R.VOREADOU

identity arrow of the shape 1) and each constant (if such exist) is re-

placed by the name of the formal arrow lj ; let h be the resulting word.

Starting again from S, replace by the shape 3 x (resp. 3^ ) the variable

(in S) which was replaced by the name of x in forming h, and leave all the

other variables and constants in S as they are; let R (resp.T) be the re-

sulting shape. With name h, domain R and codomain T we define a formal ar-

row h:R -5- T between shapes. We then define Jc to be the class of all for-

mal arrows (between shapes) gotten in this way, for all shapes S and all

X£j£-oo . The elements of JC will be called formal instances of elements

o — —

of {a,a" ,b,b~ ,c,d,e,l} between shapes; the elements of JfQ- XQ0 will be

the expanded such instances.

Guided by the fact that every ordinary allowable natural transforma-

tion can be gotten as a composite of a (composable) string of (simple or

-1 -1

expanded) instances of elements of {a,a ,b,b ,c,d,e,l} , we proceed to

define the class X of the formal analogs of these composites.

We define Ot to be the class of all triads

( hn...h2h1 , S, T )

such that: h ...h?h1 is a word of finite length in which every h^ is the

A A

name of a formal arrow h- between shapes, each h. is an element of 3C ,

dA. = 3 h.

1

, S = 9 h-1and T = 3-h . The elements of X will also be

1 l o l+ l ' o I n

called formal arrows between shapes; in particular, they will be called

strings of formal instances of elements of

{a,a-1 ,b,b-1

,c,d,e,l} . If k =

( hn...h2h1 , S, T )eJT , we define 3Qk = S , d±k = T .

With composition of elements of Jf defined in the obvious way, i.e.

the composite of

h . . .h-h.:T -^- R and k . ..k^k. : S -*- T

n 2 1 mil

being

hn"-h2hlkm---k2kl:S~ R

it is clear that shapes and strings of formal instances of elements of

{a,a ,b,b~ ,c,d,e,l} form a category % . (In practice, when there is no

danger of confusion, we often denote arrows of U by their names only.)

The desired closed category #~ of shapes and "abstract allowable nat-

ural transformations" is gotten from U when we introduce additional condi-

tions (on the arrows of % ) which will establish