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Chapter 4 Homology and Cohomology Functors
CHAPTER 4
HOMOLOGY AND COHOMOLOGY FUNCTORS
Introduction In this chapter we introduce the most useful class of structure preserving functors from Y to graded abelian groups. These are exact functors that take coproducts to coproducts in the covariant case and to products in the contravariant case-homology and cohomology functors respectively. In Section 1 we begin with some of the elementary properties of these functors. Then in Section 2 we consider the extent t o which these functors are determined by their action on finite spectra. For homology functors the connection is the strongest possible-this presents us with a basic limitation of these functors. For cohomology functors the situation is more complex and we study at length the functor o n Y strongly determined by a cohomology functor restricted to 9.In particular we show that such a functor takes coproducts t o products and is almost exact. Finally in Section 3 we derive the major representability theorems. We show that cohomology functors, exact functors on 9 and homology functors are all representable. In addition, for each of these there is a representability result for the corresponding natural transformations.
1. Basic properties of homology and cohomology functors
It is a standard element of the methodology of category theory that a category be studied via functors from it to a more familiar category. In the case of a graded (additive) category a natural choice is functors to Ab,, the category of graded abelian groups. Of greatest value are those functors that preserve the maximal amount of structure so we might require that a functor take exact triangles to exact sequences and coproducts to coproducts or products depending on variance-these are 45
46
HOMOLOGY AND COHOMOLOCY FUNCTORS
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the functors that, in effect, satisfy the Eilenberg-Steenrod axioms (minus the axiom for a point) together with the wedge axiom [47]. As for the one remaining element of basic structure, the smash product, there is an obvious analog in Ab*, the tensor product over Z, so we might further require that the functor under consideration preserve that too. This last restriction is quite severe for, among other things, though the smash product is exact, the tensor product is not so in general. With these comments in mind we define H : Y+ Ab, t o be a homology functor if H covariant, exact and if the canonical map U H(X,)-+ H ( UX,) is an isomorphism. And H : Y + A b , is a cohomology functor if it is H(X,) is an contravariant, exact and if the canonical map H ( UX,)+ isomorphism. For example, for X in Y let X,(Y) = T,(X A Y) and X * ( Y ) = [ Y, XI*, then X, and X* are the homology and cohomology functors represented by X. An important special case is X = S, here T , = S, is the (stable) homotopy functor and T *= S* is the (stable) cohomotopy functor. In Theorems 16and 11we will prove that in fact every homology and cohomology functor is so representable. Because homology and cohomology functors act nicely with respect t o the triangulated and coproduct structures of Y such functors can be expected t o also act nicely with respect t o derived structure. For instance consider a sequence X, +X2+. . .-+ wcolim X, = X.
n
PROPOSITION 1. (a) If H is a homology functor then the natural map colim H(X,)+ H ( X ) is an isomorphism. (b) If H is a cohomology functor then there is a short exact sequence O+lim' H(X,)+ H(X)-+lim H(X,)+O. PROOF. These results are immediate from application of H to the exact triangle defining wcolim X,. 0 In the presence of the representability theorems Proposition 1 is just a rephrasing of Proposition 3.4. A useful application of Proposition 1 is that homology and cohomology functors are 'determined' by their values o n sphere spectra, precisely:
PROPOSITION 2. If 7 : H +K is a natural transformation of homology or cohomology functors such that v ( S ' ) is an isomorphism for all r then 7 is a natural equivalence. PROOF.We will consider only the case of H and K cohomology functors the other case being if anything easier. Note first that the diagram
CH.
BASICPROPERTIES OF FUNCTORS
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commutes because composing with any projection get the commuting diagram
47
n K(X,)+
K(X,) we
For X in Y let X' +X 2 + . . .-+ X be a crude cellular tower. If X = X" for some n we see that v ( X ) is an isomorphism by a simple induction starting with the fact that v(U S') = v ( S ' ) is an isomorphism. For a general X, application of Proposition 1 gives
n
0-Iim' 0-
H(Xr)-H(X)-lim
H(X')-0
1
1T ( X )
1
liml K(X')-
K(X)-
lim K(X')-O
So by the 5-lemma v ( X ) is an isomorphism. 0 As an application of Proposition 2 we have the following, (b) being a weak form of the Kunneth formula.
COROLLARY 3. Let H be a homology or cohomology functor. (a) If H ( 9 ) = 0 for all r then H ( X ) = 0 for all X . (b) If H ( X ) = 0 then H ( X A Y )= 0.
PROOF. (a) is immediate from Proposition 2. And (b) is immediate from (a) since if H is a homology (resp. cohomology) functor then so is H ( X A ). 0 Another application is t o a universal coefficient theorem and a Kunneth formula in the following context. A spectrum X is a ring spectrum if there are maps m : X A X + X and i : S + X such that the composites X = S A X"-X A X A X and X = X A S I n i \ X A X A X are the identity. And a spectrum Y is an X-module spectrum if there is a map n : X A Y + Y
48
-
HOMOLOGY AND COHOMOLOGY FUNCTORS
such that the composite Y = S A Y
X
A
[CH.
4, 1
Y -li Y is the identity and
commutes. Then 7r,(X) is a ring and the functors X,, X*, Y,, Y* all take values in the category of 7r,(X)-modules. That is, the product of f :S"+Xandg:Sb+Xisgiven byS"+b=Sa/\Sbfng,X/(X~Xand for instance the product of f : S" + X and g : sbU+ Y (in Y*(U)) is given by satbU= S" A s b U X A Y 4Y. COROLLARY 4. Let X be a ring spectrum and Y a n X-module spectrum. (a) If 7r.J Y) is a flat 7r,(X)-module then there is a natural isomorphism X * ( W ) O T * ( X ) ~ * ( Y ) +Y*(W). (b) If m*(Y) is a n injective m,(X)-module then there is a natural isomorphism Y*( W)+ Hom,,cx)(X,( W), 7r,( Y)). PROOF.The following diagrams define the indicated natural transformations: (a) S ; ~ S A S - , ( W A X ) A Y ; ~ W A ( X A Y ) ~ W A Y , (b) S ~ X WA- X A Y A Y . The conditions on 7r,(Y) imply that X * ( )@,.(~)7r*(Y) is a homology functor and Hom,,(x)(X( ), 7r*(Y)) is a cohomology functor. The corollary is then immediate from Proposition 2. 0 There is a further variant of Corollaries 3 and 4 that will be needed later. PROPOSITION 5. If X is a ring spectrum and Y is a n X-module spectrum thenX,(U)=O impliesthat Y * ( U ) = O = Y*(U).
PROOF.We are assuming that U A X = 0. Therefore U A X A Y = 0. But the structure map X A Y + Y is an epimorphism (split by Y = S A Y*X A Y) and therefore U A Y = 0, i.e. Y,(U) = 0. For the other part let f : U + Y be an element of Y*(U). Then f can be factored as
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4,21
and thus f
FINITELY DETERMINED FUNCTORS
49
= 0.
Note that all that is required of Y in Proposition 5 is that the map Y A X be a monomorphism. Y
2. Finitely determined functors In studying the structure of Y a problem of obvious significance is that of analyzing the relationship between Y and the subcategory S. Central to this is the functor, introduced in Chapter 2, that assigns to each X in Y a category A (X)whose objects are maps f : X, +X with X, E 9 ' (the small-in fact countable+quivalent subcategory of 9 of Proposition 3.11) and whose morphisms are commuting diagrams
X
with f and g in A(X).And to i : X + Y, A assigns the natural transformation A (i) :A (X)+A (Y) given by A (i) of f :X, +X being if and A ( i ) of
being
Y
'"/" y h x,-x,.
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HOMOLOGY AND COHOMOLoGY FUNCTORS
[CH.
4, 2
So A is a functor taking values in a category of categories. More precisely A takes values in the category of filtered categories. That is, given f : X, +X and g : X, +X in A ( X ) then the diagram
is in A ( X ) . And given
X
x, 7 x 0 in A ( X ) there is an exact triangle X , s X , l X , + s X , in 9’ and it follows that g factors through k via X , + X . Then k is a morphism in A ( X ) and ki = kj. In Chapter 5 we will consider the extent to which a spectrum is determined by its diagram category. Here we turn t o the analogous question for homology and cohomology functors. That is, given H : Y+ Ab, to what extent do the values taken by H evaluated on finite spectra determine the values taken on arbitrary spectra? This can be made more precise by the introduction of a functor on Y derived from H I 9 which can then be compared to H itself. If H is a homology functor then we define a functor H by letting f i ( X ) = colimAcx)H(X,), the colimit as in Appendix 1. And for f : X + Y, Hcf) : A ( X ) + A( Y ) is the map induced by the natural transformation A 0 again as in Appendix 1. Then the maps H ( f ) : H(&)+ H ( X ) for f in A ( X ) , induce a map from the colimit which defines a natural transformation x : H + H. Further XIS is an equivalence and with it we will identify HI9 and HIS. This follows from the fact that for X in 9 there is an equivalence e : X o + X in A ( X ) which is therefore terminal in A ( X ) . For then f i ( X ) = H ( X o )and via this x ( X ) is just the isomorphism H(e). Similarly if H is a cohomology functor then we define H by letting H ( X ) = H(X,), the limit as in Appendix 1. And H ( f ) is the map induced by A 0 as in the appendix. And the maps Hcf) : H ( X ) + H(X,) for f in A ( X ) induce a map to the limit which defines a natural transformation cp : H +H. Further as in the covariant case cp9 l is an equivalence of the two functors so restricted. It is left for the reader to verify that for H either a homology or cohomology functor, H is additive.
CH. 4,
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FINITELY DETERMINED FUNCTORS
51
In the covariant case we have the following theorem originally proved by Milnor [93] (unstably).
THEOREM 6. For a homology functor H, x : H + H is an equivalence. PROOF.Applying Proposition 2 it suffices to show that is a homology functor, that it is exact and commutes with coproducts. First note that by Proposition A1.7 fi(X)satisfies: (i) for any a E A(X) there is a map f : X,+X in A(X) and b E A(Xa) with H ( f ) ( b )= a, (ii) for any f :Xa X in A (X)and a E f i ( X a )with fiCf)(a)= 0 there is a morphism
X
in A(X)with A ( h ) ( a )= 0. To prove exactness consider an exact triangle X A Y 4 Z z s X .It suffices to show that fi(X)+ fi(Y)+ A(Z)is exact. Since fi(g) fiCf)= A(gf)=O we are left to show that kerA(g)Cirn&(f). So consider a E ker fi(g). By (i) we have i : Ya+.Y in A (Y)and b E &(Ya)with f i ( i ) ( b )= a and by (ii) we have
-
in A (2)with A ( k ) ( b )= 0. This gives the diagram
x
t n
Y
t
-z
-sx t
t
x y zYa-Z,-sXy with the bottom row exact and n any fill-in map. Since the bottom row is in 9’ and HIP = H I P we get b = &(rn)(c) and thus a = A ( f ) [ H ( n ) ( c ) ] . To prove that H commutes with coproducts consider a family {Xy} y E r. There is a natural map p r : LI,-~~(X~)-+A(LI,-XY). Since A is
HOMOLOGY AND COHOMOLWY FUNCTORS
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additive this map is an isomorphism if is a commuting diagram
[CH.
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r is finite. For an arbitrary r there
LI A ( X ’ ) LA(LI X ’ ) r
i-
t-
qT
colim pA
A(LI x’)
colim LI A(xy)-colim ACr
AC I‘
A
A
where the colimits are over the finite subsets of r and 9 is the natural map. So it suffices t o show that 9 is an isomorphism. To see that 9 is monic consider an element x in colim A(UAX’). Ultimately it is represented by a map f : U + & X y with U in 9 and an element u in H ( U ) . If 9 ( x ) = 0 then there is a diagram
u-LIX’ A
gs
.1
v-LIXY with g in 9’ and H ( g ) ( u )= 0. Since V is small h factors through a finite coproduct &XY with A C B. But (f, u ) and (h, H ( g ) ( u ) )represent the same element in colim f i ( u A X y ) .Therefore x = 0. To see that 9 is onto consider y in f i( &X ’ ) . It is represented by a pair (i, w ) with i : W-,&X” and w E H ( W ) . As above i factors through a finite coproduct which gives a representative of an element in colimAfi(uA Xy) mapping t o y . 0 This result is extremely important in that it shows that in the strongest That possible sense a homology functor is determined by its action o n 9. is, given H on 9 we can recapture H on all of Y by letting H ( X ) = colimncx,H(X,). This is very useful but it is also a significant restriction because it implies that certain phenomena cannot be detected using homology functors. For instance
COROLLARY 7 . Iff :X then H c f ) = 0.
+ Y is an f -phantom m a p and H
is a homology functor
In the contravariant case we cannot expect H + A t o be an equivalence in general for if f : X + Y is an non-trivial f-phantom map
a.4, 21
FINITELY
DETERMINED FUNCTORS
53
then &(f) = 0 for any cohomology functor H but in particular Y*(f)-#0. However we will see in Corollary 5.17 that q ( X ) : H ( X ) + f i ( X ) if not always an isomorphism is always an epimorphism. For the present let us concentrate on the following: arguing as in Theorem 6 H + fi would be an equivalence if fi were a cohomology functor, with this in mind how close does fi come t o being a cohomology functor? First observe that H acts well with respect t o minimal weak colimits.
PROPOSITION 8. Let F : A +9’ be a diagram over a directed category with minimal weak colimit X. Then the induced map H ( X ) + limn H(X,) (X, = F ( a ) ) is an isomorphism.
PROOF.An element x of limn fi(X,) is an assignment t o each f : W + X, in A(X,) of an element xcf) E H ( W ) such that for any commuting diagram Xa
ft
.
ts
U L V H ( i ) x ( g )= xcf). Then define x’ in &(X) as follows: for f : W + X in A (X) we have the isomorphism limn [ W, X,] + [ W, X ] so f factors as
x,-x W
and we define x’cf) = x ( g ) . The element x’ is well-defined, for given two factorizations-which since A is filtered we can regard as through the same X,-gl, g2 : W + X, there is a commutative diagram
and therefore x ( g , )= x ( g ) = x(g2).By a similar argument we see that x’ is in fact an element of A ( X ) and by definition it maps t o x. It is further clear that x’ is the unique element with this property completing the proof. 0
HOMOLOGY AND COHOMOLOGY FUNCTORS
54
[CH.
4, 2
In particular, since an arbitrary coproduct is the colimit of its finite coproduct summands, the following is easily proven. COROLLARY 9. H takes coproducts to products. So H must fail to be a cohomology functor by failing to be exact. However following Adam [7]we will now show that f i does satisfy a partial exactness condition. The work involved in doing this will be substantial but beyond clarifying the nature of H this result is needed to prove the representability of homology functors. THEOREM 10. If X A Y i Z A sX is an exact triangle and X = with Xu in 9then H ( X )+H ( Y )-+&(Z) is exact.
u,,X,
PROOF.We begin with the special case of X in 9. Given y in H ( Y )with H ( i ) ( y )= 0 we must construct z in H ( Z ) with Hu)(z)= y . This requires a coherent choice of elements z(f) in H(Za)corresponding to the maps f : Za +Z in A (Z).To define an appropriate z(f) consider the diagram
with the upper sequence an exact triangle in 9’. Let g : ‘1 + ’ be a fill-in map. Then by assumption H(il)(y(g))= 0 and therefore Y k ) = H ( y l ) ( z )for some z in H(Za).Of course z is not unique so consider the coset C ( f )= z + H ( k f ) H ( s X )in H(Za). LEMMA: C(f)is independent of the fill-in map. PROOF. If g‘ is another fill-in map then g - g’ factors through i. From this it follows that y ( g ) = y ( g ‘ ) . 17
Further given
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in A (Z) it is easily checked that H ( h ) induces a map h * : C(g)-+ Ccf).S o the problem is to choose one representative zcf) from each Ccf) so that H ( h ) z ( g ) =2 0 . To do this it will be necessary to know more about the category % whose objects are the CWs and morphisms the maps h*.
