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/ ]

Quantales are interesting examples also from the viewpoint how they may be used for many-valued truth in logic. There are certain mechanisms where "points can be recovered", but in my view this is not a sufficient justification for working only algebrally in logic, or algebraically in topology for that matter. Propositional two-valued logic is a bit similar. The boolean values are not pointfree but "term-free", if the analogy is allowed. We can do something with propositional logic, but representation of "condition" or "state" requires "insideness, in the sense of inside points". Points as such representing states to me makes no sense in practice. We can surely create fancy examples, but we cannot define things like "asthma" or logically differentiate between Alzheimer's and vascular dementia using pointfree topology. Pointfree or not, term-free or not, I think it is important to justify freeness whenever the calculation machinery allows it, but at the same time refrain from being overenthusiastic about pointfreeness in the sense of "I can work totally without points". Such things I would call not just pointfree but indeed pointless, in particular as such pointlessness kind of intentionally shuts out any possibility for real-world application. Some parts of theoretical mathematics is about seductive tricks, and some mathematicians fall for it. Potential practicality of even "deepest theoretical theory" keeps feet on the ground, even if practicality is not realizable or desirable. But my view is that we must keep "real-world applicability" at least as a "general burden" in the sense that all science must useful, in one way or another. Science should never be just "aus liebe zur Kunst". Patrik On 2023-01-27 19:55, Pedro Resende wrote: > 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, >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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

Dear colleagues, I met Bill only relatively recently at ct99 in Coimbra, so, for instance, I never saw him bringing up his political views in a public mathematical discussion, but I have witnessed many times his generosity and efforts to give deep and honest advice to colleagues of all ages, in that peculiar way of his which was both self confident and yet devoid of arrogance. He was also very patient. I have very fond memories of our conversations during ct meetings, at the Bristol seminar he organized , at the México workshops organized by Quico, and also in Arcadia with Fatima and Silvana. Motivated by Bill’s interest on o-minimal structures we once attended an important summer school on Model Theory. When the organizers realized Bill was there they made a public announcement celebrating his presence and invited him to chair a session. We Category Theorists were indeed privileged to share an epoch, or part of one, with Bill’s profound intellect. His results will influence our subject for many years. Still, I would like to encourage younger generations to continue studying his work, including his messages to this list, where they will find a treasure of remarkable brightness, full of fascinating ideas still to be pursued. Yours, Matías [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

The Department of Computer Science at the Vrije Universiteit Amsterdam offers an open position as assistant professor in Theoretical Computer Science, in the area of logic or semantics, broadly construed. https://workingat.vu.nl/ad/assistant-professor-tcs-tenure-track-career-track/b860v9 Deadline for application is March 14, 2023. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Colleagues, It’s very sad to hear about Bill Lawvere’s passing. I haven’t seen him since a meeting at Colgate several years ago, where he came with his son, who I believe is a math teacher. Bill was of course a great mathematician who changed the way we think about logic and the foundations of mathematics, but he was also a kind and generous person. When I was a student of Saunders at Chicago, Bill often visited for the weekend and lectured, discussed ideas with me, and encouraged me. I suppose he was there to talk to Ieke, who visited often then, because he and Saunders were working on the book. But Bill was always happy to patiently explain things to me and to suggest things for me to work on. When he later came to CMU for a seminar, I told him he would have a bit more time than a usual lecture, up to 2 hours. The room had large blackboards on 3 walls, and he used all 3 of them, moving desks out of the way and telling the audience to turn their chairs. After 3 hours I had to cut him off - he had only heard the part about having more time! I am very grateful for the patience and generosity that Bill showed me as a student, and for his mathematical ideas which have fascinated and motivated me ever since then. I also recall that Saunders had the very highest opinion of Bill. Best wishes, Steve [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear colleagues, The Linguistics department of University College London (UCL) is recruiting a lecturer (permanent academic post) in Computational Linguistics. For more details of the position, see here: https://www.ucl.ac.uk/work-at-ucl/search-ucl-jobs/details?jobId=3786&jobTitle=Lecturer%20in%20Linguistics%20(Computational%20Linguistics) Best, Mehrnoosh [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

The New York City Category Theory Seminar This semester there is a special three-part lecture series on TQFT and Computation. These lectures take place on February 8, 15 and 22 in The Graduate Center. See below. Department of Computer Science <http://cs.gc.cuny.edu> Department of Mathematics <http://math.gc.cuny.edu/> The Graduate Center of The City University of New York <http://www.gc.cuny.edu/> THIS SEMESTER, SOME TALKS WILL BE IN-PERSON AND SOME WILL BE ON ZOOM. Time: Wednesdays 07:00 PM Eastern Time (US and Canada) IN-PERSON INFORMATION: 365 Fifth Avenue (at 34th Street) map <http://maps.google.com/maps?sourceid=navclient&ie=UTF-8&rlz=1T4GFRC_enUS206 US206&q=365+Fifth+Avenue,+ny&um=1&sa=X&oi=geocode_result&resnum=1&ct=title> (Diagonally across from the Empire State Building) New York, NY 10016-4309 Room 5417 (not the usual Room 6417) The videos of the lectures will be put up on YouTube a few hours after the lecture. ZOOM INFORMATION: https://brooklyn-cuny-edu.zoom.us/j/89472980386?pwd=Z3g3Q3h3V1dQUmg2ZlVGU1Rw SEhMZz09 Meeting ID: 894 7298 0386 Passcode: NYCCTS Seminar web page. <http://www.sci.brooklyn.cuny.edu/~noson/Seminar/index.html> Videoed talks. <https://www.youtube.com/channel/UCNOfhimbNwZwJO2ltv1AZOw/videos> Previous semesters. <http://www.sci.brooklyn.cuny.edu/~noson/Seminar/Previous%20Semesters.html> researchseminars.org page. <https://researchseminars.org/seminar/Category_Theory> Contact N. Yanofsky <mailto:noson@sci.brooklyn.cuny.edu> to schedule a speaker or to add a name to the seminar mailing list. _____ _____ Spring 2023 _____ _____ * Speaker: Igor Baković, University of Osijek, Croatia. * Date and Time: Wednesday February 1, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Enhanced 2-adjunctions. * Abstract: Whenever one has a class of objects possessing certain structure and a hierarchy of morphisms that preserve structure more or less tightly, we are in an enhanced context. Enhanced 2-categories were introduced by Lack and Shulman in 2012 with a paradigmatic example of an enhanced 2-category T-alg of strict algebras for a 2-monad and whose tight and loose 1-cells are pseudo- and lax morphisms of algebras, respectively. They can be defined in two equivalent ways: either as 2-functors, which are the identity on objects, faithful, and locally fully faithful, or as categories enriched over the cartesian closed category F, whose objects are functors that are fully faithful and injective on objects. Lack and Shulman called objects of F full embeddings, but we will call them "enhanced categories" because they are nothing else but categories with a distinguished class of objects, which we call tight.The 2-category F has a much richer structure besides being cartesian closed; there are additional closed (but not monoidal) structures, and we show how 2-categories with a right ideal of 1-cells as in 2-categories with Yoneda structure on them can be presented as categories enriched in F in the sense of Eilenberg and Kelly. Since Lack and Shulman were mainly motivated by limits in enhanced 2-categories, they didn't further develop the theory of enhanced (co)lax functors and their enhanced lax adjunctions. The purpose of this talk is to lay the foundations of the theory of enhanced 2-adjunctions and give their examples throughout mathematics and theoretical computer science. _____ * Special Topic: TQFT and Computation, First Lecture. * Speaker: Mikhail Khovanov, Columbia University. * Date and Time: Wednesday February 8, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Universal construction and its applications. * Abstract: Universal construction starts with an evaluation of closed n-manifolds and builds a topological theory (a lax TQFT) for n-cobordisms. A version of it has been used for years as an intermediate step in constructing link homology theories, by evaluating foams embedded in 3-space. More recently, universal construction in low dimensions has been used to find interesting structures related to Deligne categories, formal languages and automata. In the talk we will describe the universal construction and review these developments. _____ * Special Topic: TQFT and Computation, Second Lecture. * Speaker: Mee Seong Im, United States Naval Academy, Annapolis. * Date and Time: Wednesday February 15, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Automata and topological theories. * Abstract: Theory of regular languages and finite state automata is part of the foundations of computer science. Topological quantum field theories (TQFT) are a key structure in modern mathematical physics. We will interpret a nondeterministic automaton as a Boolean-valued one-dimensional TQFT with defects labelled by letters of the alphabet for the automaton. We will also describe how a pair of a regular language and a circular regular language gives rise to a lax one-dimensional TQFT. _____ * Special Topic: TQFT and Computation, Third Lecture. * Speaker: Joshua Sussan, CUNY. * Date and Time: Wednesday February 22, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Non-semisimple Hermitian TQFTs. * Abstract: Topological quantum field theories coming from semisimple categories build upon interesting structures in representation theory and have important applications in low dimensional topology and physics. The construction of non-semisimple TQFTs is more recent and they shed new light on questions that seem to be inaccessible using their semisimple relatives. In order to have potential applications to physics, these non-semisimple categories and TQFTs should possess Hermitian structures. We will define these structures and give some applications. _____ * Speaker: Jens Hemelaer, University of Antwerp. * Date and Time: Wednesday March 15, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: EILC toposes. * Abstract: In topos theory, local connectedness of a geometric morphism is a very geometric property, in the sense that it is stable under base change, can be checked locally, and so on. In some situations however, the weaker property of being essential is easier to verify. In this talk, we will discuss EILC toposes: toposes E such that any essential geometric morphism with codomain E is automatically locally connected. It turns out that many toposes of interest are EILC, including toposes of sheaves on Hausdorff spaces and classifying toposes of compact groups. _____ * Speaker: Jim Otto. * Date and Time: Wednesday March 29, 2023, 7:00 - 8:30 PM. IN PERSON TALK * Title: P Time, A Bounded Numeric Arrow Category, and Entailments. * Abstract:We revisit the characterization of the P Time functions from our McGill thesis. 1. We build on work of L. Roman (89) on primitive recursion and of A. Cobham (65) and Bellantoni-Cook(92) on P Time. 2. We use base 2 numbers with the digits 1 & 2. Let N be the set of these numbers. We split the tapes of a multi-tape Turing machine each into 2 stacks of digits 1 & 2. These are (modulo allowing an odd numberof stacks) the multi-stack machines we use to study P Time. 3. Let Num be the category with objects the finite products of N and arrows the functions between these. From its arrow category Num^2 we abstract the doctrine (here a category of small categories with chosen structure) PTime of categories with with finite products, base 2 numbers, 2-comprehensions, flat recursion, & safe recursion. Since PTime is a locally finitely presentable category, it has an initial category I. Our characterization is that the bottom of the image of I in Num^2 consists of the P Time functions. 4. We can use I (thinking of its arrows as programs) to run multi-stack machines long enough to get P Time.This is the completeness of the characterization. 5. We cut down the numeric arrow category Num^2, using Bellantoni-Cook growth & time bounds on the functions, to get a bounded numeric arrow category B. B is in the doctrine PTime. This yields the soundness of the characterization. 6. For example, the doctrine of toposes with base 1 numbers, choice, & precisely 2 truth values (which captures much of ZC set theory) likely lacks an initial category, much as there is an initial ring, but no initial field. 7. On the other hand, the L. Roman doctrine PR of categories with finite products, base 1 numbers, & recursion (that is, product stable natural numbers objects) does have an initial category as it consists of the strong models of a finite set of entailments. And is thus locally finitely presentable. We sketch the signature graph for these entailments. And some of these entailments. Similarly (but with more complexity) there are entaiments for the doctrine PTime. _____ * Speaker: Walter Tholen, York University. * Date and Time: Wednesday April 19, 2023, 7:00 - 8:30 PM. ZOOM TALK. * Title: What does “smallness” mean in categories of topological spaces? * Abstract: Quillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as the class of finite discrete spaces, or just the empty space , as the examples and remarks in the existing literature may suggest? In this talk we will demonstrate that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces, such as those of T_i-spaces (i= 0, 1, 2), can be quite challenging and may lead to unexpected surprises. In fact, we will show that there are significant differences in this regard even amongst the categories defined by the standard separation conditions, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of Top. (Based on joint work with J. Adamek, M. Husek, and J. Rosicky.) _____ * Speaker: Dusko Pavlovic, University of Hawai‘i at Mānoa. * Date and Time: Wednesday April 26, 2023, 7:00 - 8:30 PM. ZOOM TALK. * Title: Program-closed categories. * Abstract: > Let CC be a symmetric monoidal category with a comonoid on every object. Let CC* be the cartesian subcategory with the same objects and just the comonoid homomorphisms. A *programming language* is a well-ordered object P with a *program closure*: a family of X-natural surjections CC(XA,B) <<--run_X-- CC*(X,P) one for every pair A,B. In this talk, I will sketch a proof that program closure is a property: Any two programming languages are isomorphic along run-preserving morphisms. The result counters Kleene's interpretation of the Church-Turing Thesis, which has been formalized categorically as the suggestion that computability is a structure, like a group presentation, and not a property, like completeness. We prove that it is like completeness. The draft of a book on categorical computability is available from the web site dusko.org. _____ * Speaker: Gemma De las Cuevas, University of Innsbruck. * Date and Time: Wednesday May 3, 2023, 7:00 - 8:30 PM. ZOOM TALK. * Title: A framework for universality across disciplines. * Abstract: What is the scope of universality across disciplines? And what is its relation to undecidability? To address these questions, we build a categorical framework for universality. Its instances include Turing machines, spin models, and others. We introduce a hierarchy of universality and argue that it distinguishes universal Turing machines as a non-trivial form of universality. We also outline the relation to undecidability by drawing a connection to Lawvere’s Fixed Point Theorem. Joint work with Sebastian Stengele, Tobias Reinhart and Tomas Gonda. _____ * Speaker: Arthur Parzygnat, Nagoya University. * Date and Time: Wednesday May 17, 2023, 7:00 - 8:30 PM. IN PERSON TALK. * Title: Inferring the past and using category theory to define retrodiction. * Abstract: Classical retrodiction is the act of inferring the past based on knowledge of the present. The primary example is given by Bayes' rule P(y|x) P(x) = P(x|y) P(y), where we use prior information, conditional probabilities, and new evidence to update our belief of the state of some system. The question of how to extend this idea to quantum systems has been debated for many years. In this talk, I will lay down precise axioms for (classical and quantum) retrodiction using category theory. Among a variety of proposals for quantum retrodiction used in settings such as thermodynamics and the black hole information paradox, only one satisfies these categorical axioms. Towards the end of my talk, I will state what I believe is the main open question for retrodiction, formalized precisely for the first time. This work is based on the preprint https://arxiv.org/abs/2210.13531 and is joint work with Francesco Buscemi. _____ _____ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Steve, I've been following this thread with interest, though I've never worked on the subject. In trying to understand what it's actually all about, I've come to the conclusion that it's "copoint topology". Is that Crazy? Cheers, Bob ________________________________ From: Steven Vickers <s.j.vickers.1@bham.ac.uk> Sent: January 23, 2023 9:47 AM To: categories list <categories@mta.ca> Subject: categories: Re: Terminology for point-free topology? 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 <pedro.m.a.resende@tecnico.ulisboa.pt> Sent: Monday, January 23, 2023 11:44 AM To: ptj@maths.cam.ac.uk <ptj@maths.cam.ac.uk> Cc: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; categories list <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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] --_000_YQXPR01MB2646265C1D8882E80428F9F5E5C99YQXPR01MB2646CANP_ Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable <html> <head> <meta http-equiv="Content-Type" content="text/html; charset=Windows-1252"> <style type="text/css" style="display:none;"> P {margin-top:0;margin-bottom:0;} </style> </head> <body dir="ltr"> <div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);" class="elementToProof"> <br> </div> <div id="appendonsend"></div> <hr style="display:inline-block;width:98%" tabindex="-1"> <div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> Robert Pare <R.Pare@Dal.Ca><br> <b>Sent:</b> January 24, 2023 8:19 AM<br> <b>To:</b> Steven Vickers <s.j.vickers.1@bham.ac.uk><br> <b>Cc:</b> categories@mta <categories@mta><br> <b>Subject:</b> Re: categories: Re: Terminology for point-free topology?