Re: grothendieck toposes and grothendieck abelian categories

Re: grothendieck toposes and grothendieck abelian categories

[Peter Freyd]

Bill Lawvere writes:

    With the experience of the last decades, it should be possible to
    explain how Zermelo???s theorem "in a base topos" was sufficient to the
    need of his friend Reinhold Baer. (Lifting the basic case to modules
    and sheaves is routinely done with suitable right adjoint functors.)

I think everybody loved Reinhold Baer. I first spent time with him one
week in Warsaw. A few years later I helped him secure his (retirement)
apartment in Zurich (by renting it for 6 months). And somehow I just
never got around to telling him the easy way to construct injective
modules. The suitable right-adjoint functors -- that Bill mentions --
tell one how to construct an injective cogenerator for R-modules but
neither the construction nor the proof of injectiviy need mention
anything about functors: it???s pretty easy to show that any divisible
abelian group is injective (with one use of Zorn???s lemma): after
deciding whether it???s left or right R-modules you???re talking about,
take the set of abelian-group maps from  R  to  Q/Z  and view it as an
R-module. Done. And no more need of the axiom of choice. The R-module
homomorphisms from  M  to this construction are in easy one-to-one
correspondence with the abelian-group homomorphisms from  M  to  Q/Z.
(With a little work you can do the same for the category of additive
functors from a small abelian category to the category of abelian
groups. No ordinal numbers in sight.)

As for Bill???s question:

    For deriving functors, Godement used resolving sheaves more concrete
    than injective ones, but Grothendieck preferred the latter. When do
    they exit?

Well, in the Forward of the 2003 TAC version of "Abelian Categories" I
wrote (on page -15):

    Pages 131-132: The very large category _B_ (Exercise 6A) -- with a
    few variations -- has been a great source of counterexamples over
    the years. As pointed out above (concerning pages 85-86) the
    forgetful functor is bi-continuous but does not have either adjoint.
    To move into a more general setting [to obtain a topos _C_ instead
    of an abelian category], drop the condition that  G  be a group and
    rewrite the convention to become  f(y) = 1_G for  y  not in  S
    (and, of course, drop the condition that  h  be a homomorphism --
    it can be any function). The result is a category that satisfies
    all the conditions of a Grothendieck topos except for the
    existence of a generating set. It is not a topos: the subobject
    classifier would need to be the size of the universe. If we require,
    instead, that all the values of  f  be permutations, it is a topos
    and a boolean one at that. Indeed, the forgetful functor preserves
    all the relevant structure (in particular, Omega  has just two
    elements). In its category of abelian-group objects  Ext(A,B)  is a
    proper class iff there???s a non-zero group homomorphism from  A  to
    B  (it needn???t respect the actions), hence the only injective
    object is the zero object (which settled a once-open problem [years
    ago] about whether there are enough injectives in the category of
    abelian groups in every elementary topos with natural-numbers
    object).

           http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf

On the subject of the need for choice, while there take a look at the
very last item in my 2003 Forward (page -14). The embedding theorem
for abelian categories is an easy consequence of an embedding theorem
for regular categories and for that there???s a choice-free proof.



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