LEMMA. (a) Each morphism is a surjection of sets. (b) %OP is a filtered category, i.e. (i) given Ccf) and C(g) in V there is a diagram C ( f ) t C(h)-* C(g) in %, (ii) given i*, j * : C(f)+ C ( g ) there is a k* : C(h)+ C ( f ) with i*k* = j*k*. (c) There is at most one morphism from C ( g ) to C(f).
PROOF.(a) Consider h* : C(g)+ Ccf) and z in Ccf).There is a commuting diagram
with rows exact. For w in C ( g ) arbitrary H(j,)(w)=y ( g , ) and HCi,)H(h)(w)= H ( j 2 ) ( ~ ) . Therefore z - H ( h ) ( w )= H ( k 2 ) ( ~=) H ( h ) H ( k ) ( x )and so z is in im h*. (b) For the first part we have C ( g )t C ( f l g ) + C ( f )with the obvious maps. As for the second part we are given
Z
So if Z , ~ Z , & ~ + s Z is , exact then f factors as hk. Then k * : C(h)-, Ccf) is as desired. (c) This is immediate from (a) and (b)(ii). Cl The category % is not quite tractable enough so we enlarge it by with the same objects as those of V and a defining a category
HOMOLOGY AND COHOMOL~GY FUNCTORS
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[CH.
4, 2
morphism k : Ccf)-. C ( g ) for each commuting diagram of sets
Then
g has the following properties.
LEMMA. (a) There is at most one morphism from C ( g ) to Ccf). (b) Each morphism is a surjection of sets. (c) Given Ccf) and C ( g ) there is a C ( h ) and morphisms C ( g ) C ( h)+ CV). (d) There are only countably many equivalence classes of equivalent objects in g. +
PROOF.(a)-(c) are immediate from (a)-(c) of the preceding lemma. (d) It will suffice t o show for f, g : 2, +Z in A (2)that if kf = k g then Ccf) and C ( g ) are equivalent (in g) for the set of maps [Z,, s X ] is countable (that is obj 9 'is countable and for U, V finite spectra [ U, v] is countable). But if kf = k g then we have Y
k -z -sx
---
tr ya and
I
tf
z,
m
k
II sx
Y -2 -sx ts' t g I1 I m Y, 2, sx where m
=
kf
= kg.
And we have
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with nil = f and nj2 = g’. Let H ( l ’ ) ( z )= y ( n ) and z1= H ( i l ) ( z )and z2= H(i2)(z).Then H(l)(zl) = y ( f ‘ ) and H(l’)(z2) = y ( g ’ ) . Therefore
sets up the equivalence between Ccf) and C(g). 0 It is now an easy matter to show that lime C(f) is nonempty for it follows from the lemma that there is a cofinal sequence C, +- C2+- * * * of set surjections in this category. But an element { z c f ) } of this limit is in particular a choice for each f : 2,+ Z in A ( Z ) of an element ~ ( fin) C ( f )C H ( 2 , ) such that for
Z
in A ( Z ) , H ( h ) z ( g )= z(f). that is an element z in f i ( z )with f i ( j ) ( z )= y. We turn now to the general case, that is, an exact triangle X, with X , E B and y E fi( Y) with H ( i ) ( y ) = 0. X, Y A Z A For f c A let C ( f )C obj A ( Z ) be the set of maps f : 2, + Z such that kf factors through the inclusion s&X,G s ~ A X , .Consider the family of pairs (c z) where f C A and z assigns to each f : 2,- Z in C ( f ) an element ~ ( fE) H ( 2 , ) such that (a) given
su,
Z
commuting with f, g in C ( f ) we have H ( h ) ( z ( g ) )= 2 0 , (b) z(jf) = y ( f ) for f : 2, + Y (which of course implies that f E C ( f ) for all 0. Partially ordering these pairs in the obvious way we extract a maximal element (cz). To complete the proof we must show that f = A for then z is an element of A(Z)and A ( j ) ( z ) = y as desired.
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HOMOLCGY AND COHOMOLOGY FUNCTORS
So suppose that there is an index consider the diagram
Q
EA -
k
i’
T.il
t m‘ R
i”
t m“
(let 0 = r U {a})and
II
k’
si’
Y-Z‘-sUx,-sY
II
2
Si
Y i _ Z - s U x , -As Y
II
r
[CH. 4,
T.
II
i2
k”
Si“
Y-Z”-sUX,-sY r
where Z’ is the weak pullback of k and il, and Z” is the weak pullback of k‘ and &so the diagram commutes and the TOWS are exact triangles. Using z we define an element z“ of fi(2”)by letting z”(f) = z(m’m‘’n. Then fi(j”)(z”) = y . Since Z“ is the weak pullback (see Proposition 3.1) we have the commuting diagram
i’
LIXp-Y
n
with the top row an exact triangle. Then fi(&p)(z”)= fi(f‘i’)(z’’)= f i ( i ’ ) ( y ) = 0 since i’ factors through i. And since p is an epimorphism (and therefore splits) f i ( i , ) ( z n ) = 0. But X, E 9 so applying the special case considered above it follows that there is an element z‘ in fi(Z‘)with f i ( m ” ) ( z ’ )= z”. Define 2’ on C(0)as follows: given f :Z, --+ 2 in C ( 0 ) there is a commuting diagram
Z-sUX,
J‘
A
m’
t
Zk’-SUX,
z>
and consequently a map g : Z,+Z’ completing the diagram. Let z l ( f )= z ’ ( g ) . Although the fill-in map g is not unique the resulting class z ’ ( g ) is. For if g l , g2 : Z +Z’ are two fill-in maps then k‘(gl - g2) = 0 = m‘(gl - g2) imply that gl - g2 = fig3 and therefore z ’ ( g l )- z‘(g2) = z’(g1- g2) = z’(j‘ig3)= fi(j‘)(zf(ig3))= y(ig3) = fi(iI(y(g3)) =o.
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59
Further zI satisfies (a). For if we have
Z
with f , g in C ( 0 )and g’ : Z, +Z’ is a fill-in for g then g‘h is a fill-in for f. And then H(h)(zl(g))= H(h)(z’(g’)) = z’(g’h)= zlcf).And z1also satisfies (b) for if f :Z, + Y then zl(jf) = zl(m‘jf) = z’(j’f)= H(j’)(z’cf)) = ycf). Finally zl is an extension of 2, i.e. considering f : 2, +Z in C(T)there is fill-in g :Z, +Z” and zl(f) = z‘(”’g) = H(m”)(z’(g)) = z “ ( g )= z(f). Therefore (0,zl) sits above (r,z ) in the partial ordering, which is a contradiction. 0
3. Representability theorems We come now to a major characteristic of homology and cohomology functors, namely their representability. We begin with the well-known cohomology representability theorem of Brown [35].