</font> <div> </div> </div> <style type="text/css" style="display:none"> <!-- p {margin-top:0; margin-bottom:0} --> </style> <div dir="ltr"> <div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)"> Dear Steve,</div> <div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)"> <br> </div> <div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)"> I've been following this thread with interest, though I've never worked</div> <div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)"> on the subject. In trying to understand what it's actually all about, I've</div> <div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)"> come to the conclusion that it's "copoint topology". Is that Crazy?</div> <div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)"> <br> </div> <div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)"> Cheers,</div> <div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)"> <br> </div> <div class="x_elementToProof" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255)"> Bob<br> </div> <div id="x_appendonsend"></div> <hr tabindex="-1" style="display:inline-block; width:98%"> <div id="x_divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" color="#000000" style="font-size:11pt"><b>From:</b> Steven Vickers <s.j.vickers.1@bham.ac.uk><br> <b>Sent:</b> January 23, 2023 9:47 AM<br> <b>To:</b> categories list <categories@mta.ca><br> <b>Subject:</b> categories: Re: Terminology for point-free topology?</font> <div> </div> </div> <div class="x_BodyFragment"><font size="2"><span style="font-size:11pt"> <div class="x_PlainText">CAUTION: The Sender of this email is not from within Dalhousie.<br> <br> Dear Pedro,<br> <br> Of course, that's the very reason why I wanted to transfer it to the style of working without points.<br> <br> 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.<br> <br> 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.<br> <br> Do you think there's a less derogatory term for the style of reasoning without points?<br> <br> All the best,<br> <br> Steve.<br> <br> ________________________________<br> From: pedro.m.a.resende@tecnico.ulisboa.pt <pedro.m.a.resende@tecnico.ulisboa.pt><br> Sent: Monday, January 23, 2023 11:44 AM<br> To: ptj@maths.cam.ac.uk <ptj@maths.cam.ac.uk><br> Cc: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; categories list <categories@mta.ca><br> Subject: Re: categories: Re: Terminology for point-free topology?<br> <br> 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… :)<br> <br> Pedro<br> <br> <br> [For admin and other information see: <a href="http://www.mta.ca/~cat-dist/">http://www.mta.ca/~cat-dist/</a> ]<br> </div> </span></font></div> </div> </body> </html> --_000_YQXPR01MB2646265C1D8882E80428F9F5E5C99YQXPR01MB2646CANP_-- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Vaughan, The idea of generic point as dense point looks misconceived to me. I'll say more below. First, however, can I check we're in line with intro/extrovert? When I discussed the circle, was I using them with the same metaphorical content as you had in mind? More below. Steve. ________________________________ From: Vaughan Pratt <pratt@cs.stanford.edu> Sent: Monday, January 23, 2023 11:17 PM 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? In my case the problem I'm having with Chu(E,k) is that I have no intuition about E-enriched categories. A given Chu space (a,r,x) would have a and x each be an object of E, with x a frame defined by inclusion. What I can't picture is r: a x x → k. A good picture for internal logic in E is often to imagine E as sheaves over some B, with the objects as local homeomorphisms to B. That might sound a bit point-set, but Joyal and Tierney showed how to make sense of it for general E. Then geometric constructions, such as binary product a x x, are calculated fibrewise. The subobject classifier k is not geometric, but your maps r are just subobjects of a x x, and those too can be seen fibrewise. Actually, frames aren't geometric either, but J&T showed that the internal frames correspond to more general bundles over B. When you build in the right conditions on r, it should correspond to a bundle map, from the local homeomorphism for a to the more general bundle for x. That is why we're approximating a bundle by a local homeomorphism. There is a best possible approximation, the discrete coreflection, or the "set of points" as calculated internally in E, but my example with B = Sierpinski shows it can be badly deficient even for straightforward bundles. I must stress that the ability to treat bundles as internal spaces is a wonderful feature of point-free topology, something to be treasured, so one shouldn't be tempted to treat such straightforward bundles as pathological on the grounds of their "non-spatiality". "In fact the generic point in the topos of sheaves, on its own, is enough for most purposes." Would that be the dense point? No. You'll see more discussion of this in my arXiv notes, but perhaps I can explain the generic point in computer science terms as a formal parameter. Suppose you're working in a programming language L, and you write a procedure with a formal parameter x of type T. Within the scope of the declaration x:T, you are no longer working in pure L, but in L with an element of T freely adjoined. Call it L[x:T]. The freeness lies in the fact that whatever you construct with x can be transported, by substitution, to any actual parameter a:T. Note that a may itself be in the scope of some other formal parameter, hence in some L[x':T']. Substituting a for x transports any construction in L[x:T] into one in L[x':T'], so, in some sense you get a homomorphism L[x:T] -> L[x':T']. Now think of x as "generic", in that it has no properties other than what follows from being of type T, and the actual parameters a as being more specific. The same idea applies to the generic point in a classifying topos. Instead of types and elements, we have geometric theories and models. For instance T might be the geometric theory of Dedekind sections, and then a model is a real number. The classifying topos Set[x:T] is the geometric mathematics of Set with a model of T freely adjoined - that is pretty much what the universal property of classifying toposes tells us. If we have a model a of T, and it could be in some other topos, say Set[x':T'], then by substitution we get a functor Set[x:T] -> Set[x':T'] that preserves geometric structure (essentially: colimits and finite limits). It has a right adjoint, and hence we get a geometric morphism. That's a generalized point of Set[x:T] - its generalized points are equivalent to models of T. Note also that, for a locale, that classifying topos is equivalent to the topos of sheaves. I hope this helps clarify the fact that the generic point does not live in Set, but in a different topos Set[x:T] that is more or less a syntactic construct. Its sufficiency as a point on its own lies in the fact that it can be instantiated as any other more specific point. I can't see any way in which it is helpful to see it as a "dense" point. (Are you trying to think of it as a "thickened" point where a circle meets a tangent? I don't think that analogy goes anywhere.) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

" If I understand properly what you're suggesting, it's to replace (Set, 2) by some (E, k). That's not going to solve the issue with lack of points that I was talking about. If anything, it makes it worse, because without choice it's harder to find points." In my case the problem I'm having with Chu(E,k) is that I have no intuition about E-enriched categories. A given Chu space (a,r,x) would have a and x each be an object of E, with x a frame defined by inclusion. What I can't picture is r: a x x → k. "In fact the generic point in the topos of sheaves, on its own, is enough for most purposes." Would that be the dense point? "But then, if you wanted to adapt the Chu spaces somehow to allow that in, you might as well take it on its own." The idea of a dense point seems right. However maybe there's a simple way to approach it in Chu(Set,2). My earlier question was, how many points are needed to support representing a local as a Chu space over 2? It seems to me that a dense set of points (suitably defined) would suffice, e.g. in a flat n-dimensional space, D^n where D consists of the dyadic rationals. The intersection of the line y = x with the unit circle in the upper right quadrant would presumably be the dense point at (1,1)/√2. But Euclidean geometry is just real algebraic geometry limited to degree 2, suggesting replacing D by the field of constructible numbers. You wouldn't need the notion of a dense point because all Euclidean constructions would produce a real point, but space would still be just as countable as D^n. Space would still have atomless parts, namely the uncountably many points created by Dedekind or Cantor (the latter using Cauchy sequences) that aren't needed for Euclidean geometry. For real algebraic geometry, just drop the limit of 2 on degree of polynomials. And drop "real" for general algebraic geometry. All this is to permit locales to be represented as Chu spaces over 2, namely by requiring