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. On Wed, 21 Oct 1998, Philippe Gaucher wrote:
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.