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 29-Jan-2002 20:56:41 -0400,2784;000000000000-00000000