R.VOREADOU

Next, in any closed category C , we distinguish four processes of

construction of arrows of C from given arrows of C :

1) given f:A —- B and centrals u:X —• A and v:B -- Y, form the

composite vfu:X —- Y

2) given f :A -- B and g: C -*- D , form fSg:ASC -* BSD

3) given f:ASB -- C , form rr(f) :A -*- [B,C]

4) given f:A -• B and g:CSD —- E , form the composite

([B,C]KA)ED (1Bf)g)1 ([B,C]KB)ED e331 CM) -%+• E ,

which we denote by g(fKl), so that f:[B,C]HA —--C denotes

the composite e(lKf); this involves the object C which does not

show up in the notation f, but this does not cause any real

difficulty in the way we use , because C is always understood

from the context.

If C is a category whose objects are shapes (resp. allowable func-

tors), then we introduce a notion of rank for the objects and the arrows

of C . Rank is defined by the following rules: for shapes (resp. allowable

functors), r(l) = 1 (resp. r(lc) = 1 ), r(I) = 0 , r(SST) = r(S)+r(T) ,

r([S,T]) = r(S)+r(T)+l ; for arrows, r(h:S--T) = r(S)+r(T) . An arrow

h:S - T of C is called trivial if r(h) = 0 .

Especially useful for arguments by induction on rank is the "cut eli-

mination property" of classes of arrows of a closed category C , defined

as follows: A class B of arrows of C has the cut elimination property if

for every h:T —~ S in (8 , at least one of the following is true:

(i) h is central

(ii) h is of the form

T_^A8B £Bg ,CBD-I*.S

with x,y central, f and g non-trivial, f and g in &

(iii) h is of the form

T v ^ \ [B,C]_Z^S

with y central and f in (B

(iv) h is of the form

T _2L^([B,C]KA)BD fM CSD -!. S

with x central and f,g in &