Products redone

Products redone

[Peter Freyd]

Kai Bruennler asked:

  Is there a binary product in the category of sets and functions 
  that is "strictly associative", i.e.

  A x (B x C) = (A x B) x C   and
  the associativity isomorphisms are equal to the identity?

I gave a construction using the axiom of choice. Well, we can do
better. Fix on an ordered-pair construction, say Weiner's, and denote 
its values as  <x,y>. Let  l  be the left-coordinate function, r the
right-coordinate function. 

Fix on a sequence of finite ordinals, say von Neumann's. Define,
inductively, an n-TUPLE as something of the form  <x,y>  where  y  is
a (n-1)-tuple if  n>1  else any old set if  n=1. (There are no 
0-tuples.) Note that the same set can be an n-tuple for any number of
values for  n. But if  x  is an  n-tuple and we are given  n  then
each of it's  n  coordinates is well-defined:

        x_1 = lx
        x_2 = l(rx)
...
    x_{n-1} = l(r(r(...r(rx)...)))
        x_n = r(r(r(...r(rx)...)))

where there are  i-1  applications of  r  used for  x_i, one 
application of  l  used for  x_i  if  i<n  and no application of  l
used for  x_n.

A trick of this construction of binary products is that we define 
products twice. Given  n  sets  a_1, a_2,...,a_n  define their n-fold
PRE-PRODUCT as the set of pairs of the form  <x,n>  where  x  is an  
n-tuple such that  x_i \in  a_i  each relevant  i.

If two sets  x  and  y  arise as pre-products, that is, if there are
positive ordinals  m  and  n  and  sets  a_1, a_2,..., a_m, b_1, b_2,
...,b_n  such that  x  is the pre-product of the  a's  and  y  is the
product of the  b's, then we define their PRODUCT as the (m+n)-fold 
pre-product of  a_1, a_2,..., a_m, b_1, b_2,...,b_n.

Define  F(x) =  x  if  x  is a pre-product else { <y,1> | y \in x}.
Define the product of arbitrary  x  and  y  as the product of  F(x)
and  F(y).