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(0, m) thus w(n, = wo(n) Qo(n)+ 4>(0, @(O, p(n')). p(nl)). We We check check that this does does as as is is inin@(n,n') nl) = tended: tended:
weo(N)(n)
= w(n, 0) = wo(n)
whilst
w(f x I)(m, m') = w(f(m), J(f)(m')) = wo(f(m)) + 4>(0, pJ(f)(m')) - 4>(m,O) + 4>(0, m') = 4>(m, m'). 0
Remark. If j(M) :: M M -----+ + J(M) J ( M ) is is a monomorphism monomorphism for for each each M M Remark. f , cofibrations cofibrations will be monomorphisms monomorphisms for for as JJ(f)j(M) then as (f)j (M) == j(N) j (N) f, for all all M gives gives a perfectly theory, however taking J(M) J ( M ) == 0 for such a theory, acceptable monad (even if it is one one that is is a bit boring!) and clearly acceptable fact, in that case cofibrations cofibrations need not be monomorphisms monomorphisms -- in fact, in that case case, every every morphism morphism is is a cofibration! cofibration ! case, Cofibrations Trivial Cofibrations class of morphism morphism that we study will be the trivial cofibraThe next class tions. This will quickly quickly give us an understanding of the structure of tions. each of the important classes classes of morphisms in this type of theory. each earlier discussion, discussion, for for the cylinder enriched enriched with cubical filler filler In the earlier conditions, we saw that, that, there, there, trivial cofibrations cofibrations were were strong deformadeformaconditions, retractions. Many of the conditions conditions of the non-additive non-additive Theorem tion retractions. (6.9) are no longer longer valid in this additive setting, but that result gives gives (6.9) 66
us some idea of what to expect. Lemma (8.3). If f : M -----t N is a trivial cofibration, then f is a split monomorphism, i. e. there is some r : N -----t M, with r f = I d M , and moreover fr
~
Id N .
Proof. Suppose f is a trivial cofibration with homotopy inverse 9 : N -----t M and homotopies H : gf ~ Id M and K : fg ~ Id N . We
extend H : M x [ -----t Mover N x I using the homotopy extension property of f : This gives us a homotopy f:NxI-----tM
such that feo(N)
= 9 and f(f x 1) = H.
Set r
= fel(N),
we calculate
rf: rf
=
fel(N)f
= f(f x I)el(M)
Hel(M) =
Id M
as required, so r splits f. Of course f : 9 ~ r so ff : fg ~ fro We saw earlier that the additive cylinders have natural involutions so the relation of homotopy is symmetric., in fact ffi(N) : fr ~ fg. Now K : f 9 ~ I d N and additive cylinders come with a natural subdivision, so the relation of homotopy is transitive and fr~fg~IdN
implies fr
~
I d N , completing the proof.
0
Proposition (8.4). A morphism f : M -----t N is a trivial cofibration if and only if f is a split monomorphism with complementary summand in N, some object K for which j(K) is a split monomorphism. Proof. Suppose f : M
N is a trivial cofibration, then from Lemma (8.3), f is a split monomorphism and N ~ M EB K for some subobject K of N. We also saw that I d N ~ fr, where r : N -----t M is -----t
the splitting. We denote this homotopy by H : NEB J(N) 67
-----t
N, so
H(n,O) = nand H(n,j(N)n) = fr(n). If we write n = (m, k), then fr(m, k) = (m, 0), whilst
j(N)(n) = (j(M)(m),j(K)(k )). We therefore have
=
H(n,j(N)(n)) - H(n,O) = fr(n) - n = fr(m, k) - (m, k) = (m,O) - (m, k)
H(O,j(N)(n))
-(0, k) . Set h(x) = projK H((O, 0), (0, - x)) one gets a morphism from J(K) to K such that hj(K)(k) = k, i.e. splitting j(K). Conversely if N ~ MffJK and f : M - f N is the first inclusion then f is a cofibration since J (1) is split by J (r), where r is as before. The splitting, r, gives a homotopy inverse to f . One of the two homotopies is trivial, the other is constructed from the splitting of j(K) by reversing the algebraic argument used in the first part of this proof. 0
Remark. This is a good place to note the simplicity of the process of combining additive homotopies. Aspects of this have been in both of the above proofs and we extract them here as they make life much easier if they are understood. (i) Suppose H : f
~
H(m,O)
9:M
-f
= f(m),
and so H is determined by
N then
H(m ,j(M)(m))
f and h : J(M)
-f
= g(m) N given by
h(x) = H(O,x). In fact H(m, m') = f(m) + h(m') and h(j(M)(m)) = g(m) - f(m). Because of this the reverse homotopy, Hi(M) : 9 ~ f is very simply
Hi(M)(m, m') = g(m) - h(m') Exercise. Check that this is a homotopy: 9
~
f , as claimed.
(ii) Now suppose H: fo ~
h
is determined by h : J(M) 68
-f
N
K:
II ~ h
is determined by k : J(M)
then define L: MffiJ(M)
--+
--+
N
N by L(m,m') = lo(m)+h(m')+k(m').
Exercise. Check that L: 10
~
h.
The Additive Factorisation Lemma
As we have very simple full descriptions of cofibrations and trivial cofibrations it is very easy to prove that with respect to an additive cylinder as given above, the factorisation of morphisms as cofibrations followed by homotopy equivalences holds.
Proposition (8.5). Given any morphism 1 : M --+ N, 1 = pfi f where if is a cofibration and pf a homotopy equivalence, homotopy inverse to a trivial cofibration. Proof. The idea of the proof is similar to that given earlier for the non-additive case but here some details simplify and so we will give a separate proof. The mapping cylinder of 1 is given, as above, by the pushout
M
_---=--1__
eo(M)1
N lit
M ffi J(M) --:7f-f- M f
so M f ~ N ffi J(M) with it = incN : N --+ N ffi J(M) and 7ff(m, m') = (f(m), m'). The morphism it is a trivial cofibration by Proposition (8.4) , since j(J(M)) : J(M) --+ J2(M) is naturally split by the multiplication f..l : J2 --+ .J of the monad J. The morphism Pf : Mf --+ N is the projection onto the summand, N, hence Pf is right inverse to it, Pfit = Id N, and Pf is a homotopy equivalence. We set if = 7ffel(M) so if(m) = (f(m),j(M)(m)) elementwise. It is clear that Pfif = I, so it only remains to check that if is a cofibration. Using Proposition (8.2), we check to see if J(i f ) is a split monomorphism: 69
J(i,) : J(M) -- J(N (J) J(M)), but p(M)Jj(M) =
IdJ(M)
so the composite
J(N (J) J(M)) -- J2(M) L
J(M)
can easily be checked to split J(i,). (Use the exercise on how near J is 0 to being additive.) Homotopy Equivalences There are several easy consequences of the definition of additive homotopy equivalences that are useful in what follows . These are left as exerCIses.
Exercises. (i) If 1 and 9 are homotopy equivalences and gl is defined, prove that 9 1 is a homotopy equivalence. (ii) If f' is homotopy inverse to the homotopy equivalence I, prove that f' is a homotopy equivalence. (iii) If 1 ~ 9 and 1 is a homotopy equivalence, prove that 9 is a homotopy equivalence. (In each case, give details of how the necessary homotopies are defined. For (ii), the proof is more or less immediate from the definition.) (iv) If 1 is a homotopy equivalence and 9 is such that Ig is defined and is a homotopy equivalence, prove that 9 is a homotopy equivalence, giving the construction of the homotopies. Finally (v) If 9 is a homotopy equivalence and 1 is such that 19 is defined and is a homotopy equivalence, prove that 1 is itself a homotopy equi valence. Some of these results will be used in the next proposition which gives a complete description of homotopy equivalences in this additive theory.
Proposition (8.6) (cf. Hilton (1965)). A morphism 1 : M ~ N is a homotopy equivalence if and only if there is a factorisation of f in the form
70
M
,M
(f)
K
where K is such that j (K) : K
o
~'
N
-----t
projN
(f)
J (M) - - - , N
J (K) is a split monomorphism.
Proof. Since incM is a trivial cofibration, projN is a homotopy inverse to a trivial cofibration and () is an isomorphism, any j with such a factorisation must be a homotopy equivalence. Conversely if j is a homotopy equivalence, Proposition (8.5) gives a factorisation j = Plil with PI = projN, a homotopy equivalence, but then i I must also be a homotopy equivalence and, as it is a cofibration, it must be a trivial cofibration, hence by Proposition (8.4), N (f) J(M) must be isomorphic to M (f) K with j (K) a split monomorphism, i.e.
M ~N
as required.
(f)
J(M) = M
-----t
M
(f)
K ~N
(f)
J(M)
0
Dold 's Theorem in the Additive Case
The conditions that had to be imposed on a cylinder for the proof of Dold's theorem (6.3) are not satisfied for the general cylinder in the additive setting considered here, but the general structure of the proof is the same. Where filler conditions were used however, we use instead the detailed description of cofibrations and the neat way in which homotopies combine. The definition of homotopy under A, ~,is the same and the following Lemma (8.7) is the analogue of Proposition
(6.2). Lemma (8.7). (i) Suppose
A
!~
M
j
,N ,
9
are two maps j, 9 : i
-----t
i' and : j ~ 9 is given by 71
+
II> (x , x') == f(x) @(x, f (x) +
then then II>@ is is aa homotopy homotopy under under A A if if and and only only if if
Ax[
A i'
i x I[
Mx [
II>
·N
commutes commutes but but lI>(i @(ixx I)(a, I ) ( a ,a') == lI>(ia, @(ia,J(i)a') J(i)al) == f(ia) f (ia) + +
Exercise. Exercise. (a) (a) Show Show that if II> @ :: fo fo ~ 21 h fl then then lI>i(M) @i(M) :: h fl ~ = fo isis also also under A A.. (Use (Use the the description description of the the reverse reverse homotopy homotopy given given in in the the Remark (8.4) and and then then apply apply part (i) (i) of the the Lemma.) Lemma.) Remark after after Proposition (8.4) A
A
=A
(b) (b) Show Show that if if fo fo ~ 21 hand fl and h fl ~ 21 12 f 2 then then fo fo~ 12, f2, by using the the same same Remark 0 Remark as as in in (a) (a),, but this this time time for for the the composed homotopy. homotopy.
(2.11).. The next step step is is to to prove prove an an additive additive version of Proposition Proposition (2.11) The In fact fact that that result did did not use use any any special special properties properties of the the cylinder cylinder In so the the proof could could be be carried carried over over word word for for word, word, however however an an additive additive so is quite quite 'fun' so so we we have have included a separate proof. proof. proof is (8.8). If in in a diagram Proposition (8.8). A
/\
x---· y f
72
in an additive category C with cone monad, J , we have fi c:: j and i is a cofibration, then there is a morphism g : X ~ Y, f c:: 9 and 9 : i ~ j, that is gi = j. Proof. Suppose q, : A E9 J(A)
~
Y, q,: fi c:: j, so
q,(a, a') = fi(a)
+ cp(a')
where cp(j(A)a) = j(a) - fi(a), cp : J(A) ~ Y. As i is a cofibration, there is a splitting p : J(X) ~ J(A) for J(i). Let w(x,x') = f(x) + cp(px') , then w(x,O) = f(x) and we define g(x) = w(x,j(X)x) = f(x)+cp(pj(X)x). Thus W : f c:: 9 and it remains to check that gi = j :
gi(a)
W(ia,j(X)ia)
=
+ cp(pj(X)i(a)) fi(a) + cp(pJ(i)j(A)a) fi(a) + cp(j(A)a) fi(a) + j(a) - fi(a)
=
j(a)
=
fi(a)
since j(X)i = J(i)j(A)
as required.
0
This follows exactly the same lines as the proof of Proposition (2.11) but the extra precision in how to extend homotopies makes the calculations explicit. Leading to Dold's theorem, the key Lemma (6.4) is next on our agenda to attack.
Lemma (8.9). Suppose 9 : i
~
i with i a cofibration, i : A ~ X,
and 9 c:: I dx , then there is a morphism g' : i ~ i such that g' 9 ~ I dx . Proof. Suppose q, : 9 c:: Id x , q,: X E9 J(X)
~
X, q,(x, x') = g(x)
+ cp(x'),
so cp(j(X)x) = x - g(x). Let q,' : A E9 J(A) ~ X be given by q,'(a, a') = i(a) + cp(J(i)a') and use a splitting p : J(X) ~ J(A), for J(i) to extend q,' over X x I with starting value I dx ; explicitly define 73
W(x, x') = x + cp(J(i)p(x')) and then set g'(x) = w(x,j(X)x) = Wel(X)(X). This gives:
g'(x) = x + cp(J(i)pj(X)x) and of course, '11 : I dx ~ g' .
If x = ia ,
g'i(a) = i(a) + cp(J(i)pj(X)i(a)) = i(a) + cp(J(i)pJ(i)j(A)(a))
+ cp(J(i)j(A)(a)) i(a) + i(a) - gi(a)
= i(a) =
= i( a),
,.
.
I.e. 9 t =
.
to
Set'll' : g'
~
I dx for the reverse homotopy so then
w'(g x I)(x, x') = g'g(x) - cp(J(i)pJ(g)(x')). We have w{gxI)
g'g call the composite homotopy,
4>
g - - I dx ;
r, so by our earlier remarks
r(x,x') = g'g(x) - cp(J(i)pJ(g)(x')) + cp(x') . We leave it as an exercise that, indeed, r(x,O) = g'g( x) and but
r
r(x , j(X)x) = x,
is a homotopy under A
r(x , x') = g' g(x) + ,(x') where ,(x') = cp(x') - cp(J(i)pJ(g)(x')) and again a simple calculation - left as an exercise - shows that ,J(i) = 0. 0
74
Theorem (8.10). Suppose C zs an additive category with cone monad J. Let
A
/\
X - - - , X'
f
be a commutative diagram in C in which i and i' are cofibrations. Then if f is a homotopy equivalence, it is a homotopy equivalence under A. Proof. The proof of the reduction of Theorem (6.3) to Lemma (6.4) cannot be bettered. Each of the components of that proof can be replaced by their additive versions proved above. 0
Given the success of proving an additive version of Dold's theorem it is natural to expect that other results, the Relativity Principle and the Gluing Theorem, would also adapt to the additive case. They do but do not seem to do so for a general additive cylinder. The problem is that pushouts of cofibrations do not seem to be necessarily cofibrations. We shall reexamine this in section 4 of the next chapter.
