Subject: categories: A situation in search of terminology

From: Vaughan Pratt <[email-masked]>
Date: Thu, 29 Apr 2004 07:02:08 -0700

I'm looking for language advice on dealing with the following "St. Ives"
problem, as a 2-categorical down-sizing of the 4-categorical original.

I have a 2-category CAT^{\pm} (arising from a 2-monad C\pm1 analogous to
the monad X+1) that is disconnected in the sense that it is representable
as a sum of connected 2-categories.  Each summand has homcats that are
similarly disconnected, being representable as a sum of connected homcats.

If I take a summand 2-category, and from each of its homcats coherently
take a summand 1-category, I end up with a 2-category with an initial and
a final category serving as respectively the syntax and the semantics of
a certain 2-theory (of linearly distributive but not self-dual categories
of dually typed objects, but that's another story I'm hoping Myles Tierney
will like better than the Chu story).

I don't think it's caused by my doing anything wrong--the situation appears
to have arisen naturally and I've found no way of making it go away.

Has this situation arisen before?  If so, what is the proposed terminology
for this kind of finality and its associated uniqueness?  If not, I'm happy
to make up my own conventions, but I don't want to reinvent the wheel here.

Vaughan Pratt