Re: Grothendieck: more complete URL

[Colin McLarty <colin.mclarty@case.edu>]

I would not be surprised if Grothendieck's Tohoku proof that all AB5
categories have enough injectives is straightforwardly constructive --
with the unhelpful limitation that from a constructive viewpoint
nearly no categories satisfy the AB5 axioms.  Especially the last
axiom, 5, requires completeness in a strong sense which is tailored to
make each step of the classical Baer proof work.

The classical corollary, that all sheaf categories (i.e. sheaves of
modules)  have enough injectives, includes saying that all module
categories have enough injectives.  So it cannot be more constructive
than that.

I have not looked at constructive forms of the result, or relativizing
the whole to any base topos.

best, Colin



2010/9/9 Steve Vickers <s.j.vickers@cs.bham.ac.uk>:
> Dear Colin,
>
> Can I ask a technical question about foundations here?
>
> Does the "enough injectives" result assume a classical base theory? The
> classical result for module categories uses choice, and my understanding
> is that the result for sheaf categories uses Barr covers to make
> available the classical result.
>
> I wonder if there's an unequivocally constructive formulation.
>
> Regards,
>
> Steve.
>

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