You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Posina Venkata Rayudu (Jul 07 2023 at 15:49):Dear All,
I just thought of sharing with you my gratitude towards Professor F. William Lawvere and his inspirational seriousness of purpose:
https://conceptualmathematics.substack.com/p/professor-f-william-lawvere
Thanking you,
Yours truly,
posina
Notification Bot (Jul 07 2023 at 15:54):Posina Venkata Rayudu has marked this topic as resolved.
Notification Bot (Jul 09 2023 at 05:02):Posina Venkata Rayudu has marked this topic as unresolved.
Posina Venkata Rayudu (Jul 09 2023 at 05:02):History and Philosophy of Mathematic Reform
Posina Venkata Rayudu (Jul 09 2023 at 05:05):Posina Venkata Rayudu said:
Being the main contradictor that Professor F. William Lawvere is (according to Grothendieck ;) I am sure he would be happy to hear you contradicting his understanding of the history of mathematics education (as usual, it's just what I think ;)
Jencel Panic (Jul 12 2023 at 23:11):Conceptual Mathematics is one of the greatest books ever!
Josh Lalonde (Aug 01 2023 at 13:00):Hi everyone, apologies if this has been posted before but I was wondering if anyone has any links or references to biographical materials about F. William Lawvere. I'm particularly interested in his political engagement, but anything about his life would be helpful. Thanks in advance!
Notification Bot (Aug 01 2023 at 13:02):A message was moved here from #general > Lawvere bio materials? by Morgan Rogers (he/him).
Matt Earnshaw (Aug 01 2023 at 13:44):Here's a newspaper article covering his ousting from Dalhousie 19710122-DalhousieGazetteonWLawvere.pdf
Matt Earnshaw (Aug 01 2023 at 13:55):Unrelated, but just today I have scanned Introduction to Linear Categories and Applications which may be of interest to the pedagogues and philatelists out there. These are seemingly lecture notes for a course that would usually be called Linear Algebra. Some of the material later appeared in the books.
John Baez (Aug 01 2023 at 14:17):Philatelists are stamp collectors. Is this book of interest to them?
Matt Earnshaw (Aug 01 2023 at 14:22):Figuratively speaking.
Todd Trimble (Aug 01 2023 at 14:31):Matt Earnshaw said:
Figuratively speaking.
I still don't get it.
John Baez (Aug 01 2023 at 14:40):Maybe "bibliophiles"?
Matt Earnshaw (Aug 01 2023 at 14:48):Some years ago I received some lesser spotted Lawvere documents in an email entitled "among stamp collectors". I wasn't confused by the title. The psychological structures behind any form of collecting are surely at work in building this archive.
John Baez (Aug 01 2023 at 15:15):Okay, got it.
Kyle Wilkinson (Aug 01 2023 at 15:55):Josh Lalonde said:
Hi everyone, apologies if this has been posted before but I was wondering if anyone has any links or references to biographical materials about F. William Lawvere. I'm particularly interested in his political engagement, but anything about his life would be helpful. Thanks in advance!
I do recall an account in a book I have. Unfortunately, between attending this conference and drafting a prospectus, it'll be some time before I can track it down. But if I recall correctly, it's an account of getting him hired at Dalhousie. I speculate there's a connection with his eventual development of functorial semantics, too. I wish there were a "remind me" feature here.
Todd Trimble (Aug 01 2023 at 16:36):Matt Earnshaw said:
Some years ago I received some lesser spotted Lawvere documents in an email entitled "among stamp collectors". I wasn't confused by the title. The psychological structures behind any form of collecting are surely at work in building this archive.
To a degree, perhaps, but collectors may have very different motivations. For many, it's the pleasure of hunting down items because they are rare and for no other reason. Or because they are curiosities or just plain weird. Or because they belong in any well-rounded collection. Or because they have a particular special meaning or value for the individual.
Remember: "no question is dumb".
Jason Erbele (Aug 01 2023 at 16:50):Kyle Wilkinson said:
I wish there were a "remind me" feature here.
