From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5002 Path: news.gmane.org!not-for-mail From: claudio pisani Newsgroups: gmane.science.mathematics.categories Subject: Re: Fundamental Theorem of Category Theory? Date: Mon, 22 Jun 2009 12:31:48 +0000 (GMT) Message-ID: Reply-To: claudio pisani NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1245678946 11733 80.91.229.12 (22 Jun 2009 13:55:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 22 Jun 2009 13:55:46 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Mon Jun 22 15:55:44 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 1MIjzs-0005q1-00 for gsmc-categories@m.gmane.org; Mon, 22 Jun 2009 15:55:44 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MIjN9-0000jk-TE for categories-list@mta.ca; Mon, 22 Jun 2009 10:15:44 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5002 Archived-At: The Yoneda Lemma is in fact a particular case of the reflections of Cat/X i= n discrete (op)fibrations over X (the reflection of an object x:1->X gives = the slice X/x -> X, which corresponds to the representable X(-,x)); another particular case (X=3D1), gives the components of a category. The above reflections are a consequence of the "comprehensive" factorizatio= n systems (final functors, discrete fibrations) and (initial functors, disc= rete opfibrations) on Cat. It turns out that several aspects of category theory can be developed in an= y finitely complete category C with two factorization systems properly rela= ted (the main axiom is "reciprocal stability"). Thus category theory can be indeed founded on (a generalization of) the Yon= eda Lemma; in particular, in this perspective, universal properties inside = C depend on the universal properties which follow from the factorization sy= stems. Best regards Claudio Pisani =0A=0A=0A [For admin and other information see: http://www.mta.ca/~cat-dist/ ]