categories: Re: Terminology for point-free topology?

[Pedro Resende <pedro.m.a.resende@tecnico.ulisboa.pt>]

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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