n the pushout pair of pushout diagram X __v__ Z
"]y    , YI"' v'
v'is v' is also a retract in i n S. S.
406
incx
Proof. v is of the form X ~ X EB K with K stably cofree, hence VI
is of the form Y
incy ~
Y EB K and so
VI
is a retract in S.
0
One of the main difficulties in handling maps from S is that although each has at least one decomposition X~XEBK~YEBL~Y
with K and L stably cofree, it also has numerous other similar decompositions with the corresponding K, L non stably cofree  just add an arbitrary Cinjective to both K and L and amend the isomorphism accordingly, it is extremely unlikely that the resulting injectives will be stably cofree. To get round this difficulty we need a canonical way of achieving such a factorisation. We have already considered one way of getting factorisations in our earlier section I.8 on the general additive homotopy theory: Taking a map f:X~Y
we have (up to isomorphism) if: X ~ YEBC(X)
with if(x) = (f(x),j(x)). If f is a homotopy equivalence then if is a trivial cofibration so there is some Cinjective K and an isomorphism
X EB K : I Y EB C(X) such that
f
= pfi f = pryBincx·
Such a decomposition will be called a canonical decomposition. It is natural to hope that since C(X) is cofree, K will be stably cofree if (and only if) f is in S. This is, in fact, the case as the next theorem shows.
407
Theorem (1.5). A morphism f zs in S if and only if in any canonical decomposition, ~~
X
t
X E9 K
8
~
Y E9 C(X)
FY
Y,
t
of f, K is stably cofree . Proof. The result in one direction is trivial. So we assume and shall prove that K must be stably cofree. Suppose f has a decomposition
f is in S
W
XtXE9J~YE9LtY
with J, L stably cofree. Then adding C(X) to both central terms we obtain
X
t
X E9 .J E9 C(X)
WEf)Id t
Y E9 L E9 C(X)
t
Y.
Similarly from a canonical decomposition, one can obtain
X
t
X E9 K E9 L
8Ef)Id t
Y E9 C(X) E9 L
t
Y.
There is thus an isomorphism :
X E9 J E9 C(X)
t
X E9 K E9 L
given by
however it is easily checked that if 0 1 is given by the matrix
then incx(x) = af(x) which is not necessarily x, so does not necessarily have an isomorphism in its top left corner and we cannot as yet apply the result that was quoted earlier (from Bass) to obtain an isomorphism between K E9 Land J E9 C(X).
408
This difficulty is easily surmounted. surmounted. If ()0 is given by
then
a! + {3(}12 =
Id
so so an application of the automorphism
0 0) 0
Id (
o
Id 0
(}12
Id
to X X EB $ .J 7 EB $ C(X) C ( X ) before application of cI> @ gives gives an isomorphism cI>' @' which in its dx in the top left corner poits matrix representation has IIdx sition. sition. Thus, Thus, using the lemma in Bass (1968) (1968) once once more, more, we find find an isomorphism isomorphism
J EB C(X) 95. K EB L,
J,Land L and C(X) C ( X ) are are stably cofree, cofree, we conclude conclude that K K is is also also as J, and as stably cofree. 0 cofree. between a canonical decomposition decomposition of We next look at relationship bet~een composite and the the composite of two two canonical decompositions. a composite t Z Suppose we have !f : X X    t Y, Y, g: g :Y Y + Z with canonical decomdecompositions positions

incx
X
+
Y
+
(J
X EB J
:* Y EB C (X)
Y EB K
:* Z EB C (Y)
pry
+
Y
and incy
lit
prz
+
Z.
Using the the same same method as as before before we we can form form aa composite composite decomposidecomposiUsing tion
X
incx
+
X EB J EB K
7~
Z EB C(Y) EB C(X)
with 409
prz
+
Z
Now look at the canonical decomposition of ggf, f , say X
incx
+
~
X EEl L ~ Z EEl C(X)
prz
+
Z.
Working somewhat as in the proof of (1.5), (1.5), we form a new decomposition .
X ~ X EEl L EEl C(Y)
~E9Idc(y)
~
prz
) Z EEl C(X) EEl C(Y)
+
Z
and hence an isomorphism
( : X EEl L EEl C(Y) ~ X EEl J EEl K,
being the composite
Amending (( as as before, we can invoke invoke Lemma 3.3 3.3 of Bass (1968) (1968) used before, to obtain an isomorphism L EEl C(Y) ~ .J EEl K.
We We have now only only a small small amount of work left to to prove: prove: (1.6). If f, f , 9g are are such such that that gf g f is is defined defined and and is is in SS Proposition (1.6). then ff E E SS if and and only only if 9g E E S. S.
Proof. If If gf gf E E S, S, then then in in the the canonical decomposition, decomposition, given given above, Proof. LL is is stably stably cofree. cofree. Hence Hence if either either JJ or or K K isis stably stably cofree, cofree, so so is is the the (*). other by (*). 0

we mention that if {fa :: XO' X, ~ yO'} Y,) isis aa family family of mormorFinally we phisms such that phisms in in SS such
is defined defined then then decompositions decompositions of of the the individual individual fa fa together together give give aa dedeis 410
composition of f with the relevant objects stably cofree. (It was for this reason that we have given a somewhat strange definition of cofree.)
Theorem (1. 7). The class S is precisely the class of simple equivalences in A (relative to the cylinder defined by (G,j , f..L)). Proof. Since el(X) is always in S and the generating processes stay always within S by the above results we have that every simple equivalence is in S. On the other hand given any stably cofree K and suppose K EB G(L) ~ G(M) then clearly ~~
X~XEBK
and
Fx
XEBK~X
are simple  if one requires the use of EBG(La) etc. in place of G(L), it is necessary to use (S 4) in order to prove that incx : X + X EB K etc. are simple, however this presents no real difficulty. For instance,
is the simple equivalence jo : X EB K  t Mo with Mo the mapping cylinder of the zero map L ~ X EB K. Thus we must have that all 0 maps in S are simple and so the result follows. We thus have a complete description of the simple equivalences in this generic case of an additive injective type cylinder structure. In the next section we use this description to calculate E(X) in this case.
