categories: complete Galois groups

[Clemens Berger <Clemens.BERGER@univ-cotedazur.fr>]

Dear all,

    may I ask a question which isn't really at the heart of the discussion
but nonetheless touches upon one aspect of point-free topology (and is
related to the Morita-equivalence mentioned by Pedro) ?

    It is well-known that connected, locally connected, Boolean
Grothendieck toposes are equivalent to classifying toposes of localic
groups. It is also well-known that the adjunction between locales and
spaces does not carry over to induce an adjunction between localic
groups and topological groups. Yet there are subclasses of groups where
such an adjunction on group objects exists and underpins a
Morita-equivalence of the corresponding classifying toposes, namely
profinite groups and prodiscrete groups (these two are treated more or
less explicitly in SGA1/3, resp. SGA4).

More recently (in work of Noohi, Bhatt-Scholze and Caramello) an even
larger class of topological groups has been outlined where such an
adjunction should exist and the corresponding ``Galois representation
theorem'' should hold, namely the class of ``complete Galois groups''.

I use the term ``Galois group'' for a topological group whose topology
is generated (in an obvious way) by its open subgroups. Since every open
subgroup is also closed, the underlying space of a Galois group is
zero-dimensional.

Galois groups form a reflective subcategory of the category of all
topological groups, and the classifying topos of a topological group is
equivalent to the classifying topos of its Galois group reflection.
Thus, from a Morita-equivalence perspective, it is enough to study
Galois groups.

A Galois group is said to be complete if it is complete for its
two-sided uniformity. Bhatt and Scholze show that a Galois group G is
complete iff the induced morphism from G to the automorphism group
Aut(p_G) of the canonical point p_G of BG is an isomorphism. They show
more precisely that for any Galois group G, the induced morphism
G->Aut(p_G) is completion w/r to the two-sided uniformity of G. They
finally show that a locally connected Boolean Grothendieck topos with a
conservative ``tame'' point p is equivalent to BAut(p).

My question is: what is the corresponding picture on the localic group
side ? How can we characterise ``intrinsically'' toposes that are of the
form BG for a complete Galois group G ? This would somehow distinguish
those cases where point set topology still gets his hands on, from those
where we are forced to use localic techniques.

Two concluding remarks:

(1) Banaschewski (following Kriz) showed that a topological group G is
complete for its two-sided uniformity iff the frame of opens of G is a
cogroup in frames. So, in particular, we get a zero-dimensional localic
group out of any complete Galois group. Does this induce a Morita
equivalence of the respective classifying toposes ?

(2) Perhaps the generic example of a complete Galois group, which is not
prodiscrete, is the group Sigma_N of permutations of the (discrete) set
of natural numbers, endowed with the compact-open topology. The
classifying topos BSigma_N is the celebrated Schanuel topos of nominal
sets. So, to some extent, all this is about how to apply Galois
theoretical ideas to toposes that behave like the topos of nominal sets.

All the best,
                 Clemens.



Le 2023-01-27 18:55, Pedro Resende a ??crit??:
> Hi Steve,
>
> Sorry for the radio silence, it???s been a hectic week.
>
> Concerning your question about a less derogatory expression??? I think I
> like `algebraic reasoning??? versus `point-based reasoning??? (which to me
> sounds better than `pointwise', I don???t know why).
>
> This is analogous to commutative algebra versus algebraic geometry.
>
> In any case, am I right that it seems to be somewhat consensual (in
> this thread) that `pointfree topology??? is the appropriate terminology
> for the kind of topology that *can* (but not necessarily has to) be
> studied without reasoning in terms of points?
>
> Incidentally, in my mind the `pointfree' terminology should also apply
> to more general notions, such as quantales, or at least some classes
> of them. For instance, inverse quantal frames are `the same' as
> localic etale groupoids, and they have associated etendues.
>
> Best wishes,
>
> Pedro
>
>> On Jan 23, 2023, at 1:47 PM, Steven Vickers <s.j.vickers.1@bham.ac.uk>
>> wrote:
>>
>> Dear Pedro,
>>
>> Of course, that's the very reason why I wanted to transfer it to the
>> style of working without points.
>>
>> That's slightly unfair, in that in many cases of reasoning
>> algebraically, without points, it's not at all clear how to do it
>> pointwise.
>>
>> You and I have certainly experienced that in our work on quantales,
>> which are much more purely algebraic gadgets. Our approach via localic
>> suplattices (algebras for the lower hyperspace monad) gives a more
>> point-free approach to the subject, but it takes effort - I think
>> you'll agree - to work with the hyperspaces in a pointwise manner.
>>
>> Do you think there's a less derogatory term for the style of reasoning
>> without points?
>>
>> All the best,
>>
>> Steve.

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]