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. 30-Jan-2002 09:02:37 -0400,2558;000000000001-00000000