From: Philippe Gaucher <gaucher@irmast1.u-strasbg.fr>
To: categories@mta.ca
Subject: Re: cogenerator in omegaCat ?
Date: Wed, 21 Oct 1998 18:45:26 +0200 [thread overview]
Message-ID: <199810211645.AA06093@irmast1.u-strasbg.fr> (raw)
> 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.
next reply other threads:[~1998-10-21 16:45 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
1998-10-21 16:45 Philippe Gaucher [this message]
1998-10-21 19:36 ` Michael Barr
-- strict thread matches above, loose matches on Subject: below --
1998-10-21 23:36 Ross Street
1998-10-21 22:47 Carlos Simpson
1998-10-20 9:33 Philippe Gaucher
1998-10-21 14:54 ` F W Lawvere
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