This proof really fell through the cracks. The argument from injective Abelian groups to injective R-modules was known by 1960. The last piece was in place in 1974 with Barr's theorem on coverings with AC. The proof may be published somewhere but I can't find it and Peter suggests it is not. Its absence works mischief. Eisenbud COMMUTATIVE ALGEBRA (p. 621) proves modules (n Set) have enough injectives and then sends readers off to Hartshorne for the Godement construction to prove modules over topological spaces have enough injectives. But he has in effect already proved it for all Grothendieck toposes! He merely has to send readers off to Mike's paper or Peter's (1977) book for the AC result. While Eisenbud states results explicitly for module categories over Sets, he frames them to hold much more generally and he sends readers to sources for that generality. I am glad to hear it will be in the Elephant. best, Colin PS thanks to Carsten Butz for reminding me of Andreas Blass "Injectivity, projectivity, and the axiom of choice" (Trans. AMS Volume 255, November 1979) for both history and a proof that choice will be required. 2010/10/11 Prof. Peter Johnstone <P.T.Johnstone@dpmms.cam.ac.uk>:
Dear Colin,
Yes, the argument is correct, and it'll be in volume 3 of the Elephant. I don't know why it hasn't been published elsewhere.
Peter
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