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.