Vaughan Pratt <pratt@cs.stanford.edu> Wednesday, March 5, 2008 8:32 am wrote, with much else:
On a related note, a careful reading of Max Kelly's "Basic Concepts of Enriched Category Theory" reveals that it is thoroughly grounded in Set,as I pointed out in August 2006 in my initial Wikipedia article on Max. I gave some thought to how one might eliminate Set from the treatment,without much success, and concluded that Max's judgment there was spot on.
Without addressing this particular issue I want to say I appreciate the phrase in the article: "the explicitly foundational role of the category Set." I take it this is Vaughan's? Various people including Sol Feferman promote the view that if you use "sets" then you are admitting that you use ZF and not some categorical foundations. Vaughan's phrase goes aptly against that: If you use sets, then you use sets, but there is no reason it cannot be on categorical foundations. He does not say it *is* on categorical foundations, and that is fine in the context. He reminds people that it *could* be. best, Colin