From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1473 Path: news.gmane.org!not-for-mail From: Jiri Adamek Newsgroups: gmane.science.mathematics.categories Subject: Re: Functorial injective hulls Date: Thu, 30 Mar 2000 10:56:04 +0200 (MET DST) Message-ID: <200003300856.KAA17187@lisa.iti.cs.tu-bs.de> 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 1241017853 31395 80.91.229.2 (29 Apr 2009 15:10:53 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:10:53 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Mar 30 09:14:16 2000 -0400 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id JAA24388 for categories-list; Thu, 30 Mar 2000 09:11:10 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: ELM [version 2.5 PL2] Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 23 Xref: news.gmane.org gmane.science.mathematics.categories:1473 Archived-At: Bill's question concerning minimal functorial injective extensions seems very interesting. Bill's comment was: > 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 ? I have two different answers: 1. NO in case of Pos (and order-embeddings): there does not exist a minimal pair (F,f) consisting of an endofunctor F of Pos whose values are complete lattices and a natural transformation f: Id -> F whose components are order-embeddings 2. YES in case of Set (and monomorphisms): the embedding Id -> Id + K, where K is the constant functor with value 1 , is minimal. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx