Re: Comma category

From: Todd Trimble (trimble1_at_optonline.net)
Date: 10/10/04 

Date: Sun, 10 Oct 2004 22:11:05 +0000 (UTC)

On 10 Oct 2004, noone wrote:
>> What motivates you to ask these questions?
>
>Thanks Todd for your very helpful answers to my questions. Here my
>motivation was just to have a better understanding of comma categories in
>term of universal definition. Indeed it is often introduced as an
>elementary definition.

You're welcome. After reading your letter, I think I have a better
sense of what you were aiming for -- see below.

>
>> We can think of the comma category construction (for functors
>> targeted at E) as a functor of the form
>>
>> Cat/E x Cat/E ~= Cat/(E x E) --> Cat,
>> by pulling back along <dom, cod>: E^2 --> E x E,
>>
>> F|G ----------> C x D
>> | |
>> | pullback | F x G
>> | |
>> E^2 ----------> E x E,
>> <dom, cod>
>>
>> where E^2 is the category of arrows (and commutative squares
>> between them) in E. But this is not a right adjoint (either
>> as a functor on pairs (F, G), or as a functor in one of the
>> arguments F, G with the other held fixed). For example, the
>> binary product of F, F' in Cat/E is given by pullback F x_E F',
>> but (F x_E F')|G is not isomorphic to F|G x F'|G, so the
>> functor -|G does not preserve products.
>
>Let 1-->2<--3 be a diagram (the one used to see a pullback as a special
>case of limit). Let Diag : Cat --> Cat^(1-->2<--3) be the diagonal
>functor. We have, for any object C-F->E<-G-D of Cat^(1-->2<--3), a
>representation
>
> Cat(_, F|G) =~ Cat^(1-->2<--3)(Diag _, C-F->E<-G-D)
>
>By Parameterized Representability Theorem, this induces a unique functor
>(up to isomorphism) from Cat^(1-->2<--3) to Cat, which is therefore right
>adjoint to Diag. Do you agree with that?
>

Sorry, not quite. You're getting warmer though. :)

If X is any category, then a representing object for

           a |--> X^{(-)->(0)<-(+)} (Diag(a), x -f-> z <-g- y)

is just the pullback of f and g. However, F|G is not the
pullback of F and G. It *is* however a pullback of the
diagram
                   F dom cod G
                C ---> E <--- E^2 ---> E <--- D

and while we're at it we might as well cast this in the
language of weighted limits: take V = Cat (where V-categories
are now *2-categories*), let J be the category (-) -> (0) <- (+)
(considered as a 2-category by viewing the hom-sets as discrete
categories), and let F: J --> V be the functor (or 2-functor)
into Cat which sends (+) and (-) to the terminal category 1,
(0) to the category 2 = (0 -> 1), the arrow (-) -> (0) to the
inclusion i_0: 1 --> 2 valued at 0, and the arrow (+) -> (0)
to the inclusion i_1: 1 --> 2 valued at 1.

Then given a 2-category X, and a 2-functor G: J --> X (viz.,
a diagram of 1-cells C -F-> E <-G- D in X), a limit of G
w.r.t. the weight F defined above is called a *comma object*
F|G in X. I believe it is also called a *lax pullback* of F
and G, although I tend not to like such terminology, as "lax"
is overused, sometimes confusingly so.

Perhaps this is the type of statement you were aiming for?

Todd Trimble

Quantcast