* Re: cogenerator in omegaCat ?
@ 1998-10-21 16:45 Philippe Gaucher
1998-10-21 19:36 ` Michael Barr
0 siblings, 1 reply; 6+ messages in thread
From: Philippe Gaucher @ 1998-10-21 16:45 UTC (permalink / raw)
To: categories
> 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.
^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: cogenerator in omegaCat ?
1998-10-21 16:45 cogenerator in omegaCat ? Philippe Gaucher
@ 1998-10-21 19:36 ` Michael Barr
0 siblings, 0 replies; 6+ messages in thread
From: Michael Barr @ 1998-10-21 19:36 UTC (permalink / raw)
To: categories
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.
>
>
^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: cogenerator in omegaCat ?
@ 1998-10-21 23:36 Ross Street
0 siblings, 0 replies; 6+ messages in thread
From: Ross Street @ 1998-10-21 23:36 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: cogenerator in omegaCat ?
@ 1998-10-21 22:47 Carlos Simpson
0 siblings, 0 replies; 6+ messages in thread
From: Carlos Simpson @ 1998-10-21 22:47 UTC (permalink / raw)
To: categories
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.
^ permalink raw reply [flat|nested] 6+ messages in thread
* Re: cogenerator in omegaCat ?
1998-10-20 9:33 Philippe Gaucher
@ 1998-10-21 14:54 ` F W Lawvere
0 siblings, 0 replies; 6+ messages in thread
From: F W Lawvere @ 1998-10-21 14:54 UTC (permalink / raw)
To: categories
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.
>
>
^ permalink raw reply [flat|nested] 6+ messages in thread
* cogenerator in omegaCat ?
@ 1998-10-20 9:33 Philippe Gaucher
1998-10-21 14:54 ` F W Lawvere
0 siblings, 1 reply; 6+ messages in thread
From: Philippe Gaucher @ 1998-10-20 9:33 UTC (permalink / raw)
To: categories
Dear all,
Does it exist a cogenerator in the category of (strict) omega-categories ?
pg.
^ permalink raw reply [flat|nested] 6+ messages in thread
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