their set of states to be a locale where the order is given by set inclusion. But if "point-free" means replacing the concrete notion of the point (1,1)/√2 itself with the generic dense point located there, I no longer see how to represent a locale as a Chu space of any kind. If there's better (e.g. more pedagogically suitable) language than what I used above, I'm all ears. Vaughan On Mon, Jan 23, 2023 at 5:25 AM Steven Vickers <s.j.vickers.1@bham.ac.uk> wrote: > Dear Vaughan, > > I've worked point-free on midpoint algebras (my paper that [-1,1] is an > Escardo-Simpson interval object), and I think I understand what you're > getting at there. > > Here's a particular mathematical problem I've been looking at, to check > whether my thinking on extrovert/introvert is in line with yours. > > Now we have satisfactory point-free accounts of exp and log, can we do the > same with trigonometry? > > That boils down to defining group homomorphisms from R to the circle group > S^1 (viewed as a sublocale of C = R^2). It may be that a good way to do > that is (1) (introvert?) define a midpoint homomorphism from [0, 1] to a > region of S^1 close to 1, with 0 mapping to 1, and then (2) (extrovert?) > use the homomorphism property to extend to the whole of R. > > I'll turn now to the Chu spaces. If I understand properly what you're > suggesting, it's to replace (Set, 2) by some (E, k). That's not going to > solve the issue with lack of points that I was talking about. If anything, > it makes it worse, because without choice it's harder to find points. > > The problem lies, rather, in the fact that the Chu space relies on pairing > two *sets*. I'm perfectly happy to allow "set" to mean object in some > chosen base topos. However, insisting on a set on the points side smashes > too much topological structure to work well in general. As my example with > Sierpinski showed, it in effect forces you to approximate bundles with > local homeomorphisms, and that can leave you with nothing useful that is > available for the points side of the Chu space. > > When you switch to generalized points, there are now enough. In fact the > generic point in the topos of sheaves, on its own, is enough for most > purposes. But then, if you wanted to adapt the Chu spaces somehow to allow > that in, you might as well take it on its own. > > Hope that helps, > > Steve. > ------------------------------ > *From:* Vaughan Pratt <pratt@cs.stanford.edu> > *Sent:* Sunday, January 22, 2023 9:32 PM > *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? > > Hi Steve, > > "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." (Your reply to me here > of Jan. 17) > > Thanks for that and your accompanying remarks , Steve. > > Space is both extroverted (Euclid's relatively clear Postulate 2 that a > finite straight line can be produced) and introverted (Euclid's vaguer > Definition 2 that a line (segment) is breadthless length). > > From a Topological Systems/Chuish perspective, I wonder if the extroverted > nature of space is best appreciated through points and its introverted > nature through states. > > After all, we have Hoelder's 1901 notion of a linearly ordered group for > the former (and the free such on one generator will be the integers and > hence both abelian and Archimedean), while we have the > Pavlovich-P-Freyd-Leinster notion of the continuum as a final coalgebra, > which can be as small as the unit interval if you stick to midpoint > algebras (rather than continued fractions as Dusko and I did in 1999) and > as such ideal for filling in the gaps between consecutive integers. > > That Euclid's Definition 2 is vaguer than his Postulate 2 is consistent > with the applicability of free algebras to the extroverted nature of space > appearing much earlier than that of final coalgebras to its introverted > nature. > > These thoughts came to me after spending a few weeks mulling over a > conversation I had with my classmate (1962-5) Ross Street about our common > but independently arrived at interest, decades ago, in what Ross calls > "efficient" constructions of the reals. > > And along a different line of thought, is Chu(Set,2) the right category > for topological systems, or might there be some advantage to Chu(E,k) > where E is the appropriate topos for the application at hand, or perhaps > just the free topos, and k its subobject classifier? > > Best, > Vaughan > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I am incredibly saddened at the passing of Bill Lawvere. In my first term of graduate school in 1974, I found myself in Bill’s Algebra class and, looking back, it was the single most fortuitous and defining moment in my career. I was enthralled and captivated by a new view of mathematics, learning, and scholarship that opened my eyes and would serve as a fingerpost directing me towards my future path. Whatever modest successes I may have had as a mathematician I owe to Bill’s teachings, example, and mentorship. Others will speak more astutely than I ever could to his many, many contributions and accomplishments; what I will continue to cherish is his kind friendship and also his unmatched generosity of spirit. As I now reflect on Bill, what comes to mind is the definition of the word *mensch* - a person of integrity and honor, of kindness and consideration, someone to admire and emulate. Kimmo Rosenthal On Tue, Jan 24, 2023 at 4:51 PM Alberto Peruzzi <attipg2011@gmail.com> wrote: > Dear all, > > a very sad news: Lawvere’s family asked me to provide you with the message > below. > Alberto Peruzzi > > > > > Francis William Lawvere (known to most as Bill) died peacefully yesterday > (January 23, 2023) at sunrise, surrounded by family at his home in Chapel > Hill, North Carolina after a long period of illness. He was born in Muncie, > Indiana on Feb. 9, 1937 and has family, friends and colleagues around the > world. No services will be held at this time, however several memorial > celebrations of life and scientific work are anticipated. > > Correspondence with family can be sent to his wife Fatima at > fatimaylawvere@gmail.com <mailto:fatimaylawvere@gmail.com> or to > 111 East Winmore Ave., Chapel Hill, NC 27516 USA. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

To whom it may concern, Dear Sir, Please, be so kind as to include the Announcement of the Tbilisi Summer School in your e-mail list. I hope, you will find all necessary information in the Announcement itself. Best regards, George Chikoidze Contact Person of the Centre on Language, Logic, Speech at the Tbilisi State University Address: Dpt. of Language Modelling Inst. of Control Systems Georgian Academy of Sciences 34, Gamsakhurdia Av. 380060 Tbilisi Georgia Phone: + 995 32 38 21 36 Fax: + 995 32 94 23 91 E-mail: chiko@contsys.acnet.ge ANNOUNCEMENT Tbilisi Summer School in Language, Logic, and Computation 29th August - 8 September 2000 Tbilisi, Georgia The Georgian Centre for Language, Logic, and Speech, based at the Tbilisi State University, will host Tbilisi Summer School, the main purpose of which is to make the students and young scholars acquainted with the modern state of affairs in the mentioned fields of science, and - at the same time - to further contacts and scientific collaboration between Western and Eastern Scholars. LECTURERS Jurij Apresjan, Moscow Matthias Baaz, Vienna Pascal Boldini, Paris Marina Glavinskaja, Moscow Michel Parigot, Paris Carl Vogel,Dublin Andrej Voronkov, Manchester COURSES - Foundations of linguistic semantics (Ju.Apresjan) - On the generalisation of proofs and calculations (M.Baaz) - Type theories for semantics and cognition (P.Boldini) - Semantics of aspect (M.Glavinskaja) - Proofs as programs (M.Parigot) - Cognitive constraints on linguistic theory. (C.Vogel) - Logical foundations of deductive databases. (A.Voronkov). SCHEDULE Tuesday, August 29 12.00 - 13.00 Opening of the School 18.00 - Banquet Wednesday, August 30 10.00 - 13.00 M. Parigot. Proofs as programs 13.00 - 14.00 Lunch 14.00 - 17.00 Ju. Apresjan. Foundations of linguistic semantics Thursday, August 31 10.00 - 13.00 M. Parigot. Proofs as programs. 13.00 - 14.00 Lunch 14.00 - 17.00 Ju. Apresjan. Foundations of linguistic semantics. Friday, September 1 10.00 - 13.00 M. Glavinskaja. Semantics of aspect. 13.00 - 14.00 Lunch 14.00 - 17.00 C. Vogel. Cognitive constraints on linguistic theory. Saturday, September 2 Recreation (sight-seeing in Tbilisi and environs, Sunday, September 3 excursion). Monday, September 4 10.00 - 13.00 M. Glavinskaja. Semantics of aspect. 13.00 - 14.00 Lunch 14.00 - 16.00 C. Vogel. Cognitive constraints on linguistic theory. Tuesday, September 5 10.00 - 14.00 A. Voronkov. Logical foundations of deductive databases. 14.00 - 15.00 Lunch 15.00 - 17.00 C. Vogel. Cognitive constraints on linguistic theory. Wednesday, September 6 10.00 - 12.00 A. Voronkov. Logical foundations of deductive databases. 12.00 - 14.00 M. Baaz. On the generalization of proofs and calculations. 14.00 - 15.00 Lunch 15.00 - 17.00 P. Boldini. Type theories for semantics and cognition. Thursday, September 7 10.00 - 12.00 M. Baaz. On the generalisation of proofs and calculations. 12.00 - 14.00 P. Boldini. Type theories for semantics and cognition. 14.00 - 15.00 Lunch Friday, September 8 10.00 - 12.00 M. Baaz. On the generalization of proofs and calculations. 12.00 - 14.00 P. Boldini. Type theories for semantics and cognition. 14.00 - 15.00 Lunch 15.00 - 16.00 Closing of the School. 