categories: Re: colimits of categories

[Michael Batanin <mbatanin@ics.mq.edu.au>]

on 31/1/02 6:27 AM, david carlton at carlton@math.stanford.edu wrote:


> For what it's worth, I can show that Decat can't be a left adjoint (in
> the relevant sense).  If it were, its right adjoint would be a functor
> F from Set (a 1-category trivially extended to a 2-category) to 1Cat
> such that, for all categories C and sets S, the category Hom_1Cat(C,
> FS) is equivalent to the set Hom_Set(Decat(C),S) (thought of as a
> discrete category).
> 

Sorry, when I wrote about Decat as a left adjoint I thought about
n-groupoids rather than n-categories. Steve Lack has already clarified the
situation.

But now I understand you are asking about weak (or pseudo) colimits. They
exist in 1-Cat and 2-Cat and can be expressed in terms of appropriate
weighted colimits. I never saw a paper about pseudocolimits in 3-Cat (here
we can use Gray-categories instead of general tricatgories). They must be
expressed as weighted colimits as well, or a codescent object of a
simplicial Gray-category. I'd like to have a reference if such a paper
already exists.

Michael Batanin.