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?
Categorically speaking this question is undecidable. The question has different answers for equivalent copies of Set. Isbell points out (reported in CTWM, end of VII-1) that if Set (or any subcategory thereof containing a countably infinite set) is skeletal, on-the-nose associativity is impossible. Stacy Finkelstein in her thesis (or at least in a talk on Tau Categories that I recall as being based on her thesis) gave a subcategory of Set consisting of ordinals up to w^w and their (order-ignoring) functions with an on-the-nose product. In the course of the discussion following my question of 3/11/96 to this list about the relative ease of defining set membership and composition in terms of each other, I posted a similar construction (on 3/14/96) for the whole of Set (more precisely, for a subcategory of Set consisting of those sets that can be well-ordered, more precisely yet Ord(inals) and their (order-ignoring) functions). (I learned about Stacy's construction shortly thereafter.) These latter versions of Set are of course not skeletal by virtue of distinct ordinals (w, w+1, etc.) being isomorphic, necessary by Isbell's observation. Whereas my set-membership question and its subsequent lengthy discussion were I gather appreciated by many, the reactions to the on-the-nose product I posted as part of it varied from indifference to outright hostility. On reflection these reactions, coming from category theorists, are entirely consistent with the categorical undecidability of whether Set admits on-the-nose product. Vaughan Pratt