75
II
Homotopical Algebra
The The term 'Homotopical Algebra' was was the the title title of a set set of lecture notes algebra, in in tes by D. D. Quillen (1967). (1967). The The basic ideas ideas of homological algebra, which which methods methods from from topological homology homology theory theory such such as as chain comcomplexes, long long exact exact sequences, sequences, etc., etc., were were applied applied to to problems problems in in algebra, algebra, had been widely widely known known since since the the publication of of the the fundamental fundamental book by Cartan and and Eilenberg Eilenberg (1956), (1956), reinforced by by MacLane's 'Homology' (1963) (1963) and and Grothendieck's T6hoku T6hoku paper (1957). (1957). Using Using similar similar methods but in a non-additive setting Dold (1958), Dold-Puppe thods in non-additive setting (1958), Dold-Puppe (1958, (1958, 1961) 1961) and and others others had shown shown the the applicability of homotopy theoretic theoretic arguments arguments to to algebra, algebra, hence homotopical algebra. algebra. The The theory theory as as presented sented by Quillen Quillen was was an an abstract form form of homotopy theory theory and and was was designed designed for for application application in in deformation deformation theory theory and and the the cohomology cohomology of of algebras It was was developed developed from from three three basic basic types types of of algebras (Quillen (Quillen (1970)). (1970)). It example: example:
(i) (i)
topological homotopy homotopy theory theory topological
(ii) (ii)
simplicia1homotopy homotopy theory theory simplicial (Kan, Moore Moore and and others) others) (Kan,
(iii) (iii)
the theory theory of of derived derived categories categories the (Grothendieck, Verdier) Verdier) (Grothendieck,
The aspect aspect of of homotopy homotopy theory theory applied applied to to algebra algebra was was mirrored mirrored by by an an The aspect of of algebra algebra applied applied to to homotopy homotopy theory theory in in the the third third basic basic source source aspect
76
for the theory, Quillen's paper (1969) on rational homotopy theory.
1. Quillen Type Theories Quillen's initial contribution was to introduce the notion of a model category. This attempted to axiomatise the main features common to the motivating examples mentioned above. We recall the definition.
Definition (1.1). A model category is a category, C, together with three distinguished classes of morphisms, we, col, fib , called weak equivalences, cofibrations, and fibrations, respectively. A cofibration (fibration) is said to be a trivial cofibration (trivial fibration) if it is also a weak equivalence. The weak equivalences , cofibrations, and fibrations are assumed to satisfy the following axioms (M 0) - (M 5). (M 0) C has finite limits and colimits. (M 1) Given a commutative diagram of solid arrows in C
where i is a cofibration, p is a fibration, and where either i or p is trivial, then the dotted arrow exists so that the triangles commute. (M 2) (a) Any map I may be factored f = pi where i is a trivial cofibration and p is a fibration. (b) Any map f may be factored f = pi where i is a cofibration and p is a trivial fibration.
77
(M 3) (a) Any isomorphism is a cofibration. The composite of cofibrations is a cofibration. If
A
·B
(1.2)
J
X---y is a pushout diagram in C and i is a cofibration, then j is a cofibration. (b) Any isomorphism is a fibration. The composite of fibrations is a fibration. If
(1.3)
A
B
is a pullback diagram in C and p is a fibration, then q is a fibration. (M 4) (a) If (1.2) is a pushout diagram in C and i is a trivial cofibration, then so is j. (b) If (1.3) is a pullback diagram in C and p is a trivial fibration, then so is q. (M 5) Any isomorphism is a weak equivalence. For two morphisms in C
if any two of / ,g, and g/ are weak equivalences, then so is the third. Starting from the notion of a model category a rich abstract homo78
topy theory has been developed by Quillen (1967) including classical topics such as cofibration and fibration sequences, and Toda brackets. Cofibration and fibration sequences are discussed here in Chapter IV.
Remark. (a) The notion of a model category is self dual, that is, if C is a model category, then the dual category cop is also a model category, where the roles of cofibrations and fibrations are interchanged. (b) By (M 0) a model category C has an initial object e and a terminal object t . An object A of a model category C will be called cofibrant if the unique morphism eA : e ~ A is a cofibration, and fib rant if the unique morphism tA : A ~ t is a fibration. Example. Str(Zlm (1972) has proved that the category Top has the structure of a model category where the elements of cof are the closed cofibrations (i.e. closed continuous maps which are cofibrations in the sense of (I.2.5)(b) with respect to the canonical cylinder Ion Top) , the elements of fib are the fibrations (i.e. continuous maps which are fibrations in the sense of (I.3.2)(b) with respect to the canonical co cylinder P on Top), and the elements of we are the homotopy equivalences (see (1.2.4)). Every object is cofibrant and fibrant. Introducing the notion of a category of fibrant objects K.S . Brown (1973) has shown how one can obtain a large part of Quillen's theory by using fibrant objects only. We shall dualise his axioms to handle a theory of cofibrant objects.
Definition (1.4). Let C be a category with finite coproducts and an initial object e. Assume that C has two distinguished classes we, cof, of morphisms , called weak equivalences and cofibrations. A cofibration is said to be a trivial cofibration if it is also a weak equivalence. We define a cylinder object for an object B in C to be some object B x [ together with maps eo, el : B
~
B x I , a: B x I
~
B
such that a is a weak equivalence and the induced map eo
+ el
: BuB ~ B x I 79
is a cofibration satisfying
the codiagonal morphism V'8: BUB
---+
B.
We will call C a category of cofibrant objects if the following axioms (C 1) - (C 5) are satisfied.
(C 1) Any isomorphism is a weak equivalence. For two morphisms in C
if any two of is the third.
f, 9 and 9 f are weak equivalences, then so
(C 2) Any isomorphism is a cofibration. The composite of cofibrations is a cofibration.
(C 3) Given any diagram
with i a cofibration, the pushout XU B exists A
A---U-_ B
i\
\J
X---XUB v
A
and j is a cofibration. If i is trivial, so is j .
(C 4) For each object B, there is at least one cylinder object (B x I, eo, el, o)
80
(C 5) For any object B, the unique map eB : e --+ B is a cofibration. (We will say that an object B is cofibrant, if eB is a cofibration.)
+ eI) = \1 B, we have aeo = a(eo + et)io = Id B
Remarks 1. Since a( eo
where io : B --+ BuB is the inclusion into the first 'cofactor'. Thus by (C 1), eo is a weak equivalence. In fact eo (and eI) are trivial cofibrations since
eo = (eo
+ eI)io
and io is the map induced in the pushout
e
eB
B
eB I B
IiI ,BUB 20
and thus is a cofibration, hence eo is a cofibration. 2. It is important to realise that B x I is not always the result of forming a product of the object B with some object I; it may, in fact , not be functorial in B, although eventually we shall assume that it can be so chosen.
Example. If C has a Quillen model category structure (see (1.1)), the full subcategory of cofibrant objects in C is a category of cofibrant objects in the sense of definition (1.4) . Remark . More recently, Baues (1989) has introduced another method of generating a cofibration structure on a category. Adding one essential axiom (axiom on fibrant models) to those of K.S. Brown, Baues introduces the notion of a cofibration category which allows one to handle K.S . Brown type theories where the objects are not necessa-
81
rily cofibrant. In full detail, a cofibration category in the sense of Baues is a category, C, together with two distinguished classes of morphisms, we, col, called weak equivalences and cofibrations such that the axioms (C 1), (C 2) , (C 3) of K.S . Brown and the following axioms (C 3') , (FA) , (FM) are satisfied.
(C 3') Given any diagram
with i a cofibration, the pushout XU B exists A
A
il
_----=u=____
B
Ij
X---XUB
v
A
and j is a cofibration. If u is a weak equivalence, so is v. (FA)
Factorisation axiom: Any map I may be factored I = pi where i is a cofibration and p is a weak equivalence.
(FM) Axiom on fibrant models : For each object X in C there is a trivial cofibration X -- RX where RX is fibrant. (A trivial cofibration is a cofibration which is also a weak equivalence. An object R of C is fibrant , if each trivial cofibration i : R - - Q admits a retraction r : Q - - R , i.e. ri = Id R .) Note that Baues has a discussion on how the axioms (C 3) and (C 3') are interrelated (see Baues (1989), Lemma (1.4)). In particular, a category of cofibrant objects satisfies (C 3'). We return to this in the next section when we have proved certain results on mapping cylinders in this context. 82
Other theories of much the same type as the above have been studied by other authors, for instance Heller (1968, 1970, 1972) and Shitanda (1984). We now relate our approach to abstract homotopy theory starting from the notion of a cylinder functor to K.S. Brown's approach via cofibration structures.
Definition (1.5). Let I = (( ) x I, eo, el, a) be a cylinder on a category C. We say that I is a generating cylinder for a cofibration structure on C if C together with weak equivalences taken to be homotopy equivalences (1.2.4) and cofibrations as in (1.2.5)(b), has the structure of a category of cofibrant objects (1.4) in such a way that, for any object X in C,
is a cylinder object for X.
Remark. The terminology generating for this type of situation is suggested by Grothendieck's (1984) study of homotopy structures. It is important to note that not all categories of cofibrant objects need have generating cylinders. Theorem (1.6). Let C be a category which has push outs and an initial object, e. Let 1= (( ) x I, eo, eI, a) be a cylinder on C which satisfies DNE(2) and E(3,1,1) and in which ( ) X I preserves pushouts and the initial object, then I is generating. Proof. Since C has pushouts and an initial object, C has finite coproducts. We note that ( ) x I preserves finite coproducts since it is assumed to preserve pushouts and the initial object. We have to verify the axioms (C 1) - (C 5) of a category of cofibrant objects (1.4). Axiom (C 1) is an easy exercise, (C 2) follows from Proposition (1.2.6), (C 3) from Proposition (1.2.7), Theorem (1.6.9), Remark (1.6.6), and Corollary (1.6.12). For any object X of C, 83
(X x I, eo(X) , el(X), £7(X)) is a cylinder object by Proposition (1.5.8) and Proposition (1.5.10). Thus (C 4) holds. Finally, every object of C is cofibrant. This can easily be seen from the definition, since ( ) x I is assumed to preserve the initial object. This gives (C 5) . Thus I is generating. 0
Example. The canonical cylinder T on the category Top of topological spaces is generating and induces on Top the structure of a category of cofibrant objects. Baues (1989) has also an approach to abstract homotopy theory based on the notion of a cylinder. He introduces what he calls an I- category.
Definition (1.7). An I-category consists of a category C with an initial object, e, a distinguished class , co!, of morphisms in C called cofibrations and a cylinder I = (( ) xl, eo, el, (7) on C in our notation (1.2.1). These are supposed to satisfy the following axioms .
(I 1) Cylinder axiom: - stating that I is a cylinder on C. (I 2) Pushout axiom: Given any diagram X~A~B with i a cofibration, the pushout X U B exists A
A--_u-_ B
Ij
(1.8)
X - - - , XUB
v
A
and j is a cofibration. Moreover, the functor ( ) x I carries the pushout diagram (1.8) into a pushout diagram and preserves the initial object.
84
(I 3) Cofibration axiom: Any isomorphism is a cofibration. For any object X the unique map ex : e - - X is a cofibration. (Thus xu y = Xu Y exists.) Moreover, i : A __ X e
in co! satisfies the homotopy extension property (1.2.5)( a) with respect to any object of C (i .e. a cofibration in an I-category is a cofibration in our sense (1.2.5)(b)).
(14) Relative cylinder axiom: If i : A - - X is a cofibration and one forms the pushout AUA
i Ui
--=--.::"--'----
X UX
eo(A)+eJ(A)j
j
A x l - - - X U A x lUX A
A
then the natural map
j:XUAxlUX __ Xxl A
A
induced by i x [ and eo(X) + el(X) is a cofibration. (I 5) The interchange axiom: For all objects X there is a map T: (X x 1) x I - - (X x 1) x I
with
Teo(X x 1) = eo(X) x I T(eo(X) x I) = eo(X x I) for c = 0, l. With these axioms Baues (1989) (Theorem 1.3.3) proves that C together with the class co! and weak equivalences taken to be homotopy equivalences with respect to I, is a cofibration category where all objects are fib rant and cofibrant. As they stand above, the axioms contain the extra structure given by the class co!, which is not determined by
85
( ) x ] since (I 3) does not say every morphism which satisfies the homotopy extension property is in cof. As Baues comments if one requires that cof is exactly the class of morphisms satisfying the HEP then the axioms reduce to conditions on I. His result then shows that I is generating under these conditions. The difficulties with these axioms are that (I 5) is not satisfied in some interesting cases and (I 4) seems hard to verify as it is too far removed from a property of the cylinder that can be checked directly. However, axiom (I 4) can be described as a certain kind of cubical homotopy extension property in dimension 2. If ( ) x [ is assumed to preserve pushouts and the initial object, then the natural map j:XUAx]UX~Xx[
A
A
satisfies the HEP if and only if the following condition holds for any object Y of C. For any morphisms of C -X : A x ]2 ~ Y,
a,
such that
aeo(X) -X(eo(A x 1))
=
= a(i x 1) ,
=
ael (X)
-X(ec:(A) x 1)
=
x 1) ,
E
= 0, 1,
there exists a morphism of C
A:X
X [2 ~
A(i
X [2) = -X
Y
such that
A(eo(X x 1))
= a,
A(eo(X) x 1)
=
A(el(X) x 1)
=
This property for a map i : A ~ X has been considered, under the notation HEP'(2), by Kamps (1978 a) studying operations on homotopy sets. It also appears ~n Shitanda's paper (1989) on abstract homotopy theory. Furthermore It has been used implicitly by Schon (1992) in his formal approach to the union lemma for cofibrations due to Lillig (1973) 86
stating that if A and B are closed subspaces of a topological space X such that the inclusions A --+ X, B --+ X and A n B --+ X are cofibrations, then the inclusion AU B --+ X is a cofibration. Diagramatically, this condition can be exhibited by the following picture.
12 , Y)
C(X
X
C(A
X [2, Y)
,,(iX/10
It should be noted that in the category Top with the canonical cylinder T , a map i : A --+ X satisfies the homotopy extension property HEP'(2) if i x 1d[o,l) : A x [0,1]
--+
X x [0,1]
is a cofibration (which in turn follows from the fact that i : A --+ X is a cofibration). This can be proved using a homeomorphism of pairs of spaces between
([0,1] x [O,I],{O} x [0,1]) and
([0, 1] x [O,I],{O} x [O,I]U[O,Ij x {O,I}) 87
(see (see R. R. Brown Brown (1988), (1988), p. p. 253, 253, Baues Baues (1989), (1989), pp.. 29). 29).
2. 2. Homotopy Homotopy Theory Theory in in Categories Categories of of Cofibrant Cofibrant Objects Objects In In this this section section we we will will handle handle the the theory theory of of categories categories of of cofibrant cofibrant objects. We develop the theory in full generality in much the same objects. We develop the theory in full generality in much the same way way as K.S. Brown Brown (1973) (1973) does does for for the the dual dual case case and and provide provide proofs proofs for for the the as K.S. convenience convenience of of the the reader. reader. In be aa category category of of cofibrant cofibrant objects objects (1.4). (1.4). In the the following following let let CC be The m a p p i n g cylinder cylinder (see (see section section 1.2) 1.2) adapted adapted to to the the The notion notion of of aa mapping present present situation situation will will be be aa basic basic tool tool in in what what follows follows..
