Re: Generalizing Comma Categories

[Michael Shulman <shulman@math.uchicago.edu>]

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).
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]