There is a "Starred messages" feature that could potentially be used for reminders.
Josh Lalonde (Aug 01 2023 at 17:26):Matt Earnshaw said:
Here's a newspaper article covering his ousting from Dalhousie 19710122-DalhousieGazetteonWLawvere.pdf
Thanks this is great!
Josh Lalonde (Aug 01 2023 at 17:27):Kyle Wilkinson said:
Josh Lalonde said:
Hi everyone, apologies if this has been posted before but I was wondering if anyone has any links or references to biographical materials about F. William Lawvere. I'm particularly interested in his political engagement, but anything about his life would be helpful. Thanks in advance!
I do recall an account in a book I have. Unfortunately, between attending this conference and drafting a prospectus, it'll be some time before I can track it down. But if I recall correctly, it's an account of getting him hired at Dalhousie. I speculate there's a connection with his eventual development of functorial semantics, too. I wish there were a "remind me" feature here.
No rush, but if you do find the book at some point and remember my question, please post it here.
Kyle Wilkinson (Aug 08 2023 at 21:50):@Josh Lalonde
So, I decided I needed a break and found the source I was thinking of. Let me start by saying I had my dates a little mixed up in my head. This was after Lawvere's time at Dalhousie. Also, I for some reason thought his Functorial Semantics was later work, when it was actually part of his dissertation. I think it was published later. In any case, this account is immaterial to that work, though both people were alike in spirit with respect to work on categories and syntax/semantics.
The other person was a biologist I know some here are familiar with given the relationship he had with category theory: Robert Rosen. If you haven't heard of him, I recommend at least doing a quick search. As far as I am able to discern from literature, he was the first to publish works on applications of category theory outside of mathematics itself. His first such seems to have been in 1958, so pretty early. He also studied the topic under both MacLane and Eilenberg, though transferred out of math to earn his doctorate in biology instead.
Ultimately, he would develop a model of life (to be distinguished in kind from non-life) using some basic category theory, in addition to lots of writing on abstract "modeling relations" (formal syntax-semantics maps), measurement, and feedforward systems which he called "anticipatory". There have been mixed reviews on his use of math, since he did not always write it in a rigorous manner, especially when using it to develop his more philosophical ideas. The really interesting part to me IS the philosophy of science work he did, though. In many ways, he (in spirit) anticipated some of the work I see happening recently in this community, especially as relates to questions of compositionality and emergent/generative features, though he used different terms. It's exciting to see math really making formal headway into some of these old questions.
Anyway, this does relate back to Lawvere! It is possible they knew of each other due to both having been at Columbia with Eilenberg (at different times, though. Rosen left in 1957 and Lawvere arrived in 1960). Rosen would start at SUNY-Buffalo and move to Dalhousie in 1975. Lawvere, of course, was at Dalhousie first and in 1974 moved to SUNY-Buffalo. But they did in fact know each other when Rosen by his account went to bat for the "eminent category theorist" whom he helped get hired at SUNY, just before he left. He doesn't name Lawvere, but I can't imagine any other conclusion.
So, I don't know if that is helpful to you. I just happen to have been interested in Rosen's work, and recalled that passage. I guess I recalled it being more relevant than it turned out to be once I found the source again. The source, by the way, is "Old Trends and New Trends in General System Research," a memorial lecture he gave for von Bertalanffy. https://doi.org/10.1080/03081077908960904
Here is the excerpt, which follows a brief historical account of the development of category theory. One of his points in the talk was that category theory is a natural language for formally describing Bertalanffy's general system theory (for instance like Mesarovic has tried to do):
"It is interesting to note that Category Theory has elicited exactly the same reaction within the community of pure mathematicians that system-theoretic approaches have generated in the community of experimental or empirical scientists. Initially, category theory was dismissed out of hand by many mathematicians as at best a cumbersome language for stating well-known facts about algebraic topology. Later, when it became clear that category theory could answer important open questions in a variety of areas, it was conceded to be of some limited usefulness for specific applications, but still of no significance as an independent branch of mathematics. This attitude is still prevalent; indeed, just a few years ago, I served on the Executive Committee of the Department of Mathematics at SUNY Buffalo, when that department was in the process of hiring an eminent category theorist. A substantial group of senior faculty tried to block this appointment on the grounds that category theory was not real mathematics; they were dissuaded only with difficulty from writing to other departments to collect opinions on whether category theory was mathematics or not."