Definition --t Y Definition (2.1). (2.1). Let Let ff ::XX -d Y be be aa map map in in C. C. IfIf
(X ( X xX I,1,eo, eo,ele l,,0") a) isis aa cylinder cylinder object object for for XX and and X_----'f__, Y
(2.2)
eoj
jjt
X x [--- Mf 7rf
pushout in in C, C, then then Mf Mfisis called called aa mapping mapping cylinder cylinderof of f.f . isis aa pushout
1)that that the the structure structure map map We have have already already proved proved ((1.4), ((1.4),Remark Remark 1) We Xx [ 1of of aa cylinder cylinder isis aa trivial trivial cofibration. cofibration. Hence Hence by by axioms (C (C 4), 4), (C (C 3)3) of of aa category category of of cofibrant cofibrant objects objects any any map map ff has has axioms jf: :YY --+ Mf in mappingcylinder. cylinder. Furthermore, Furthermore,by by (C (C3), 3),the the map map jf aa mapping --t M in f the mapping cylinder diagram (2.2) is a trivial cofibration. the mapping cylinder diagram (2.2) is a trivial cofibration. Themaps maps IIdy Y and and ff0"a ::XX Xx [ 1- - - t YY satisfy The dy : :YY ---t --t Y satisfy eo --t X X eo :: XX -+
-
fO"eo = f = I dy f
and so sotogether together define defineaa map mapPf pf : :MMf satisfying and f - - - t YY satisfying 88
PJ'Trf
= fa
and Pflt
= Id y .
Then, by (C 1), Pf is a weak equivalence. We set if = 'Trfel : X ~ M f . We then have pfi f
= Pf'Trfel = fael = f,
thus obtaining a factorisation (mapping cylinder factorisation)
X
f
Y
\}
(2.3)
Mf
of
f.
Proposition (2.4) (Factorisation Lemma). If C is a category of cofibrant objects and f is a map in C, then f = Pfif, where Pf zs left inverse to a trivial cofibration and if is a cofibration. (In particular, Pf is a weak equivalence.) Proof. It only remains to prove that if is a cofibration. To see this we construct if in a different fashion. The pushout diagram
ey l y---yuX Zy
shows by (C 5) and (C 3) that ix is a cofibration. f induces a map f U I dx : X U X ~ Y U X and, as one can easily verify, we get a pushout diagram
xuX
f UId x
eo + ell X x [
·yuX h
'Trf 89
Mf
where h is the map with components
jf : Y
~ Mf
and
7rfel
: X ~ Mf,
so h is a cofibration by (C 4) and (C 3). Finally we note if
= 7rf el = 7rf(eo + el)i l = hix
which, being the composite of two cofibrations, is itself a cofibration by
(C 2).
f
0
Remark. If f is a weak equivalence, then in the factorisation = Pfif of Proposition (2.4) so is if.
There is a canonical way of assigning a homotopy category Ho(C) to a category C of cofibrant objects by formally inverting all arrows in C which are weak equivalences. The general construction is that of forming a category of fractions (see Gabriel-Zisman (1967)) . We recall the explicit construction given by Bauer-Dugundji (1969) . Set theoretic problems which might arise (see Pfenniger (1989)) will not be discussed here. Let C be a category, and let L be any class of its morphisms. For simplicity we shall assume that L contains all isomorphisms of C and is closed under composition. Then Bauer and Dugundji construct a category which we will denote by C(L- I ) together with a functor TJ : C ~ C(L- I )
such that the following properties hold: (CF 0) The category C(L- I ) has the same objects as C and TJ is the identity on objects. (CF 1) If u E L, then TJ(u) is invertible in C(L- I ). (CF 2) If 'Y : C ~ D is any functor to any category D such that 'Y( u) is invertible for each u E L, then there exists a unique functor'f : C(L- I ) ~ D such that 'Y = 'fTJ.
90
The morphisms of C(L:C(C-')1 ) are obtained from chains, m, of morphisms of C of the form II e e ----t
Ul
f--
e
----t
In e e ... .. e ----t
Un
f--
e
C , by factoring out by an equivalence equivalence relation where the u; U i are all in L:, defined as as follows. follows. Two such chains m, m' are equivalent equivalent if there is a finite sequence
of chains in which each mi is obtained from m mi-1 mi+l) by one i-l (or from mi+d of the following following moves: (a) A 'pushout' operation :: replace (a) e
I
----t
e
a
f--
e
9
----t
e
fi
f--
in mi-1 (or mi+l) by uf
vP
. + . t .
if there is a commutative diagram in C 9
a
e-e-e
with
v p in L:C .. vf3
(b) (b) A 'pullback' 'pullback' operation :: replace
.'0.+.'.'f.
P
in mi-1 (or mi+l) by 91 91
s
e
gv
OIU
ef--e--te
if there is a commutative diagram in C
f
(J
e-e-e
with au in L:. (c) Deletion or insertion of redundant terms: replace e~e~e~e III
mi-l (or mi+d in which a is an identity morphism by e..!..Le
or a segment 01 f (J ef--e--tef--e
by e
(OI(J
e
if f is an identity. (This makes sense, since L: is assumed to be closed under composition.) Conversely one can factorise any map and insert an identity thus reversing this basic move . We note that TJ maps a morphism the chain
f :A
--t
B of C to the class of
Definition (2.5). If C is a category of cofibrant objects , then the homotopy category, Ho(C), of C is defined to be the category of fractions C(L:- 1) where L: is the class of weak equivalences. Thus we have a natural universal functor TJ :
C - - t Ho(C) 92
satisfying (CF 0) - (CF 2). In particular, '" inverts each s E L::. In the following let C be a given category of cofibrant objects. In order to be able to use the homotopy theory of C to the full , it is usually necessary to be able to describe simply the maps in Ho( C) and to say when ",U) = ",(g). The first approximation to this is clearly to use the cylinder objects to define a notion of homotopy between maps.
Definition (2.6). Two maps f,g : A if for some cylinder object
B in C are homotopic
for A there is a map H:Ax [ - B
with f = HeQ , 9 = H el. One writes f ~ 9 or H : f (between f and g) .
~ g.
The map H is called a homotopy
Homotopic maps become equal in Ho(C) .
Proposition (2.7). If f ~ 9 : A -
B, then ",U) = ",(g).
Proof. Since
we have
similarly
",(CT)",(el) = IdA. Since "'( CT) is invertible by (CF 1), CT being a weak equivalence, we obtain
hence 93
where H H : If
~ II
9g is a homotopy.
[7 0
First we note that cylinder objects in the sense of Quillen (d. (cf. (1.4)) can be pieced together to show that ~ I Iis an equivalence relation on C(A,B). C(A, B).
Exercise. Exercise. The setting is a category of cofibrant objects, C. (i) - - t B, B, then If I ~If. (i) Show that if If :: A --t f. (ii) If (B x I, eo, el, a) is a cylinder object for B, show that (ii) ( B I,eo, el, u) for B,
(B x I, e~, e~ , a) where eb e~ = B. Hence prove = el and e~ ei == eo is also a cylinder object for B. that if fI ~ E9 g then 9g ~ Ef f.. (iii) I,eO,el,a) I',e~ e~,a') (B x I, eo, el, u) and (B ( B x I!, eb,, ei, ul) are two cylinder (iii) Suppose (B B,, prove that objects for B
(B
X [" , e"0, e"1,
a")
is a cylinder object for B where B x I" I" is obtained by gluing el of B B xI I to eo eb in B x I' 1'.. More precisely B x [" 1" is given by the pushout of B x I ~ B
-=L B
x I'.
(The detailed definition of e: , e1 ey and a" u" is left to you.) (The e~, you.) Hence prove that I ~Iis a transitive relation on C(A, B) . C(A, B). (Quillen's result is his Lemma 3 of Quillen (1967) 1.) (1967),, 1. I. §fj 1.) We now describe the behaviour of the homotopy relation with respect to composition of maps. The following following is almost trivial.
=
Proposition (2.8). (2.8). If If H H :: f ~ 9g :: A arbitrary then uH ~ ug :: A A ----t - t C. u H : uf uf E
--t
0
B and u : B Band
-
--t
C is C
The usual other part of this type of result involving precomposition by say v : D ---t - t A does not follow follow easily unless cylinders can be chosen I ) :: fIvv I However, a functorially (in which case H(v x 1) ~Igv, gv , of course). However, functorially form of this result still holds without this assumption. We include it as assumption . form functorial ' already. its proof shows that cylinders are 'nearly functorial' 94
We start with a preparatory lemma.
Lemma (2.9). Any commuting diagram A _---'u=_____ B
al
Ib
C
D
v
with a a cofibration and b a weak equivalence, can be factored in the form
A--u-..... B
It~
al
C - - - - B' - - -....., D s
p
with t a trivial cofibration. Proof. We apply (C 3) and obtain a pushout
A--_u-_ B
al C
la' CUB
u'
A
with a' a cofibration. Since bu = va there is an induced map
d:CUB-----tD A
such that du' = v and da' = b. Applying the Factorisation Lemma (2.4) to d gives d = pi with p a weak equivalence and i a cofibration. Defining t = ia', s = iu', we have pt
= pia' = da' = band 95
ps
= piu' = du' = v,
and t is a trivial cofibration by (C 1) and (C 2).
Proposition (2.10). Given any map u : B
0
C, cylinder objects
---t
(B x I , eo(B), el(B), (J(B» for Band
for C, then we can find a cylinder object
(C x
[1 ,e~(C),e~(C),(JI(C»
for C, a trivial cofibration t : C x I
---t
C x T' and a map
u:BxI---tCxT' such that the follow ing diagram commutes. uUu BuB -----=-==--=---C uC
eo(B) + el(B)1
" eo(C) +
B x l (J(B)
e\(7
CxT'
I
~C) + e,(C) CxI
t
(JI(~
AC)
B----------~C
u
Thus cylinders are functorial up to weak equivalences.
Proof. Apply Lemma (2.9) to the square BUB
eo(C)u + el(C)u
--:.........:.------.:~~.
eo(B)+ e1 (B)1
C x I (J(C)
B x [------- C
u(J( B )
The result follows almost immediately.
96
0
We can now prove the precomposite result. Proposition (2.11). If f ~ 9 : B arbitrary then for any cylinder object
----t
C and u : A
----t
B
2S
(A x I, eo, el, cr) for A, there is a trivial cofibration, t : C
H : tf u
~
t gu : A x I
----t
C' and a homotopy
----t
C'.
Proof. Let h : f ~ 9 where h : B x I ----t C is the given homotopy. Apply Proposition (2.10) to u and the specified cylinder object for A, to obtain a cylinder object
(B X [', e~, e; , cr')
for B, a map
u :A
x [
B x I and a trivial cofibration
----t
t' : B x I
----t
B x I'.
The pushout diagram
t' B x I ---'--B x l'
hI
lh'
C-----C' t yields a trivial cofibration t by (C 3). Now look at h'u : A x I h'ueo
----t
C',
= h'e~u = h't'eou = theou = tfu
and similarly h'uel = tgu
so H = h'u : tfu
~
tgu as required.
0
We return now to the description of maps in Ho( C) and of deciding when 'f/U) = 'f/(g) for maps f,g in C. In order to achieve this aim, we approximate the homotopy category Ho(C) by a certain quotient category 7rC of C. We factorise 'f/ : C ----t
97
Ho( C)
as a composite TJ
C
' Ho(C)
\) 7rC
where 7r is the projection. The passage from 7rC to Ho( C) is more easily described than that from C to Ho( C) directly. In order to define the category 7rC and establish its properties we need a series of auxiliary results. However, the ideas involved will be of independent interest.
Definition (2.12). Given a diagram in C Bt-!-A~C
and a cylinder object
(A
X
!, eo, el, (1)
for A, a double mapping cylinder Mu ,v of u and v is a colimit of the diagram
A
A
y~ B
y~ Ax!
C
This means we have a diagram A
A
y~ B
y~ Ax!
C
(2.13)
k
JB
98
Jc
such that the following holds:
(a) keo = jBu and kel = jev (b) If j~ : B such that
--+
D, k': A x I
k'eo
--+
D, jc: C
--+
D are any maps
= j~u and k'el = jcv
then there exists a unique map Z:
Mu,v
--+
k
k'
D
such that .
.,
ZJB = JB, Z =
.
.,
,ZJe = Je·
This colimit clearly exists if C has all finite colimits j if its colimits are restricted to those assured by axiom (C 3), one has merely to use (C 3) twice to obtain this colimi t by means of the two stage construction via the mapping cylinder of u described by the diagrams (I.7.8), (I.7.9) in a special case. The details are left to the reader as an exercise.
Exercise. (a) Let
be a given diagram in C and let
(A x I,eo,eI,a) be a cylinder object for A . Suppose the following diagrams are pushouts eo
A
A---=------ AxI
ul
el
AxI
In~
Inu vI
B - - - - - Mu
C
ju
.,
Jv
99
M'v
2u
A -----'--- Mu
IV'
Vi G
B----M'
M
Jc
JB
where
Prove that
M together with v'ju, V'7r u, Jc resp ect i vel y
M' together with jB, U'7r~, u'j~ is a double mapping cylinder of u and v. Moreover, if
Mu---- M" Po is a pushout, prove that
M" together with Poju, Po 7ru =
Pl7r~, pd~
is a double mapping cylinder of u and v. (b) Let (2.13) be a double mapping cylinder diagram. Prove that
AuA u
eo
+ el , A x I
u vi
BUG
Ik jB
+ jc
100
Mu,v
is a pushout.
Proposition (2.14). Given a diagram B~A~C
the map ic : C - - - t Mu ,v into the double mapping cylinder is a cofibration. Futhermore, if u is a weak equivalence, jc is a trivial cofibration. Exercise. Prove Proposition (2 .14). HINT: Adapt the arguments used to prove Proposition (1.7.12). Definition (2.15). A homotopy commutative square in C is a diagram of the form
A--v-_C
ul
Ix
B
such that yu ~ xv. If a homotopy h : yu a curved arrow:
~
y
,D
xv has been specified this will be marked by
A _ _v_...., C
"I ~ B
Ix y
D
Theorem (2.16). Given any pair of maps in C B~A~C
there is a homotopy commutative diagram
101
B - -w- - B ' with a specified homotopy h : A x I - - t B', h: wu ::= tv. Moreover: (a) if u is a weak equivalence, then so is tj (b) if
A _--'v'---_. C
uJ
V
x
B - -y- · D
zs another homotopy commutative square with a specified homotopy 9 : A x I - - t D, g: yu::= xv, then there is a unique map z:
such that zw
= y,
9
= zh
B'
and zt
--t
D
= x.
Proof. Take B' = Mu,'IJ the double mapping cylinder. This comes equipped with maps jB : B - - t Mu,'IJ k : A x I - - t Mu,v and jc : C - - t Mu ,'IJ as in (2.13). Define w
= jB,
t
= jc,
h
= k.
Then h : wu ::= tv . By (2 .14) t is a weak equivalence if u is, and (b) follows directly from the universal property of the coli mit diagram (2.13) . 0 Remark. As we have mentioned in Chapter I, the double mapping cylinder is an example of a homotopy coli mit , in fact of a homotopy pushout. Edwards and Hastings (1976) have some results about the existence of homotopy limits and colimits in Quillen model categories, Theorem (2.16) suggests that general homotopy colimits may exist in categories of cofibrant objects. Anderson (1978) has a discussion of this. We will address some of these questions later in Chapter V. 102
We have as an immediate corollary of (2.16), one part of the description of morphisms in Ho(C).
