COHERENCE AND NON-COMMUTATIVE DIAGRAMS IN CLOSED CATEGORIES
variance. If we identify shapes with allowable functors via CJV , then the
allowable graphs (between shapes) of [11] (in [11] they are defined direct-
ly, by certain inductive rules) "are" the graphs of the allowable natural
transformations; in other words, an allowable graph £:S T between
shapes is the graph obtained when, in the graph of an allowable natural
transformation h:pv(S) —*~ fv(T) , we replace every 1„ of the domain and
codomain by 1. As in [11],we easily see that the category (UG of shapes
and allowable graphs makes sense and is a closed category.
We now define a class JC of (formal) arrows between shapes, by de-
fining that, for any shapes A,B,C ,
a
"ABC
*ABC
bA
\ l
CAB
dAB
eAB
1 A
(AHB)HC -
AH(BHC) -
AHI -*• A
A -*- AHI
AHB -*
A -*- [B,AHB]
[A,B]HA -^ B
A —- A
AH(BHC)
(AHB)SC
are (formal) arrows between shapes (i.e. we define each such arrow as the
formal triad formed by its name, its domain and its codomain), and requir-
ing that JV consist precisely of all such formal arrows a
ABC *ABC
b,
~AB
:LR , eAr , 1A for all shapes A,B,C. The elements of
will be called formal simple instances of elements of {a,a ,b,b ,c,d,e,l]
between shapes. In particular, for any shape A, 1A:A ~^"A will be called
the formal identity arrow of A. It will always be clear from the context
whether the symbol 1 denotes an identity arrow or a shape. We will often
omit the shape-subscripts on the names of elements of OT , as being clear
from the context.
We next need the formal analog of the expanded instances of a, a" ,
b, b , c, d, e and (trivially) 1.
Let S be a shape. Suppose that, in S, precisely one variable is re-
placed by the name of an element x of Jt , each one of the other vari-
oo '
ables (if such exist) is replaced by the name of the formal arrow 1- , (the
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