Ross Street wrote:
Part of what Bob Paré was arguing, I believe, was that we should be flexible (pun intended) about what "small" means. If "small" means "finite" then FinSet is not "essentially small". Also, "small" could mean "no more than one element".
Thanks, Ross. Hopefully Bob will phrase it that way next time. ;) If 2 is the usual symmetric monoidal closed category with objects 0 and 1 and only non-identity morphism 0 --> 1, then Chu(2,1) has four objects while Chu(2,0) has only three, but both are self-dual. The CEO of search engine company Cuil (Old Irish for knowledge) had finite categories of this kind in her 1997 Ph.D. thesis. What got me started on my previous message was that Bob was calling these "syntactic" when to me they were semantic. If by "syntactic" he meant "finite," or more generally less than some specified ordinal, then I have no problem with that, other than that I'd prefer he be specific about the ordinal rather than vaguely saying "syntactic." Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]