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