From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8101 Path: news.gmane.org!not-for-mail From: "Dr. Cyrus F Nourani" Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on factorization systems Date: Mon, 5 May 2014 10:34:41 +0000 (UTC) Message-ID: References: Reply-To: "Dr. Cyrus F Nourani" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1399405450 856 80.91.229.3 (6 May 2014 19:44:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 6 May 2014 19:44:10 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue May 06 21:44:05 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1WhjSN-000162-1T for gsmc-categories@m.gmane.org; Tue, 06 May 2014 19:47:07 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44370) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1WhIs1-00063Z-11; Mon, 05 May 2014 10:23:49 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1WhIry-0006Xj-Hc for categories-list@mlist.mta.ca; Mon, 05 May 2014 10:23:46 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8101 Archived-At: Orthogonal areas: Factorizations on product topologies on fields with adjunctions can have interesting applications to explore. There are a chapter or two at A Functorial Model Theroy that can be applicable. http://appleacademicpress.com/title.php?id=9781926895925 Best regards. Akdmkrd.tripod.com DE cyrusfn@alum.mit.edu Apr 21, 2014 09:00:21 AM, tholen@mathstat.yorku.ca wrote: Michael, > >The formally correct answer to your "one place" question is certainly >"No", but here are a few suggestions. > >As a primer on orthogonal factorization systems (E,M) without epi-mono >constraints in a published book I would recommend Section 14 of the >Adamek-Herrlich-Strecker book, simply because (unlike many other >accounts of the topic) it is free of redundant requirements. > >In my view, that section is, however, not the best in terms of >discussing closure of M (and E) under limits (colimits). In a paper >with John MacDonald (LNM 962,Springer 1982, pp 175-1982) we showed the >equivalence of: > >i. (E,M) orth f.s. in C; >ii. E (considered as a full subcat of C^2) is coreflective in C^2, and >E is closed under composition; >iii. M is reflective in C^2, and M is closed under composition. > >As Im and Kelly (J. Korean Math. Soc. 23 (1986) 1-18) pointed out, >reflectivity in C^2 leads to all the desired limit stability properties >of the class M in C. This approach to orth. f.s. is taken in the first >chapter of my book with Dikranjan on Closure Operators (Kluwer 1995). >(There, however, we assume for "convenience" M to be a class of monos, >but, as is pointed out there, the essential proofs all work in >generality.) > >There is also the important aspect of considering an orth. f.s. (E,M) >as an Eilenberg-Moore (!) structure with respect to the monad C |--> >C^2 in CAT, for which I would refer you to my paper with Korostenski >(JPAA 85 (1993) 57-72) and with George Janelidze (JPAA 142 (1999) >99-130). > >Sorry, certainly not just one place, especially since the above >references don't do justice to tons of other contributions. And things >get even more complicated if we talk historical firsts, which would >start with Mac Lane (Bull. AMS 56 (1950))... > >Regards, > >Walter > > >Quoting Michael Barr akapbarr@gmail.com>: > >> First let me explain that our math dept email system has been down for ten >> days and there is no indication when it will be back, although our sysop >> has been working on it day and night. I will circulate an announcement >> when it is running again. Meantime, use this address. >> >> Is there one place that develops all the properties of factorization >> systems? We are especially interested in the non-strict case, that is in >> which the right factor needn't be epic, nor the left factor be monic, but >> the unique diagonal fill-in condition holds. >> >> Michael >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]