Subject: categories: Re: Comma categories

From: [email-masked] (Ross Street)
Date: Tue, 20 Oct 1998 10:26:56 +1000

>Does any body know if comma categories have been defined in
>enriched contexts?

Lawvere's La Jolla paper, where general comma categories were introduced,
showed how to construct them from pullbacks and a "cylinder" (or "arrow
object") construction.  John Gray (SLNM p. 254) showed that cylinder is a
universal notion which a 2-category may or may not have. I pointed out
[Fibrations and Yoneda's lemma in a 2-category, Lecture Notes in Math. 420
(1974) 104-133;     MR53#585] that finite completeness for a 2-category
should mean that it have pullbacks, a terminal object, and cylinders (a
similar idea was in my PhD thesis for differential graded categories which
are finitely complete when they admit pullbacks, a zero object and
"suspension"). Finite completeness for 2-categories is further analysed in
[Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8
(1976) 149-181; MR53#5695].

More generally, finite completeness for a V-category  A (= a category with
homs enriched in  V) means that its underlying category has finite ordinary
limits, which are preserved by representables  A(a,-)  into  V,  and that
it admits cotensoring by the "finite" objects of  V.  There is some choice
about what you mean by "finite" object in  V  however "finitely
presentable" is often the right thing.  Sometimes, as in the case of  V =
Cat, the finite objects are generated by a few finite objects - that is why
"cylinder" plays the important role in 2-categories (it is the finite
generating object, cotensor with which is cylinder).

So why am I going on about finite limits in 2-categories?  Well, Lawvere's
construction shows that comma objects exist in any finitely complete
2-category.  Comma objects are particular finite limits just like
pullbacks.

In particular, there is a 2-category  V-Cat  of V-categories, V-functors
and V-natural transformations.  It is certainly complete (as a 2-category)
for any decent  V.   So, indeed, it is well known that comma objects (or
comma V-categories) exist.  They have their uses but NOT for the wonderful
use that Lawvere put them to:  Lawvere provided a formula for left (right)
Kan extensions of ordinary functors which involves taking a colimit (limit)
over a comma category. [Indeed, more is true; see my definition of
"pointwise Kan extension" in "Fibrations and Yoneda's lemma in a
2-category".]  However, this formula does not work even for additive
categories (= categories enriched in the monoidal category of abelian
groups).

Regards,
Ross
http://www.mpce.mq.edu.au/~street/