2. The Group E(X) We now have all the information necessary to start to calculate the groups E(X) . In this section we prove a general result: Suppose f : X + Y and 9 : X  t Z are two homotopy equivalences. In order to determine if < f > = < 9 >, we need only look at the case when f and 9 are trivial cofibrations (cf. (VI.4.2)), hence we shall assume
f:X+XEBK g:X+XEBL are the two inclusions of X as direct summands. 411
Now suppose
/XalK X
s
~ XfJJL
is a commutative diagram with s simple. (By (VI.4.13) it is sufficient to consider this case as it generates the other equivalence relation involving homotopy commutative diagrams.) Since s is simple, it has the form X fJJ K
()
t
X fJJ K fJJ M ~ X fJJ L fJJ N
t
X fJJ L
with M, N stably cofree. Since sf = g we must have that
s(x,O) = (x,O) or, in other words, that e must restrict to the identity map on the direct summand X. Thus we can easily see that < f > = < g > if and only if there exist stably cofree objects M and N satisfying K fJJ M
~
L fJJ N.
This observation is the essence of the proof of the next theorem. Before stating it, we need to define a group analogous to the reduced projective class group of algebraic K theory. Initially we make no restriction on the size of injectives concerned, but will see later that for nontrivial results such a restriction is often necessary. Let A be the basic abelian category in which we are working and as usual C = (e,j, J.L), the additive monad used to define the injective type cylinder. Let Io(A; C) denote the abelian group with generators the isomorphism classes, [K], of relative Cinjectives, K, in A and with relations
(1) (2)
[K] + [M] = [K fJJ M] [K] = 0, if K is cofree.
Io(A;C) will be called the relative Cinjective class group of A.
412
Remark. The analogy with the reduced projective class group should be obvious. To define that group, one uses finitely generated projectives and divides out by the free modules. We next prove a connection between E(X) and Io(AiC).
Theorem (2.1). If E(X) is the group of simple homotopy types of X then there is an isomorphism (J :
E(X)
?
Io(AiC)
for all X. Moreover the isomorphism is independent of
x.
Proof. We shall need some more information on the structure of Io(AiC) to start with. CLAIM
1. If K is stably cofree, [K] = O.
CLAIM 2. If K and L are related by K E8 M stably cofree, then [K] = [L].
~
L E8 N with M , N
Clearly the first claim implies the second as
[K] = [K]
+ [M]
= [K E8 M] = [L EB N] = [L]
+ [N]
= [L]
and the first is equally easy since there are cofree M and N with K E8 M = N hence [K] = [K EB M] = o. Next we define (J. If < j > is representable by the trivial cofibration j : X ? X EB K then (J < j > = [K]. If < j > = < g > then by the comments preceding the definition of Io(AiC), and also claim 2,
(J<j>=(J. If [K] is any element of Io(AiC), (J
<X
incx +
X EB K > = [K],
so (J is bijective with inverse
[K]I+ < X
?
X E8 K > .
If < j >, < 9 > E E(X) then by (VI.2.10), we need only work out the pushout
413
X..:....fXEElK
k
9
and note
< f > + < 9 > = < g' f > . Thus 0(
< f > + < 9 >) =
[K EEl L] =
0
< f > + 0 < 9 > .
Again the inverse
[K]
f+
<X
t
X EEl K
preserves addition. That 0 does not depend on X is clear.
> 0
This last point can be illustrated in another way. Suppose
f :X is any map and 9 : X
t
t
Y
X EEl K then f* < 9 > = < gf
>
where gf = qf it with qf as in the pushout,
thus
ig : X
t
X EEl K EEl C(X),
up to isomorphism, is the inclusion of X, so qf : Y EEl C(X)
t
Y EEl C(X) EEl K EEl C(X)
is also, up to an automorphism of its codomain, the inclusion of Y EEl C(X). We thus have gf is the inclusion of Y into 414
Y EB C(X) EB C(X) EB K and incy
< gl > = < Y ~ Y EB K > . The induced morphism,
<X
t
f*, is thus the isomorphism
XEBK
>~
and is independent of f. The functor E is thus essentially a constant functor. We finish this subsection with a slightly more precise construction and description of Io(A;C). This will enable us to decide when two relative Cinjectives determine the same element of Io(A;C). Suppose that A has only setwise many isomorphism classes of relative Cinjectives and form the free abelian group, F, generated by this set. Let R be the subgroup generated by the elements (a) < K EB M in A
>  < K >  < M >, K, M relative Cinjectives
(b) < K >, if K is cofree. Then Io(A;C)= FIR. (Here < K > denotes the isomorphism class determined by the relative Cinjective K and [K] = < K > +R.)
Theorem (2.2). Given two relative Cinjectives K and M zn A, [K] = [M] if and only if there is a relative Cinjective T and cofree U, V with
Proof. Clearly given such an isomorphism [K] = [M]. Conversely if [K] = [M] then we have < K >  < M > E R so there exist S;, S;', TJ, TJ' relatively Cinjective and cofree Uk, VI such that
< K >  < M >=
 [L:( < Tj EB Tj' > 
[~( < S~ EB S:' > 
< Tj >  < Tj'

< S:'
Uk > 
»]
L: < VI > . 1
Gathering terms with like signs on one side gives +E+E+E+E i
j
j
I
=< M > + E < s~ > + E < Sr > + E < Tj ffi Tj' > + E < Uk > . i
i
j
k
Writing
we note, that as F is free , the isomorphism classes of K , S~ ffi S~', TJ and Tj' must be a permutation of those of M , S;, S~', and TJ ffi TJ' so there is an isomorphism
as required.
0
Corollary (2.3). Given a relative Cinjective K, [K] = 0 in Io(A;C) if and only if there exist a relative Cinjective T and cofree objects U, V with
Remark. If one has finitely generated modules then cancellation of T will often be possible. In such a case the condition ' [K] = 0' is exactly 'K is stably cofree'. 3. Examples In this section we handle four examples in which Io(A ; C) = O. These examples have been chosen as being fairly elementary, i.e. they need only algebraic results which are relatively well known or accessible in the 'standard literature'.
