Re: Small is beautiful

[Vaughan Pratt <pratt@cs.stanford.edu>]

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

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