At the very least, it is interesting to piece together these random connections from the past! Also, there is much to be gleaned by digging through the old texts, finding forgotten threads of research.
Josh Lalonde (Aug 11 2023 at 21:03):@Kyle Wilkinson thanks, this is very interesting. Rosen is someone I've had on my radar for a while but never looked into in any detail.
Posina Venkata Rayudu (Aug 22 2023 at 11:51):Speaking of Rosen, I thought you might find my corrections (of some related misunderstandings published in Neuron) grandiously entitled:
On making sense of science
https://disqus.com/home/discussion/cell-press/dialogue_across_chasm_are_psychology_and_neurophysiology_incompatible_neuron/
of some interest. I look forward to your critique (unvarnished ;)
Posina Venkata Rayudu (Aug 22 2023 at 12:18):Josh Lalonde said:
Hi everyone, apologies if this has been posted before but I was wondering if anyone has any links or references to biographical materials about F. William Lawvere. I'm particularly interested in his political engagement, but anything about his life would be helpful. Thanks in advance!
I just thought you might find:
F. William Lawvere (1937–2023): A lifelong struggle for the unity of mathematics
https://euromathsoc.org/magazine/articles/143
of some interest.
The above slide (due to Professor Peter Johnstone, unless I'm mistaken) is within couple of days of the publication of Professor F. William Lawvere's 2007 Axiomatic Cohesion (http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf), which can be traced to his 1991 Future of Category Theory (cf. categories of Being and Becoming; https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1991-some-thoughts-on-the-future-of-category-theory.pdf). I can't readily recollect any earlier discussion of Being/unity/cohesion vs. Becoming/change/variation. Please correct me if I'm mistaken (which is more often the case than not ;) Thank you!
Josh Lalonde (Aug 30 2023 at 13:00):Posina Venkata Rayudu said:
Josh Lalonde said:
Hi everyone, apologies if this has been posted before but I was wondering if anyone has any links or references to biographical materials about F. William Lawvere. I'm particularly interested in his political engagement, but anything about his life would be helpful. Thanks in advance!
I just thought you might find:
F. William Lawvere (1937–2023): A lifelong struggle for the unity of mathematics
https://euromathsoc.org/magazine/articles/143of some interest.
The above slide (due to Professor Peter Johnstone, unless I'm mistaken) is within couple of days of the publication of Professor F. William Lawvere's 2007 Axiomatic Cohesion (http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf), which can be traced to his 1991 Future of Category Theory (cf. categories of Being and Becoming; https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1991-some-thoughts-on-the-future-of-category-theory.pdf). I can't readily recollect any earlier discussion of Being/unity/cohesion vs. Becoming/change/variation. Please correct me if I'm mistaken (which is more often the case than not ;) Thank you!
Thank you very much, I'll check these out.
Tobias Fritz (Jul 23 2025 at 02:23):I'm happy to announce that the Lawvere Archives now feature several new posthumous publications by Bill:
I have only been involved in preparing the first; the other three have been prepared by Francisco Marmolejo.
fosco (Jul 23 2025 at 05:39):What beautiful items to add to one's reading list!
David Michael Roberts (Jul 23 2025 at 06:19):It is unusually brief because it is a fragment of an introduction to an Appendix of a much larger document that was classified as SECRET. The origin of this document is closely intertwined with historical events that occurred in the mid-1960s.
:eyes:
fosco (Jul 23 2025 at 06:40):David Michael Roberts said:
It is unusually brief because it is a fragment of an introduction to an Appendix of a much larger document that was classified as SECRET. The origin of this document is closely intertwined with historical events that occurred in the mid-1960s.