(a) Localisation Theoretic Homotopy Suppose L = (L,7jJ,/L) is a localising monad (d. IlL5 , Relative Injective Type Theories, Example (c)). We saw earlier that in this case each L injective, K , was an image of L (i.e. K was always isomorphic to L(K)) hence all Linjectives are cofree and Io(R Mod; L) = O.
416
(b) I dempotentM onadic Homotopy This is essentially as in (a) above. If C = (C,j,f.L) is an additive monad on A, then C is idempotent if f.L is an isomorphism. This occurs in localisations and also in those rare cases where the injective envelope is functorial. Suppose K is Cinjective and C is an idempotent monad, we show that K ~ C(K) as follows. There is some K' with KEEl K' ~ C(I 1) and Go, G1 , G2
:
1) >
E
and natural transformations 1/0 : Fo
>
F1 , 1/1 : Fl
>
F2,
1/~:
Go
>
G1 , 1/; : G1 > G2
then several composite natural transformations can be formed
Fo
Go
I1/~
11/0
Fl
C
1)
G1
E.
11/;
11/1
G2
F2
Method 1: Compose vertically to get 1/11/0 : Fo and then horizontally to get
>
F2 and
1/;1/~
: Go
>
G2,
(1/; 1/~) . (1/11/0) . Method 2: Compose horizontally first to get 1/~ . 1/0 : GoFo 1/; ·1/1 : G 1 F1 > G 2 F2 , and then horizontally to get
>
G1 F1 and
(1/; . 1/1)( 1/~ . 1/0) . The Godement interchange law states that these two composites are equal: (1/; 1/~) . (1/11/0) = (1/; . 1/1)( 1/~ . 1/0). Remarks: The rule can be thought of as being a rule of 2dimensional algebra. It occurs in an abstract form in the definitions of 2category theory (Catenriched category theory) but also in work on composing squares in the double groupoids related to the crossed module techniques developed by Brown and Higgins. We use it and give a further discussion of it in Chapter IV, Proposition (1.10). DUAL OR OPPOSITE CATEGORY
Given any category C, the dual or opposite category COP is defined to have:
= Ob(C) COP(A, B) = C(B, A).
 the same objects as C, Ob(COP)  for A, B objects in COP,
It will be convenient to write j"P in COP for the morphism that is
f in C.
 composition induced from that in C : j"P : A > B, gOP: B > C in are the same as f : B > A g : C > B in C so compose to give fg .
433
cop
DUALITY PRINCIPLE (not a formal statement)
If a concept, definition or result involves purely categorical conditions and methods, then there is a dual concept obtained by reversing all arrows in diagrams, and with valid dual results. Even the proofs dualise! Example: Pushout square (cf.) of
B...'!A~C IS
a square
A __a_ _'B (3
I"
C'D 8
,a
such that (i) = 8(3. (ii) If given any other commutative square
A
_=a~_.
B
11
pi
C   ...·D' 8' involving (a, (3) there is a unique morphism ~ : D and ~8 = 8'. This dualises to: Pullback square (cf.) of
is a square
A ...,_..::a_ _ (3
B
I"
CD 8
a,
such that (i) = (38. (ii) If given any other commutative square
434
t
D' such that
~,
="
A
_._:::.Oi_ _
~I
B
11
C ··D' 8' involving (Oi, (3) there is a unique morphism ~ : D' and 8~ = 8'.
+
D such that
,~
="
Product (cf.) and coproduct (cf.) are dual. Monomorphism (cf.) and epimorphism (cf.) are dual. For instance a monomorphism j"P : A + B in c op will 'really' be an epimorphism f : B + A in C. As (Cop)O P = C which is the primary case and which the dual often depends on 'taste' . Remarks: The reader who has not met categorical duality before should not worry over much about it. You get used to it. The treatment we give in section II.3 is designed with the 'nondualised' reader in mind. COMMUTATIVE DIAGRAM
We do not need this in great formality so content ourselves with some simple examples and a glance at the general idea. We work in an arbitrary category,
C. A triangular diagram
in C is commutative if (3Oi
= ,.
A square diagram
A _=Oi::......___... B
I~
11
C'D 8 in C is commutative if (3Oi = 8,. In general a diagram in C is a collection of objects and morphisms linking them to form a directed graph. The diagram is commutative if given any two objects A, B in the diagram and two paths along morphisms starting at A and ending at B the composites of these two paths are equal. In other
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words however you go from A to B following 'arrows' and composing as you go the answer will always be the same. ISOMORPHISM AND EQUIVALENCE OF CATEGORIES A functor F : C + D is called an isomorphism of categories if there is a functor G : D + C such that the composite functors GF : C + C and FG : D + D are the identity functors on the respective categories. In practice isomorphism of categories is less useful than equivalence of categories: a functor F : C + D is called an equivalence of categories if there is a functor G : D + C and natural isomorphisms
." : GF
+
Ide , .,,': FG
+
Id v .
MONOMORPHISM, EPIMORPHISM, ISOMORPHISM In group theory, the term 'monomorphism' is synonymous with ' 1  l' or 'has trivial kernel'. Both these ideas use elements. In a general category the objects may not have 'elements' and the concept looks slightly different . A morphism 0' : A + B in a category C is a monomorphism if, given any two morphisms j3, j3' : C + A such that 0'j3 = O'j3' then it must be the case that j3 = j3' . Note that in the category of sets, taking C to be a singleton set (that is with exactly one element) gives a neat way of saying that 0' is 1  1. The notion of epimorphism is dual : A morphism 0' : A + B in a category C is an epimorphism if, given any two morphisms j3 , j3' : B + C such that j30' = j3'O', then it must be the case that j3 = j3'. A morphism 0' : A + B in a category C is an isomorphism if there is a morphism j3 : B + A such that j30' = IdA and 0'j3 = I dB . If 0' : A + B is an isomorphism, then the corresponding j3 is completely determined by 0' . It is usually written 0'1 and is called the inverse of 0'. The term ' invertible morphism' is also used as a synonym for isomorphism . Weaker forms of inverse are sometimes used, namely left (right) inverse. These are mentioned under SPLIT MONOMORPHISM, and SPLIT EPIMORPHISM . Warning: Even in quite usual categories such as that of rings, there are epimorphisms that are not surjective, for instance the inclusion of the integers , 7L. , into the rational numbers, !Il, is both a monomorphism and an epimorphism, but of course is not surjective and moreover is not an isomorphism . Thus 'monic' plus 'epic' does not imply 'iso'. This is discussed more fully in standard texts on category theory.
