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