Corollary (2.17). Every morphism in Ho(C) may be written as a left fraction 1](ttl1](f) where t is a weak equivalence .
Proof. From the explicit construction of Ho( C) as the category of fractions C(E- 1 ) given earlier, one can deduce (see Bauer-Dugundji (1969)) that an arbitrary morphism of Ho(C) has the form 1]( un)-l " 1](fn)
0
•
••
0
1]( Ul)-l "
1](h),
where m,
is a chain of morphisms of C such that the weak equivalences. Given a segment
Ui
are all in E, the class of
··· ---tB?A~C+-- ·· ·
where U is a weak equivalence, we can find, by (2.16)(a), a homotopy commutative square A--V-_C U
I
It
B - -w- - B '
such that t is a weak equivalence. Applying the functor 1] :
C ---t Ho(C),
by (2.7) we obtain a commutative square in Ho(C)
1](V)
A---C
~(U)I B
I~(t) 1]( w) 103
B'
and hence we can replace ".,(v)".,(ut 1 by ".,(tt 1".,(w)) and then compose with adjacent maps to simplify, making use of the fact that ~ is closed under composition. Induction on the 'length' n of the chain, m, now gives the result. 0
Remark. This proof is, of course, well known and similar proofs can be found in many places in the literature (e.g. Gabriel- Zisman (1967), Hartshorne (1966)). Proposition (2.18). Given j, g : A - - t B with a weak equivalence t : A' - - t A such that ft ~ gt then there is a weak equivalence s : B - - t B' such that sf ~ sg. Proof. Suppose H : A' x I - - t B is the homotopy H : jt ~ gt. Now let Mt,t be the double mapping cylinder of t and t. Thus we have a colimit diagram
A'
A'
y~ A' x [
A
~A
Y
k
Jo
J1
Mt,t The triple (1, H, g) defines a map J : Mt ,t - - t B and (1 dA , tu(A'),I dA ) gives i : Mt ,t ---t A by the universal property of the colimit. The maps ]0,11 : A ---t Mt,t are weak equivalences by (2.14) and so i]o = 1dA implies i is a weak equivalence. We factor i as in the Factorisation Lemma (2.4), so we can represent i as Mt ,t
i'
---t
D
P
---t
A
where i ' is a weak equivalence and hence a trivial cofibration, and p is a weak equivalence. As Mt,t can also be defined by the pushout 104
A'UA'
tU tl AUA
eo
+ el .
A'xI
I Mt,t , jo + jl jo
jl
. Mt,t
+
jojo + jljl is a cofibration. This implies that i' (jo
+ h) : A U A
-; D
is a cofibration. Noting that
pi'(jo + jl) h) = = V V' A, pi'(j0 A, we have that ((D i'jo, i'i'h D , iljo, jl,,p) p) is a cylinder object for A. Now let B' B' be given by the pushout Now
J Mt ,t ---=--- B
i'l
Is
D----B'
J'
thus defining s and J' with s E L: C. Finally
J'i'jo = sJjo = sf, J'i'jl = sJjl = sg shows is the required homotopy. shows that J' is
-
roJ
0
In order to define define the category category 1rC, TC, we introduce the following following relation on C (A, ( A ,B) B ) where A, A, B are are objects of C. C.
Definition ; B are Definition (2.19). (2.19). If f,g :: A -+ are maps of C, C, we say, ff is is equivalent equivalent to to 9g (written (written ff N g), g ) , if there is is a weak equivalence equivalence tt :: B tf ~ B - ; D with withtf ~ tg. t g .
-
roJ
Lemma is Lemma (2.20). (2.20). The The relation relation is an an equivalence equivalence relation on on C(A, B) which is compatible with composition. C ( A ,B ) is compatible roJ
Proof. Proof. The The relation relation
-
roJ
is is clearly clearly reflexive reflexive and and symmetric. 105
In order to show that", is compatible with composition let
f,g: A
---t
B
be maps such that f '" 9 and u : B - - - t C, v: E - - - t A arbitrary. Then there exists a weak equivalence t : B - - - t D with tf ~ tg. By (2.11) there is a weak equivalence t : D - - - t D such that
Itfv
~
Itgv.
This proves fv '" gv. By (2.16)(a) there is a homotopy commutative diagram
D---D' u'
i.e. t'u ~ u't, where t' is a weak equivalence. By (2.11) there is a weak equivalence til : D' - - - t D" such that
t"t'uf Then t"u't
~
till : D"
Dill such that
---t
t"u'tf·
t"t'u by (2 .8) . By (2.11) there is a weak equivalence tlllt"u'tg
Then r
~
~
tlllt"t'ug.
= tlllt"t' is a weak equivalence such that by (2 .8), r( uf)
tlllt"t'uf '" tlllt"u'tg
~
~
tlllt"u'tf tlllt"t'ug = r(ug),
hence uf '" ug. By similar manipulations one can prove that '" is transitive. The details are left to the reader. 0
Notation. We will write 'lrC for the quotient category Cj "" obtained by replacing each C (A, B) by C (A, B)j '" . The equivalence class of a map f : A - - - t B in C will be denoted by [fl E 'lrC (A, B).
106
Proposition (2.21). There is a unique functor
r/ : 7rC
---+
Ho( C)
mapping the objects identically and sending [I] E 7rC (A, B) to
",(1) E Ho(C)(A, B) such that the diagram C
",
'Ho(C)
'\) 7rC
commutes where 7r is the projection. Proof. We have only to show that r/ is well defined. The rest is clear. Let f, 9 E C (A, B) such that I '" g. Then there is a weak equivalence t : B ---+ D with tf ~ tg. We apply", and obtain ",(t)",(I)
= ",(tf) = ",(tg) = ",(t)",(g)
by (2.7), hence ",(1) = ",(g), since ",(t) is invertible.
0
Now the transition from 7rC to Ho(C) via r/ has good formal properties in the following sense. If we say that a class [t] in 7rC is a weak equivalence if t itself is a weak equivalence, then Ho( C) is obtained from 7rC by formally inverting the weak equivalences in 7rC. Furthermore, by Theorem (2.16)(a) and Proposition (2.18) we have the following result.
Theorem (2.22). The class 7r I: weak equivalences in 7rC admits a 0 calculus of left fractions in the sense of Gabriel-Zisman (1967). We exploit this fact following the ideas in Gabriel-Zisman (1967) except that the conditions occur in 7rC not in C itself. For objects A, B of C we define a new set
[A, B] = l!p 7rC(A, C) where the direct limit is taken over the category with objects the weak equivalences 107
[t] : B
-----t
and in which a map from [t] to [t'] : B in rrC
C C' is a commutative triangle
-----t
(2.23)
C -[-a]-' C' (Thus denoting this category by B [A, B] = colim (B
1 W.E.,
1 W.E.
~
we have 7rC(A ,-)
rrC
---t
Set)
where c is the codomain restriction functor.) Explicitly, the elements a of [A, B] are equivalence classes of pairs ([1], It]) representing a diagram in rrC A
B
~A C
(2.24)
with tEL: , the equivalence relation being generated as follows. If t, t' E L: and (2.23) is a commutative triangle in rrC then ([j],[t]) as before is equivalent to ([a][j], It']). In fact , we have the following direct description of the equivalence relation (d. Gabriel-Zisman (1967),1.2.3) . If A
B
~A C
A
and
~Al c'
B
are diagrams in rrC with t, t' E L: then ([1], It]) is equivalent to ([1'], It']) if and only if there exists a commutative diagram in rrC
108
A
B
~Y.~A] G [til] G'
A
~/Ia1 Gil such that til E I: and [a][f] = [a'][f']. There is an induced function 'Y: [A, B] --+ Ho(C) (A, B)
sending the equivalence class of ([f], [tD as in (2.24) to TJ(ttITJ(J) and (2.17) states that this is surjective. We shall show that 'Y is also injective. This will show amongst other things that TJ(J) = TJ(g) if and only if there is a weak equivalence t such that tf ~ tg (see Proposition (2.26) below). The easiest way to do this is , in fact , to show that the map 'Y is a natural isomorphism in one step (thus incidentally reproving (2.17) in the process). The proof uses the universal property (CF 2) of the pair, (Ho(C), TJ) : any functor F : C --+ D , which inverts all 8 E I: must factor uniquely through TJ , i.e. F must be
C ~ Ho(C) ~ D for some F'. We first form yet another category 7rC with objects the objects of C and with
7rC (A, B) = [A, B]. If we can show that 7rC has the same universal property as Ho( C) then we are done. There are however several points to make precise. Firstly we have to show 7rC is a category by defining a suitable composition. If a E [A, B] and {3 E [B, G], then one has a diagram
A
~ [f] ~
B [t] /
B'
/
~ [g] ~ 109
G G'
/' / [8]
with with s,s ,tt EE 2:. C . By By (2.16)(a), (2.16)(a),we we can can complete complete this this to to give give
ABC
~y~~ B' c'
~A1 D
where [t'][g]and and t't' EE 2:C.. where [g'][t] [g'] [t]== [t'][g] We f], [t'sD. We can can thus thus represent represent fJa pa by by ([g' ([g'f], [t's]).Using Using aa different different represenrepresentative or fJp does does not not change change the the equivalence equivalence class class of of the the result, result, so so tative for for aa or the the composition composition isis well well defined. defined. into aa ItIt isis then then routine routine to to check check that that this this composition composition makes makes 7rC into category. category. Our Our second second point point isis that that the the universal universal property property isis really really that that of of the the pair, (Ho(C),1]), q),and and hence hence we we must must produce produce aa functor functor pair, (Ho(C),
rr : C ----+ 7rC. This + BB in in CC to to the the This isis easily easily defined defined by by sending sending aa map map ff :: AA ----+ equivalence equivalenceclass classof of ([t], ([f],[IdBD· [IdB]). We Wenow now check check that that (7rC, r;) 7)has has the therequired requireduniversal universal property. property.
(x,
Theorem (2.25). (a) (a)1ft If t EE2:, E, then thenrr(t) ~ ( tisis ) an an isomorphism. isomorphism. T h e o r e m (2.25). +VV isis aa functor functor such such that thatB( 0(t) is an an isomorphism isomorphism for for (b)IfIf B0::CC----+ (b) t) is - each eacht t EE2:, C, then then there there isis aa unique unique7J0::7rC TC ----+ -+ VV satisfying satisfying7Jrr $7==B.0.
Thus7rC Zand andHo(C) Ho(C)are areequivalent equivalent categories categories(relative (relativeto toC, C,i.e. i.e. the the Thus equivalence equivalencecommutes commuteswith with1]qand andrr). 7). -4B B isis aa weak weak equivalence, equivalence, then then the the equivaequivaProof. (a) (a)IfIf t t : :AA ----+ Proof. lence class class ofof ([t], ([t],[I[ dB]) I ~ B has ]has ) an an inverse inverse which which can can be be represented represented by by lence ([IdB], [t]),since since by by the the above above description, description, composition composition ofof ([I ([IdB], [t]) ([I dB], [tD, dB], It]) and([t], ([t],[I[IdB]) gives([t], ([t],[tD [t])which whichisisequivalent equivalenttoto ([IdA], ([IdA], whereand dB]) gives [I[IdA]) dAD wherecomposingininthe theother otherway way we weobtain obtain([Id ([IdB], [IdB]). ], [IdBD. asascomposing B (b)Given GivenB,0,define define7J(A) g(A)==B(A) 0(A)on onobjects objects and and ifif aaisisrepresented represented (b) by ([f], It]) define by ( [fI [tl) define
110
This is easily checked to be independent of the choice of representatives and clearly satisfies 7J' 0 37fj == (). 0.
Proposition If ff,, g 9 :: A ---t B are maps in C, P r o p o s i t i o n (2.26). (2.26). If +B C, then
-
77(1) rl(f = = 77(9) rl(g)
in Ho( C) if B ---t + C such Ho(C) if and only if if there is a weak equivalence equivalence t : B that tf t f ~ tg.
-
Proof. The if-part if-part is a reformulation of the fact that the functor 77' Ho( C) of (2.21) 7' :: 7rC KC ---t Ho(C) (2.21) is well defined. defined. the function Y maps the Conversely, assume 77(1) = 77(9)· ~ ( g ) Now . function 'y q(f) = ([f], [IdB]) ~ ( f ) . since equivalence class of ([!],[Id ]) to 77(1). Hence, since'Y is injectiy B dB]) is equivalent to ([g], [I dB])' Thus by the explicit desve, ([!], ([f], [I [IdB]) ([g], [IdB]). cription of the equivalence relation, relation, there exists a weak equivalence t' : B ---t B' such that
-
[t' g]. = [t' [t'gl. [t'ff]I =
This means we have a weak equivalence equivalence til t" : B' tilt' f
which completes the proof.
~
-
---t
C such that
tilt' 9
0
We will need some of these results later when we compare our version of abstract homotopy theory and the homotopy category it defines with the quotient category obtained by inverting simple equivalences. We conclude this section by indicating how a gluing gluing theorem for weak equivalences equivalences can be derived from the axioms axioms of a category of cofibrant objects. objects.
Theorem If C (2.27). If C is a category of of cofibrant objects, objects, then the T h e o r e m (2.27). Gluing Theorem equivalences holds: T h e o r e m for weak equivalences
111
Let ho
Ao
12
Bo
~ Al
I~
hI
. Bl
g2
(2.28)
f~
h2
A2
. B2
~A
h
g;
~·B
be ere the hand squares are be a commutative cube cube in in C wh where the leftleft- and rightright-hand are push outs and II pushouts fl and gl are are cofibrations. T hen h e n if h oo),hI) h l , h2 h z are are weak equivalences, equivalences, so is is h .
In he Gluing heorem (2.27) In order to to prove tthe Gluing T Theorem (2.27) we we need the following following analogue roposition (1.7.4) analogue of PProposition (1.7.4)..
P roposition (2. 29 ). Let C be Proposition (2.29). be a category category of cofibrant cofibrant objects objects.. Sup Sup-pose
A (2.30)
_-,-f_
B
[j
i[
X -- -·y 9
i s aa pushout in in which which ii is is aa cofibration cofibration and and f is i s aa weak weak equivalence equivalence.. is h e n 9g isis aa weak weak equivalence. equivalence. TThen In particular, particular, axiom axiom (C (C 3') 3') (s (see the discussion discussion of of aa cofibration cofibration cacaIn ee the tegory in in section section 1) 1 ) holds holds in in any any category category of of cofibrant cofibrant objects. objects. tegory We postpone postpone the the proof proof of of PProposition (2.29) and prove prove Theorem Theorem We roposition (2 .29) and (2 .27) first. (2.27) first. Reduce TTheorem (2.27) to to the the special special case case where where 12 f2 and and xer cise. Reduce heorem (2.27) EExercise. g2 g2 too too are are cofibrations. cofibrations. 112
HINT: Adapt the arguments used to prove the corresponding reduc-
tion of Theorem (1.7.1) to the present situation. Note that one uses axiom (C 3') which holds by Proposition (2.29) and the Factorisation Lemma (2.4). We now sketch the proof of the special case of Theorem (2.27). Thus we assume that in diagram (2.28) lI,h,9I,92 are cofibrations and ho, hI, h2 are weak equivalences. Consider the following diagram which decomposes the cube (2.28). (We have borrowed the diagram and its lay-out from Baues (1989), p. 85.) The diagram is constructed in such a way that all squares are pushouts. The pushouts exist by axiom (C 3), since II and hare cofibrations.
