Dear all, Does it exist a cogenerator in the category of (strict) omega-categories ? pg.
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. Bill ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* On Tue, 20 Oct 1998, Philippe Gaucher wrote:
Dear all,
Does it exist a cogenerator in the category of (strict) omega-categories ?
pg.
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.
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.
In response to Ph. Gaucher's question: I (try to, at least...) treat this question for weak $n$-categories in my preprint ``Limits in $n$-categories'', available on the xxx preprint server as alg-geom 9708010. If I understand correctly, the set-theoretical problem you raise is the same as the one encountered in section 5 of my preprint. The conclusion is that the (weak) $n+1$-category $nCAT$ is closed under direct limits. It seems that coproducts of strict $n$-categories, if they exist, cannot actually be the ``right'' ones because in that case, every weak $n$-category would be equivalent to a strict one. I haven't made this argument rigorous, though. ---Carlos Simpson PS what is a ``comma category'' or ``comma object''?
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.
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
participants (5)
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carlos＠picard.ups-tlse.fr -
F W Lawvere -
Michael Barr -
Philippe Gaucher -
street＠mpce.mq.edu.au