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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) +
. 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.
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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.
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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 .
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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.
440
If lim F exists for all F in CV then
k : C --+ CV has a right adjoint (d.) since
CV(kc, F) ~C(C, limF) . Dually if k has a left adjoint (d.), then C has all 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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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*
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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
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- 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
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(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
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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