18.00 Banquet STUDENTS Besides local students, the School also welcomes students from abroad. The Participation fee for these students will be $120 (wich includes excursion, banquet, reprints, etc.).Foreign students will be comfortably accommodated with Georgian families (with two meals) for $40 per day. LOCATION AND SIGHTSEEING TOURS Georgia is the ancient country situated between Black and Caspian seas, the Caucasus and Turkey. It is the country of the Golden Fleece, the myth of Argonauts, Jason and Medea, and Prometheus, chained to the Caucasus mountains. Tbilisi - capital of Georgia - has more than 1 million in habitant. It is situated some 100-150 km to the south of main Caucasus ridge, in the beautiful valley of the river Mtkvari, surrounded by the green slopes of the Caucasus spurs. The city has a long (1500 year old) history and abounds in historical and cultural memorials. Georgia is famous for its high quality wines, exquisite cuisine and cordial hospitality. The main site of the Symposium, Tbilisi State University, is the chief centre of education in the country, and has several outstanding scholars in science, art and politics among its graduates. During the School there will be an excursion over the famous Georgian Military Road, which leads us through the ancient capital of Georgia - Mtskheta, with its abundant architectural and historical monuments - and which brings us across the main ridge of the Caucasus by the Cross Pass. The destination of the envisaged trip is the mountain resort Kazbegi with its Trinity Church situated on the top of high peak facing the second mountain (after Elbrus), the peak of the Caucasus - Mkinvarcveri (Glacier - mountain). TRAVEL INFORMATION Tbilisi can only be reached by air. If there are no direct trip between your point of departure and Tbilisi, the most convenient connections are via Istambul, Frankfurt or Moscow. ORGANISING COMMITTEE: T.Khurodze (Chair, Pro-rector of Tbilisi State University) R.Asatiani (Institute of Oriental Studies) N.Chanishvili (Tbilisi State University) G.Chikoidze (Institute of Control Systems) K.Pkhakadze (Institute of Applied Mathematics) Kh.Rukhaja (Institute of Applied Mathematics). For additional information, please, use the following address: George Chikoidze Dept. of Language Modelling Inst. of Control Systems Georgian Academy of Sciences 34, K. Gamsakhurdia 380060 Tbilisi Georgia Phone: +9 9532 382136 E- mail: chiko@contsys.acnet.ge

[-- Attachment #1: Type: text/plain, Size: 2261 bytes --] Spam detection software, running on the system "ciao.gmane.io", has identified this incoming email as possible spam. The original message has been attached to this so you can view it or label similar future email. If you have any questions, see @@CONTACT_ADDRESS@@ for details. Content preview: Dear all, a very sad news: Lawvereâ€™s family asked me to provide you with the message below. Alberto Peruzzi Francis William Lawvere (known to most as Bill) died peacefully yesterday (January 23, 2023) at sunrise, surrounded by family at his home in Chapel Hill, North Carolina after a long period of illness. [...] Content analysis details: (6.8 points, 5.0 required) pts rule name description ---- ---------------------- -------------------------------------------------- 0.0 DKIM_ADSP_CUSTOM_MED No valid author signature, adsp_override is CUSTOM_MED 0.0 URIBL_BLOCKED ADMINISTRATOR NOTICE: The query to URIBL was blocked. See http://wiki.apache.org/spamassassin/DnsBlocklists#dnsbl-block for more information. [URIs: mta.ca] 1.0 FORGED_GMAIL_RCVD 'From' gmail.com does not match 'Received' headers 0.2 FREEMAIL_REPLYTO_END_DIGIT Reply-To freemail username ends in digit (attipg2011[at]gmail.com) 1.1 DATE_IN_PAST_03_06 Date: is 3 to 6 hours before Received: date 0.7 LOCALPART_IN_SUBJECT Local part of To: address appears in Subject 0.0 FREEMAIL_FROM Sender email is commonly abused enduser mail provider (attipg2011[at]gmail.com) 0.2 HEADER_FROM_DIFFERENT_DOMAINS From and EnvelopeFrom 2nd level mail domains are different 0.0 FREEMAIL_FORGED_FROMDOMAIN 2nd level domains in From and EnvelopeFrom freemail headers are different 1.0 FREEMAIL_REPLYTO Reply-To/From or Reply-To/body contain different freemails 2.5 SPOOFED_FREEM_REPTO Forged freemail sender with freemail reply-to [-- Attachment #2: original message before SpamAssassin --] [-- Type: message/rfc822, Size: 1920 bytes --] From: Alberto Peruzzi <attipg2011@gmail.com> To: categories@mta.ca Subject: categories: Bill Lawvere Date: Tue, 24 Jan 2023 18:21:46 +0100 Message-ID: <E1pKRBN-0006DI-3r@rr.mta.ca> Dear all, a very sad news: Lawvere’s family asked me to provide you with the message below. Alberto Peruzzi Francis William Lawvere (known to most as Bill) died peacefully yesterday (January 23, 2023) at sunrise, surrounded by family at his home in Chapel Hill, North Carolina after a long period of illness. He was born in Muncie, Indiana on Feb. 9, 1937 and has family, friends and colleagues around the world. No services will be held at this time, however several memorial celebrations of life and scientific work are anticipated. Correspondence with family can be sent to his wife Fatima at fatimaylawvere@gmail.com <mailto:fatimaylawvere@gmail.com> or to 111 East Winmore Ave., Chapel Hill, NC 27516 USA. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I am devastated by the news that Bill Lawvere passed away yesterday. Truly, we have lost one of the giants of our subject. I imagine everyone on this list will have their own special memories of him; here are a few of mine. It was fifty years ago this month that I first corresponded with Bill, sending him the first draft of my paper `The associated sheaf functor in an elementary topos' (which he accepted for publication in JPAA). I still have the reply he sent me: re-reading it brought tears to my eyes as I remembered the very considerable trouble he had taken to help and encourage a young author who was then totally unknown to him, by giving me excellent advice on how to sharpen the presentation of the paper. We met a few months later, at the Aarhus Open House in May 1973; it was there that I aroused Bill's anger by presuming to give a talk about John Conway's theory of numbers and games, which Bill thought (wrongly) to be antithetical to his very strong political views. But the storm passed, and Bill remained a staunch supporter of my work (though I could always rely on him for frank criticism of any imperfections in it). When, in September 2004, Jiri Adamek suggested to a group of senior category-theorists that there was a need for a `steering committee' to ensure the continued smooth running of international CT conferences, there was lengthy discussion about almost every aspect of his proposal; but one thing which nobody queried was his suggestion that Bill should be invited to chair the committee. Happily, Bill accepted the invitation (on condition that I should agree to be the secretary), and he provided us with wise and thoughtful guidance until he stepped down in 2010. The organizers of the 2007 CT meeting in Carvoeiro decided to make the celebration of Bill's 70th birthday a theme of the conference, and they invited me to give a laudatory talk summarizing Bill's mathematical achievements. It was an invitation I accepted with considerable trepidation, and even 24 hours before the talk was due I was unsure how I was going to do justice to all aspects of Bill's work. But in the event it went well, and Bill approved of it; he was particularly amused that I'd included a quote from Karl Marx on one of my slides (he thought, incorrectly, that I hadn't been aware of its source). But there's no need for me to recall anything I said in that talk; Bill himself did the job of summarizing his work much better, in an interview with the Coimbra mathematicians which is available online. Reading it again today, I'm struck once more by the extraordinary mathematical mind of a man I've been privileged to know for almost fifty years. RIP Bill; we'll all miss you. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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 <pedro.m.a.resende@tecnico.ulisboa.pt> Sent: Monday, January 23, 2023 11:44 AM To: ptj@maths.cam.ac.uk <ptj@maths.cam.ac.uk> Cc: Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk>; categories list <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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Vaughan, I've worked point-free on midpoint algebras (my paper that [-1,1] is an Escardo-Simpson interval object), and I think I understand what you're getting at there. Here's a particular mathematical problem I've been looking at, to check whether my thinking on extrovert/introvert is in line with yours. Now we have satisfactory point-free accounts of exp and log, can we do the same with trigonometry? That boils down to defining group homomorphisms from R to the circle group S^1 (viewed as a sublocale of C = R^2). It may be that a good way to do that is (1) (introvert?) define a midpoint homomorphism from [0, 1] to a region of S^1 close to 1, with 0 mapping to 1, and then (2) (extrovert?) use the homomorphism property to extend to the whole of R. I'll turn now to the Chu spaces. If I understand properly what you're suggesting, it's to replace (Set, 2) by some (E, k). That's not going to solve the issue with lack of points that I was talking about. If anything, it makes it worse, because without choice it's harder to find points. The problem lies, rather, in the fact that the Chu space relies on pairing two *sets*. I'm perfectly happy to allow "set" to mean object in some chosen base topos. However, insisting on a set on the points side