THEOREM 11. If H : Y + A b * representable.
is a cohomology functor then H is
PROOF.By Proposition 2 it will suffice to construct a spectrum X and a natural transformation 7 : X * +H such that q(S’) is an isomorphism for all r. Observe that for y E H ( Y ) there is a natural transformation vy : Y*+H defined by vy(W ) ( f = ) H ( f ) ( y ) .In particular for W = S’ this gives a map r,(Y)+ H(S’). We will define X as a minimal weak colimit and 9 as vx for an element we will construct in H ( X ) .To begin, let Xo = S; the coproduct over all y in H(S‘) for all r and let xo in H ( X o ) = n H ( S ; ) be n y . Then qw(S’) : r , ( X ) + H ( S ‘ ) is epic for all r. Inductively assume that we have and elements x, in H(X,) with constructed X o 4 X l & - .*X,, H(i,,,-,)(x,,,)= x,-~ and such that ker v,(S‘) C ker r,(i,). Let X,,+l be defined by the exactness of ~ S ; ~ X n ~ X n + l +where s ~ S the ; coproduct is over all y in ker vxn(S’)for all r. Then H(j)(x,,)= 0 so there is an element x,,+~ in H(X,,+,) with H(i,,)(x,,+l)= x,,. Further ker vmm(S‘) C ker rr(i,,)and T ~ ~ + ~is( S epic ’ ) since vxm(S’) is epic. Therefore
-
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HOMOLOGY AND COHOMOLOGY FUNCTORS
[CH. 4,
3
we have constructed the diagram Xo+ X , + * . . . Let X = wcolim X,. Since H ( X ) + lim H ( X , ) is epic there is an element x in H ( X ) mapping t o x,, for all n. From this we get colim ml(X,,)-
m,(X)
and it follows from the observations made above that v,(S') isomorphism for all r. 0
is an
A standard consequence of the representability of cohomology functors is the representability of the stable natural transformations of cohomology functors. A natural transformation 8 : H + K is stable if the following diagram commutes:
-
H(sX) 12
sH(X)-
K(sX) 11
sK(X)
Let SNT(H, K) denote the stable natural transformations from H t o K .
COROLLARY 12. If H and K are cohomology functors with representing spectra X and Y respectively then there is an isomorphism of SNT(H, K ) and [ X , Y ] .
PROOF.Since H (resp. K) is naturally equivalent t o X * (resp. Y*), SNT(H, K ) = SNT(X*, Y*). Then SNT(X*, Y * )= NT(Xo, Yo), see Appendix 1, page 443 = [ X , Y ] by Yoneda's lemma (Lemma A1.3).
0
There is an important variant of Theorem 11 that among other things will imply the representability of homology functors.
THEOREM 13. If H : 9-*Ab* is a contravariant exact functor then H is representable in 9,i.e. there is a spectrum X (in 9)and a natural equivalence 7 : [ ,XI* +H.
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REPRESENTABILITY THEOREMS
61
PROOF.We define H : 9’-Ab* as above (i.e. A(X)= limn(x)H ( X , ) ) and then using Proposition 8, Corollary 9 and especially Theorem 10 we can duplicate the procedure used to prove Theorem 11. That is, given Y in Y and y E A(Y) there is a natural transformation vY : [ , Y ] * + H (the left-hand side restricted to 9) defined by qy(W ) ( f )= f i ( f ) ( y ) . And as in Theorem 11 it suffices to construct X and x E f i ( X ) such that ~ ~ is(an9 ) isomorphism for all r. For it is easy to see that if 7 : H + K is a natural transformation of two exact functors on 9 and ~ ( 9 is an ) isomorphism for all r then 7 is an equivalence. But the properties of proved above give what we need to carry through the construction of X and x of the earlier representability proof. 0 Given two contravariant exact functors defined on 9, H and K, let us consider the relationship between maps of the representing objects and natural transformations of the functors. If X represents H and Y represents K there is of course a map [X, Y]+SNT,(H, K). However as opposed t o the situation in Corollary 12 we cannot expect this map to be an isomorphism in general. For if f : X + Y is a non-trivial f-phantom map then by definition it will induce the zero natural transformation of the represented functors. T o make this more precise let [X, YIr be the quotient of [ X , Y ] by the subgroup of f-phantom maps. Then there is a factoring
and we will prove below that [ X , Y]f+SNT,(H, K) is an isomorphism giving the desired relationship between the representing objects and the natural transformations. First we have a representability result for fi.
PROPOSITION 14. The natural equivalence on 9, 7 : [ ,XI + H, extends to a natural equivalence on Y,+j: [ , X I , + 12. PROOF.Let X and x E A ( X ) be as in Theorem 13, then we define 7jx : [ , XIf + by &(f) = f i ( f ) ( x ) .It is easily seen that 7jx is well-defined and monic. To see that it is an epimorphism suppose that we are given Y in Y and y E f i ( Y ) . Then observe that the construction of X in = Y @ X and ql= (y, x ) . Theorems I 1 and 13 could have begun with That is, we can construct a tower Xo+ X , +. - .+wcolim X, = X’ and
62
HOMOLOGYAND C O H O M O L o G Y FUNCTORS
[CH.
4, 3
elements x, E f i ( X , ) giving x' in f i ( X ' ) such that vx.: [ ,X'l- H is an isomorphism and if f Ig :X0+ X' is the composite then fiCflg ) ( x ' ) = xo. Therefore
and g is an equivalence and hence &(g-'f)
= y.
0
THEOREM 15. If H and K are contravariant exact functors on 9represented by X and Y respectively then [ X , Y],+SNT,(H, K) is an isomorphism.
PROOF.It is easy to see that a map from X to Y induces the zero transformation precisely when it is an f-phantom map, so we must show that a stable natural transformation comes from a map of the representing spectra. Identifying H (resp. K ) with [ , X I * (resp. [ , Y ] * )we have by Proposition 14 a map 8 : R(X)+SNT,([ ,XI*, K ) given by 8(x)(f) = x(f). Now suppose that we are given a stable natural transformation T : [ ,XI*+ K, we want to define x E R ( X ) with 8(x) = T. But for f : U + X i n A ( X ) , ~ ( f ) E K ( U ) s o l e tx ( f ) = ~ ( f ) . T h e n x E k ( X ) a n d e ( x ) = 7. 0 We turn now to the covariant case proving a representability theorem due to Adams [7]. THEOREM 16. If H : Y-+Ab, is a homology functor then H is representable.
PROOF. By Theorem 6 there are natural equivalences colimA(y)H(Ya)+ H ( Y ) and colimA(y)P*( Y, A X ) + P*( Y A X ) so it suffices t o find X and a natural equivalence of H and P*( A X ) restricted t o 9. But the Spanier-Whitehead duality functor D of Chapter 1 sets up a correspondence between homology and cohomology functors defined on 9, i.e. D H ( U )= HD(U).Applying Theorem 13 there is a spectrum X and a natural equivalence DH = [ ,XI". And for U in 9 there is an equivalence natural in U : [D(U ) ,X I , = P*( U A X ) . Then since D2= I on 8 we get the desired natural equivalence (on 9):
CH.
REPRESENT ABILITY THEOREMS
4, 31
63
Natural transformations of homology functors are similarly represented. Let H and K be homology functors represented by X and Y respectively. THEOREM 17. There is an isomorphism of SNT(H, K ) and [ X , Y ] ,
PROOF.The restriction map SNT,(H, K)+SNT9(H, K) is an isomorphism with inverse given by 7 ( X )= colim,,(,) ~ ( x ,Spanier-Whitehead ). duality gives rise to an isomorphism SNT9(H, K) = SNT9(DH, D K ) and by Theorem 15 this latter group is isomorphic to [ X , Y ] , Combining these observations we get the desired isomorphism of SNT,(H,K) and [ X YIP 0 COROLLARY 18. If H is a homology functor represented by X and Y then X and Y a r e equivalent.
PROOF.The natural equivalence of 7r*( A X ) and 7r*( A Y ) gives rise to f maps X e Y with 7r,cfg) = 1 and 7 r , ( g f ) = 1. Therefore 7r,(f) is an I3
isomorphism and hence f is an equivalence. 0 By Theorem 11 and 16 there is a natural correspondence between homology and cohomology functors pairing those functors represented by the same spectrum. However in general there will be no simple relationship between the values taken by corresponding functors. For instance, there are spectra W, X, Y, Z such that (1) W , ( X ) # 0 and W * ( X )= 0, (2) Y , ( Z )= 0 and Y * ( Z )# 0. For an example of (1) consider Theorem 16.16. And an example of (2) is given in Chapter 6.
