Re: Grothendieck: more complete URL

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

Colin McLarty wrote:

>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.

Does this depend on how the axiom is phrased?
An elementary version says that an Ab5 category
is an cocomplete abelian category such that,
given any object X, subobject A of X,
and directed system (B_i)_i of subobjects of X,
A \cap \sum_i B_i = \sum_i (A \cap B_i).
The obvious proof that modules form an Ab5 category
works in the internal language of any topos with NNO.

It is less clear to me whether it follows from this
that filtered colimits preserve exact sequences,
which is the more abstract form of axiom 5.


--Toby


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