smashes too much topological structure to work well in general. As my example with Sierpinski showed, it in effect forces you to approximate bundles with local homeomorphisms, and that can leave you with nothing useful that is available for the points side of the Chu space. When you switch to generalized points, there are now enough. In fact the generic point in the topos of sheaves, on its own, is enough for most purposes. But then, if you wanted to adapt the Chu spaces somehow to allow that in, you might as well take it on its own. Hope that helps, Steve. ________________________________ From: Vaughan Pratt <pratt@cs.stanford.edu> Sent: Sunday, January 22, 2023 9:32 PM 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? Hi Steve, "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." (Your reply to me here of Jan. 17) Thanks for that and your accompanying remarks , Steve. Space is both extroverted (Euclid's relatively clear Postulate 2 that a finite straight line can be produced) and introverted (Euclid's vaguer Definition 2 that a line (segment) is breadthless length). From a Topological Systems/Chuish perspective, I wonder if the extroverted nature of space is best appreciated through points and its introverted nature through states. After all, we have Hoelder's 1901 notion of a linearly ordered group for the former (and the free such on one generator will be the integers and hence both abelian and Archimedean), while we have the Pavlovich-P-Freyd-Leinster notion of the continuum as a final coalgebra, which can be as small as the unit interval if you stick to midpoint algebras (rather than continued fractions as Dusko and I did in 1999) and as such ideal for filling in the gaps between consecutive integers. That Euclid's Definition 2 is vaguer than his Postulate 2 is consistent with the applicability of free algebras to the extroverted nature of space appearing much earlier than that of final coalgebras to its introverted nature. These thoughts came to me after spending a few weeks mulling over a conversation I had with my classmate (1962-5) Ross Street about our common but independently arrived at interest, decades ago, in what Ross calls "efficient" constructions of the reals. And along a different line of thought, is Chu(Set,2) the right category for topological systems, or might there be some advantage to Chu(E,k) where E is the appropriate topos for the application at hand, or perhaps just the free topos, and k its subobject classifier? Best, Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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 > On Jan 21, 2023, at 7:42 PM, ptj@maths.cam.ac.uk wrote: > > I was wondering how long it would be before someone in this thread > referred to my `point of pointless topology' paper! Perhaps not so many > people know that the title was a conscious echo of an earlier paper > by Mike Barr called `The point of the empty set', which began with the > words (I quote from memory) `The point is, there isn't any point there; > that's exactly the point'. > > As Steve says, to fit that title I had to use the word `pointless', but > on the whole I prefer `pointfree'; it carries the implication that you > are free to work without points or to use them (in a generalized sense), > as you prefer. > > Peter Johnstone > > On Jan 21 2023, Steven Vickers wrote: > >> Dear David, >> >> Yes, and it's an excellent paper with a witty title for which only >> "pointless" would do. >> >> I particularly like what Peter said when explaining the significant >> difference in the absence of choice (such as in toposes of sheaves), and >> that "usually it is locales, not spaces, which provide the right context >> in which to do topology". >> >> He went on to say, >> >> "This is the point which ... Andre Joyal began to hammer home in the >> early 1970s; I can well remember how, at the time, his insistence that >> locales were the real stuff of topology, and spaces were merely figments >> of the classical mathematician's imagination, seemed (to me, and I >> suspect to others) like unmotivated fanaticism. I have learned better >> since then." >> >> This is all part of the argument for using a reformed topology, but there >> is nothing particular there about the pointwise style of reasoning for >> it. Hence we are still left with the question of how to reference the two >> concepts, the reformed topology and the reasoning without points. >> >> Would you call Ng's paper with me pointless? Points are everywhere in it. >> (Of course, there's the separate issue of whether it was pointless in the >> sense of not worth the trouble. But an important feature of the style is >> that it forces you to be careful to distinguish between Dedekind reals >> and 1-sided (lower or upper) reals, and in Ng's thesis this uncovered >> unexpected roles of 1-sided reals in the account of Ostrowski's Theorem >> and the Berkovich spectrum. So there is a bit of payoff.) >> >> Best wishes, >> >> Steve. >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Hi Steve, "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." (Your reply to me here of Jan. 17) Thanks for that and your accompanying remarks , Steve. Space is both extroverted (Euclid's relatively clear Postulate 2 that a finite straight line can be produced) and introverted (Euclid's vaguer Definition 2 that a line (segment) is breadthless length). From a Topological Systems/Chuish perspective, I wonder if the extroverted nature of space is best appreciated through points and its introverted nature through states. After all, we have Hoelder's 1901 notion of a linearly ordered group for the former (and the free such on one generator will be the integers and hence both abelian and Archimedean), while we have the Pavlovich-P-Freyd-Leinster notion of the continuum as a final coalgebra, which can be as small as the unit interval if you stick to midpoint algebras (rather than continued fractions as Dusko and I did in 1999) and as such ideal for filling in the gaps between consecutive integers. That Euclid's Definition 2 is vaguer than his Postulate 2 is consistent with the applicability of free algebras to the extroverted nature of space appearing much earlier than that of final coalgebras to its introverted nature. These thoughts came to me after spending a few weeks mulling over a conversation I had with my classmate (1962-5) Ross Street about our common but independently arrived at interest, decades ago, in what Ross calls "efficient" constructions of the reals. And along a different line of thought, is Chu(Set,2) the right category for topological systems, or might there be some advantage to Chu(E,k) where E is the appropriate topos for the application at hand, or perhaps just the free topos, and k its subobject classifier? Best, Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I was wondering how long it would be before someone in this thread referred to my `point of pointless topology' paper! Perhaps not so many people know that the title was a conscious echo of an earlier paper by Mike Barr called `The point of the empty set', which began with the words (I quote from memory) `The point is, there isn't any point there; that's exactly the point'. As Steve says, to fit that title I had to use the word `pointless', but on the whole I prefer `pointfree'; it carries the implication that you are free to work without points or to use them (in a generalized sense), as you prefer. Peter Johnstone On Jan 21 2023, Steven Vickers wrote: >Dear David, > > Yes, and it's an excellent paper with a witty title for which only > "pointless" would do. > > I particularly like what Peter said when explaining the significant > difference in the absence of choice (such as in toposes of sheaves), and > that "usually it is locales, not spaces, which provide the right context > in which to do topology". > >He went on to say, > > "This is the point which ... Andre Joyal began to hammer home in the > early 1970s; I can well remember how, at the time, his insistence that > locales were the real stuff of topology, and spaces were merely figments > of the classical mathematician's imagination, seemed (to me, and I > suspect to others) like unmotivated fanaticism. I have learned better > since then." > > This is all part of the argument for using a reformed topology, but there > is nothing particular there about the pointwise style of reasoning for > it. Hence we are still left with the question of how to reference the two > concepts, the reformed topology and the reasoning without points. > > Would you call Ng's paper with me pointless? Points are everywhere in it. > (Of course, there's the separate issue of whether it was pointless in the > sense of not worth the trouble. But an important feature of the style is > that it forces you to be careful to distinguish between Dedekind reals > and 1-sided (lower or upper) reals, and in Ng's thesis this uncovered > unexpected roles of 1-sided reals in the account of Ostrowski's Theorem > and the Berkovich spectrum. So there is a bit of payoff.) > >Best wishes, > >Steve. > > ________________________________ From: David Yetter <dyetter@ksu.edu> > Sent: Friday, January 20, 2023 3:06 AM To: I.Moerdijk@uu.nl > <I.Moerdijk@uu.nl>; Steven Vickers (Computer Science) > <s.j.vickers.1@bham.ac.uk> Cc: categories list <categories@mta.ca> > Subject: Re: categories: Re: Terminology for point-free topology? > > I seem to recall from back in my days as a grad student or new PhD that > Peter Johnstone wrote a paper entitled "The Point of Pointless Topology". > Just in honor of