Since h is a cofibration and ho is a weak equivalence, by axiom (C 3') the arrow labelled CD is a weak equivalence, hence @ is a weak equivalence by (C 1), since h2 = @o CD is a weak equivalence. Since 91 is a cofibration, by (C 3) the composite horizontal arrow. +-- • +-- • is a cofibration, hence @ is a weak equivalence by (C 3'), since @ is a weak equivalence. Similarly, since h is a cofibration, f~ is a cofibration, hence @ is a weak equivalence, since hI is a weak equivalence. It follows that h = @ 0 @ is a weak equivalence, as desired. 0 113
Proof o/Proposition (2.29) (cf. Baues (1989), I,§1b). First we decompose the given pushout (2.30) using the mapping cylinder Mf of / and the double mapping cylinder M f •i of / and i. Therefore we choose a cylinder object
(A x I,eO,el,a) for A and a mapping cylinder diagram for / , i.e. a pushout of the form eo
A-~-AxI
j~f
Ij
B----Mf Jf
This gives rise to the mapping cylinder factorisation of /
f
= Pfif ,
where if = 7rfel is a cofibration and Pf is a weak equivalence which is left inverse to it. Recall that it is a trivial cofibration by axiom (C 3) , since eo is a trivial cofibration. We obtain a diagram
where the left- and hence the right-hand square is a pushout and the lower row is the map g. Since / is assumed to be a weak equivalence, so is if. Hence Jx is a trivial cofibration by axiom (C 3) , in particular jx is a weak equivalence. Consider the pushout
B
Jf
j
Mf
jl y
N Jf 114
This pushout exists, since j I is a cofibration. Furthermore, since j I is a trivial cofibration, so is it by (C 3). In particular, it is a weak equivalence. Let
A
eo
il
X
A
-Axl
Ax!
I~~
il
I~i Ji
el
X
Mi
"/
1i
M!t
be pushout diagrams, i.e. the first diagram is a mapping cylinder diagram for i and the second diagram can be viewed as a modified mapping cylinder diagram for i. Then we have an induced map 'Y:Mi--+N
such that 'YJi = JIg and 'Y~i = J~I'
Since ji and j I are weak equivalences, in order to show that 9 is a weak equivalence it remains to show that 'Y is a weak equivalence. Note that
is a pushout. Furthermore, ~i and ~~ are cofibrations since i is assumed to be a cofibration. Now consider a pushout diagram ~i
Ax l - - - - M i
n:1
IT;
Mi --r-!---'" P t
115
This pushout exists by (C 3) giving rise to cofibrations ri, ri. Let
f3:P--X be the induced map such that
f3 ri = Pi and f3ri = P~, where Pi is the weak equivalence appearing in the mapping cylinder factorisation of i constructed in (2 .3) and the weak equivalence P~ is obtained in a similar way. We apply the Factorisation Lemma (2.4) to f3 and obtain a factorisation of f3
P
f3
X
~J Q
where k is a cofibration and q is a weak equivalence. Defining P~ = kri ,
Pi = kri'
we obtain trivial cofibrations Pi, p~. Finally note that there is a commutative cubical diagram
(2 .31) 7r' I
Mj,i - - - - - - - S
Pi
~Mi -----------------. ~Q P~
with an induced map 8 which satisfies 116
jx = 8j~
such that the top, the bottom, the right-hand and the left-hand side of the diagram are pushouts. Since jx and j; are weak equivalences, so is 8. We apply (C 3) first to the bottom square of (2.31) , then to the right-hand square of (2 .31) and obtain that 'Y is a weak equivalence. D
Exercise. Show that the axioms (C 1), (C 2), (C 3') and (FA) of a cofibration category imply axiom (C 3). We should point out that many of the arguments in this section become much simpler if we restrict ourselves to the special case that the cofibration structure is induced by a cylinder (see Theorem (1.6)), but there are interesting and useful cofibration structures which do not arise in this way.
Remark on Quotients and Fractions. For a category, C, with a generating cylinder I the 'natural' homotopy category is the quotient category hC = Cj
~
(see the remark after (1.2.4)). The question arises, how this category is related to the 'artificial' category of fractions
where I:h.e. is the class of homotopy equivalences defined by the cylinder structure (see (1.2.4)). Consider the projection functor 7r:C~hC
(see the exercise after (1.2.4)) and the universal functor 'fJ: C ~Ho(C)
(see (2.5)) . Since 7r inverts all homotopy equivalences, 7r factors through Ho(C) :
117
An argument similar to the proof of (2.7) (see also (VI.3.5)) shows that if f ~ g, then ",(f) = ",(g). Thus we obtain an induced functor hC
---+
Ho(C),
which is inverse to (3. As a result we get that hC and Ho( C) are isomorphic.
3. Duality In the classical homotopy theory of topological spaces one has a heuristic principle known as Eckmann-Hilton duality which assigns to a theorem involving homotopy equivalences, cofibrations and fibrations of spaces (see the example following Definition (1.1)) a 'dual' statement by the following nalve recipe: - Invert all the arrows. - Replace fibrations by cofibrations and vice versa. - Leave homotopy equivalences unchanged.
Example. Starting from Dold's theorem (D)
(D) If A
!~
X---X'
f
is a commutative diagram in Top where i and i' are cofibrations, then if f is a homotopy equivalence, f is a homotopy equivalence under A. 118
we obtain the following Eckmann-Hilton dual (D*). (D*) If EI
9
. E
~J B
is a commutative diagram in Top where p and pI are fibrations, then if 9 is a homotopy equivalence, 9 is a homotopy equivalence over B. However, the Eckmann-Hilton dual of a theorem is not necessarily true and in case it is (as for example the Eckmann-Hilton dual (D*) of Dold's theorem (D) (see Dold (1963), (6.1))) a separate proof has to be provided. On the other hand, abstract homotopy theory, i.e. homotopy theory valid in an arbitrary C allows the usual categorical duality. If a theorem TH(C) is true in any category C, then it is also true in the opposite category coP. We may reinterpret TH( COP) in the category C thus obtaining a dual theorem THOP( C) valid in C without necessity of a separate proof. In order to illustrate the procedure we dualise our categorical version, Theorem (1.6.3), of Dold's theorem. As we have mentioned earlier (see section 1.3) a cylinder on the opposite category cop is a co cylinder
p = ((
)1,cO ,c I,S)
on C, i.e . we have a functor
together with three natural transformations CO, CI : ( )1 __
such that cos =
CIS
Ide, s: Ide __ (
= Ide ·
119
)1
Given a co cylinder , P, on a category C, we now list, with respect to P, those notions of homotopy theory which are dual to the basic notions of homotopy theory with respect to a cylinder. Definition (3.1). Two morphisms I,g : X ---+ Y in C are said to be homotopic (with respect to P) (f ~ g), if there is a homotopy ¢ between 1 and 9 (¢: 1 ~ g), i.e. a morphism ¢ : X ---+ yl in C with co(y)¢ = I, cl(y)¢ = g. As in Definition (1.2.4) the notion of being homotopic gives rise to the notion of homotopy equivalence (with respect to P). Dual to the notion of homotopy induced by a cylinder in a category of objects under a fixed object (see Definition (1.6.1)) we have a notion of homotopy induced by the co cylinder P in a category CB of objects over a fixed object B of C. Definition (3.2). If
I, 9
: p' ---+ pare morphisms over B where B then 1 is homotopic over B to 9 (with respect to P) (f ~ g), if there is a homotopy ¢ over B between 1 and
p:E
---+
B, p' : E'
---+
B
9 (¢ : I~g), i.e. a homotopy ¢: E' B
---+
El , ¢ : 1 ~ 9 in C and
pl¢ = s(B)p', i.e. the diagram
El
E'
jpl
P'j B
s(B)
Bl
commutes. As in Definition (1.6.1)(b) the notion of being homotopic over B gives rise to the notion of homotopy equivalence over B. For the notion of fibration with respect to P which is dual to the notion of cofibration with respect to a cylinder we refer to Definition (1.3 .2). 120
We now describe the cubical structure induced by a co cylinder P on
C, i.e. we have an induced functor QP : cop X C __ Cub into the category Cub of cubical sets defined by
where we have written yIO = y, yII = yI, and yIn = (yIn-i)I
defining the iterated co cylinder functor ( )In. In order to make precise the face and degeneracy operators w~ resp. (~ involved in QP(X, Y), firstly we define maps (for i = 1,' . " n;
j=1,"',n+1) w~(Y)
: yIn __ yIn-I,
(~(Y):
yIn __ yIn+1
by and (~(Y) = (s(yIn+1-i»Ij-l.
Now in QP(X, Y) we have w~
= C(Idx,w~(Y»
: C(X, yIn) __ C(X, yIn-I)
for w = 0,1 and (~= C(Idx,(~(Y»: C(X,yIn) __ C(X,yr+I).
The definition of the functor QP on maps should be clear.
Definition (3.3). A co cylinder P on a category C is said to s~tisfy the Kan condition E(n,v,k) (0 ::; v ::; I, 1 ::; k ::; n) if for any objects X, Y of C, the cubical set QP(X, Y) satisfies the Kan condition E(n, v, k) (see (1.5.3». If for some n, P satisfies E(n, v, k) for all v = 0,1, k = 1,' .. ,n, we say that P satisfies the Kan condition E(n).
121
We leave it to the reader to write out the obvious definition of the Kan conditions NE(n, v, k), DNE(n, v, k), NE(n), DNE(n) for a cocylinder P. We are now in a position to state the categorical dual of the categorical version, Theorem (1.6.3), of Dold's theorem.
Theorem (I.6.3)OP. Suppose P is a cocylinder on a category C which satisfies DNE(2,1,1) and E(3,1,l) and in which ( )1 preserves weak pullbacks. Let
E'
9
,E
~J B
be a commutative diagram in C where p', pare fibrations (with respect to P). Then if 9 is a homotopy equivalence (with respect to P), 9 is a homotopy equivalence over B.
Example. If the assumptions of Theorem (1.6.3)OP hold for the canonical co cylinder W on the category Top given by the path space construction (see Example 1 in section 1.3) then, by specialisation to the category, then, by specialisation to the category Top, Theorem (1.6.3)OP shows that the Eckmann-Hilton dual (D*) of Dold's theorem (D) is true. In fact, as we shall see, the properties required for Ware automatically induced by the properties of the canonical cylinder I on Top since the cylinder functor ( ) X [0,1] on Top is left adjoint to the path space functor ( )[0,1]. This type of questions will be dealt with next. In the category Top of topological spaces the canonical cylinder
1=(( ) x [0,1],eo,e},a) and the canonical co cylinder W = (( )[0,1],£0,£1,8) 122
are related via the canonical adjunction a : ( ) x [0, 1] ---l ( )[0,1)
consisting of the natural family of bijections a: Top (X x [0,1], Y)
----t
Top (X, y[O,l))
defined by
«a(¢»)(x))(t) = ¢>(x, t) where X , Yare topological spaces and ¢> : X x [0,1] ----t Y is a homotopy. The natural transformations eO,e1,a, resp. [o , [1,S, are related by the formulae
[i(Y)a(¢»
= ¢>ei(X)
(i
= 0,1)
(3.4)
a(fa(X)) = s(Y)f where ¢> : X x [0, 1]
----t
Y is a homotopy and
f :X
----t
Y is a map.
In the general case we have the following definition.
Definition (3.5). Let C be a category, let I = « ) x I, eo, e1, a) be a cylinder and P = « )I,[O, [ l , S) be a co cylinder on C. We say that I is left adjoint to P (written I ---l P ) if there is an adjunction a : ( ) x I ---l ( )1,
i.e. a family of bijections a: C(X x I, Y)
----t
C(X, yI)
natural in X and Y, where X and Yare objects of C, such that the above conditions (3.4) hold, where ¢> : X X [ ----t Y is a homotopy and f : X ----t Y is a map. In this case we write a : I ---l P. We call (I,P) an adjoint cylinder/co cylinder pair on C.
123
Remark. If 1= (( ) x I, eo, el, 0") is a cylinder on a category C,
( )1 : C -----t C is a functor and a : ( ) x I -I ( )1 is an adjunction, then we can define natural transformations
by the equations
a- l (Id x l)ej(X1) (i a(O"(X))
= 0, 1)
such that I -I defined as (( ) I , co, Cl , s) is a co cylinder on C and I is left adjoint to 1-1 (see Mitchell (1965), V, 2, Kamps (1972 a), (7.9)). Proposition (3.6). Let (I,P) be an adjoint cylinder/cocylinder pair on a category C. Then two maps of C are homotopic with respect to I if and only if they are homotopic with respect to P. Similarly, if B is an object of C, two morphisms over B are homotopic over B with respect to I if and only if they are homotopic over B with respect to P. It follows that in this case the notions of homotopy equivalence (resp.
homotopy equivalence over B) with respect to I and with respect to P coincide. Proof. Let a : I -I P be an adjunction. If f, g : X -----t Yare maps of C and ¢ is a homotopy with respect to I between f and g, i.e. ¢ : X x I -----t Y is a map such that
¢eo(X) = f and ¢el(X) = g, then by (3.4) a( ¢) : X
-----t
Y 1 is a map such that
co(Y)a(¢) = ¢eo(X) = f and cl(Y)a(¢) = ¢eI(X) = g, i.e. a( ¢) is a homotopy with respect to P between f and g. Conversely, again by direct application of (3.4), we see that a homotopy with respect to P is transformed into a homotopy with respect to I using the inverse map a-I of a. 124
Similarly, if
f, 9 : p'
---t
p :E
pare morphisms over B, where ---t
B, p' : E'
---t
B
and ¢ is a homotopy over B with respect to I between f and 9 (in particular p¢ = p' a( E')) then, as we have proved, a( ¢) is a homotopy with respect to P between f and 9 and we have, by naturality of a and
(3.4) p1a(¢)
= a(p¢) = a(p'a(E')) = s(B)p',
i.e. a( ¢) is a homotopy over B. Conversely, if 'It : E' ---t EI is a homotopy over B with respect to P, i.e. pI'lt = s(B)p', we have
a(p(a- 1 ('lt))) = pI(a(a- 1 ('lt))) = pI'lt = s(B)p' = a(p'a(E')), hence, by bijectivity of a,
p(a- 1 ('lt)) = p'a(E'), I.e. a- 1 ('lt) is a homotopy over B.
0
We recall that the definition of a fibration with respect to a co cylinder which has been given in Definition (1.3.2) (b) can be exhibited by the following type of diagram (1.3.3).