SPLIT MONOMORPHISM, SPLIT EPIMORPHISM
A morphism 0' : A   t B in a category C is a split monomorphism if there is a morphism j3 : B + A such that j30' = IdA. In this case j3 is called a splitting of 0' . A morphism 0' : A + B in a category C is a split epimorphism if there is a morphism j3 : B + A such that 0'j3 = I dB .
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involving (a, (1), there is a morphism ~ : D ~5
+
D' such that
~,
="
and
= 5'.
1£ the morphism ~ is unique with this property, then the weak pushout is a pushout. If in a category C, any pair (a, (1) of morphisms with common domain has a (weak) pushout, we say that C has (weak) pushouts. Suppose F : C + 'D is a functor which is such that if A
a
pi C
5
'B
l'
'D
is a pushout square in C, then FA
Fa
IF1
FPI FC
, FB
F5
FD
is a pushout square in 'D, then we say that F preserves pushouts. A similar sense is attached to 'F preserves weak pushouts'. Remarks: (i) The dual notion is called a pullback (resp. weak pullback) . A pullback in C is the same as a pushout in coP. (ii) The uniqueness in the above is part of the universal property of pushouts. Other constructions involving universal properties include products, coproducts and pullbacks, and more generally limits, colimits and adjoints. The uniqueness clause implies that the construction is unique up to isomorphism, so, for instance, if two different constructions give two pushouts of the same 'fork' then the resulting objects will be isomorphic by an isomorphism compatible with the other morphisms in the two pushout squares. COPRODUCTS
Given a pair of objects A , B, in a category C, a coproduct of A and B is an object of C, denoted AU B together with morphisms iA: A
+
AU B
iB : B
+
AU B
with the following universal property: given any object C of C and morphisms f : A+C
g : B+C
there is a unique morphism h : Au B + C such that hi A = f , hi B = g. (The notation h = f + 9 will often be used.) The universal property easily implies that any two coproducts of A and B 438
are isomorphic in such a way as to be compatible with the 'inclusions'. It is thus usual to refer to 'the coproduct' of A and B in most situations. If for any pair of objects, A, B, of C, their coproduct exists in C then we say C has pairwise coproducts. If C has pairwise coproducts, then given any nonempty finite family {A l ,' .. , An} of objects of C, one can form a coproduct of the family in a fairly obvious sense extending the above. This notion can also be extended to infinite families but we have used this very rarely in the book. If the family of objects is empty, a coproduct of the family is an initial object
(d.). If A = B in the above then there is defined an important morphism called the co diagonal
\1 A : A U A + A. This is given by the universal property taking f = 9 = IdA : A + A. To interpret what it does consider the case of Sets in which AuB is the disjoint union of A and B . The codiagonal \1 A 'folds' the two copies of A together mapping them both onto A. The defining equations for \1 A are
\1 Ail
= IdA,
\1 Ai2
= IdA
where i l and i2 are 'inclusions' into the two 'cofactors'. A functor F : C + V preserves coproducts if given any A, B in C such that AuB exists then F(AUB) is the coproduct in Vof F(A) and F(B) so that iF(A) = F(iA) and iF(B) = F(i B). In some situations there may seem to be a natural choice of coproduct and it is tempting to pick such a coproduct as 'the' coproduct. The above statement must then be interpreted carefully as it only implies F(A U B) ~ F(A) U F(B) and equality is not implied. This can lead to difficulties and usually it is better to work with a general coproduct described using the universal property rather than specifying a chosen one. Remark: If a category C has (finite) coproducts (including that of the empty family) and pushouts, then it has all (finite) colimits (d.) . If C has an initial object (d.) and has pushouts, then C has finite colimits. PRODUCTS
The dual notion of coproduct is product. This gives for a pair of objects A, B an object denoted An B or A x B depending on the context, and projection maps PA : A x B + A, PB : A x B + B. The precise statement of the universal property is left to you . The extension from a product of a pair of objects to that of a family is routine as is the dualisation of comments made above. A prop duct of the empty family is a terminal object (d.). Dual to the codiagonal construction given above, one has the diagonal morphism
439
~A :
For Sets,
~A(a)
A
+
A x A.
= (a, a) and so is exactly a diagonal.
A useful fact: If C is a category and {Ai : i E A} is some family of objects whose coproduct Ai exists in C, then for any object B
U
i EA
C( UAi, B) ~ iEA
II C(Ai' B), i EA
the product of the 'homsets'. This is just another way of writing the universal property. The dual situation leads to an isomorphism C(A,n B i ) ~ i EA
where
II C(A, Bi), iEA
nBi denotes the product in C of a family {Bi : i E I} of objects of iEA
C. Here and elsewhere we have used n for the product in a general category whilst IT is reserved for use in Sets and other similar categories such as that of abelian groups . We have also tried to use the more suggestive notation E!) for the coproduct (direct sum) in abelian categories whilst U will be used in the general case. LIMITS AND COLIMITS
Suppose C is a category and 1) is a small category so we can form the category CD of diagrams of type 1) in C. There is for any object C in C an obvious constant diagram of type 1) with C everywhere and all morphisms being the identity on C. More precisely we have a functor ke : 1) + C with ke(d) = C for all din Ob(1)) and if (J : d + d', kc((J) = Ide . This gives an object ke in CD and on varying C a functor
k :C
+
CD
Suppose now that F : 1) + C is any given diagram. A limit of F is an object C in C and a natural transformation 1) : ke + F such that given any other similar set of data, (C', 1)' : kel + F), there is a unique morphism Cl' : C' + C with 1)k", = 1)'. This interprets as saying that C gives the 'best approximation' to F (from the left) by a constant diagram . The standard type of universality argument shows that if (C, 1)) and (C' , 1)') are both limits for F then C ~ C' in such a way that 1) and 1)' correspond. We write C = lim F. If lim F exists for all F in CD and for all small categories 1), then C, is said to have all limits (or to be complete) . If lim F exists for all F in CD for all finite categories 1), then C has finite limits , it is finitely complete. A very useful result is that if C has a terminal object (d.) and has pullbacks , then C has finite limits.
