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
vii
Previous Page Next Page