From: street@mpce.mq.edu.au (Ross Street)
To: categories@mta.ca
Subject: Re: Comma categories
Date: Tue, 20 Oct 1998 10:26:56 +1000 [thread overview]
Message-ID: <199810200024.KAA27351@macadam.mpce.mq.edu.au> (raw)
>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/
next reply other threads:[~1998-10-20 0:26 UTC|newest]
Thread overview: 12+ messages / expand[flat|nested] mbox.gz Atom feed top
1998-10-20 0:26 Ross Street [this message]
-- strict thread matches above, loose matches on Subject: below --
2009-09-24 22:37 Steve Lack
2009-09-24 20:23 Tony Meman
2007-11-08 1:32 Robert L Knighten
2007-11-08 0:05 Bill Lawvere
2007-11-05 12:21 claudio pisani
2007-11-02 16:12 wlawvere
2007-10-31 15:20 Uwe Egbert Wolter
1998-10-20 21:11 F W Lawvere
1998-10-19 16:14 Manuel Bullejos
1998-10-19 17:19 ` Vaughan Pratt
1997-07-01 18:13 comma categories categories
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