440
If lim F exists for all F in CV then
k : C + CV has a right adjoint (d.) since
CV(kc, F) ~C(C, limF) . Dually if k has a left adjoint (d.), then C has all Vindexed colimits. We leave the reader to dualise the other terms and definitions above. We write colim F for the colimit of F. A functor F : C + C' is said to be right exact if it preserves finite colimits , that is if X : V + C is a diagram with V finite, then there is a natural isomorphism
F (colim X)
~
colim F X.
If F preserves the initial object and pushouts then it is right exact, and, of course, conversely. The dual notion is left exact. EXPLICIT CONSTRUCTION OF LIMITS AND COLIMITS
The properties of the limit of a diagram really are all given by the universal property, but it is sometimes of use to have an explicit description of how the limits are 'built' . The typical case is the construction of a limit for a set valued functor . Let F : V + Sets be a diagram in Sets . Form the product
P(F)
=
II{F(d) : d E Ob(V)} .
If x E P(F), write Xd for its component or coordinate in F(d) . There is a function Pd : P(F) + F(d) which given x returns the value Xd, Pd(X) = Xd . Now consider L(F) ~ P(F) determined by the condition x E L(F) if and only if given a : d + d' in V (and thus giving F(a) : F(d) + F(d') in Sets) , F(a)(xd) = Xd' , This L(F) is the object part of limF, the projection maps from L(F) to F( d) are the restrictions of the Pd projection maps of
P(F) . Thus a limit can be constructed as a subset of the product of all the objects in the diagram. Dually a colimit can be constructed as a quotient of the coproduct of all the objects in the diagram. The quotient is given by an equivalence relation . INITIAL AND TERMINAL OBJECTS
An object t in a category C is a terminal object if for each object C of C, there is a single unique morphism from C to t . An object i in a category C is an initial object if for each object C of C, there is a single unique morphism from i to C.
Remarks: (i) In Sets, any singleton set is a terminal object and the empty 441
set is an initial object . (ii) The usual universal argument shows that terminal objects where they exist are unique up to isomorphism. Similarly for initial objects. (iii) The term final object is often used as an alternative for 'terminal object' . (iv) Let i be an initial object in C. Then, if AU B together with iA : A + Au B , iB: B + Au B
is a coproduct of A and B, the square Z
·B
I
A
Ii"
ZA
'AuB
is a pushout , and vice versa. Dually, a terminal object in a category allows one to describe products as pullbacks. ADJOINT FUNCTORS Suppose F : C + V and G : V + C are two functors then F is said to be left adjoint to G (and G is said to be right adjoint to F) if there is a natural isomorphism
BC,D : V(FC,D) ~ C(C,GD) (that is, natural in both C and D). For example such an adjoint pair arises if G is a functor that forgets structure such as the forgetful functor from the category of groups to that of sets that forgets the structure of a group leaving just its underlying set. This forgetful functor has a left adjoint given by the functor that, given a set, returns the free group on that set. Such a freeforget adjoint pair is typical of the examples that we will need . An adjoint pair (F, G) can be specified in several ways . In particular setting D = FC, we get a special morphism B'c,FC(I d FC ) : C + GFC which gives a natural transformation I de + GF called the unit of the adjunction. Dually taking C = GD and using BGb D(Id aD ) : FGD + D gives a natural transformation FG + I dv called c~unit of the adjunction . These are interrelated by triangular diagrams (see for instance MacLane (1971) for a full discussion .). These diagrams are related to those in the definition of a monad (d.) and any adjoint pair generates both a monad and a comonad. As mentioned under LIMITS AND COLIMITS, when it exists the Vindexed limit functor lim: CV + C is right adjoint to the constant diagram functor k : C + CV , similarly colim : CV + C when it exists is left adjoint to k. Another connection between limits/colimits and adjoints is the easily proved result that a left adjoint preserves any colimits that exist whilst a right adjoint always preserves any limits . 442
MONAD
A monad on a category C consists of a functor T : C + C (an endofunctor) together with natural transformations TJ : I de + T called the unit of the monad and fL : T2 + T called the multiplication. These are required to satisfy two laws, usually expressed by the commutativity of two diagrams:  the associativity of fL :
commutes,  the left and right identity laws for the unit :
commutes. Remarks: (i) Any adjoint pair (F, G) with unit TJ : I de + GF and co unit
c: : FG
+
I dv defines a monad on C with T = GF,
Ide
TJ:
+
T, fL = Gc:F : T2
+
T.
(ii) A monad on cop is a comonad on C and consists of an endofunctor T : C + C, counit , c: : T + Id and comultiplication T + T2 satisfying the dual diagrams . ADDITIVE CATEGORY
In some categories such as that, Ab, of abelian groups, each set of morphisms, Ab(A , B) , has a natural structure of an abelian group. This happens in such a way that the composition Ab(A, B) x Ab(B, C) + Ab(A, C) is linear in both variables
(J + g) 0 h = f 0 h + 9 0 h a (b + c) = a b + a c. 0
0
0
(This example is enriched over Ab in much the same way as in section lIlA where we consider categories enriched over the category of simplicial sets.) Any category A in which each A(A, B) is an abelian group and each composition is bilinear (as above) is an Abcategory or preadditive category.
