Subject: Injectives and choice

From: categories <[email-masked]>
Date: Mon, 8 Dec 1997 15:41:42 -0400 (AST)

Date: Mon, 8 Dec 1997 13:11:00 -0500 (EST)
From: Colin McLarty <[email-masked]>

        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?

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From: categories <[email-masked]>
Date: Mon, 8 Dec 1997 17:25:12 -0400 (AST)

Date: Mon, 8 Dec 1997 15:01:48 -0500 (EST)
From: Andreas Blass <[email-masked]>

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