From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2445 Path: news.gmane.org!not-for-mail From: Natalie Newsgroups: gmane.science.mathematics.categories Subject: duality theory Date: Fri, 19 Sep 2003 07:43:00 +0400 Message-ID: <1631960559.20030919074300@myrealbox.com> Reply-To: Natalie NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018667 4194 80.91.229.2 (29 Apr 2009 15:24:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:27 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Sep 19 14:48:19 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2003 14:48:19 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A0PKL-0004Si-00 for categories-list@mta.ca; Fri, 19 Sep 2003 14:45:53 -0300 X-Priority: 3 (Normal) Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 19 Original-Lines: 62 Xref: news.gmane.org gmane.science.mathematics.categories:2445 Archived-At: I'm interested in duality theory for an arbitrary category, especially for "classical" algebraic categories( e.g., SEMI, the category of semigroups and their homomorphisms). I want to obtain such result: to build general "dualization algorithm" (for varieties), and "to hang" fundamental operations and identities at each step of this "algorithm", where they arise. But I haven't possibility to get books (I'm think these books would help me) such as Borceux, "Categorical Algebra"; Clark/Davey, "Natural Dualities for the Working Algebraist"; Johnstone, "Stone Spaces"; Manes, "Algebraic theories" and more others. IS THERE DUALITY THEORY FOR THE CATEGORIES described above? ------------------------------------------------------------- My ideas in this direction are restricted only by the next: 1. Using factorization systems(in particular, via congruences lattice) for the category of algebras(but HOW in general situation, without special methods?) 2. Using inclusion of the category TH^op (considering as theory in the sense of (Barr/Wells)'s "Toposes, triples and theories") in the category MOD(TH) of models for this theory. 3. Via iso of categories (SET^(W))^op = CABA_(W^op) (for given endofunctor( or, narrow concept, functor part of triple) W on SET). 4.(main!!) Via generalization of the standart duality example (ComRing1)^op ~=~ AffSchemes What is the role of Birkhoff's subdirect representation theorem for algebras in the construction of the topological space SPEC, how we can construct (in general situation) the sheaf of algebras on this space? And the main: what the grounds of this construction( if it is possible)? How to prove directly the duality between algebraic and geometric theories ( if it is available)? -------------- The next questions/exersices parallels this "algorithm": A. The best test for this general theory --- to apply it for the well-known duality (ComRing1)^op ~=~ AffSchemes, mentioned above. B. If (4.) is available, how we can in general terms to obtain the equivalence between the category CABA_(W^op) in (3.) and the correspondent category given by construction in (4.)? C.(deeper) How the duality theory connect algebra, logics and topology? D. What is this "algorithm in terms 2-categories?" ------------------------------------------------------------- Natalie natalie_reznik@myrealbox.com