R.VOREADOU
the closed structure, independent of any particular carrier of this struc-
ture. A formal theory of the background, in which a wide class of coher-
ence problems (including the problem for closed categories) can be stated
from this point of view, is given in [9].We will follow the spirit of [9]
here, without introducing its terminology and technical details. The idea
is to replace allowable functors by their "shapes", i.e. formal words in-
volving all the properties needed to define the functors they "represent",
but not mentioning the category y , and then consider certain formal ar-
rows between such shapes as "abstract allowable natural transformations"
("abstract allowable natural transformation" meaning that which remains of
an allowable natural transformation, when we do not mention the category y
with respect to which it is originally defined). We thus form a closed cat-
egory & which is free with one (non-constant) generator and whose arrows
are all canonical, and we state the coherence question for closed catego-
ries as the coherence question for this $~, i.e. as a question of testing
equality of arrows of £". More precisely, here we proceed as follows:
Shapes are defined as in [11],inductively, by the rules: 1 and I
are shapes; if S and T are shapes, then SST and [S,T] are shapes. The l's
that appear in a shape are called variables (they are considered as differ-
ent, by the order in which they appear in the shape); the l's are called
constants. It is now clear that there is a mapping ^v between the class of
all shapes and the class of all allowable functors with respect to a closed
category V ; the image under fv of a shape S is the allowable functor ob-
tained when, in S, every variable is replaced by the functor lv and every
constant is taken to mean the constant functor I. As in [11],with each
variable in a shape S is associated its variance (in S) which is positive
or negative, according as the functor fV(S) is covariant or contravariant
in the argument corresponding to the variable in question.
A graph £:S —- T between shapes ([11]) is an involution on the dis-
joint union of the sets of variables of S and T, and is illustrated by
linkages between corresponding variables; its main properties are that
i) no variable is linked with itself, ii) mates that are both variables
in the same shape S or T are of opposite variances, and iii) if a variable
in S is linked with a variable in T, these two variables are of the same
viii
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