In an adjoint cylinder/co cylinder situation fibrations with respect to the cocylinder can be characterized by means of the cylinder in the expected way. Dually, a cofibration with respect to the cylinder can be characterized by means of the co cylinder. 125
Proposition (3.7). Let (I,P) be an adjoint cylinder/cocylinder pair on a category C. Then a map p : E ----+ B ofC is a fibration with respect to P if and only if the following homotopy lifting property with respect to the cylinder I holds: in each commutative diagram of solid arrows (1.3.4) y_....::.f_ _ E
eo(Y)
/"
/ /'
P
/
YxI
B
1;
there is a diagonal extending the diagram commutatively.
The proof is straightforward and obtained by similar manipulations to those in the proof of Proposition (3.6) . 0 If '(I,P) is an adjoint cylinder/co cylinder pair on a category C and a : I ---l P is an adjunction then, for any objects X, Y of C, by iteration the bijection a:
C(X
x
I, Y)
----+
C(X , yI)
gives rise to bijections
inducing an isomorphism between the cubical set QI(X, Y) associated to the cylinder and the cubical set QP (X, Y) associated to the cocylinder. To be precise, one has to renumber the face and degeneracy operators in one of the cubical sets, say QP (X, Y), i.e. one has to replace w~, (~by w~, (~defined by wi
n
= wnn + 1- i '
1j
'',n
=
I' n+2-j
"':,n
.
This implies that the cylinder I satisfies the Kan condition E( n) (resp. NE(n), DNE(n)) if and only if the co cylinder P satisfies E(n) (resp. NE(n), DNE(n)) . We now turn to Quillen type theories. If we dualise Theorem (1.6) we obtain the following result . 126
Theorem (1.6)OP. Let C be a category which has pullbacks and a terminal object, t. Let P = « )1, co , C}, s) be a cocylinder on C which satisfies DNE(2) and E(3,1,1) and in which ( )1 preserves pullbacks and the terminal object. Then C together with weak equivalences taken to be homotopy equivalences with respect to P and fibrations as in (1.3.2) (b) has the structure of a category of fibrant objects in the sense of K.S. Brown (see K.S. Brown (1973) or (re)dualise Definition (1.4)) .
In this case we say that the co cylinder is generating. In an adjoint cylinder/co cylinder situation we obtain the following result.
Theorem (3.8). LetC be a category which has push outs, pullbacks, an initial object and a terminal object. Let (I,P) be an adjoint cylinder/cocylinder pair on C such that I satisfies DNE(2) and E(3). Then both the cylinder and the cocylinder are generating, i.e. defining weak equivalences to be homotopy equivalences and cofibrations and fibrations as before (see (1.2.5)(b), (1.3.2)(b)), C has the structure both of a category of cofibrant and of fibrant objects. In particular, Theorem (3.8) applies to the category Top of topological spaces. Proof. Theorem (3.8) follows from Theorem (1.6) and its dual Theorem (1.6)OP together with our discussion of Kan conditions in an adjoint situation. The cylinder functor ( ) x I being left adjoint to the cocylinder functor ( )1, push outs and the initial object are automatically preserved by ( ) x I and pullbacks and the terminal object are preserved by ()I. 0
We now relate the adjoint cylinder/co cylinder situation to Quillen's axiom system. We shall need a notion which describes a certain simultaneous homotopy extension and lifting property. 127
Definition (3.9). Let C be a category with a cylinder I = (( ) x I, eo, e1, (J).
Let (i,p) be a pair of maps of C, where i : A - - X, p: E - - B. We say that (i,p) has the covering homotopy extension property (CHEP) iffor any morphisms
:AxI--E, 'IjJ:XxI--B, I:X--E such that
p = 'IjJ(i x 1), eo(A) = Ii and 'ljJeo(X) = pi there is a morphism q, : X x I - - E such that
q,(i x I) = , pq, = 'IjJ and eo(X) = f. Example. Str(l!m (1966) has proved that in the category Top any pair (i, p) where i is a closed cofibration and p is a fibration satisfies the CHEP. Further examples have been listed in Berrick-Kamps (1985), ( 4.6). Definition (3.10). Let (I,P) be an adjoint cylinder/co cylinder pair on a category C. We say that (I,P) is weakly generating, if C together with weak equivalences, cofibrations and fibrations defined as in Theorem (3 .8) satisfies Quillen's axioms (M 0), (M 1) , (M 3) , (M 4), and (M 5) of a model category (see (1.1)) together with a weak form , (M 2'), of axiom
(M 2). (M 2') (a) Any map I may be factored I = pi where i is a homotopy equivalence and p is a fibration. (b) Any map I may be factored I = pi where p is a homotopy equivalence and i is a cofibration. The following proposition deals with Quillen's axiom (M 1) in an adjoint situation.
Proposition (3.11). Let (I,P) be an adjoint cylinder/cocylinder pair on a category C such that I satisfies DNE(2) and E(3). Suppose 128
that any pair pair oj of morphisms (i, ( i ,p) p ) where i is a cofibration and p is a fibration satisfies the CHEP. fibration Then Then Quillen's axiom (M ( M 1) 1 ) holds. holds.
Proof (cf. Hastings (1974)). (1974)). Given a commutative diagram in C ProoJ(cf. A i
_-=--J_ E
j
X
jP 9
B
-
where ii is a cofibration, p is a fibration, we have to show that there is a diagonal h : X -----+ E extending the diagram commutatively if either ii or p is trivial. We restrict ourselves ourselves to the case where ii is a trivial cofibration the dual situation where p is a trivial fibration being left to the reader. By Theorem (1.6.9) (1.6.9) a trivial cofibration is a strong deformation reX -----+ A A,, i.e. ri == IdA, IdA, and a tract. Hence there is a retraction Tr : X homotopy under A,
-
.
A
H =Idx, H:: ir zr ~Idx, i.e. a homotopy H : ir
~ E
H:: X xX I IIdd x ,, H
-----+ ---t
X,, such that X
H(i x I) = ia(A) .
-
Consider
-----+ E , gH gH:: XX x [1 -----+ B, fJa(A) u ( A ) : A x [1 --t
and
Then we have
gia(A) = ~f pJa(A), gH(i x II)) = giu(A) u(A),
= fJ = fJa(A)eo(A) a(A)eo(A)= = (Jr)i ( f r ) i,, and
ggH H eo(X) e o ( X )= = gir = = pp(Jr). (fr).
129
fr -+ E. Jr:: XX -----+ E.
Since (i, p) is assumed to satisfy the CHEP there is a morphism F:XxI~E
such that F(i x I) = fa(A), pF = gH, Feo(X) = fro
Define h : X
~
E by h = Fel(X) . Then we have
ph = pFel(X) = gHel(X) = gId x = 9 hi = Fel(X)i = F(i x I)el(A) = fa(A)el(A) =
f.
0
We now collect our results to obtain the following theorem.
Theorem (3.12). Let C be a category with finite limits and colimits. Let (I,P) be an adjoint cylinder/cocylinder pair on C such thatl satisfies DNE(2) and E(3). Suppose that any pair of morphisms (i,p) where i is a cofibration and p is a fibration satisfies the CHEP. Then (I,P) is weakly generating. Proof. The theorem follows from Proposition (3.11), Theorem (I.5.11) and its dual, and Theorem (3.8). 0
Remark. As we have indicated in section I.3, there is an alternative way of doing abstract homotopy theory in a category C with a cylinder I based on the notion of fibration defined by the homotopy lifting property with respect to the cylinder (see for instance Kamps (1968), (1969), (1972 a)) . One could ask when I is a generating cylinder for a fibration structure on C. The result would be a theorem similar to Theorem (1.6). Using Kan conditions one would be able to prove the axioms for a category of fibrant objects except for the dual of axiom (C 4) (see (1.4)). Note that in this case no assumption on the preservation of (co )limits by the cylinder functor will be necessary. FUrthermore, there is an even more sophisticated way of doing abstract homotopy theory using the more general notion of a (semi )cubical homotopy system which is defined to be a functor
Q : cop x C ~ Cub 130
such that Q(X, Y)o = C(X, Y). The notion being self dual, in this approach the notions of cofibration and fibration (resp. homotopyextension property and homotopy lifting property) are strictly dual to each other (see for instance Kamps (1970), (1972 b) , (1973 a,b), (1978 a,b), Berrick-Kamps (1985)). We end this section with a few remarks on richer structure on cocylinders. Let P = « )I,£O,£1,s) be a co cylinder on a category C.
Definition (3 .13). A natural involution on P is a natural transformation
such that i
0
i = Id, £oi = £1, £li = £0 and is = s .
The dual form of a subdivision is more like a composition. Let C denote the pullback functor ( )101 Xeo ( )1.
This means
is a functor together with two natural transformations p, T : C
--t (
)1
such that for each object X of C we have a pullback diagram in C C(X)
p(X) , Xl
7()()j XI
joo()() ----,
X
.
£l(X)
Clearly, C exists if C is assumed to have pullbacks.
131
Definition (3.14). A natural composition transformation on P is a natural transformation c : C ---+ ( )I
such that
COC
= cOT,
cIC
= cIP.
Definition (3.15). A natural comultiplication on P is a natural transformation such that (a)
(( )I,cl,m) is a comonad,
i.e. for every object X of C one has two commutative diagrams
Xl
m(X)
m(x)j (XI)I
(XI)I
jm(x (m(X))I
1
)
((XI)I)I
and
Xl
ym~ Xl (b)
co(X 1)m(X)
Cl(X I )
(XI)I
, Xl (Cl(X))1
= s(X)co(X) = (co(X))lm(X).
Definition (3.16). A natural interchange transformation on P is a natural transformation T : ( ) 12
---+ ( ) 12
such that for each object X of C
co(X1)T(X)
= (cO(X))I,
(co(X))IT(X)
= cO(X1) ,
cl(X1)T(X)
= (cl(X))1
(cl(X))IT(X)
132
= cl(XI) .
Although many examples of co cylinders do have one or more of these extra pieces of structure, there seems to be no overall pattern that can be observed. As a result we will not make use of such structure here , however in any given family of examples, assumption of such structure may shorten proofs and strengthen results, so the reader is advised, if tackling a new context, to check to see if any of these structures is available throughout their setting.
4. Additive Abstract Homotopy and Categories of Cofibrant Objects In our earlier discussion of additive abstract homotopy theory, we noted that any cylinder, I , defined by a cone monad, J, already defines enough well behaved structure for Dold's theorem to be relatively easy to prove. The same, however, is not true for other quite similar looking results such as the Relativity Principle and the Gluing Theorem. Having now introduced and studied various Quillen type theories of abstract homotopy theory, and the notion of generating cylinder, it is natural to ask if cylinders on abelian, or more generally additive, categories are generating and that will be the main question examined in this section. The picture is complex. An arbitrary general cylinder certainly defines notions of homotopy equivalence and cofibration (defined by the homotopy extension property) and as a starting point we note:
Proposition (4.1). Let I be an arbitrary cylinder defined by a cone monad ,J = (J,j , J.l) on an additive category, C, having finite colimits, then the resulting classes of homotopy equivalences (taken as weak equivalences) and cofibrations, satisfy axioms (C 1), (C 2) and (C 5) of Definition (2.4) . Furthermore the pushout of a trivial cofibration is a trivial cofibration. Proof. (C 1) This was discussed in the exercises on homotopy equivalences (after Proposition (1.8.5)). The characterisation of homotopy equivalences given by Proposition (1.8 .6) shows that isomorphisms are homotopy equivalences - but , of course, a one-line proof with trivial 133
homotopies will do that! (C 2) Recall the result of Proposition (1.8.2): a morphism
f:M--'>N is a cofibration with respect to I if and only if
J(f) : J(M)
--'>
J(N)
is a split monomorphism. It is then immediate that any split monomorphism and thus any isomorphism, is a cofibration. If f and 9 are cofibrations, with 9f defined then as J(9f) = J(9)J(f) and both J(9) and J(f) are split monomorphisms , so must be J(9f), i.e. 9f is a cofibration. (C 5) As 0 --'> M is a split monomorphism, so is J(O) --'> J(M) and every object is cofibrant. Finally we check that part of (C 3) that states that the pushout of a trivial cofibration is a trivial cofibration. We use Proposition (1.8.4) which gives that / : M --'> N is a trivial cofibration if and only if / is a split monomorphism with complementary summand in N, some subobject K for which j(K) is a split monomorphism. Consider such a trivial cofibration, f : M --'> M EEl K , then pushing-out along 9 : M --'> M', the corresponding induced morphism is /' : M' --'> M' EEl K and so is also a trivial cofibration. 0 The situation over axiom (C 4) and the rest of (C 3) is more confused, and complex. There is little point in an abstract theory in which the principal 'examples' fail to satisfy the axioms. Sometimes in examples J is an additive functor - but not always, and in fact in the main and, in some sense motivating, example, it is not. There are however many interesting examples with .J additive and amongst those we have the family of relative injective monads. With the additional assumption of the additivity of J, (C 4) is easy to prove. For the injective family of monads, J is not additive, but (C 4) is still valid, however a separate proof is needed. The situation for (C 3) is similar. If J is a relative injective type monad then (C 3) is fairly simply proved, but the proof is not valid without some alteration, for the injective type cone monads. It does not seem possible to obtain (C 3) and (C 4) without some extra conditions on the cone functor. 134
Proposition (4.2). Suppose I is given by an additive cone monad J = (J,j,J.l) (i.e. J is an additive functor) then (C 4) is satisfied.
Proof. We have to check that a(M) : M EEl J(M) ~ M is a homotopy equivalence and that eo + el : M EEl M ~ M EEl J(M) is a cofibration. The first of these is true without assumptions on the cone, and can be safely left to the reader. For the second statement, we have to check that
J(eo
+ el) : J(M) EEl J(M)
~
J(M) EEl J2(M)
is a split monomorphism. (Note: here we are using the additivity of J to conclude
J(M EEl M) ~ J(M) EEl J(M) and J(M EEl J(M» ~ J(M) EEl .J2(M).) This morphism is given elementwise by:
J(eo(M»(m) + J(el(M»(m/) = (m + ml, Jj(M)(m /» however
Jj(M) : J(M) ~ J2(M) is split by J.l(M) : J2(M) ~ J(M) by the monad properties of J = (J,j, J.l) (d. section 1.8) and because of this J2(M) ~ J(M) EEl K for some K, with the inclusion of the first summand given by Jj(M) up to this isomorphism. Thus J(eo + el) can be thought of as
J(M) EEl J(M)
~
J(M) EEl J(M) EEl K
(m, m/)
1--+
(m + ml, ml, 0)
which is clearly a split monomorphism.
0
We will return to (C 4) in the non-additive injective case, shortly.
