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