From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/886 Path: news.gmane.org!not-for-mail From: street@mpce.mq.edu.au (Ross Street) Newsgroups: gmane.science.mathematics.categories Subject: Re: Comma categories Date: Tue, 20 Oct 1998 10:26:56 +1000 Message-ID: <199810200024.KAA27351@macadam.mpce.mq.edu.au> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241017290 28105 80.91.229.2 (29 Apr 2009 15:01:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:01:30 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Tue Oct 20 21:12:28 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA00763 for categories-list; Tue, 20 Oct 1998 20:18:23 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Sender: street@macadam.mpce.mq.edu.au Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 51 Xref: news.gmane.org gmane.science.mathematics.categories:886 Archived-At: >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/