From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1460 Path: news.gmane.org!not-for-mail From: F W Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: Functorial injective hulls Date: Thu, 23 Mar 2000 14:50:57 -0500 (EST) Message-ID: References: <1000322181057.ZM23382@pascal.math.yorku.ca> Reply-To: wlawvere@ACSU.Buffalo.EDU NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017846 31351 80.91.229.2 (29 Apr 2009 15:10:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:10:46 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Mar 24 04:44:33 2000 -0400 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id QAA22968 for categories-list; Thu, 23 Mar 2000 16:31:48 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: <1000322181057.ZM23382@pascal.math.yorku.ca> Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 68 Xref: news.gmane.org gmane.science.mathematics.categories:1460 Archived-At: But by contrast, functorial injective resolutions do exist, usually by some sort of double-dualisation monad. What if the "hull" or minimality requirement is imposed on the process qua functor instead of at each object? Do such functors exist ? ***************************************************************** F. William Lawvere Mathematics Dept. SUNY Buffalo, Buffalo, NY 14214, USA 716-829-2144 ext. 117 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ***************************************************************** On Wed, 22 Mar 2000, Walter Tholen wrote: > Peter, > > Jirka Adamek had prepared a draft response to your earlier remark that a poset > with top element should disprove the assertion in the Abstract of our paper > (with Herrlich and Rosicky) which he had circulated. His response is attached > below, slightly edited by me - hence I take full responsibility for its > contents. > > Our proof of the Theorem adds only one twist to the proof you have just > circulated: monomorphisms get substituted by an absolutely ARBITRARY class H of > morphisms; H-injective then indeed means that the contravariant hom sends H to > epis; and H-essential is as you described as well (: an h in H such that g.h is > in H only if g is in H). We are able to compensate for the loss of mono through > condition 1, while condition 2 obviously replaces your (epi&mono is iso). For > full details, please consult the paper. > > Best wishes, > Walter. > > > ============================================================================= > Dear Peter, > The precise result we prove in our paper is the following: > > Theorem. Let H be a class of morphisms in a category C such that > 1. all H-injective objects form a cogenerating class, and > 2. the class of all H-essential morphisms which are epimorphic > is precisely the class of isomorphisms of C . > Then C cannot have natural H-injective hulls (i.e. they cannot > form an endofunctor together with a natural transformation from Id) > unless every object in C is H-injective. > > The abstract we have given in our posting was meant to be an abbreviation of > this precise statement. While condition 1 holds true for the set H of all > (mono)morphisms in a poset with top element, condition 2 fails. > > Best regards, > J.A., H.H., J.R., W.T. > > > xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx > alternative e-mail address (in case reply key does not work): > J.Adamek@tu-bs.de > xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx > > > > >