Thanks for all the responses to my colimits question; I greatly appreciate them, and it will take me a while to digest them. I think I should be a bit more precise as to what I'm asking for, so let me try again: First, here's what I mean by a colimit of categories: let I be a 1-category, trivially extended to a weak 2-category with only identity 2-morphisms. (Or we could even have I be an arbitrary weak 2-category; I'm not too worried about that right now, but eventually I'd like an answer.) And let F be a weak functor from I to the 2-category 1Cat (which happens to be strict, but we think of it as a weak 2-cat.) Then: I want a 1-cat colim(F) such that, for all 1-cats C, the categories Hom_1Cat(colim(F),C) and Hom_(1Cat^I)(F,diag^I(C)) are equivalent, where by diag^I(C) I mean the constant functor from I to 1Cat^I sending all objects of I to C and all morphisms to identity morphisms. So I'm not looking at the set of functors from colim(F) to C: I don't really care whether or not that's equivalent to the set of functors from F to diag^I(C). I want an equivalence of categories. I'm fairly sure that some of the responses that I got answer this question; I'll look up the references and come back if I have more questions, but I'm provisionally happy with that for now. Here's the second question: Once we've constructed this, we can ask under what conditions the sets Decat(colim(F)) and colim(Decat(F)) are naturally bijective. (I guess there's a natural map from colim(Decat) to Decat(colim).) It's true for filtered index sets; is it true for general index categories? Also, we can generalize these questions to the setting of F be a functor from I to nCat, by which I mean the weak (n+1)-category of all n-categories; then we have a functor of n-categories that we want to represent. Since the notion of "weak n-category" is a matter of some debate, I'm willing to take it for granted that such a colimit does exist. Then: In what context (e.g. for what index categories) do we expect the (n-1)-categories Decat(colim(F)) and colim(Decat(F)) to be equivalent? Or the sets Decat^n(colim(F)) and colim(Decat^n(F))? Again, a definitive answer to that last question is unlikely since it would depend on having a firm grasp of weak n-categories (and of nCat), but I'm curious what people's instincts are. 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). In fact, we can't even have Hom_Set(Decat(C),S) equal Decat(Hom_1Cat(C, FS)). One way to see this is to first let C be the discrete category with two objects, which shows that the cardinality of Decat(FS) is just the cardinality of S. So we might as well assume that the objects of FS are just S and that no two objects are isomorphic. But then set C to be the category 0 -> 1; use this to select a morphism f:s->t for any s,t in S, and then to show that the composite of f:s->t and g:s->t is an isomorphism, so all objects are isomorphic after all. David Carlton carlton@math.stanford.edu 31-Jan-2002 23:01:36 -0400,969;000000000001-00000000