Subject: categories: Bruennler question

From: Peter Freyd <[email-masked]>
Date: Wed, 17 May 2000 10:46:52 -0400 (EDT)

Kai Bruennler asks:

  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?

The answer is yes if you're willing to use a lot of choice. Perhaps 
the quickest construction is to assume a well-ordering on the universe
with the property that  x < y  whenever  x \in  y. Then define the
pair  l:AxB --> A, r:AxB --> B  by stipulating that  AxB  is a von
Neumann ordinal "lexicagraphically ordered" by  l  and  r, that is,

          (lx < ly)  or  (lx = ly  and  rx < ry)
whenever
          (x \in  y)  and  (y \in  AxB).