categories: Re: colimits of categories

[Steve Lack <stevel@maths.usyd.edu.au>]

David Carlton writes about colimits of categories. Other
people have answered many of his questions, but not yet:

 > 
 > 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.)

If by n-Cat you mean strict n-categories and strict n-functors, then
here is a response to the ``colimit'' part of the question. 

First of all, you can't generalize from filtered colimits to arbitrary
colimits, even in the case 1Cat-->0Cat. Write Cat for 1Cat and Set for 0Cat,
and P:Cat-->Set for the decategorification functor, which sends a category
to its set of isomorphism classes of objects. This functor P preserves
filtered colimits, as you said (or see below), and clearly also preserves 
coproducts, but doesn't preserve the coequalizer
        f       q
    A ----> B ----> C
      ---->
        g

where C is the free-living isomorphism, with objects 0 and 1, and two
mutually inverse non-identity arrows 0-->1 and 1-->0;
where B has objects 0 and 1, and arrows freely generated by 0-->1 and
1-->0, and where q is the evident quotient;
where A has two objects 0 and 1 and arrows freely generated by 0-->0
and 1-->1, and where f and g are the evident functors.

Alternatively, observe that Cat is locally finitely presentable (lfp), so that
if it preserved all colimits it would have a right adjoint, and prove
that it does not have one.

On the other hand, decategorification P:n-Cat-->(n-1)-Cat does
preserve filtered colimits for any n. To see this, write n-Catg
for the full subcategory of n-Cat consisting of those n-categories
in which all n-cells are invertible. The inclusion I:n-Catg --> n-Cat
has a right adjoint R which forgets all non-invertible n-cells.
(It also has a left adjoint.) Now RI=1 and PIR=P,
so that P will preserve filtered colimits if PI and R do so. But
PI:n-Catg ---> (n-1)-Cat has a right adjoint D, which regards an
(n-1)-category as an n-category with no non-identity n-cells. Thus
PI preserves all colimits, and P will preserve filtered colimits
provided R does so. 

If V is locally finitely presentable, then so is V-Cat [G.M.Kelly 
and Stephen Lack, V-Cat is locally presentable or locally bounded if V is 
so, Theory Appl. Cat. 8:555-575, 2001]. From the equation
(n+1)-Cat=(n-Cat)-Cat and the fact that 0-Cat(=Set) is lfp, it follows by
induction that n-Cat is lfp for every n. Similarly from the
equation (n+1)-Catg=(n-Catg)-Cat and that fact that 1-Catg(= the category
Gpd of groupoids) is lfp, it follows that n-Catg is lfp for every n. Now 
R:n-Cat-->n-Catg is a right adjoint functor between lfp categories, so will 
preserve filtered colimits if and only if its left adjoint I:n-Catg-->n-Cat 
preserves finitely presentable objects.

For an object G of V, write 2_G for the V-category with objects 0 and 1,
and homs 2_G(0,0)=2_G(1,1)=I, 2_G(1,0)=0, and 2_G(0,1)=G. By the Kelly-Lack 
paper, the finitely presentable objects of V-Cat are the closure under
finite colimits of the V-categories of the form 2_G for G a finitely 
presentable object of V. It follows that I:(n+1)-Catg --> (n+1)-Cat
will preserve finitely presentable objects if I:n-Catg ---> n-Cat does
so. Thus it remains only to show that I:Gpd-->Cat preserves finitely
presentable objects, or equivalently that R:Cat-->Gpd preserves filtered
colimits.

There are various ways to do this. One could use the description of
filtered colimits in Cat given in the Kelly-Lack paper to show that
IR preserves filtered colimits, and deduce that R does so. Alternatively
one could show that the ``free-living isomorphism'' (called C above)
is finitely presentable in both Gpd and Cat, and constitutes a
strong generator of Gpd, and deduce that I preserves finitely presentable
objects.

Similarly, P:n-Cat-->(n-1)-Cat will preserve whatever limits PI 
preserves, and once again an inductive argument shows that PI will
preserve whatever limits PI:Gpd-->Set preserves (products, for instance).

Steve Lack.