Subject: categories: Re: cogenerator in omegaCat ?

From: Philippe Gaucher <[email-masked]>
Date: Wed, 21 Oct 1998 18:45:26 +0200

> 	No, it seems not since a co-generator for omega cat
> would surely give rise to one for cat in particular, but such
> does not exist. This contrasts with the situation for the
> "larger" universe of simplicial sets.  A category of "small"
> sets is a kind of approximation to a co-generator, but each
> enlargement of the meaning of "small" creates new categories
> which are not co-generated.

The argument sounds reasonable. Before this question, I was 
convinced of the existence of this cogenerator. I have to find
something else for the lemma I would like to prove...

Since it does not exist, I have another questions (I suppose well-
known) and any reference abou the subject would be welcome : 

How does one prove the cocompleteness of omegaCat (small & strict) ?
The only idea of proof I had in mind until this question was : omegaCat
is obviously complete (and the forgetful functor towards the category of Sets 
preserves projective limits), and well-powered and a cogenerator 
=> the cocompleteness (Borceux I, prop 3.3.8 p 112).

Without cogenerator, how can one prove the cocompleteness ? The explicit 
construction of the colimit seems to be very hard : the forgetful
functor towards Set does not preserve colimits because the 
underlying set of the colimit might be bigger than the colimit of the
underlying sets. Every time two n-morphisms are identified in the 
colimit of the underlying sets, p-morphisms (with p>n) might be "created"
by the colimit.

Thanks in advance for any answer. pg.

---

From: Michael Barr <[email-masked]>
Date: Wed, 21 Oct 1998 15:36:24 -0400 (EDT)

I imagine that omega-categories, however defined, will be a locally
c-presentable category (c=cardinal of the continuum) in the sense of
Gabriel-Ulmer, equivalently complete and c-accessible in the sense of
Makkai-Pare and hence cocomplete.  In other words, the colimit will grow
but not by much.  Actually, aleph_1 is all you are really going to need.