COHERENCE AND NON-COMMUTATIVE DIAGRAMS IN CLOSED CATEGORIES

while the graph of the obvious instance

(ly81Y)Bly — lY®(lyHly)

of c is illustrated by

(1VB1V)81V -^1V8(1VB1V)

'" C-H-=£=£U-

The complete formal details about graphs and natural transformations, as

needed here, are given in [11]. The graph of a natural transformation h is

denoted by [~h . By definition, two natural transformations h,hf:F —-G with

the same domain and the same codomain are equal if and only if i) Th = Th'

and ii) for every choice of arguments for the functors F and G according

to the graph [" h = Th' (i.e. with equal arguments corresponding to each

linkage via the graph), the corresponding components in y of h and h' are

the same. It follows that equality of graphs is a necessary condition for

equality of natural transformations. Moreover, [11] observes that graphs

of composable instances of elements of {a,a ,b,b ,c,d,e,l} can be com-

posed, and therefore the graph of any allowable natural transformation can

always be found.

Finally, [11] answers the question (II) by giving a wide class of

situations where equality of graphs suffices for the equality of allowable

natural transformations h,h':F —-G, namely: h = h? if and only if |~ h =

Thf, provided that the allowable functors F and G, written as words in lv

I, S, [,] (as described above), do not involve any word of the form [X,Y]

where no lv appears in Y and at least one lv appears in X (such F and G

are called proper in [11]). There are examples ([11],[2]) which show that

Th = Th' does not always imply h = h! in case the domain and the codomain

of h (and hf) are not both proper; these examples are discussed in §1 of

the first part of the present paper.

We may observe that the notion of graph, as used in the coherence

theorem of [11],is essentially independent of the closed category y , for

which question (II) is formulated, and the result of [11] is immediately

applicable with respect to any closed category, while question (II) can be

repeated for any closed category.

However, the essense of the coherence question is a question about

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