443
(Any ring is an Abcategory with one object.) A preadditive category A is an additive category if A has a zero object and direct sums. A zero object in A is an object 0 such that for each object A of A there is a unique morphism 0 + A and a unique morphism A + 0 so 0 is both an initial and a terminal object . The composite A + 0 + B in A(A, B) gives the zero for the abelian group structure on A(A, B). It is easily shown that for any A, AU 0 and A x 0 exist and are isomorphic to A. If A U B exists in A with A ~ Au B ~ B then there is also a morphism A U B + A U 0 ~ A which is the identity on the A part and the zero morphism on the B part, similarly for B . This can be used to show Au B is also the product of A and B . In this case Au B is written A Ell B and is called the direct sum of A and B . ADDITIVE FUNCTOR If A and B are additive categories, then a functor F : A if for each, A, A' in A, the function
FA,A': A(A, A')
+
+
B is additive
B(FA,FA')
is a homomorphism of abelian groups . It is then easy to show that for such an F, F(O) = 0 and F(A Ell A') ~ F(A) Ell F(A') so F preserves direct sums . DIRECT SUM As mentioned briefly under ADDITIVE CATEGORY, if A is an additive category, the direct sum of two objects A and B is an object A Ell B together with morphisms
iA : A iB : B
+ +
A Ell B A Ell B
making (A Ell B, i A, iB) a coproduct, and morphisms
PA : A Ell B PB : A Ell B
+ +
A B
making (A Ell B ,PA,PB) a product. These maps are related by equations
PAi A = IdA PAi B = 0
PBi B = IdB PBiA = 0
(so iA is a split monomorphism)
and
iAPA
+ iBPB = IdA(JJB .
In situations where notation is in danger of being overloaded we write incA ,
projA etc. for i A, PA. It is worth noting that, as mentioned under ADDITIVE CATEGORY the fact that (A Ell B , iA, iB) is a coproduct together with the existence of a' zero object implies the existence of the 'projections' PA , PB making (A Ell B,PA,PB) a product and satisfying the equations. Dually a product structure will give you the coproduct structure. 444
DIRECT SUMMANDS AND SPLIT MONOMORPHISMS IN AN ADDITIVE SETTING If C ~ A EB B as above then A is a direct summand of C. If i : A > C is a morphism such that there is a right inverse r : C > A, ri = IdA, and kernels (d.) exist in A, then C ~ A EB B for some object Band B is called the complementary summand of the split monomorphism . (If it is not known if all kernels exist then B may still exist, of course and the term will still be used.) KERNELS AND COKERNELS Suppose C is a category with a zero object 0, so 0 is both a terminal and an initial object for C, then in any C(A, B) there is a distinguished element namely the zero map
A
>
0 > B.
(We looked at this in ADDITIVE CATEGORY, but the additivity is not strictly needed there.) If I : A  > B is a morphism in C, then a kernel of I is the limit of the diagram
I
A===:: B
o
'
thus a kernel of I is an object K er I and a morphism k : K er I > A with the two properties (i) Ik = 0 and (ii) if I : L > A is any morphism so that II = 0, then there is a unique morphism I' : L > K er I such that kl' = I. As an example, if C = Ab, a kernel for I : A > B is the inclusion of the usual kernel Ker I = {a : I(a) = O}. If I : L > A is any morphism satisfying II = 0 then the unique morphism I' just sends x to l(x), but with I( x) considered as an element of K er I not merely of A . Dually given I, a cokernel of I is a coli mit of the above diagram so it is an object Coker I and a morphism c : B > Coker I satisfying ci = 0 and if d : B > D satisfies dl = 0 then there is a unique d' : Coker I > D so that die = d. Taking C = Ab, Coker 1= B/1mI where 1mI
= {f(a)
: a E A}.
Both kernels and cokernels are of course, unique up to isomorphism. ABELIAN CATEGORIES An abelian category A is an additive category with kernels and cokernels so that every monomorphism is a kernel and every epimorphism is a cokernel. The main features to note are: (i) each 'homset' A(A, B) is an abelian group with composition a 'bilinear' morphism; (ii) A has a zero object; (iii) A has pairwise products that are also coproducts;
445
(iv) every morphism has kernel and a cokernelj (v) every monomorphism is a kernel, every epimorphism is a cokernel. INJECTIVE AND PROJECTIVE OBJECTS
An object P in a category C is said to be projective if given any epimorphism a : A + B in C and any map f : P + B, there is a morphism J : P + A such that aJ = f. This can be rephrased in a neat way: Given any morphism a : A + Band an object C of C there is an induced mapping C(C,a) : C(C, A)
+
C(C, B)
given by composition: g 1+ ago An object P is projective if and only if for any epimorphism a : A + B, C(P, a) is a surjection, i.e. is onto. Dually an object Q is injective if given any monomorphism a : A > B, the induced mapping C(a,Q) : C(B,Q) + C(A,Q),h 1+ ha, is a surjection, thus any map from A to Q extends to one defined on B. Remark: The above makes sense in any category, but we will use these ideas mostly in abelian categories. As an example in Ab, any free abelian group is projective whilst the quotient ~ /71.. of the additive group of rational numbers by the subgroup of integers is injective. GENERATING SETS OF OBJECTS, GENERATORS AND COGENERATORS
In the category of sets, the singleton set, 1, has the extremely useful property that it can be used to detect differences between functions in the following sense: if h, hi : X + Yare two functions and hihi, then there is some map x : 1 > X such that hx i h'x . Of course this merely says that if hihi then there is some x E X such that h(x) i h'(X), but it can be abstracted to make sense in other categories. In a category C a set, S, of objects is a generating set if for any h, hi : A + B in C with hihi, there is some C E S and x : C > A such that hx i h'x. If S consists only of one object, G, say then G is said to be a generator for C. For instance 71.. is a projective generator for Ab, the category of abelian groups . Dually one has the notion of a cogenerating set which can detect differences by maps to objects from the set. The dual of a generator is then a cogenerator. In the study of abelian categories, injective cogenerators are very important. CARTESIAN CLOSED CATEGORY
This is briefly discussed in the text (111.4). ENRICHED CATEGORY
Various cases of such are discussed in the text in the early part of 111.4.
