From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/558 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: Injectives and choice Date: Mon, 8 Dec 1997 17:25:12 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017035 26262 80.91.229.2 (29 Apr 2009 14:57:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:57:15 +0000 (UTC) To: categories Original-X-From: cat-dist Mon Dec 8 17:25:15 1997 Original-Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id RAA07508; Mon, 8 Dec 1997 17:25:12 -0400 (AST) Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:558 Archived-At: Date: Mon, 8 Dec 1997 15:01:48 -0500 (EST) From: Andreas Blass Even to prove that there is a non-zero injective abelian group needs a little bit of choice, but only a little. (By contrast, the statement that every divisible abelian group is injective is equivalent to the axiom of choice.) The details are in my paper "Injectivity, projectivity, and the axiom of choice" (Trans. Amer. Math. Soc. 255 (1979) 31--59). Andreas Blass > Date: Mon, 8 Dec 1997 13:11:00 -0500 (EST) > From: Colin McLarty > > > Grothendieck's proof that every AB5 category has enough injectives > uses the axiom of choice (actually Zorn's lemma--which John Bell points out > to me is significantly weaker than choice in toposes). And the proof in > Johnstone's TOPOS THEORY that the category of Abelian groups over any > Grothendieck topos has enough injectives uses Barr's theorem: Every > Grothendieck topos is covered by one that satisfies the axiom of choice. > This theorem itself assumes the axiom of choice in the base topos (i.e. the > one over which the others are Grothendeick). > > Are there any good results showing how necessary the axiom of > choice, or Zorn's lemma, is to these results? >