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.
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