that I've always favored "pointless topology" as the > term for the theory of locales and sheaves on locales. > >Best Thoughts, >David Y. > >________________________________ >From: Steven Vickers <s.j.vickers.1@bham.ac.uk> >Sent: Wednesday, January 18, 2023 6:12 AM >To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl> >Cc: categories list <categories@mta.ca> >Subject: categories: Re: Terminology for point-free topology? > >This email originated from outside of K-State. > > >Dear Ieke, > > Thanks for mentioning that. It's a beautiful paper, both in its results > and in its presentation, and one I still return to. > > Another place where I think you were even more explicit was in "The > classifying topos of a continuous groupoid I" (1988), where you said - > > "... in presenting many arguments concerning generalized, "pointless" > spaces, I have tried to convey the idea that by using > change-of-base-techniques and exploiting the internal logic of a > Grothendieck topos, point-set arguments are perfectly suitable for > dealing with pointless spaces (at least as long as one stays within the > 'stable' part of the theory)." > > (Would you still say that "pointless" and "point-set" are the right > phrases there? I'm proposing "point-free" and "pointwise".) > > On the other hand, in your book with Mac Lane, those ideas seemed to go > into hiding. In fact I explicitly wrote "Locales and toposes as spaces" > as a guide to reading the points back into the book. > > My first understanding of these pointwise techniques came in the 1990's, > as I developed the exposition of "Topical categories of domains". That > was before I knew those papers of yours, but I felt right from the start > that I was merely unveiling techniques already known to the experts - > though I hope you'll agree I've been more explicit about them and > particularly the nature and role of geometricity. > > I still don't know as much as I would like about the origin and history > of those techniques. It would certainly improve my arXiv notes if I could > say more. > > Might they even have roots in Grothendieck? I once saw a comment by Colin > McLarty to the effect that (modulo misrepresentation by me) Grothendieck > was aware of two different lines of reasoning with toposes: by > manipulating sites concretely, or by using colimits and finite limits > under the rules corresponding to Giraud's theorem. I imagine that as > being something like the distinction between pointless and pointwise. > >Best wishes, > >Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

THE FOURTEENTH INTERNATIONAL TBILISI SYMPOSIUM ON LOGIC, LANGUAGE AND COMPUTATION 18-22 September, 2023 Telavi, Georgia Website: https://events.illc.uva.nl/Tbilisi/Tbilisi2023/ *********************************************************************** FIRST CALL FOR PAPERS The Fourteenth International Tbilisi Symposium on Logic, Language, and Computation will be held 18-22 September 2023 in Telavi, located in the Kakheti region, Georgia, north-east of Tbilisi. The Programme Committee invites submissions for contributions on all aspects of logic, language, and computation. Work of an interdisciplinary nature is particularly welcome. Areas of interest include, but are not limited to: * Natural language syntax, semantics, and pragmatics * Linguistic typology and semantic universals * Language evolution and learnability * Variability in language * Sociolinguistics * Historical linguistics, history of logic * Natural logic, inference and entailment in natural language * Natural language processing * Distributional and probabilistic models of information, meaning and computation * Logic, games, and formal pragmatics * Logic and cognition * Logics for artificial intelligence and computer science * Knowledge representation * Foundations of machine learning * Formal models of multiagent systems * Logics for social networks * Logics for knowledge, belief, and information dynamics * Computational social choice * Information retrieval, query answer systems * Constructive, intuitionistic, modal and other non-classical logics * Algebraic and coalgebraic logic and semantics * Categorical logic * Models of computation PROGRAMME The programme will include the following tutorials and a series of invited lecturers. *Tutorial speakers* Language: Peter Sutton (UPF, Barcelona) Logic & Computation: Frank Wolter (University of Liverpool) *Invited speakers* Language: - Heather Burnett (CNRS-LLF, Paris) - Stephanie Solt (ZAS, Berlin) Logic & Computation: - Balder ten Cate (University of Amsterdam) - Nina Gierasimczuk (Danish Technical University) WORKSHOPS There will be two workshops embedded in the conference programme: "The Semantics of Hidden Meanings" Organisers: Heather Burnett (CNRS-LLF, Paris), Stephanie Solt (ZAS, Berlin), Peter Sutton (UPF, Barcelona) For more details see the workshop webpage: https://sites.google.com/view/hidden-meanings and "Learning and Logic" Organisers: Balder ten Cate (University of Amsterdam) and Aybüke Özgün (University of Amsterdam). More details will soon be made available via the TbiLLC website (see top). PROGRAMME COMMITTEE Samson Abramsky (University College London, UK) Philippe Balbiani (CNRS & University of Toulouse, FR) Guram Bezhanishvili (New Mexico State University, USA) Nick Bezhanishvili (University of Amsterdam, NL) Olga Borik (Universidad Nacional de Educación a Distancia, ES) Zoé Christoff (University of Groningen, NL) Agata Ciabattoni (TU Wien, AT) Milica Denić (Tel Aviv University, IL) David Gabelaia (Javakhishvili Tbilisi State University, GE) Berit Gehrke (Humboldt-Universität zu Berlin, DE, co-chair) Jim de Groot (Australian National University, AU) Helle Hvid Hansen (University of Groningen, NL, co-chair) Daniel Hole (University of Stuttgart, DE) Rosalie Iemhoff (Utrecht University, NL) Maarten Janssen (Charles University, Prague, CZ) Clemens Kupke (University of Strathclyde, UK) Temur Kutsia (Johannes Kepler University Linz, AT) Mora Maldonado (CNRS & University of Nantes, FR) Louise McNally (Universitat Pompeu Fabra, Barcelona, ES) Lawrence Moss (Indiana University, USA) Aybüke Özgün (University of Amsterdam, NL) Mehrnoosh Sadrzadeh (University College London, UK) Viola Schmitt (Humboldt-Universität zu Berlin, DE) Todd Snider (Heinrich Heine University of Düsseldorf, DE) Ana Sokolova (University of Salzburg, AT) Luca Spada (University of Salerno, IT) Yasutada Sudo (University College London, UK) Jakub Szymanik (University of Trento, IT) Carla Umbach (University of Cologne, DE) Marcin Wągiel (Masaryk University, Brno, CZ, and University of Wrocław, PL) Fan Yang (Utrecht University, NL) Malte Zimmermann (University of Potsdam, DE) Sarah Zobel (University of Oslo, NO) IMPORTANT DATES Submission deadline: Fri 17 March 2023 Notification: Wed 24 May 2023 Final abstracts due: Mon 26 June 2023 Early Registration deadline: Fri 30 June 2023 Late Registration deadline: Mon 24 July 2023 Symposium: 18-22 September 2023 SUBMISSION INFO Authors can submit an abstract for presentation at the symposium of up to 3 pages excluding references, and max 4 pages including references. Abstracts should report on original, unpublished work. Submissions should be done via the EasyChair conference system here: https://easychair.org/conferences/?conf=tbillc2023 All accepted abstracts will be compiled into an informal proceedings volume which will be made available electronically in advance of the symposium. PUBLICATION INFORMATION After the symposium, authors of accepted abstracts will be invited to submit a full-length paper to the post-proceedings of the symposium, which will be published in the LNCS series of Springer. The full-length submissions will undergo a new single-blind peer review process. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear David, Yes, and it's an excellent paper with a witty title for which only "pointless" would do. I particularly like what Peter said when explaining the significant difference in the absence of choice (such as in toposes of sheaves), and that "usually it is locales, not spaces, which provide the right context in which to do topology". He went on to say, "This is the point which ... Andre Joyal began to hammer home in the early 1970s; I can well remember how, at the time, his insistence that locales were the real stuff of topology, and spaces were merely figments of the classical mathematician's imagination, seemed (to me, and I suspect to others) like unmotivated fanaticism. I have learned better since then." This is all part of the argument for using a reformed topology, but there is nothing particular there about the pointwise style of reasoning for it. Hence we are still left with the question of how to reference the two concepts, the reformed topology and the reasoning without points. Would you call Ng's paper with me pointless? Points are everywhere in it. (Of course, there's the separate issue of whether it was pointless in the sense of not worth the trouble. But an important feature of the style is that it forces you to be careful to distinguish between Dedekind reals and 1-sided (lower or upper) reals, and in Ng's thesis this uncovered unexpected roles of 1-sided reals in the account of Ostrowski's Theorem and the Berkovich spectrum. So there is a bit of payoff.) Best wishes, Steve. ________________________________ From: David Yetter <dyetter@ksu.edu> Sent: Friday, January 20, 2023 3:06 AM To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl>; Steven Vickers (Computer Science) <s.j.vickers.1@bham.ac.uk> Cc: categories list <categories@mta.ca> Subject: Re: categories: Re: Terminology for point-free topology? I seem to recall from back in my days as a grad student or new PhD that Peter Johnstone wrote a paper entitled "The Point of Pointless Topology". Just in honor of that I've always favored "pointless topology" as the term for the theory of locales and sheaves on locales. Best Thoughts, David Y. ________________________________ From: Steven Vickers <s.j.vickers.1@bham.ac.uk> Sent: Wednesday, January 18, 2023 6:12 AM To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl> Cc: categories list <categories@mta.ca> Subject: categories: Re: Terminology for point-free topology? This email originated from outside of K-State. Dear Ieke, Thanks for mentioning that. It's a beautiful paper, both in its results and in its presentation, and one I still return to. Another place where I think you were even more explicit was in "The classifying topos of a continuous groupoid I" (1988), where you said - "... in presenting many arguments concerning generalized, "pointless" spaces, I have tried to convey the idea that by using change-of-base-techniques and exploiting the internal logic of a Grothendieck topos, point-set arguments are perfectly suitable for dealing with pointless spaces (at least as long as one stays within the 'stable' part of the theory)." (Would you still say that "pointless" and "point-set" are the right phrases there? I'm proposing "point-free" and "pointwise".) On the other hand, in your book with Mac Lane, those ideas seemed to go into hiding. In fact I explicitly wrote "Locales and toposes as spaces" as a guide to reading the points back into the book. My first understanding of these pointwise techniques came in the 1990's, as I developed the exposition of "Topical categories of domains". That was before I knew those papers of yours, but I felt right from the start that I was merely unveiling techniques already known to the experts - though I hope you'll agree I've been more explicit about them and particularly the nature and role of geometricity. I still don't know as much as I would like about the origin and history of those techniques. It would certainly improve my arXiv notes if I could say more. Might they even have roots in Grothendieck? I once saw a comment by Colin McLarty to the effect that (modulo misrepresentation by me) Grothendieck was aware of two different lines of reasoning with toposes: by manipulating sites concretely, or by using colimits and finite limits under the rules corresponding to Giraud's theorem. I imagine that as being something like the distinction between pointless and pointwise. Best wishes, Steve. ________________________________ Hi Steve, A very early illustration of the strategy of using points in pointless topology is in my paper with Wraith (published 1986). I just looked at it again, and the strategy is explicitly stated in the introduction : "the strategy is to use adequate extensions of the base topos available from general topos theory, which enable one to follow classical arguments about points of separable metric spaces rather closely. Although both approaches are equivalent, we will follow the second one, because it shows more clearly the interplay between general topos theory and arguments (somewhat similar to those) from topology" We used it to prove an actual theorem. Of course I used this strategy much more often, e.g. in my two 1990 papers with Joyal. Ieke [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I seem to recall from back in my days as a grad student or new PhD that Peter Johnstone wrote a paper entitled "The Point of Pointless Topology". Just in honor of that I've always favored "pointless topology" as the term for the theory of locales and sheaves on locales. Best Thoughts, David Y. ________________________________ From: Steven Vickers <s.j.vickers.1@bham.ac.uk> Sent: Wednesday, January 18, 2023 6:12 AM To: I.Moerdijk@uu.nl <I.Moerdijk@uu.nl> Cc: categories list <categories@mta.ca> Subject: categories: Re: Terminology for point-free topology? This email originated from outside of K-State. Dear Ieke, Thanks for mentioning that. It's a beautiful paper, both in its results and in its presentation, and one I still return to. Another place where I think you were even more explicit was in "The classifying topos of a continuous groupoid I" (1988), where you said - "... in presenting many arguments concerning generalized, "pointless" spaces, I have tried to convey the idea that by using change-of-base-techniques and exploiting the internal logic of a Grothendieck topos, point-set arguments are perfectly suitable for dealing with pointless spaces (at least as long as one stays within the 'stable' part of the theory)." (Would you still say that "pointless" and "point-set" are the right phrases there? I'm proposing "point-free" and "pointwise".) On the other hand, in your book with Mac Lane, those ideas seemed to go into hiding. In fact I explicitly wrote "Locales and toposes as spaces" as a guide to reading the points back into the book. My first understanding of these pointwise techniques came in the 1990's, as I developed the exposition of "Topical categories of domains". That was before I knew those papers of yours, but I felt right from the start that I was merely unveiling techniques already known to the experts - though I hope you'll agree I've been more explicit about them and particularly the nature and role of geometricity. I still don't know as much as I would like about the origin and history of those techniques. It would certainly improve my arXiv notes if I could say more. Might they even have roots in Grothendieck? I once saw a comment by Colin McLarty to the effect that (modulo misrepresentation by me) Grothendieck was aware of two different lines of reasoning with toposes: by manipulating sites concretely, or by using colimits and finite limits under the rules corresponding to Giraud's theorem. I imagine that as being something like the distinction between pointless and pointwise. Best wishes, Steve. ________________________________ Hi Steve, A very early illustration of the strategy of using points in pointless topology is in my paper with Wraith (published 1986). I just looked at it again, and the strategy is explicitly stated in the introduction : "the strategy is to use adequate extensions of the base topos available from general topos theory, which enable one to follow classical arguments about points of separable metric spaces rather closely. Although both approaches are equivalent, we will follow the second one, because it shows more clearly the interplay between general topos theory and arguments (somewhat similar to those) from topology" We used it to prove an actual theorem. Of course I used this strategy much more often, e.g. in my two 1990 papers with Joyal. Ieke [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

*** 2nd International Conference on Homotopy Type Theory (HoTT 2023) *** Monday 22nd May - Thursday 25th May, 2023 *** Carnegie Mellon University, Pittsburgh (USA) This is the first call for papers for the 2nd International Conference on Homotopy Type Theory (HoTT 2023). Abstracts (no more than 2 pages in A4 format) should be submitted via Easychair, using the following link. https://easychair.org/conferences/?conf=hott2023 <https://easychair.org/conferences/?conf=hott2023> Submissions open on 3rd February 2023 and close on 3rd March 2023. Invited speakers of the conference are: Julie Bergner (University of Virginia, USA) Thierry Coquand (Chalmers University, Sweden) András Kovács (Eötvös Loránd University, Hungary) Anders Mörtberg (Stockholm University) Further information on the conference can be found at the website: https://hott.github.io/HoTT-2023/ <https://hott.github.io/HoTT-2023/> With best regards, Steve Awodey (Chair of the organising committee) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear all, Applications for this position are now live: https://mq.wd3.myworkdayjobs.com/en-US/CareersatMQ/details/Lecturer-in-Math= ematics_R000010406-1 Applications close 5th March. As well as our head of school, you should also feel free to contact me with enquiries of a formal or informal nature. (I am part of the hiring panel). All the best, Richard Richard Garner <richard.garner@mq.edu.au> writes: > Dear all, > > This is to give you advance warning that in the new year we will be > advertising a permanent position here at Macquarie University in Sydney. > > While the position will be open to any field of mathematics, priority > will be given to excellent candidates working in category theory. As > many of you are probably aware, the group here is pretty active: we have > Steve Lack and myself as continuing academics, Ross Street, Dominic > Verity and Mike Johnson as emeritus professors, and John Power as an > honorary professor; plus a community of 11 postgraduate students and a > couple of postdocs. > > This is a Lecturer position, analogous to a UK lecturer or a US > assistant professor, with a base salary of AU$110,000-$130,000, and is a > balanced academic role with 40% research, 40% teaching and 20% service. > We expect the formal advertisement to be live at the start of February > with a closing date early in March. This rather short period is forced > upon us by federal government rules, so it seems best to get word out > now. > > Please feel free to distribute this message further, and don't hesitate > to get in touch if you have any questions about anything to do with > this. > > Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]