From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5026 Path: news.gmane.org!not-for-mail From: claudio pisani Newsgroups: gmane.science.mathematics.categories Subject: Re: no fundamental theorems please Date: Wed, 24 Jun 2009 13:52:41 +0000 (GMT) Message-ID: Reply-To: claudio pisani NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1246005103 5255 80.91.229.12 (26 Jun 2009 08:31:43 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 26 Jun 2009 08:31:43 +0000 (UTC) To: Michael Barr , categories@mta.ca, Paul Taylor Original-X-From: categories@mta.ca Fri Jun 26 10:31:36 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MK6qL-0006m5-1R for gsmc-categories@m.gmane.org; Fri, 26 Jun 2009 10:31:33 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MK5zO-0005sL-64 for categories-list@mta.ca; Fri, 26 Jun 2009 04:36:50 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5026 Archived-At: Of course, the proof that the object x:1->X (of Cat/X) has the slice X/x -= > X as a reflection in df/X (and its final object as reflecton map) is esse= ntially the same of that of the standard Yoneda lemma, and the general case= only requires a little more effort. My point is that this formulation seems to me more in the "categorical spir= it", stating a universal property that relates categories over X and discre= te fibrations. In fact the paradigm "categories, functors and natural transformations" can= be in part replaced by "categories, functors and discrete (op)fibrations";= for instance a colimit x of the object p:P -> X of Cat/X is a reflection o= f p in slices over X, where the reflection map p -> X/x in Cat/X is the col= imiting cone, and so on. Furthermore, there is a clear analogy (which can be made precise with the p= roper choice of factorization system on posets) with the reflection of the = subsets of a poset X in lower or upper sets of X (the principal sieves bein= g a particular case). Best regards Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]