446
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454
INDEX An asterisk· marks items in the GLOSSARY. Abcategory 443· abelian category 445· abstract simple homotopy theory 342 action of a group G on a set M 151 additive category 444·  cylinder functor relative to a (cone) monad 61  factorisation lemma 69  functor 444· adjoint functors 442·  cylinder/cocylinder pair 123  pair 442· alternative models of homotopy (in simplicial sets) (discussion) 184 anodyne extension 174 axiom on fibrant models (in a cofibration category) 82 base point of a pointed set 293 belongs to the same component 295 bisimplicial set 161 boundary (cubical) 21  map in a chain complex 199  morphism (in a chain complex) (formal definition) 208 (n, v, k)box 24 canonical decomposition of a morphism 407   of a morphism in an additive homotopy theory 407 cartesian closed category 154, 446"    as enriched category 231 categorical simple homotopy theory 347 category 429"  of chain complexes as an Senriched category 233

of chain complexes and chain maps 208  of cofibrant objects (K.S. Brown) 79  of crossed complexes 220  of cubical sets 22  of diagrams of type C in V 432·  of fibrant objects (K.S . Brown) 79  of fractions 90  of groupoids 151  of objects over B 33  of objects under A 33  of pointed sets 136, 294  of simplicial objects in a category 160  of simplicial sets 160  over B 430·  under A 430· 2category 269 Scategory (abbr. for simplicially enriched category) 235 cells in a groupoid enriched setting 265 celhilar maps 346 chain complex 199   (formal definition) 208  complexes 207   in an abelian category 207  homotopy 210  map 208 change of rings, additive homotopy theory (discussion) 251 characterisation of cofibrations in the additive case 65  of homotopy equivalences in the additive case 70 cochain complex 208 cocylinder 12, 119
455
 (in the category of groupoids) 154  functor 12   (for crossed complexes) 225  object in simplicial sets 165 cocylinders for chain complexes (brief discussion) 216 codiagonal 439' codomain 429' cofibrant object (in a category of cofibrant objects) 216   (in a model category) 79 cofibration 6  (in a category of cofibrant objects) 79  (in a cofibrant category) 82  (in a model category) 77  (in a category of chain complexes) (discussion) 77  in the additive case 64  axiom (in an Icategory) 84  category (Baues) 82  structure (Baues) 81 cofibre 303 cofree 403 cogenerating set 446' cogenerator 446' cokernel 445' colimit 440') 441' comonad 443' commutative diagramm 435' compatible with degeneracies 26 complementary summand 445' complete 440' component 295 composing homotopies in the additive case 68 composition 429' comultiplication of a comonad 443' cone monad in a category of chain complexes 214  on an object relative to a (cone) monad 61 congruence of paths in a cubical set 255
 relation 431' connections 206 contractible chain complex 215 coproduct 438' cosimplicial modules 161  set 161  simplicial sets 161 cotensor of a simplicial module and a simplicial set 195 cotensored Scategory 239 cotensors in a simplicially enriched category 238 counit of a comonad 443'  of an adjunction 442' covering homotopy extension property (CHEP) 128  morphism 158 crossed complex of a CWcomplex 226  complexes (formal definition) 219   (general discussion) 217  extension of groups 226  homomorphism 398  module (of groups) 398  modules (examples) 221 cubical set 20  sets 205  T complexes 205 cylinder 3  (in a category of chain complexes) 210  (in the category of groupoids) 152  axiom (in an Icategory) 84  functor 3   (for crossed complexes) 225   (on simplicial sets) 164  object (in a category of cofibrant objects) 79 degeneracy maps of a simplicial set 161   in a singular complex 160  operators (cubical) 20
456
degree of a morphism of graded objects 207  of an element in a graded object 207 diagonal morphism 439* differential in a chain complex 199      (formal definition) 208 Vindexed coli mit 441 * direct sum 444 "   of graded objects 209  summand 445" DNE conditions in a category of chain complexes (discussion) 212 Dold's theorem (discussion) 33   in the additive case 75   (statement) 36 DoldKan theorem 201 domain 429" double groupoids with connection (remark) 158  mapping cylinder 53    (in a category of cofibrant objects) 98    (use of in mapping cylinder calculus) 368 dual category 433" duality 118  principle 434*

pair of composable maps of pointed sets 294     morphisms 292  sequence of a cofibration 302   of a fibration of groupoids 296   of morphisms of abelian groups 292 exponential law (for function groupoids) 154 face maps in a singular complex 160   of a simplicial set 161  operators (cubical) 20 factorisation axiom (in a cofibration category) 82  lemma (in a category of cofibrant objects) 89   in the additive case 69 fib rant object (in a model category) 79 fibration 14, 120  (in a model category) 77  (with respect to a cylinder) 14  in the sense of Kan 176 fibre of a map of groupoids 295 filler 24  map 26  of a horn 168 final object 442* finite category 429*  CWcomplex 343 limit 440* finitely complete 440* full subcategory 430* function complex (of crossed complexes) 222   (or Hom)(of chain complexes of abelian groups) 216  complexes of simplicial sets 165  groupoid 153 functor 430* Sfunctor 321 fundamental group of Y at y 149
EckmannMaumary approach to simple homotopy theory 365 EckmannSiebenmann abstract simple homotopy theory 345 elementary contraction 344  expansion 344 enriched categories 228, 446* epimorphism 436" equivalence of categories 436"  relations as groupoids 150 exact cofibration sequence 302  fibration sequence 302  orbit sequence 299
457
 groupoid of a cubical set 257   o f Y 149   of Y under X 149   of Y under X (relative to a cylinder functor) 263 Gsimplicial sets 341 generating co cylinder 127  cylinder 83  set 446' generator 446' generation of simple equivalences 358 geometric realisation 181 gluing lemma 48  theorem 48   for weak equivalences (in a category of cofibrant objects) 111 Godement interchange law 432' , 433' graded object 207 groupoid 149  enriched categories (detailed structure) 253    (discussion) 229  exact sequence 292 oogroupoids 206 wgroupoids 206

