R.VOREADOU

transformations; their components in y are the canonical arrows of V and

they are precisely the arrows of V which exist because of the closed struc-

ture of V. The coherence question for V can be expressed as follows:

(I) When are canonical arrows with the same domain and the same codomain

equal in V, as a result of the closed structure of V (and not because

of any special property of the particular closed category V ) ?

or, equivalently:

(II) When are allowable natural transformations witn the same domain and

the same codomain equal, as a consequence of the axioms (i)-(vi) ?

Let I denote the category with a unique object I and a unique arrow

ly . Let us call allowable functors the functors generated by the identity

functor 1V:V -*- V , the constant functor I:V —•I , and repeated applica-

tions of S, [,] and composition; in other words, the allowable functors

for V are represented by the meaningful words formed by L , I, 8 and [,]

(with parentheses as needed). Then it is easy to see that the category,

whose objects are the allowable functors and whose arrows are the allowable

natural transformations between these functors, makes sense and is a closed

category (details in [11]), in which all the arrows are canonical; the co-

herence question (II) is clearly the question of testing equality of arrows

in this latter category. The problem is posed in this way in [11],where a

first answer is given to it.

At this point it is useful to remember that a natural transformation

h , in the sense it has been used in [11] and in this introduction up to

now (i.e. in the sense of [5]),consists of a family of arrows of V (the

components of h) and a graph, which indicates by linkages the pairs of

those arguments in the domain and the codomain of h, which must always be

equal in taking components of h, and with respect to which we have the nat-

urality conditions. For example, the graph of the obvious instance

of a is illustrated by

(1VB1V)B1V-^ 1VH(1VH1V)

' " r i

^ J "

r

vi