HOMOLOGY AND COHOMOLOGY FUNCTORS
Introduction In this chapter we introduce the most useful class of structure preserving functors from Y to graded abelian groups. These are exact functors that take coproducts to coproducts in the covariant case and to products in the contravariant case-homology and cohomology functors respectively. In Section 1 we begin with some of the elementary properties of these functors. Then in Section 2 we consider the extent t o which these functors are determined by their action on finite spectra. For homology functors the connection is the strongest possible-this presents us with a basic limitation of these functors. For cohomology functors the situation is more complex and we study at length the functor o n Y strongly determined by a cohomology functor restricted to 9.In particular we show that such a functor takes coproducts t o products and is almost exact. Finally in Section 3 we derive the major representability theorems. We show that cohomology functors, exact functors on 9 and homology functors are all representable. In addition, for each of these there is a representability result for the corresponding natural transformations.
1. Basic properties of homology and cohomology functors
It is a standard element of the methodology of category theory that a category be studied via functors from it to a more familiar category. In the case of a graded (additive) category a natural choice is functors to Ab,, the category of graded abelian groups. Of greatest value are those functors that preserve the maximal amount of structure so we might require that a functor take exact triangles to exact sequences and coproducts to coproducts or products depending on variance-these are 45
46
HOMOLOGY AND COHOMOLOCY FUNCTORS
[CH.
4, 1
the functors that, in effect, satisfy the Eilenberg-Steenrod axioms (minus the axiom for a point) together with the wedge axiom [47]. As for the one remaining element of basic structure, the smash product, there is an obvious analog in Ab*, the tensor product over Z, so we might further require that the functor under consideration preserve that too. This last restriction is quite severe for, among other things, though the smash product is exact, the tensor product is not so in general. With these comments in mind we define H : Y+ Ab, t o be a homology functor if H covariant, exact and if the canonical map U H(X,)-+ H ( UX,) is an isomorphism. And H : Y + A b , is a cohomology functor if it is H(X,) is an contravariant, exact and if the canonical map H ( UX,)+ isomorphism. For example, for X in Y let X,(Y) = T,(X A Y) and X * ( Y ) = [ Y, XI*, then X, and X* are the homology and cohomology functors represented by X. An important special case is X = S, here T , = S, is the (stable) homotopy functor and T *= S* is the (stable) cohomotopy functor. In Theorems 16and 11we will prove that in fact every homology and cohomology functor is so representable. Because homology and cohomology functors act nicely with respect t o the triangulated and coproduct structures of Y such functors can be expected t o also act nicely with respect t o derived structure. For instance consider a sequence X, +X2+. . .-+ wcolim X, = X.
n
PROPOSITION 1. (a) If H is a homology functor then the natural map colim H(X,)+ H ( X ) is an isomorphism. (b) If H is a cohomology functor then there is a short exact sequence O+lim' H(X,)+ H(X)-+lim H(X,)+O. PROOF. These results are immediate from application of H to the exact triangle defining wcolim X,. 0 In the presence of the representability theorems Proposition 1 is just a rephrasing of Proposition 3.4. A useful application of Proposition 1 is that homology and cohomology functors are 'determined' by their values o n sphere spectra, precisely:
PROPOSITION 2. If 7 : H +K is a natural transformation of homology or cohomology functors such that v ( S ' ) is an isomorphism for all r then 7 is a natural equivalence. PROOF.We will consider only the case of H and K cohomology functors the other case being if anything easier. Note first that the diagram
CH.
BASICPROPERTIES OF FUNCTORS
4, 11
commutes because composing with any projection get the commuting diagram
47
n K(X,)+
K(X,) we
For X in Y let X' +X 2 + . . .-+ X be a crude cellular tower. If X = X" for some n we see that v ( X ) is an isomorphism by a simple induction starting with the fact that v(U S') = v ( S ' ) is an isomorphism. For a general X, application of Proposition 1 gives
n
0-Iim' 0-
H(Xr)-H(X)-lim
H(X')-0
1
1T ( X )
1
liml K(X')-
K(X)-
lim K(X')-O
So by the 5-lemma v ( X ) is an isomorphism. 0 As an application of Proposition 2 we have the following, (b) being a weak form of the Kunneth formula.
COROLLARY 3. Let H be a homology or cohomology functor. (a) If H ( 9 ) = 0 for all r then H ( X ) = 0 for all X . (b) If H ( X ) = 0 then H ( X A Y )= 0.
PROOF. (a) is immediate from Proposition 2. And (b) is immediate from (a) since if H is a homology (resp. cohomology) functor then so is H ( X A ). 0 Another application is t o a universal coefficient theorem and a Kunneth formula in the following context. A spectrum X is a ring spectrum if there are maps m : X A X + X and i : S + X such that the composites X = S A X"-X A X A X and X = X A S I n i \ X A X A X are the identity. And a spectrum Y is an X-module spectrum if there is a map n : X A Y + Y
48
-
HOMOLOGY AND COHOMOLOGY FUNCTORS
such that the composite Y = S A Y
X
A
[CH.
4, 1
Y -li Y is the identity and
commutes. Then 7r,(X) is a ring and the functors X,, X*, Y,, Y* all take values in the category of 7r,(X)-modules. That is, the product of f :S"+Xandg:Sb+Xisgiven byS"+b=Sa/\Sbfng,X/(X~Xand for instance the product of f : S" + X and g : sbU+ Y (in Y*(U)) is given by satbU= S" A s b U X A Y 4Y. COROLLARY 4. Let X be a ring spectrum and Y a n X-module spectrum. (a) If 7r.J Y) is a flat 7r,(X)-module then there is a natural isomorphism X * ( W ) O T * ( X ) ~ * ( Y ) +Y*(W). (b) If m*(Y) is a n injective m,(X)-module then there is a natural isomorphism Y*( W)+ Hom,,cx)(X,( W), 7r,( Y)). PROOF.The following diagrams define the indicated natural transformations: (a) S ; ~ S A S - , ( W A X ) A Y ; ~ W A ( X A Y ) ~ W A Y , (b) S ~ X WA- X A Y A Y . The conditions on 7r,(Y) imply that X * ( )@,.(~)7r*(Y) is a homology functor and Hom,,(x)(X( ), 7r*(Y)) is a cohomology functor. The corollary is then immediate from Proposition 2. 0 There is a further variant of Corollaries 3 and 4 that will be needed later. PROPOSITION 5. If X is a ring spectrum and Y is a n X-module spectrum thenX,(U)=O impliesthat Y * ( U ) = O = Y*(U).
PROOF.We are assuming that U A X = 0. Therefore U A X A Y = 0. But the structure map X A Y + Y is an epimorphism (split by Y = S A Y*X A Y) and therefore U A Y = 0, i.e. Y,(U) = 0. For the other part let f : U + Y be an element of Y*(U). Then f can be factored as
CH.
4,21
and thus f
FINITELY DETERMINED FUNCTORS
49
= 0.
Note that all that is required of Y in Proposition 5 is that the map Y A X be a monomorphism. Y
2. Finitely determined functors In studying the structure of Y a problem of obvious significance is that of analyzing the relationship between Y and the subcategory S. Central to this is the functor, introduced in Chapter 2, that assigns to each X in Y a category A (X)whose objects are maps f : X, +X with X, E 9 ' (the small-in fact countable+quivalent subcategory of 9 of Proposition 3.11) and whose morphisms are commuting diagrams
X
with f and g in A(X).And to i : X + Y, A assigns the natural transformation A (i) :A (X)+A (Y) given by A (i) of f :X, +X being if and A ( i ) of
being
Y
'"/" y h x,-x,.
50
HOMOLOGY AND COHOMOLoGY FUNCTORS
[CH.