Corollary (4.3). Given an additive cone monad J = (J,j,J.l) on an additive category, C, with finite co limits, the resulting cylinder is generating if and only if the pushout of a cofibration is a cofibration.D 135
If the functor .J is right exact, then it preserves pushouts and it is then easy to check that the pushout of a cofibration is a cofibration. Unfortunately right exactness is not satisfied in many interesting examples, and yet still this cofibration condition does hold. There would seem to be two main types of example, the 'injective-type ' cone monads and th~ 'relative-injective-type' ones . They are very similar, but at the same time, sufficiently different to make a unified treatment awkward. Injective-Type Cone Monads We assume that C is an abelian category with an injective cogenerator, U. ('Injective' is used as in a homological algebra: an object I in C is injective if for any monomorphism i : M - t N , the induced morphism
C(f,I) : C(N,I)
-t
C(M, 1)
is surjective, i.e. morphisms from M to I extend along f. An object, Q, is a cogenerator if given two morphisms i, 9 : M - t N , which are distinct, i i= g, there is a morphism x : N - t Q, such that x i i= xg.) Define a functor U : C -t
Sets~P
from C to the dual of the category, Sets*, of pointed sets (cf. IV.3.5) , by U(M) = C(M, U), pointed at the zero morphism. This functor has a right adjoint (under suitable conditions on C) given by
R : Sets~P
-t
C
R(X) = II{Ux : x E X ,x
i= *}
with each Ux isomorphic to U. The adjoint pair determines a monad with functor , J = RU, on C in the usual way (see below) . This type of monad leads to each J (M) being an injective object in C. The dual situation in which C has a projective generator yields the usual free- forget adjoint pair on C (see Huber (1962) for another approach to this theory). It must be noted that although J(O) = 0, the functor is not additive. 136
Relative-Injective-Type Cone Monads Here we assume given an additive functor U : C ---+ 1) having a right adjoint R : 1) ---+ C, with, of course, 1) an additive category. The unit I de
---+
RU
c: UR
---+
Id v
1] :
and counit
satisfy the usual relations (cf. MacLane (1971)). This means that we have two commutative triangles: 1]R R -----'--- RU R
~
URU
Rc
U1]
U
EU / .
R
U
Setting J = RU, j = 1] and fl = RcU gives a monad on C. The unit/identity diagram for this monad is then obtained by applying U from the right to the left-hand triangle, and R from the left to the right-hand triangle to get: RU _--,1],-R_U__ RU RU
RU1]
RU
~R~ RU MacLane (1963) gives several additive or abelian situations in which such a monad arises. We will look at some of those in more detail in Chapter III, section 5. A typical situation is that of comparing structure along a ring homomorphism
. 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(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' > = < 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 one-one 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
<
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 non-additive 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 C-injective object of A or relative C-injective 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 : XEBK-X 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 C-injective 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 C-injective 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
X---tXE9J~YE9L---tY
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 non-trivial 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 C-injectives, 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 C-injective 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
<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-..:....f-XEElK
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
>~<
Y
---t
YEBK >
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 C-injectives determine the same element of Io(A;C). Suppose that A has only setwise many isomorphism classes of relative C-injectives 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 C-injectives
(b) < K >, if K is cofree. Then Io(A;C)= FIR. (Here < K > denotes the isomorphism class determined by the relative C-injective K and [K] = < K > +R.)
Theorem (2.2). Given two relative C-injectives K and M zn A, [K] = [M] if and only if there is a relative C-injective 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 C-injective and cofree Uk, VI such that
< K > - < M >=
- [L:( < Tj EB Tj' > -
[~( < S~ EB S:' > -
< Tj > - < Tj'
<
»] + L: < k
j
415
S~ > -
< S:'
Uk > -
»]
L: < VI > . 1
Gathering terms with like signs on one side gives
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 C-injective K, [K] = 0 in Io(A;C) if and only if there exist a relative C-injective 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 L-injectives are cofree and Io(R- Mod; L) = O.
416
(b) I dempotent-M 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 C-injective and C is an idempotent monad, we show that K ~ C(K) as follows. There is some K' with KEEl K' ~ C(I<) with inclusion of K being of course j(K). Using additivity of C we have
C(K) EEl C(K')
~ C 2 (K) ~ C(K)
with the inclusion of C(K) being C(j(K)) hence C(K') ~ Ker f.L(K) and since f.L(I<) is an isomorphism we have C(K') = O. As K' is a direct summand of C(K') we must have K' = 0 and K ~ C(K). Again Io(A; C) = O. Relating both these ((a) and (b)) to the simple homotopy theory we find that all homotopy equivalences have the form
X ~ X EEl C(K) : I Y EEl C(X)
Y
pr I
Y
so as incx is the inclusionjo : X --+ Mo of X into the mapping cylinder of the zero map 0 : K --+ X, we have that all homotopy equivalences are simple. This seems at first glance, stronger than stating Io(A; C) = o as this latter states that all homotopy equivalences relative to X are simple - for each X. Of course the two are equivalent in this additive case.
(c) Injective-Type Homotopy with A = R-Mod, R Noetherian The injective-type homotopy theory, or i-homotopy theory, of a category of modules over a Noetherian ring is simple. Suppose
A=R-Mod with R a (left) Noetherian ring. We need some results from the literature.
417
(i) R is (left) Noetherian if and only if every injective (left) R-module is a direct sum of indecomposable (injective) R-modules. (ii) R is (left) Noetherian if and only if every direct sum of injective (left) R-modules is injective. (iii) A left R-module E is an indecomposable injective if and only if there is an irreducible left ideal I of R such that E s:: E(R/1) where E( ) denotes the injective envelope. ((i) is a theorem of Matlis and Papp and (ii) is a well known result of Bass and Papp. A convenient reference for all three is Sharpe and Vamos (1972) . (i) is Theorem 4.4, (ii) is Theorem 4.1 and (iii) is Corollary 4 to Proposition 2.28.) Now suppose K is an arbitrary injective in R-Mod. We have (by
(i) )
where E j s:: E( R/ I j ) and I j is irreducible. We assume that if jI =1= j2 then Ejl '1- E h · Now construct a family of cardinals {,Bj : j E J} as follows: If {Xj is finite, let ,Bj be any infinite cardinal. If {Xj is infinite, let ,Bj = {Xj.
Then using (ii) we have that the module
Q =.EB (EB E j ) JEJ
(3j
is injective and K EB Q s:: Q. Hence [K] + [Q] = [Q] and [K] = o. Hence 10 (R-Mod; J) = 0 and we have not even chosen an injective cogenerator ! This sort of collapse seems likely to happen much of the time since it is precisely the dual of the collapse of the projective class group when infinitely generated projectives are allowed. This duality is best illustrated by showing that the projective class group occurs as an example of an Io(A; C). This is the importance of (d) below.
418
(d) Projective-Type Homotopy with A = R-Mod Projective-type homotopy theory or p-homotopy theory was mentioned in IlA and IlL5 . In order to treat its 'simple' form more fully we need to sketch a few more of the details. As was mentioned earlier, this theory is dual to i-homotopy and we will develop simple p-homotopy with this in mind. As in IlL5 , we work in AOP (and for the sake of the exposition restrict to A = R-Mod) . There is a natural choice of injective cogenerator in AOP namely the projective generator, R , of A. Now suppose M is any module, and consider the object J(M) . The functor
is given by
[1(M)
=
A OP(M, R) A(R, M)
~M
considered as a set pointed at zero, thus [1 is exactly the underlying set functor viewed from a different direction. The functor P is also identifiable as something familiar: p : Sets~P
--t
AOP
is given by
II 'Rx with Rx = R if x =I=- *, R* = 0, xE X where 'product' is taken in AOP - hence is really coproduct in A, i.e. P(X) = '
P(X) =
II
xE X
Rx = EB Rx. xEX
Thus P(X) is the reduced free R-module on the pointed set X (reduced because the submodule generated by zero is to be 'reduced to zero') . Combining these two interpretations , one gets the interpretation of J(M) as the free module on the underlying set of M , except one requires that J(O) = 0, not R; this is necessary to ensure that J is additive. 419
Thus one can develop the p-homotopy theory in an analogous fashion, obtaining characterisations of fibrations, trivial fibrations and homotopy equivalences [of course 'fibration' is 'cofibration in A OP' and 'cosimple p-homotopy in A ' is 'simple i-homotopy in AOP']. E(X) for this situation is again independent of X and is identifiable as Io(AOP; J) ; however this latter group is precisely the group of isomorphism classes of projective modules modulo the stably free ones. 'Stably free ' takes on the same meaning as in Milnor (1966) except for the lack of 'finitely generated'. Thus , by well know results , Io(AOP; J) = 0 in this case as well. The nullity or otherwise of Io(A; J) for A = R-Mod is, as yet, unknown as it is far less easy to describe general injective modules in the general case. However one is tempting to expect or even conjecture that Io(A; J) = 0 unless some finiteness restriction is placed on A .
4. Finitely Generated Relatively Injective 7l( C 2 )-Modules We clearly could do with an example where Io(A; C) is non-zero . To obtain this we turn to a relative injective setting, in fact to the situation of a category of finitely generated modules over a (finite) group G and in particular to the simplest possible non-trivial case of G = C 2 , the cyclic group of order 2. Before we restrict to that case however, we will briefly examine the general case. The results we obtain will be fairly unsophisticated as we do not want to introduce a lot of ring or module theory into a book on abstract homotopy and simple homotopy, especially so near the end. Suppose that G is a group and 7l( G) the corresponding group ring over the integers. If we write as usual Ab for the category of abelian groups, then the category of 7l( G)-modules is the same as Ab G the category of functors from G, considered as a groupoid with one object, to Ab. As usual there is an adjunction U
Ab~AbG .
R Here U forgets the G-action and R = Homz(71(G), -) (see Hilton and Stammbach (1971) Ch. VI, § 11). The unit of the adjunction is given, for a G- module M, by 420
W = W(M) : M -----. RU(M) with W(m)(x) = xm for m E M, x E G. (Hilton and Stammbach use the term 'coinduced' for modules of the form R(A) for A in Ab; we will continue to use 'relatively cofree' or more briefly 'cofree'.) A relatively injective G-module is a direct summand of a cofree G-module. We note that as an abelian group
II
R(A) =
Ag
gEe
where each Ag ~ A and the G-action permutes the indices. To make life simple, we will assume G is finite . Later we will only need G = C 2 in any detail. Lemma (4.1). A G-module, M, is relatively injective if and only if there is an abelian group homomorphism
h : M -----. M satisfying
L
g-lh(gm) = m for all mE M.
gEe
Proof. If M is relatively injective, the counit map
M -----. RU(M) = EEl M gEe
is split as a G-module map, hence there is a family Sg :
M -----. M
of homomorphisms satisfying: for all m E M,
L
sg(gm) = m
(splitting)
gEe
and YSx
=
Sxy-l
for x, y E G
which gives G-equivariance. This G-equivariance implies that the family is specified once Sl = h is given since XSx = Sl. The splitting equation then gives 421
L:
g-lh(gm) = m
gEG
as required. Conversely given such an h, one can retrieve a splitting.
0
As a first application of this lemma, we note that, if A is an abelian group with trivial G-action, then A is relatively injective if and only if there is a homomorphism h:A----tA
satisfying nh(a) = a for all a E A where n is the order of G . Thus in this case A is relatively injective if and only if it is p-divisible and p-torsion free for all primes p dividing n. We now restrict to the case G = C2 , as our aim is not to give a complete classification of relative injectives but merely to produce a non-trivial relative-injective class group.
Proposition (4.2). Suppose A is a 7l( C2 )-module which is 2divisible and 2-torsion free as an abelian group, then A is relatively injective. Proof. First we note that 7l(C2 ) is isomorphic to 7l [x] / (x 2 - 1) , where as usual 7l[x] is the ring of polynomials over 7l, so 7l(C2 ) is generated as an abelian group by 1 and x with x 2 = 1 determining the multiplication. If A is any 7l( C2 )-module, let A+ = {a I xa = a} A-={alxa=-a}
then A + n A - ~ {a I a = -a}, the 2-torsion part of A . If A is 2-torsion free, A+ n A- = {O}. Now suppose A is 2- divisible, then given a E A there is some b E A with 2b = a. Then a = (b
+ xb) + (b -
x b) E A+
+ A-
so if A is both 2-torsion free and 2-divisible then A ~ A+ EEl A-. As A+ has trivial C2-action and is 2-divisible and 2-torsion free it will be relatively injective by our previous discussion. Consider there422
fore A- applying (4.1). We want to show there is some h- : Awith h-(a)
+ xh-(xa) =
----t
A-
a
but as xb = -b for all bE A-, this amounts to 2h-(a) = a and again as A- is 2-divisible and 2-torsion free, such .a map, h-, exists. Using h- and h+, the corresponding map for A+, we can construct 0 a suitable h : A ----t A showing that A is relatively injective. Suppose next that A is a finitely generated 7l( C 2 )-module. It is then also a finitely generated abelian group and we can split off the non-2 primary part of A, EB Ap. Provided this is a 7l( C 2 )-submodule of A, #2
it will be a direct summand of A since it is relatively injective and a direct summand of U(A). Thus to classify relative injectives completely, we would have to examine the 2-primary parts, the infinite cyclic parts (that is A / t( A) the torsion free part of A as abelian group) and also the question of whether EB Ap is indeed a submodule. We will not, #2
in fact, need a complete classification for the purposes of calculation within lo(A; C), but will need to examine some examples in detail. Starting with the torsion free part, of course, if A is torsion free as a group, then A + n A - = {O} so A + EB A-is a submodule of A. This may be a proper submodule since if a E A, we do not know much about xa even if 2a E A + EB A -. We therefore examine the cyclic module 7l( C 2 )a generated by a. To see all the possible structures it might have, we classify the ideals, I, of 7l(C2 ) then transfer this to the 7l(C2 ) / I, i.e. the cyclic modules of 7l( C2 ). Suppose 1 is an ideal in 7l(C2 ), then considering 7l C 7l(C2 ) as a subring, 1 n 7l must be of the form 7lt, the principal ideal generated by t within 7l, but then (t) = 7l(C2 )t ~ I. We have two possibilities 7l(C2 )t = I, which includes the case I = (0) the zero ideal (implying that t = 0) or 7l(C2 )t C I. In this second case we have ideals in 7l
h
= {a I a + xb E 1 for some b E 7l}
12
= {b I a + xb
and E I for some a E 7l}.
423
Since a + xb E I implies b + xa E I and conversely, II = 12 = llr. Thus there is some element of form r + x sr E I and we may assume s has minimal modulus with respect to this property, with I s I > 0. Suppose that t was not zero. Then since t + xt E I , r must divide t, so t = kr for some k E ll. The minimality of s then implies that < I s I < k otherwise I s I could be reduced further by taking off multiples of xt. Using the greatest common divisor d = (k, s), we can use the division algorithm further to reduce I s I to d, so s must divide k, say k = sl. We have r + xsr E I so multiplying by x, rs + xr E I and r s2 + X8r E I so
°
rs 2 -rElnll=llt hence r( 8 2 - 1) = mslr. The case r assume r i= in which case
°
s2 - 1 = msl
or
°
°
= corresponds to I = so we may 8
2
-
(ml)s - 1 = 0.