(with respect to a cocylinder) 13, 120  category (in a category of cofibrant objects) 92  class 5  coherence 307   (simplicial version) 317  coherent diagram (discussion) 312    in Top of type A 316   morphism 323   nerve of an Scategory 322  colimit (double mapping cylinder as) 56  commutative square (in a category of cofibrant objects) 101  equivalence 4, 120   over B 35,120   under A 34  extension property (HEP) 6    (HEP') 86  fibre 338  groups (discussion) 182  inverse 5   over B 35, 120   under A 34  lifting property (HLP) 13  limit (Bousfield and Kan) 339  limits and colimits (discussion) 334  over B 35  pullback 336  pushout 336  sequence 292  theory in a category of cofibrant objects 88  under A 34 horizontal composition of cells in a groupoid enriched setting 267   of natural transformations 432'   of track homotopy commutative squares 273 (n, i)horn (in a simplicial set) 167
homology group of a chain complex 200  sequence 292 homotopic (with respect to a cocylinder) 12, 120  morphisms 4   (in a category of cofibrant objects) 93  over B 35, 120  reI end maps 148  under A 34 homotopical algebra 76 homotopy 4  (in a category of cofibrant objects) 93
458
Icategory (Baues) 84 identity label 235  morphism 429* image of a map of pointed sets 294  of a morphism 292 induction up the skeleton 170 initial object 441* injective object 446* Cinjective object 403   of A 403 injective type additive homotopy theory (discussion) 246 interchange axiom (in an Icategory) 85  law of track homotopy commutative squares 274   in a groupoid enriched setting 268 lemma 260 inverse 436* invertible 437* involution in an additive cylinder 62 isomorphism 436*  of categories 436* isotropy group of an element 298
  NE( n) on a cylinder 27   NE(n, v, k) on a cylinder 26   (n, i) (on a simplicial set) 168  conditions and cylinders in Qrpd 157  fibration condition on an Scategory 245  fibrations 173 kernel 445*  of a map of pointed sets 294  of a morphism 292 left adjoint 123, 442"  exact functor 441*  homotopy inverse 5  inverse 437 * limit 440* locally Kan Scategory 242  weakly Kan Scategory 242 mfold left homotopy (of crossed complexes) 222 map 429* mapping cylinder 9   calculus 364   in a category of cofibrant objects 88   in the additive case 64   factorisation 10    in a category of cofibrant objects 89 model category 77 monad 60,443" monomorphism 436* Moore complex 199 morphism 429*  of crossed complexes 220  of degree r 207  of groupoids 151  over B 33  under A 33 multiplication of a monad 443*
K(M,O) the constant simplicial abelian group on an abelian group M 189 Kan complex 168   structure of simplicial modules 196  condition DNE(n) on a cylinder 26   DNE(n, v, k) on a cylinder 27   E(n) 24   E( n) in the category of Kan complexes 171   E( n) on a cocylinder 121   E( n) on a cylinder 27   E(n,v,k) 24   E(n, v, k) on a cocylinder 121   E(n, v, k) on a cylinder 26   in dimeDslOn n (on a simplicial set) 168
ncubes 459
20
nequivalence 227 nsimplex (in a simplicial object) 161 ntypes (discussion) 227 natural equivalence 430·  composition transformation (on a cocylinder) 132  comultiplication (on a cocylinder) 132  interchange transformation (on a cocylinder) 132    (on a cylinder) 18  involution (on a cocylinder) 131   (on a cylinder) 15  isomorphism 430·  multiplication (on a cylinder) 18  subdivision transformation (on a cylinder) 17  transformation 430· negative graded object 208 nerve of a category 168 nonnegative graded object 208 object 429·  group (in a groupoid) 149 operation of a group G on a set M 151 opposite category 433· orbit of a group action 298  set of a group action 298 path in a cubical set 255  in a groupoid 155  lifting property 155 pointed map of pointed sets  set 136, 293 positive graded object 208 preadditive category 443· product 439 category 431 of groupoids 152  of simplicial sets 164 projection functor 431
294
projective object 446* pullback 438* pushout 437,438*  axiom (in an Icategory) Quillen model category 77 quotient category 431
84
rank n 204 relative Cinjective 403   class group of A 412   object of A 403  cylinder axiom (in an Icategory) 85  injective type additive homotopy theory (discussion) 248 relativity principle 44 retraction 437" right action of a group G on an abelian group A 398  adjoint 442"  exact functor 441*  homotopy inverse 5  inverse 437* same simple morphism class (in EckmannSiebenmann theory) 346 saturated set 174   generated by a set B of monomorphisms 175 section 437" semi direct product groupoid 151 set of components 295 simple equivalence in an injective simple homotopy theory 402  equivalences in an abstract simple homotopy theory 360  gluing principle (S .G.P.) 361  homotopy equivalence (geometric form) 344   theory 342  isomorphism (in EckmannSiebenmann theory) 345  morphism (in EckmannSiebenmann theory) 345 460

morphisms in an abstract simple homotopy theory 360  pushout principle (S'p.P.) 359   (alternative form) 361  relativity principle (S.R.P.) 361 simplicial abelian groups 188  identities 163  modules 188   and chain complexes 199  resolution of a module 189  set 160   (formal definition) 163   (detailed description) 161  T complexes 203 simplicially enriched categories (discussion) 228   category 235   functor 321   structure on the category of simplicial modules 193 simply equivalent morphisms (alternative definition) 382 singular complex 160  simplex 159 small category 429* split epimorphism 436*  monomorphism 67) 436* splitting 436* stability group of an element 298 stably cofree 403 standard nsimplex 159 star (of an object in a groupoid) 155  bijective 155  injective 155  surjective 155 strong deformation retract 40  homotopy equivalences 275 strongly locally Kan Scategory 242 subcategory 430*
subdivision in an additive cylinder 63 sum of simple morphism classes 353  of tracks 148 suspension 303  of a graded object 208 T complex 203  of rank n 204 tensor of a simplicial abelian group and a simplicial set 193  product of crossed complexes 224   of graded objects 209 tensored Scategory 239 tensors in a simplicially enriched category 238 terminal object 441thin element 203 track of a path in a cubical set 256  homotopy category over B 285    under A 280   commutative square 271  of a homotopy 148 tree groupoid 150 triple mapping cylinder 368 trivial cofibration 40   (in a category of cofibrant objects) 79   (in a model category) 77   in the additive case 66  fibration (in a model category) 77 unit of a monad 443 of an adjunction 442* vertex group (in a groupoid) 149 vertical composition of cells in a groupoid enriched setting 266   of natural transformations 432  of track homotopy commutative squares 272
461
Vogt's lemma 275  theorem (1973) 333 weak equivalence (in a category of cofibrant objects) 79   (in a cofibrant category) 82   (in a model category) 77   in 7rC 107   of simplicial sets 182  Kan complex 169  pullback 438"  pushout 437" weakly generating 128  injective map of pointed sets 294 zero map  object
445" 444"
462