4, 2
So A is a functor taking values in a category of categories. More precisely A takes values in the category of filtered categories. That is, given f : X, +X and g : X, +X in A ( X ) then the diagram
is in A ( X ) . And given
X
x, 7 x 0 in A ( X ) there is an exact triangle X , s X , l X , + s X , in 9’ and it follows that g factors through k via X , + X . Then k is a morphism in A ( X ) and ki = kj. In Chapter 5 we will consider the extent to which a spectrum is determined by its diagram category. Here we turn t o the analogous question for homology and cohomology functors. That is, given H : Y+ Ab, to what extent do the values taken by H evaluated on finite spectra determine the values taken on arbitrary spectra? This can be made more precise by the introduction of a functor on Y derived from H I 9 which can then be compared to H itself. If H is a homology functor then we define a functor H by letting f i ( X ) = colimAcx)H(X,), the colimit as in Appendix 1. And for f : X + Y, Hcf) : A ( X ) + A( Y ) is the map induced by the natural transformation A 0 again as in Appendix 1. Then the maps H ( f ) : H(&)+ H ( X ) for f in A ( X ) , induce a map from the colimit which defines a natural transformation x : H + H. Further XIS is an equivalence and with it we will identify HI9 and HIS. This follows from the fact that for X in 9 there is an equivalence e : X o + X in A ( X ) which is therefore terminal in A ( X ) . For then f i ( X ) = H ( X o )and via this x ( X ) is just the isomorphism H(e). Similarly if H is a cohomology functor then we define H by letting H ( X ) = H(X,), the limit as in Appendix 1. And H ( f ) is the map induced by A 0 as in the appendix. And the maps Hcf) : H ( X ) + H(X,) for f in A ( X ) induce a map to the limit which defines a natural transformation cp : H +H. Further as in the covariant case cp9 l is an equivalence of the two functors so restricted. It is left for the reader to verify that for H either a homology or cohomology functor, H is additive.
CH. 4,
21
FINITELY DETERMINED FUNCTORS
51
In the covariant case we have the following theorem originally proved by Milnor [93] (unstably).
THEOREM 6. For a homology functor H, x : H + H is an equivalence. PROOF.Applying Proposition 2 it suffices to show that is a homology functor, that it is exact and commutes with coproducts. First note that by Proposition A1.7 fi(X)satisfies: (i) for any a E A(X) there is a map f : X,+X in A(X) and b E A(Xa) with H ( f ) ( b )= a, (ii) for any f :Xa X in A (X)and a E f i ( X a )with fiCf)(a)= 0 there is a morphism
X
in A(X)with A ( h ) ( a )= 0. To prove exactness consider an exact triangle X A Y 4 Z z s X .It suffices to show that fi(X)+ fi(Y)+ A(Z)is exact. Since fi(g) fiCf)= A(gf)=O we are left to show that kerA(g)Cirn&(f). So consider a E ker fi(g). By (i) we have i : Ya+.Y in A (Y)and b E &(Ya)with f i ( i ) ( b )= a and by (ii) we have
-
in A (2)with A ( k ) ( b )= 0. This gives the diagram
x
t n
Y
t
-z
-sx t
t
x y zYa-Z,-sXy with the bottom row exact and n any fill-in map. Since the bottom row is in 9’ and HIP = H I P we get b = &(rn)(c) and thus a = A ( f ) [ H ( n ) ( c ) ] . To prove that H commutes with coproducts consider a family {Xy} y E r. There is a natural map p r : LI,-~~(X~)-+A(LI,-XY). Since A is
HOMOLOGY AND COHOMOLWY FUNCTORS
52
additive this map is an isomorphism if is a commuting diagram
[CH.
4,2
r is finite. For an arbitrary r there
LI A ( X ’ ) LA(LI X ’ ) r
i-
t-
qT
colim pA
A(LI x’)
colim LI A(xy)-colim ACr
AC I‘
A
A
where the colimits are over the finite subsets of r and 9 is the natural map. So it suffices t o show that 9 is an isomorphism. To see that 9 is monic consider an element x in colim A(UAX’). Ultimately it is represented by a map f : U + & X y with U in 9 and an element u in H ( U ) . If 9 ( x ) = 0 then there is a diagram
u-LIX’ A
gs
.1
v-LIXY with g in 9’ and H ( g ) ( u )= 0. Since V is small h factors through a finite coproduct &XY with A C B. But (f, u ) and (h, H ( g ) ( u ) )represent the same element in colim f i ( u A X y ) .Therefore x = 0. To see that 9 is onto consider y in f i( &X ’ ) . It is represented by a pair (i, w ) with i : W-,&X” and w E H ( W ) . As above i factors through a finite coproduct which gives a representative of an element in colimAfi(uA Xy) mapping t o y . 0 This result is extremely important in that it shows that in the strongest That possible sense a homology functor is determined by its action o n 9. is, given H on 9 we can recapture H on all of Y by letting H ( X ) = colimncx,H(X,). This is very useful but it is also a significant restriction because it implies that certain phenomena cannot be detected using homology functors. For instance
COROLLARY 7 . Iff :X then H c f ) = 0.
+ Y is an f -phantom m a p and H
is a homology functor
In the contravariant case we cannot expect H + A t o be an equivalence in general for if f : X + Y is an non-trivial f-phantom map
a.4, 21
FINITELY
DETERMINED FUNCTORS
53
then &(f) = 0 for any cohomology functor H but in particular Y*(f)-#0. However we will see in Corollary 5.17 that q ( X ) : H ( X ) + f i ( X ) if not always an isomorphism is always an epimorphism. For the present let us concentrate on the following: arguing as in Theorem 6 H + fi would be an equivalence if fi were a cohomology functor, with this in mind how close does fi come t o being a cohomology functor? First observe that H acts well with respect t o minimal weak colimits.
PROPOSITION 8. Let F : A +9’ be a diagram over a directed category with minimal weak colimit X. Then the induced map H ( X ) + limn H(X,) (X, = F ( a ) ) is an isomorphism.
PROOF.An element x of limn fi(X,) is an assignment t o each f : W + X, in A(X,) of an element xcf) E H ( W ) such that for any commuting diagram Xa
ft
.
ts
U L V H ( i ) x ( g )= xcf). Then define x’ in &(X) as follows: for f : W + X in A (X) we have the isomorphism limn [ W, X,] + [ W, X ] so f factors as
x,-x W
and we define x’cf) = x ( g ) . The element x’ is well-defined, for given two factorizations-which since A is filtered we can regard as through the same X,-gl, g2 : W + X, there is a commutative diagram
and therefore x ( g , )= x ( g ) = x(g2).By a similar argument we see that x’ is in fact an element of A ( X ) and by definition it maps t o x. It is further clear that x’ is the unique element with this property completing the proof. 0
HOMOLOGY AND COHOMOLOGY FUNCTORS
54
[CH.
4, 2
In particular, since an arbitrary coproduct is the colimit of its finite coproduct summands, the following is easily proven. COROLLARY 9. H takes coproducts to products. So H must fail to be a cohomology functor by failing to be exact. However following Adam [7]we will now show that f i does satisfy a partial exactness condition. The work involved in doing this will be substantial but beyond clarifying the nature of H this result is needed to prove the representability of homology functors. THEOREM 10. If X A Y i Z A sX is an exact triangle and X = with Xu in 9then H ( X )+H ( Y )-+&(Z) is exact.
u,,X,
PROOF.We begin with the special case of X in 9. Given y in H ( Y )with H ( i ) ( y )= 0 we must construct z in H ( Z ) with Hu)(z)= y . This requires a coherent choice of elements z(f) in H(Za)corresponding to the maps f : Za +Z in A (Z).To define an appropriate z(f) consider the diagram
with the upper sequence an exact triangle in 9’. Let g : ‘1 + ’ be a fill-in map. Then by assumption H(il)(y(g))= 0 and therefore Y k ) = H ( y l ) ( z )for some z in H(Za).Of course z is not unique so consider the coset C ( f )= z + H ( k f ) H ( s X )in H(Za). LEMMA: C(f)is independent of the fill-in map. PROOF. If g‘ is another fill-in map then g - g’ factors through i. From this it follows that y ( g ) = y ( g ‘ ) . 17
Further given
CH.
