Re: cogenerator in omegaCat ?

[street@mpce.mq.edu.au (Ross Street)]

As to the cocompleteness of Omega-Cat, it is a result of Harvey Wolff that
V-Cat is cocomplete for decent  V.  By induction, it follows that n-Cat is
cocomplete (since (n+1)-Cat = n-Cat).  A limiting process gives that
Omega-Cat is also cocomplete.

However, a better approach is to use a result of Michael Batanin that
Omega-Cat is finitarily monadic over globular sets (a presheaf category).
It follows that  Omega-Cat  is cocomplete.

The required monad on globular sets is beautiful: it involves plane trees. See:

M. Batanin, Monoidal globular categories as a natural environment for the
theory of weak n-categories, Advances in Mathematics 136 (1998) 39-103.

R. Street, The role of Michael Batanin's monoidal globular categories,
Proceedings of the Workshop on Higher Category Theory and Mathematical
Physics at Northwestern University, Evanston, Illinois, March 1997 (to
appear).

M. Batanin, Computads for finitary monads on globular sets, Proceedings of the
Workshop on Higher Category Theory and Mathematical Physics at Northwestern
University, Evanston, Illinois, March 1997 (to appear).

M. Batanin and R. Street, The universal property of the multitude of trees,
Macquarie Mathematics Report 98/233, March 1998 (submitted).

R. Street, The petit topos of globular sets, Macquarie Mathematics Report
98/232 (March 1998; talk at the "Billfest" in Montréal, September, 1997;
submitted).

Regards,
Ross