Subject: categories: Re: cogenerator in omegaCat ?

From: [email-masked] (Carlos Simpson)
Date: Wed, 21 Oct 1998 22:47:13 GMT

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.