Here is what the site actually says: Monotone functions respect order, group homomorphisms respect the group operation, linear transformations respect linear combinations, and gangsters respect membership in the Cosa Nostra, but what morphism has ever respected membership in a set? It is sheer hubris for a relation that can't get no respect to claim to support all of mathematics. (Old argument of category theorist Mike Barr, new polemics.) That is not a bad rendition (save for the reference to Cosa Nostra) of what I actually said which was that we create these elaborate structures of well-founded trees subject to the rule that two chidren of the same leaf cannot be isomorphic. But then, unlike all other structures that we build, we make no hypothesis that functions preserve the structure. Indeed, I think a structure-preserving map must be the inclusion of a subset. And there are no non-identity endomorphisms. On Mon, 21 Aug 2006, Vaughan Pratt wrote:
I've attributed the third saying at
http://boole.stanford.edu/dotsigs.html
to Mike Barr. If there's an earlier attribution I should be using please let me know.
Vaughan Pratt