4,2]
FINITELY DETERMINED FUNCTORS
55
in A (Z) it is easily checked that H ( h ) induces a map h * : C(g)-+ Ccf).S o the problem is to choose one representative zcf) from each Ccf) so that H ( h ) z ( g ) =2 0 . To do this it will be necessary to know more about the category % whose objects are the CWs and morphisms the maps h*.
LEMMA. (a) Each morphism is a surjection of sets. (b) %OP is a filtered category, i.e. (i) given Ccf) and C(g) in V there is a diagram C ( f ) t C(h)-* C(g) in %, (ii) given i*, j * : C(f)+ C ( g ) there is a k* : C(h)+ C ( f ) with i*k* = j*k*. (c) There is at most one morphism from C ( g ) to C(f).
PROOF.(a) Consider h* : C(g)+ Ccf) and z in Ccf).There is a commuting diagram
with rows exact. For w in C ( g ) arbitrary H(j,)(w)=y ( g , ) and HCi,)H(h)(w)= H ( j 2 ) ( ~ ) . Therefore z - H ( h ) ( w )= H ( k 2 ) ( ~=) H ( h ) H ( k ) ( x )and so z is in im h*. (b) For the first part we have C ( g )t C ( f l g ) + C ( f )with the obvious maps. As for the second part we are given
Z
So if Z , ~ Z , & ~ + s Z is , exact then f factors as hk. Then k * : C(h)-, Ccf) is as desired. (c) This is immediate from (a) and (b)(ii). Cl The category % is not quite tractable enough so we enlarge it by with the same objects as those of V and a defining a category
HOMOLOGY AND COHOMOL~GY FUNCTORS
56
[CH.
4, 2
morphism k : Ccf)-. C ( g ) for each commuting diagram of sets
Then
g has the following properties.
LEMMA. (a) There is at most one morphism from C ( g ) to Ccf). (b) Each morphism is a surjection of sets. (c) Given Ccf) and C ( g ) there is a C ( h ) and morphisms C ( g ) C ( h)+ CV). (d) There are only countably many equivalence classes of equivalent objects in g. +
PROOF.(a)-(c) are immediate from (a)-(c) of the preceding lemma. (d) It will suffice t o show for f, g : 2, +Z in A (2)that if kf = k g then Ccf) and C ( g ) are equivalent (in g) for the set of maps [Z,, s X ] is countable (that is obj 9 'is countable and for U, V finite spectra [ U, v] is countable). But if kf = k g then we have Y
k -z -sx
---
tr ya and
I
tf
z,
m
k
II sx
Y -2 -sx ts' t g I1 I m Y, 2, sx where m
=
kf
= kg.
And we have
CH. 4,
2)
FINITELY DETERMINED FUNCTORS
57
with nil = f and nj2 = g’. Let H ( l ’ ) ( z )= y ( n ) and z1= H ( i l ) ( z )and z2= H(i2)(z).Then H(l)(zl) = y ( f ‘ ) and H(l’)(z2) = y ( g ’ ) . Therefore
sets up the equivalence between Ccf) and C(g). 0 It is now an easy matter to show that lime C(f) is nonempty for it follows from the lemma that there is a cofinal sequence C, +- C2+- * * * of set surjections in this category. But an element { z c f ) } of this limit is in particular a choice for each f : 2,+ Z in A ( Z ) of an element ~ ( fin) C ( f )C H ( 2 , ) such that for
Z
in A ( Z ) , H ( h ) z ( g )= z(f). that is an element z in f i ( z )with f i ( j ) ( z )= y. We turn now to the general case, that is, an exact triangle X, with X , E B and y E fi( Y) with H ( i ) ( y ) = 0. X, Y A Z A For f c A let C ( f )C obj A ( Z ) be the set of maps f : 2, + Z such that kf factors through the inclusion s&X,G s ~ A X , .Consider the family of pairs (c z) where f C A and z assigns to each f : 2,- Z in C ( f ) an element ~ ( fE) H ( 2 , ) such that (a) given
su,
Z
commuting with f, g in C ( f ) we have H ( h ) ( z ( g ) )= 2 0 , (b) z(jf) = y ( f ) for f : 2, + Y (which of course implies that f E C ( f ) for all 0. Partially ordering these pairs in the obvious way we extract a maximal element (cz). To complete the proof we must show that f = A for then z is an element of A(Z)and A ( j ) ( z ) = y as desired.
58
HOMOLCGY AND COHOMOLOGY FUNCTORS
So suppose that there is an index consider the diagram
Q
EA -
k
i’
T.il
t m‘ R
i”
t m“
(let 0 = r U {a})and
II
k’
si’
Y-Z‘-sUx,-sY
II
2
Si
Y i _ Z - s U x , -As Y
II
r
[CH. 4,
T.
II
i2
k”
Si“
Y-Z”-sUX,-sY r
where Z’ is the weak pullback of k and il, and Z” is the weak pullback of k‘ and &so the diagram commutes and the TOWS are exact triangles. Using z we define an element z“ of fi(2”)by letting z”(f) = z(m’m‘’n. Then fi(j”)(z”) = y . Since Z“ is the weak pullback (see Proposition 3.1) we have the commuting diagram
i’
LIXp-Y
n
with the top row an exact triangle. Then fi(&p)(z”)= fi(f‘i’)(z’’)= f i ( i ’ ) ( y ) = 0 since i’ factors through i. And since p is an epimorphism (and therefore splits) f i ( i , ) ( z n ) = 0. But X, E 9 so applying the special case considered above it follows that there is an element z‘ in fi(Z‘)with f i ( m ” ) ( z ’ )= z”. Define 2’ on C(0)as follows: given f :Z, --+ 2 in C ( 0 ) there is a commuting diagram
Z-sUX,
J‘
A
m’
t
Zk’-SUX,
z>
and consequently a map g : Z,+Z’ completing the diagram. Let z l ( f )= z ’ ( g ) . Although the fill-in map g is not unique the resulting class z ’ ( g ) is. For if g l , g2 : Z +Z’ are two fill-in maps then k‘(gl - g2) = 0 = m‘(gl - g2) imply that gl - g2 = fig3 and therefore z ’ ( g l )- z‘(g2) = z’(g1- g2) = z’(j‘ig3)= fi(j‘)(zf(ig3))= y(ig3) = fi(iI(y(g3)) =o.
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Further zI satisfies (a). For if we have
Z
with f , g in C ( 0 )and g’ : Z, +Z’ is a fill-in for g then g‘h is a fill-in for f. And then H(h)(zl(g))= H(h)(z’(g’)) = z’(g’h)= zlcf).And z1also satisfies (b) for if f :Z, + Y then zl(jf) = zl(m‘jf) = z’(j’f)= H(j’)(z’cf)) = ycf). Finally zl is an extension of 2, i.e. considering f : 2, +Z in C(T)there is fill-in g :Z, +Z” and zl(f) = z‘(”’g) = H(m”)(z’(g)) = z “ ( g )= z(f). Therefore (0,zl) sits above (r,z ) in the partial ordering, which is a contradiction. 0
3. Representability theorems We come now to a major characteristic of homology and cohomology functors, namely their representability. We begin with the well-known cohomology representability theorem of Brown [35].
