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.
From: pedro.m.a.resende@tecnico.ulisboa.pt > Sent: Monday, January 23, 2023 11:44 AM To: ptj@maths.cam.ac.uk <mailto:ptj@maths.cam.ac.uk> <ptj@maths.cam.ac.uk <mailto:ptj@maths.cam.ac.uk>> Cc: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk <mailto:s.j.vickers.1@bham.ac.uk>>; categories list <categories@mta.ca <mailto:categories@mta.ca>> Subject: Re: categories: Re: Terminology for point-free topology?
In addition to all the deeper reasons, `pointless’ can be taken to be derogatory, so preferably it should be used only when in tongue-in-cheek mode. At least that’s what I tell my students — just as I ask them not to say `abstract nonsense’ too enthusiastically… :)
Pedro
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