°
This only has integer solutions if ml = so s = ±l. Thus if (t) i= I , t i= 0, (t, r ± xr) with r It is a subideal of I . All elements of I must have form eJr + x sr for some eJ, sEll, (eJ - s)r
= (eJr - xsr) - s(r + xr) E In II = llt
and (eJ - s)r = nt for some nEll, but then eJr + X8r = s(r + xr) + nt which is in (t, r + xr). Similar arguments handle the case (t, r - xr) ~ I and can be adjusted for t = 0. We summarise this calculation in the following
Proposition (4.3). If I is an ideal of ll( C 2), then I has one of the following forms :
(a)
(t), t possibly zero.
(b)
(r
(c)
(t,r+xr) or(t,r-xr) withti=O, rlt .
+ xr)
or (r - xr) with r > 0. 0
Taking each type in turn we can list the different corresponding type of cyclic module. In each case we list with abelian group generators and the x-action:
424
(a) t = 0, I = (0). The module is ?l(C2 ) itself. As an abelian group it is ?l EB?l with x(a,b) = (b,a). Note that, of course, 1 is its ?l( C 2 )-generator with (1 + x) + (1 - x) = 2 an element of the corresp on ding A + EB A - . If I = (t), t i= 0, then as an abelian group ?l(C2 )/(t) ~ ?It EB ?lt again with x(a, b) = (b, a). The abelian group generators are 1 and x. If 2 t t, then ?l(C2 )/t ~ A+ EB A- generated by 1 + x and 1 - x. (b) I = (r + xr). As an abelian group, ?l(C2 )/I is ?l EB ?lr with generators 1 and l+ x. The ?l(C2)-action is x(a, b) = (-a, b+7f(a)) since (a, b) is a+b+xb, so x(a, b) = b+xa+xb or -a+(l+x )(a+b). Thus here 7f : ?l ---+ ?lr is the natural quotient map . I = (r - xr) gives a similar result: ?l EB ?lT) generators 1 and 1 - x, but x(a, b) = (a, -b - 7f(a)). (c) 1= (t, r + xr), with r It, giving ?It EB ?lr, the images of 1 and 1 + x as generators and x(a, b) = (-a, b + 7f(a)) with 7f : ?It ---+ ?lr (hence the need for the condition that r It). 1= (t, r - xr) with r It; ?It EB?lr with x(a, b) = (a, -b - 7f(a)). Having this list makes life easier. For instance if in (b) or (c), 2 does not divide r, the finite cyclic ?IT) which is an abelian group direct summand, is a submodule as is clear from the formulae for the action of x. As it is 2-divisible and 2-torsion free, it will be a relatively injective by (4.2). We will adopt the following notation: if A is an abelian group, we write A(+) for A with trivial x-action, xa = a, and A(_) for the ?l( C 2 )-module structure given by xa = -a. Then for cyclic modules of types (b) and (c) we have if r is not even, ?l(C2 )/(r + xr) ~ ?l(_) EB ?lr(+) ?l(C2 )/(r - xr) ~ ?l(+) EB ?lr(_) ?l(C2 )/(t, r + xr) ~ ?It(-) EB ?lr(+) ?l(C2 )/(t, r - xr) ~ ?It(+) EB ?lr(-).
Thus in each case there is a splitting A + EB A - . For our purposes we will not need the case of r even, but will turn straightaway to calculations within Io(A; C) . 425
Suppose A is a 2-divisible 2-torsion free abelian group and consider the class of A(+) in Io(A; C) . As such relative injectives split off from any containing module the subgroup of Io(A; C) generated by such a class is a direct summand of Io(A ; C) . First note that
so is cofree and hence [A(_)] = -[A(+)].
Suppose that for some r, r[A(+)] = 0 then setting B = A(+) EB ... EB A(+),
with r copies, we would have [B] = 0 so by (2.3) there must be ?l( C2 ) modules T , U, V with U, V cofree and
We may assume that all these relative injectives are 2-divisible and 2-torsion free so each side splits as a direct sum of the form M+ EB M-. One thus has isomorphisms
(*) (**)
B EB T+ EB V+ ~ T+ EB U+
T- EB V-
~
T- EB U-.
Since U and V are cofree, U+ ~ U- and V+ ~ V-, as abelian groups, and, again as abelian groups, we can cancel T- from (**) to get V- ~ U- and hence U+ ~ V+, but this can only happen if B ~ O. Thus [A] cannot be of finite order. Since if p =1= 2,
forms an infinite independent family of such classes , we have
Theorem (4.4). If A is the category of finitely generated ?l( C 2 )modules, then Io(A ; C) contains a countably generated fre e abelian group as direct summand. 0 So Io(A; C) is not always zero!
426
5. Open Examples and Problems
(a) The Projective Class Group It is annoying that the projective class group does not appear as a 'cosimple homotopy' group. As mentioned above, this is due to the fact that non-finitely generated projectives cause problems. How should one get around this? Here is one idea. We have not worked out all the details leaving them to you. (i) Replace a single additive co cylinder by a family of cocylinders , e.g. for a finitely generated module, M , we could assign the family of epimorphisms
{Fa
~
M: Fa finitely generated free}.
The corresponding family of co cylinders would be {M EB Fa} . Problem: investigate the analogue of naturality in this setting, then encode the analogues of the structure morphisms including comultiplications etc. (ii) Investigate the additive projective homotopy of this setting. (iii) Adapt the simple homotopy theory that corresponds to this finitely generated setting. Show (?) that the 'cosimple homotopy' group E(M) for a finitely generated R-module, M, is independent of M and is isomorphic to Ko(R), the projective class group of R.
(b) Functors on an Additive Category Suppose A is a small additive category with a projective type additive homotopy theory defined on it, given by an additive comonad (P,j, J.L). Let B be an additive category and C = Rex(A, B) , the category of right exact functors from A to B. (i) Define on C a co cylinder by FI(A)
= F(A EB P(A)) = F(A) EB F P(A),
if F : A - - B is right exact. Investigate the homotopy theory relative to this cocylinder. The interest in it is that Eckmann and Hilton (cf. Hilton (1965)) showed that the representable functors from A to Ab 427
have, in general, non-zero homotopy groups with respect to this sort of structure. In particular these homotopy groups were related to the Ext-groups. (ii) What is the corresponding simple theory? These last two open problems illustrate the potential complexity even of this additive situation. We have only scratched the surface with our calculations of additive simple homotopy groups, let alone tackling the many other types of simple homotopy theory suggested by the non-additive abstract homotopy theories. We leave this to you!
428
GLOSSARY OF TERMS FROM CATEGORY THEORY A~ a general reference for category theory one can consult MacLane (1971) and for additive and abelian categories Bass (1968) or Popescu (1973) . CATEGORY
A category, C, consists of a class, Ob(C), of objects and for each pair A, B of objects of C, a set C(A, B) of morphisms or maps with domain A and codomain B, together with for each A, B, C in Ob( C) a function c~,c : C(A , B) x C(B, C) ------> C(A, C)
called composition and written by juxtaposition ~,da , /3)
= /30' = /300'
and for each A in Ob(C), a distinguished morphism IdA E C(A , A), called the identity morphism on A . This structure is to satisfy (i) associativity: if 0' E C(A, B), /3 E C(B, C), "( E C(C, D) then
"((/30') = b/3)a (ii) identities: if 0' E C(A , B), then aIdA
= 0' = IdBa .
Remarks: (i) The basic example is the category, Sets , of sets with objects sets , morphisms being the functions between them , composition given by composition of functions and identities by the identity functions IdA (a) = a for all a EA . Other examples of categories that will be frequently used here include:
Top, a category of topological spaces and continuous maps
Ab, the category of abelian groups and homomorphisms R-Mod, the category of (left) R-modules for a ring R. (ii) A category C is small if Ob(C) is a set. (iii) A category C is finite if Ob(C) and each C(A, B) is a finite set. (iv) Some sources on category theory use the term 'hom-sets' to refer to the sets of morphisms in a category, C. This leads to a notation that is sometimes useful and we occasionally use' hom( A, B)' to denote the set of morphisms from A to B in an unspecified category. (v) If it is clear from the context which identity map is intended, I d will generally be given without a suffix.
429
A OR CATEGORY OVER E. These are defined in the text (1.6).
CATEGORY UNDER
FUNCTOR
Given two categories C and V, a functor F from C to V consists of (i) an assignment to each object C in C of an object FC in V, and (ii) for each pair A, E of objects of C, a function
F
= FA,B
: C(A, E)
--+
V(F A, FE)
compatible with composition and identities in the sense that
(a) F(f3a)
= F(f3)F(a)
if a E C(A, E), f3 E C(E, C)
(b) F(Id A) = IdFA . NATURAL TRANSFORMATION
Given categories C and V and functors F, G : C --+ V, a natural transformation TJ : F --+ G is an assignment to each object A in C of a morphism TJ(A) : FA --+ GA in V such that if a: A --+ E is a morphism in C, then the square
TJ(A) FA -:"':-:-"""GA
Fal FE
lea GE
TJ(E)
is commutative, i.e. the two composites GaTJ(A) and TJ(E)Fa are equal. If each TJ(A) is an isomorphism (cf.) then TJ is called a natural Isomorphism or sometimes a natural equivalence. SUBCATEGORY
If C is a category, a subcategory A of C consists of a class, Ob(A), of objects , which is a subclass of Ob(C) and for each pair of objects A, A' in Ob(A), a subset
A(A, A')
~
C(A, A')
such that the composition on C restricts to give a composition on A and for each A in Ob(A), IdA E A(A,A) . The subcategory A is said to be full if for each A, A',
A(A, A')
= C(A, A') .
Any full subcategory can be specified by specifying what will be the class of its objects . For example, the category of finite sets is the full subcategory of Sets determined by the class of finite sets. 430
PRODUCT CATEGORY
If C and V are categories, a new category, C x V, called the product of C and V is formed by taking
Ob(C x V)
= Ob(C)
x Ob(V)
and if (G, D), (G', D') are objects in C x V,
C x V((G, D), (G', D'))
= C(G, G')
x V(D, D') .
Composition is defined in the only way that makes sense, i.e.
(f,g)o(f',g)
=
(foJ',gog') .
It is sometimes useful to note that if C and V are small and £ is another category any functor 'of two variables'
F :C x V __ £ corresponds to a functor
F': C __ £'0 using the rule F'(G) : V __ £ is the functor F'(G)(D) = F(G, D). Here £'0 is the category of diagrams of type V in £ (d.). The passage from F to F' can be reversed setting up an equivalence of categories (d.) between £c x'O and (£'O)c . QUOTIENT CATEGORY
Suppose C is any category, and for each pair of objects A, B there is an equivalence relation RA,B defined on C(A, B) that is compatible with composition in the sense that if J,1' E C(A, B) and 9 E C(D, A), hE C(B, G) then if JRA,B1', (fg)RD,B(f'g) and (hJ)RA,C(hf') . (In this case R = {RA,B} is often called a congruence relation on C.) Form a new category Cj R with
Ob(Cj R)
= Ob(C)
and
Cj R(A, B)
= C(A, B)j RA,B'
Cj R is called the quotient category of C by the congruence R.
There is an obvious projection functor
p:C--CjR that is the identity assignment on objects and which maps a morphism in C to the equivalence class it determines, p(f) = [J]R.
431
MORPHISM
see CATEGORY. DOMAIN
see CATEGORY . CODOMAIN
see CATEGORY. VERTICAL AND HORIZONTAL COMPOSITION OF NATURAL TRANSFORMATIONS
Given Fa, FI, F2 : C diagram
---+ 1)
and a : Fa
---+
FI , (3 : FI
---+
F2, giving a
Fa
C
FI
--"'--1) ,
there is a vertical composite (3a : Fa ---+ F2 defined by: if C is an object of C, (3a(C) = (3(C)a(C) . If, : Ga ---+ GI is a natural transformation between two functors from 1) to £, then there is also a horizontal composite " a : GaFa ---+ G I FI defined by: if C is an object of C
Naturality of , and a easily gives that the alternative definition as Gla(C),Fa(C) gives the same natural transformation . This is a special case of the Godement interchange law (d.) . CATEGORIES OF DIAGRAMS
If C is a small category (d.) and 1) a category we can form a category r;c whose objects are the functors from C to 1) and whose morphisms are the natural transformations between arrows . This category is called the category of diagrams of type C in 1). This provides one way of making more precise the idea of a commutative diagram . It is also useful when defining the notion of limit and colimit . GODEMENT INTERCHANGE LAW
(This is also called the middle two interchange law.) Given categories C, 1) and £, functors
432
Any split monomorphism is a monomorphism (d.). Any split epimorphism is an epimorphism (d.). Since if {3 : B --+ A is a splitting of a split monomorphism a : A -----t B we have that
the morphism {3 is sometimes called a right inverse for a. Similarly the splitting of a split epimorphism is a left inverse . (Here we use a 'geometric' convention and it would be just as logical to reverse the words 'left' and 'right' since if {3 is the right inverse of a, (3a = IdA!) Remarks: In geometric or topological settings, a splitting of a split monomorphism is sometimes called a retraction and a splitting of a split epimorphism is often called a section. This latter is especially true if the split epimorphism is a fibration . Of course a retraction is a split epimorphism and a section is a split monomorphism . INVERTIBLE
see MONOMORPHISM, EPIMORPHISM, ISOMORPHISM RIGHT INVERSE
see SPLIT MONOMORPHISM LEFT INVERSE
see SPLIT EPIMORPHISM . PUSHOUT AND WEAK PUSHOUT
Given a pair of morphisms (a 'for~')
B~A~C in a category C, a commutative square
is called a weak pushout of (a, (3) if given any other commutative square
A
a
Pj C
B
j1' 5'
437
• D'
(-y' a
= 5'(3)
Fo, F1 , F2 : C ----> 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 2-dimensional algebra. It occurs in an abstract form in the definitions of 2-category theory (Cat-enriched 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"
C----D 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 'non-dualised' 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
435
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 .
436
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 non-empty 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 'hom-sets'. 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.
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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 V-indexed 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 free-forget 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 V-indexed 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 Ab-category or preadditive category.
443
(Any ring is an Ab-category 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 'hom-set' 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 hi-hi, then there is some map x : 1 ----> X such that hx i- h'x . Of course this merely says that if hi-hi 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 hi-hi, 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. Ab-category 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 S-enriched 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· 2-category 269 S-category (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 I-category) 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 S-category 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 CW-complex 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 I-category) 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 V-indexed 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 Dold-Kan 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* - CW-complex 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* S-functor 321 fundamental group of Y at y 149
Eckmann-Maumary approach to simple homotopy theory 365 Eckmann-Siebenmann 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 G-simplicial 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 oo-groupoids 206 w-groupoids 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 S-category 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
I-category (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* C-injective object 403 - - of A 403 injective type additive homotopy theory (discussion) 246 interchange axiom (in an I-category) 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 S-category 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 S-category 242 - weakly Kan S-category 242 m-fold 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
n-cubes 459
20
n-equivalence 227 n-simplex (in a simplicial object) 161 n-types (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 non-negative 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 I-category) Quillen model category 77 quotient category 431-
84
rank n 204 relative C-injective 403 - - class group of A 412 - - object of A 403 - cylinder axiom (in an I-category) 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 Eckmann-Siebenmann 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 n-simplex 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 S-category 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 S-category 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
