From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4065 Path: news.gmane.org!not-for-mail From: claudio pisani Newsgroups: gmane.science.mathematics.categories Subject: Re: Comma categories Date: Mon, 5 Nov 2007 13:21:13 +0100 (CET) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019697 11543 80.91.229.2 (29 Apr 2009 15:41:37 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:41:37 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Nov 5 14:45:21 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 05 Nov 2007 14:45:21 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ip6ki-0006Cl-C8 for categories-list@mta.ca; Mon, 05 Nov 2007 14:32:48 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 17 Original-Lines: 64 Xref: news.gmane.org gmane.science.mathematics.categories:4065 Archived-At: 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=20 e: ex_f f* Y/y -> Y/y=20 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=20 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, >=20 > 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? >=20 > Best regards >=20 > Uwe Wolter >=20 >=20 >=20