From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7254 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: orthogonal factorization systems Date: Sun, 15 Apr 2012 08:27:57 +1000 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1334494804 18599 80.91.229.3 (15 Apr 2012 13:00:04 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 15 Apr 2012 13:00:04 +0000 (UTC) Cc: Categories list To: Emily Riehl Original-X-From: majordomo@mlist.mta.ca Sun Apr 15 15:00:03 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.80]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1SJP3i-00083Y-WF for gsmc-categories@m.gmane.org; Sun, 15 Apr 2012 15:00:03 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42594) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SJP2Y-0000ph-Gi; Sun, 15 Apr 2012 09:58:50 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SJP2X-0000QA-8H for categories-list@mlist.mta.ca; Sun, 15 Apr 2012 09:58:49 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7254 Archived-At: Dear Emily, You are in luck. Your result is an instance of the following: PROP If C is a category bearing the orthogonal factorisation system (E,M), and T is a monad on C whose underlying functor preserves E-maps, then C^T bears the orthogonal factorisation system (U^-1(E), U^-1(M)). A full proof of which is given as Proposition 20.28 in: Abstract and concrete categories: The joy of cats (Wiley, 1990) Jiri Adamek, Horst Herrlich and George Strecker Maybe there is an older reference than this but I am not aware of such. Richard On 14 April 2012 08:09, Emily Riehl wrote: > I've placed a bet with a colleague that the following result appears in > the literature. Please help me win. > > Claim: Suppose (E,M) is an orthogonal factorization system (unique lifts) > on a symmetric monoidal category and X is a fixed monoid. If tensoring > with X preserves maps in the class E, then (E,M) lifts to an orthogonal > factorizaiton system on the category of X-modules. > > Regards, > Emily Riehl > > > > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]