categories: Re: colimits of categories

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

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