Hi, Vaughan! Thanks for your message. First, just to be clear, my proposed use of "pointless" was as a description of how you reason (without points), not of how many points there actually are (global points, in Set or S). Turning to what you wrote, I know 30 years ago you were pointing out analogies between Chu spaces and the "topological systems" (locale maps f: X -> Y with X discrete) in Topology via Logic. Is that the kind of thing you had in mind? (The Chuish space uses (X, ΩY).) Sadly, they have little bearing on what I was saying in my message, particularly with regard to bundles. The topological systems were presented as a hybrid between point-set and point-free where you could see both styles. That may or may not be good as a pedagogical device. (My own view, looking back, is that it encourages the wrong ideas.) Mathematically, their weakness is that they are no help if Y lacks global points. Classically, it is not unreasonable to view lack of global points as a pathology in the locale Y; and then the constructive tendency to lack global points appears as pathology in the logic. However, this becomes less tenable when one moves from spaces to bundles, using the fact that bundles q: Y -> B are equivalent to internal locales in the topos SB of sheaves over B. In the internal logic, the set of points is the best approximation to Y by a set (= object of SB) X. As a bundle, it is the best approximation to q by a *local homeomorphism* p: X -> B. It is very easy for decent bundles to lack such approximations for purely topological reasons. An example discussed in detail in my arXiv article is the closed point of Sierpinski, \bot: 1 -> $. Best wishes, Steve. ________________________________ From: Vaughan Pratt <pratt@cs.stanford.edu> Sent: Tuesday, January 17, 2023 1:18 AM To: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk> Cc: categories@mta.ca <categories@mta.ca> Subject: Re: categories: Terminology for point-free topology? Further to Steve's question (and yes, Happy New Year even though we're 44‰ of the way through it), has anyone looked at the options for representing locales with Chu spaces over 2 = {0,1}? (I know I should have, but I haven't, sorry about that.) Obviously the states of a Chuish locale should satisfy the frame axioms, but what happens with the points? Equally obviously, "pointless" can't mean no points at all or you wouldn't have a Chu space. So presumably it means something like fewer points. Is there canonical choice for "fewer", or are there degrees of "pointlessness"? I don't have enough intuition about "atomless parts of space" to explore this on my own without guidance from those who've gone ahead of me here. Despite the view of the continuum as a final coalgebra as Dusko Pavlovic and I organized it in 1999, and as improved on later by Peter Freyd and Tom Leinster, I still don't know which points can be removed from the continuum so as to keep it a locale. What if we enlarge Chu(Set,2) to Chu(Set,K) for some larger set K? The free Heyting algebra on 2 generators, for example? Or is no K sufficient? Vaughan Pratt On Mon, Jan 16, 2023 at 1:03 PM Steven Vickers <s.j.vickers.1@bham.ac.uk<mailto:s.j.vickers.1@bham.ac.uk>> wrote: I'm wondering if there's any consensus usage to found for "point-free" topology and related terms. I've posted a detailed discussion on https://arxiv.org/abs/2206.01113, but I can summarize the question more succinctly. It's not unusual to distinguish between two synonymous pairs: point-set/pointwise = ordinary semantics of general topology, point-free/pointless = reformed semantics of, e.g., locales or formal topology. However, that is misleading, as locale theory can be validly done using points. See, e.g., Ng-Vickers on real exp and log, https://lmcs.episciences.org/9879. The trick is to restrict to geometric constructions and to apply them to *generalized* points, to be found in arbitrary Grothendieck toposes and not just Set (or your chosen base S). Thus there are two distinctions to be made - 1 Ordinary semantics v. reformed 2 Use points v. avoid them Some terms naturally fall into place. Point-set = ordinary topology, points taken from a given set. Pointwise = use points. Point-set is a subclass of pointwise, but strict, as shown by the above example. What about pointless and point-free? I'm piloting - Pointless = avoid points (e.g. construct locale maps concretely as frame homomorphisms). There's some value judgement in my choice there, as very often the pointwise reasoning is simpler and more transparent, so there seems to be no good reason for arguing pointlessly. Point-free = reformed topology. I try to think of this as meaning that the points are liberated from their confinement to Set or S. Does anyone have comments on these, or suggestions for other phrases for the concepts? Happy New Year! Steve Vickers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]