This is an interesting and slightly tricky question. There is a notion of "comma 2-category" which has this property; I believe it was first written down by Gray. Given (strict or weak) 2-functors F:A-->C and G:B-->C betwen 2-categories (or bicategories), their comma 2-category (F/G) is defined as follows: * its objects are triples (a, b, s:F(a)-->G(b)) of an object in A, an object in B, and a morphism in C. * its morphisms from (a,b,s) to (a',b',s') are triples (u:a-->a', v:b-->b', z: s;G(v) --> F(u);s') of a morphism in A, a morphism in B, and a 2-cell in C. * its 2-cells from (u,v,z) to (u',v',z') are pairs (x:u-->u', y:v-->v') of a 2-cell in A and a 2-cell in B, such that (F(x);s').z = z'.(s;G(y)). You can check that if A and B are terminal so that F and G pick out objects c, c' of C, then (F/G) is the hom-category C(c,c') regarded as a locally discrete 2-category. There is a "dual" comma 2-category which would give you C(c,c')^op instead (assuming I didn't screw up and write down the dual version above). The tricky part is that while an ordinary comma category is expressible as a certain kind of weighted limit in the 2-category Cat, and can therefore be generalized to comma objects in other 2-categories, the comma 2-category as defined above is *not*, as far as I know, a weighted limit in the (strict or weak) 3-category 2-Cat. It does have a universal property, though: just as an ordinary comma category is equipped with a natural transformation filling a square in a universal way, the comma 2-category is equipped with a *lax* natural transformation. But there is no 3-category whose 2-cells are lax natural transformations. I believe that this universal property can be expressed as a weighted limit in Gray_lax enriched categories, where Gray_lax denotes 2-Cat with the lax version of the Gray tensor product, but I don't know any references for such a point of view. I'd be very interested to hear of other work on this question; it seems potentially important in developing a notion of exactness for 3-categories. Mike On Fri, Aug 20, 2010 at 6:46 PM, David Leduc <david.leduc6@googlemail.com> wrote:
Hi,
Thank you for all the replies to my previous questions. It is very helpful. Now, I have another (trivial) question.
If I have two constant functors Delta_X and Delta_Y that respectively returns X and Y in C, then their comma category is the discrete category C(X,Y).
How do comma categories generalize to bicategories?
When C is a 2-category, I would like the "comma category" of the "functors"(?) Delta_X and Delta_Y be the (not necessarily discrete) category C(X,Y).
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