categories: colimits of categories

[david carlton <carlton@math.stanford.edu>]

Is there a good reference for the construction of colimits of
categories?  Here, by 'categories', I mean 1-categories; and I'm
considering 1Cat as a weak 2-category.  I've been playing around with
this for the last week; I can construct limits without much trouble
(at least if the index category is a 1-category; I assume that
changing the index category to a 2-category wouldn't cause any
substantial problems), but constructing colimits seems noticeably
messier.  It's not too bad if your index category is filtered, but in
general it seems like a pain.

I frequently see statements like 'nCat is expected to have all limits
and colimits', so I assume that this has been verified in the case of
n=1 somewhere.

Also, while I'm asking, does the decategorification operation (from
nCat into (n-1)Cat) commute with colimits?  I was somewhat surprised
to see that decategorification from 1Cat into 0Cat does commute with
filtered colimits; so I'm wondering to what extent that statement
generalizes.  I.e. can I replace 1Cat by nCat, can I remove the word
'filtered', and for that matter can I replace colimits by limits?  (I
know decategorification doesn't commute with arbitrary limits of 1Cats
- indeed, that's arguably where much of the fun of higher category
theory comes into play - though I haven't thought too much about
whether or not it commutes with filtered limits.)

I have reason to hope that decategorification doesn't commute with
filtered colimits of 2-categories, but no hard evidence; trying to
check that seems like enough of a pain that I'm hoping somebody else
has done it first.  I haven't thought much about non-filtered colimits
since I can't even construct them; I'd be surprised offhand if
decategorification commuted with arbitrary colimits.  Then again, I
was surprised to see that it commuted with filtered colimits, so
clearly my intuition isn't the most reliable guide in this case.

David Carlton
carlton@math.stanford.edu