THEOREM 11. If H : Y + A b * representable.
is a cohomology functor then H is
PROOF.By Proposition 2 it will suffice to construct a spectrum X and a natural transformation 7 : X * +H such that q(S’) is an isomorphism for all r. Observe that for y E H ( Y ) there is a natural transformation vy : Y*+H defined by vy(W ) ( f = ) H ( f ) ( y ) .In particular for W = S’ this gives a map r,(Y)+ H(S’). We will define X as a minimal weak colimit and 9 as vx for an element we will construct in H ( X ) .To begin, let Xo = S; the coproduct over all y in H(S‘) for all r and let xo in H ( X o ) = n H ( S ; ) be n y . Then qw(S’) : r , ( X ) + H ( S ‘ ) is epic for all r. Inductively assume that we have and elements x, in H(X,) with constructed X o 4 X l & - .*X,, H(i,,,-,)(x,,,)= x,-~ and such that ker v,(S‘) C ker r,(i,). Let X,,+l be defined by the exactness of ~ S ; ~ X n ~ X n + l +where s ~ S the ; coproduct is over all y in ker vxn(S’)for all r. Then H(j)(x,,)= 0 so there is an element x,,+~ in H(X,,+,) with H(i,,)(x,,+l)= x,,. Further ker vmm(S‘) C ker rr(i,,)and T ~ ~ + ~is( S epic ’ ) since vxm(S’) is epic. Therefore
-
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we have constructed the diagram Xo+ X , + * . . . Let X = wcolim X,. Since H ( X ) + lim H ( X , ) is epic there is an element x in H ( X ) mapping t o x,, for all n. From this we get colim ml(X,,)-
m,(X)
and it follows from the observations made above that v,(S') isomorphism for all r. 0
is an
A standard consequence of the representability of cohomology functors is the representability of the stable natural transformations of cohomology functors. A natural transformation 8 : H + K is stable if the following diagram commutes:
-
H(sX) 12
sH(X)-
K(sX) 11
sK(X)
Let SNT(H, K) denote the stable natural transformations from H t o K .
COROLLARY 12. If H and K are cohomology functors with representing spectra X and Y respectively then there is an isomorphism of SNT(H, K ) and [ X , Y ] .
PROOF.Since H (resp. K) is naturally equivalent t o X * (resp. Y*), SNT(H, K ) = SNT(X*, Y*). Then SNT(X*, Y * )= NT(Xo, Yo), see Appendix 1, page 443 = [ X , Y ] by Yoneda's lemma (Lemma A1.3).
0
There is an important variant of Theorem 11 that among other things will imply the representability of homology functors.
THEOREM 13. If H : 9-*Ab* is a contravariant exact functor then H is representable in 9,i.e. there is a spectrum X (in 9)and a natural equivalence 7 : [ ,XI* +H.
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PROOF.We define H : 9’-Ab* as above (i.e. A(X)= limn(x)H ( X , ) ) and then using Proposition 8, Corollary 9 and especially Theorem 10 we can duplicate the procedure used to prove Theorem 11. That is, given Y in Y and y E A(Y) there is a natural transformation vY : [ , Y ] * + H (the left-hand side restricted to 9) defined by qy(W ) ( f )= f i ( f ) ( y ) . And as in Theorem 11 it suffices to construct X and x E f i ( X ) such that ~ ~ is(an9 ) isomorphism for all r. For it is easy to see that if 7 : H + K is a natural transformation of two exact functors on 9 and ~ ( 9 is an ) isomorphism for all r then 7 is an equivalence. But the properties of proved above give what we need to carry through the construction of X and x of the earlier representability proof. 0 Given two contravariant exact functors defined on 9, H and K, let us consider the relationship between maps of the representing objects and natural transformations of the functors. If X represents H and Y represents K there is of course a map [X, Y]+SNT,(H, K). However as opposed t o the situation in Corollary 12 we cannot expect this map to be an isomorphism in general. For if f : X + Y is a non-trivial f-phantom map then by definition it will induce the zero natural transformation of the represented functors. T o make this more precise let [X, YIr be the quotient of [ X , Y ] by the subgroup of f-phantom maps. Then there is a factoring
and we will prove below that [ X , Y]f+SNT,(H, K) is an isomorphism giving the desired relationship between the representing objects and the natural transformations. First we have a representability result for fi.
PROPOSITION 14. The natural equivalence on 9, 7 : [ ,XI + H, extends to a natural equivalence on Y,+j: [ , X I , + 12. PROOF.Let X and x E A ( X ) be as in Theorem 13, then we define 7jx : [ , XIf + by &(f) = f i ( f ) ( x ) .It is easily seen that 7jx is well-defined and monic. To see that it is an epimorphism suppose that we are given Y in Y and y E f i ( Y ) . Then observe that the construction of X in = Y @ X and ql= (y, x ) . Theorems I 1 and 13 could have begun with That is, we can construct a tower Xo+ X , +. - .+wcolim X, = X’ and
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elements x, E f i ( X , ) giving x' in f i ( X ' ) such that vx.: [ ,X'l- H is an isomorphism and if f Ig :X0+ X' is the composite then fiCflg ) ( x ' ) = xo. Therefore
and g is an equivalence and hence &(g-'f)
= y.
0
THEOREM 15. If H and K are contravariant exact functors on 9represented by X and Y respectively then [ X , Y],+SNT,(H, K) is an isomorphism.
PROOF.It is easy to see that a map from X to Y induces the zero transformation precisely when it is an f-phantom map, so we must show that a stable natural transformation comes from a map of the representing spectra. Identifying H (resp. K ) with [ , X I * (resp. [ , Y ] * )we have by Proposition 14 a map 8 : R(X)+SNT,([ ,XI*, K ) given by 8(x)(f) = x(f). Now suppose that we are given a stable natural transformation T : [ ,XI*+ K, we want to define x E R ( X ) with 8(x) = T. But for f : U + X i n A ( X ) , ~ ( f ) E K ( U ) s o l e tx ( f ) = ~ ( f ) . T h e n x E k ( X ) a n d e ( x ) = 7. 0 We turn now to the covariant case proving a representability theorem due to Adams [7]. THEOREM 16. If H : Y-+Ab, is a homology functor then H is representable.
PROOF. By Theorem 6 there are natural equivalences colimA(y)H(Ya)+ H ( Y ) and colimA(y)P*( Y, A X ) + P*( Y A X ) so it suffices t o find X and a natural equivalence of H and P*( A X ) restricted t o 9. But the Spanier-Whitehead duality functor D of Chapter 1 sets up a correspondence between homology and cohomology functors defined on 9, i.e. D H ( U )= HD(U).Applying Theorem 13 there is a spectrum X and a natural equivalence DH = [ ,XI". And for U in 9 there is an equivalence natural in U : [D(U ) ,X I , = P*( U A X ) . Then since D2= I on 8 we get the desired natural equivalence (on 9):
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Natural transformations of homology functors are similarly represented. Let H and K be homology functors represented by X and Y respectively. THEOREM 17. There is an isomorphism of SNT(H, K ) and [ X , Y ] ,
PROOF.The restriction map SNT,(H, K)+SNT9(H, K) is an isomorphism with inverse given by 7 ( X )= colim,,(,) ~ ( x ,Spanier-Whitehead ). duality gives rise to an isomorphism SNT9(H, K) = SNT9(DH, D K ) and by Theorem 15 this latter group is isomorphic to [ X , Y ] , Combining these observations we get the desired isomorphism of SNT,(H,K) and [ X YIP 0 COROLLARY 18. If H is a homology functor represented by X and Y then X and Y a r e equivalent.
PROOF.The natural equivalence of 7r*( A X ) and 7r*( A Y ) gives rise to f maps X e Y with 7r,cfg) = 1 and 7 r , ( g f ) = 1. Therefore 7r,(f) is an I3
isomorphism and hence f is an equivalence. 0 By Theorem 11 and 16 there is a natural correspondence between homology and cohomology functors pairing those functors represented by the same spectrum. However in general there will be no simple relationship between the values taken by corresponding functors. For instance, there are spectra W, X, Y, Z such that (1) W , ( X ) # 0 and W * ( X )= 0, (2) Y , ( Z )= 0 and Y * ( Z )# 0. For an example of (1) consider Theorem 16.16. And an example of (2) is given in Chapter 6.