From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4074 Path: news.gmane.org!not-for-mail From: Bill Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: Comma categories Date: Wed, 07 Nov 2007 20:05:04 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019702 11580 80.91.229.2 (29 Apr 2009 15:41:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:41:42 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Nov 7 20:18:28 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Nov 2007 20:18:28 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IputM-0005Vu-Lq for categories-list@mta.ca; Wed, 07 Nov 2007 20:05:04 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 27 Original-Lines: 86 Xref: news.gmane.org gmane.science.mathematics.categories:4074 Archived-At: I recently noticed that in Abstract no. 652-4 in the Notices of the AMS volume 14 (1967) page 937, John Gray advocates a systematic treatment of the calculus of comma categories and lists five operations which should be explicitly accounted for in such a calculus. He also mentions that Jon Beck contributed to that discussion. Probably John Gray's notes, if they still exist, would be a helpful guide to someone planning to write a systematic treatment as suggested recently Uwe Wolters. Bill On Mon, 5 Nov 2007, claudio pisani wrote: > > The following facts about slice categories may be > worth noticing: > > 1 In the equivalence between df/X (discrete fibrations > over a category X) and presheaves on X, the slices X/x > -> X correspond to the representable presheaves. > > 2. (Yoneda Lemma) The reflection of x:1->X (as an > object of Cat/X) in df/X is (isomorphic to) X/x (with > its terminal object as reflection map). > In particular, the full subcategory sl/X of df/X > generated by the slices over X is isomorphic to X. > > 3. For any functor p:P->X, a morphism p->X/x in Cat/X > is a cone of base p and vertex x. > > 4. So, a reflection of p->X/x of p in sl/X is a > colimiting cone. > > 5. A functor f:X->Y has a right adjoint iff the > pullback f*Y/y of any slice of Y is (isomorphic to) a > slice of X. > > 6. If ex_f -| f* : df/Y -> df/X is the "left Kan > extension" along f, then the counit > e: ex_f f* Y/y -> Y/y > is an iso for any y iff f is "dense" (aka "connected") > while it is a colimiting cone for any y iff f is > "adequate" (aka "dense"). > Using instead the adjunction > f_! -| f* : Cat/Y -> Cat/X > the counit is a colimiting cone for any y iff f is > adequate (as before), while it is an absolute colimit > iff f is dense. > > Best regards. > > Claudio > > > > --- Uwe Egbert Wolter ha > scritto: > >> Dear all, >> >> I'm looking for a comprehensive exposition of >> definitions and results >> around comma/slice categories. Especially, it would >> be nice to have >> something also for non-specialists in category >> theory as young >> postgraduates. Is there any book or text you would >> recommend? >> >> Best regards >> >> Uwe Wolter >> >> >> > > > >