COHERENCE AND NON-COMMUTATIVE DIAGRAMS IN CLOSED CATEGORIES 7
pairs of allowable natural transformations (having the same domain, the
same codomain and the same graph), which do not permit applicability of
Proposition 2 (this class is precisely the class y( introduced below), ap-
propriately enlarged to a class IV that is usable in inductive arguments
based on the cut elimination property of the allowable natural transforma-
tions (pp.xiv,xvi). Theorem 5 below shows that, indeed, a coherence theo-
rem is obtainable by excluding the elements of the class Of" .
The classes Jf and J{ are defined as follows:
0^ is the class of those pairs (h,h') of allowable natural transfor-
mations which satisfy the following conditions: h and hf have the same do-
main, the same codomain and the same graph, and they can be written in the
form as
h:S -JL^. ([B,C]KA)BD
fl51
C(3D
g
T
h ' : S - ^ - * - ( [ B
?
, C
f
] H A
f
) 1 8 D
f f
'
m
y- C
!
BD'
g
V-T
with x,x' central,
f,g,f',gT
allowable and with
1) B and B1 non-constant
2) Tf and rff not of the form
3) [B,C] associated with a prime factor of A' via x'x
4)
[B',C!]
associated with a prime factor of A via
x(xf)~
,
but Th =
[~hf
cannot be written in any of the forms "central", $ or IT .
Using conditions l)-4) above together with the properties of central
graphs (pp.xiii,xvi), the fact that every allowable k:X—- [Y,Z] is
7T(7T (k)) and Proposition B of [18,§1], we see that, in the definition of
Off , the condition "Th =
Th1
cannot be written in any of the forms "cen-
tral", S or 7T " may be replaced by "neither h nor
hT
can be written in any
of the forms "central", S or T ".
W is the smallest class of pairs of allowable natural transforma-
tions satisfying:
^fl. Jf is contained in the class
^2. if (f,f':T -• S) is in the class and u:T» -*- T, v:S -- Sf are central,
then (vfu,vf'u:T' -^- S') is in the class
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