:eyes:
the one conspiracy I will believe in
Nathanael Arkor (Jul 23 2025 at 07:31):that was classified as SECRET
By whom?
Jonas Frey (Jul 23 2025 at 07:34):@Matt Earnshaw maybe these could be added to the github repository you maintain?
Jonas Frey (Jul 23 2025 at 07:54):David Michael Roberts said:
It is unusually brief because it is a fragment of an introduction to an Appendix of a much larger document that was classified as SECRET. The origin of this document is closely intertwined with historical events that occurred in the mid-1960s.
:eyes:
The original draft of the Port Huron Statement!
Jonas Frey (Jul 23 2025 at 07:57):But seriously, I think it's explained further down in the introduction of the document, it's about a think tank job related to arms control. Sounds mysterious!
John Baez (Jul 23 2025 at 08:27):David Michael Roberts said:
It is unusually brief because it is a fragment of an introduction to an Appendix of a much larger document that was classified as SECRET. The origin of this document is closely intertwined with historical events that occurred in the mid-1960s.
Which of the papers says that?
Tobias Fritz (Jul 23 2025 at 08:40):It's from the Author Commentary to the 1962 category of probabilistic mappings paper. Bill gives explain more of the story on p.5-7 that sounds like part of a movie plot, and indicates that he was a also avery early pioneer of ACT, in a way that is reminiscent of how certain political actors in the US have been involved with ACT funding over the past ~10 years. Let me quote:
My acceptance of the job offered by the “Think Tank” in Southern California depended on an agreement that the main topic treated would be Kennedy’s Arms Control and Disarmament Agency. The preliminary interview in the Pentagon was requested by that Agency. Somewhat more precisely, the aim of the study would be: planning for the technical support of an Arms Control Treaty between the Superpowers, for example, of a reliable verification protocol to be agreed upon.
It was envisaged that a protocol would involve three tiers of verification: Space, Stratosphere, and On-site. The passage from one tier to the next would follow probabilistically from continuing observations. What would be the mathematical framework under which this whole fantasy would function? Someone described it as a “network of probabilistic mappings”. “What would that mean?” I asked myself: “It must involve diagrams in a category extending the monoid of Markov processes”, and then I produced the present document, which served as an Appendix to an Appendix of a large SECRET document.
The proposal was to study a projected system of verification and inspection for a possible Arms Control Agreement between the Superpowers. The system would be organized into the three levels: satellite surveillance, which could trigger the request for over-flight inspection, that in turn could trigger an on-site inspection. Of course, the trigger thresholds would be a matter of diplomacy, but the system as a whole would involve an elaborate network of “probabilistic mappings”.
The whole thing had to be scrutinized by the Pentagon before the Arms Control Agency could do anything. Probably, passing through so many hands increased its exposure to espionage. The leader of the group within the Think Tank stated that an important calculation to be done by the study would be the determination of the probability of the discovery of missiles concealed on the ocean floor as part of a planned circumvention of any treaty. That was also the year of the Cuban missile crisis
A few years later I came across a Russian document containing several of the results of my unpublished thesis, including the mistakes, (as well as the missing two lines that we later discovered had been missed by the typist). But there was no attribution. And in Moscow the lectures were beginning on a very similar category, called the Markov category (not without justification, of course, although I don’t believe Markov himself used categories).
I was surprised a couple of years later by being offered a job with French military intelligence. The one who transmitted that offer was a collaborator of M. Giry, which may explain why she knew about the “secret” developments in the US.
Apparently regarding the contact with the Arms Control Agency as dormant, the leaders of the Think Tank had a further proposal, disregarding their initial agreement with me: First, I should study books by Mao Tsetung and Che Guevara as a preparation for evaluating a large system designed to eliminate the guerilla threat in Vietnam. My last paychecks were for studying that proposal. Of course, I advised against it, after having verified mathematically that the proposed system was unfeasible. The last time I saw the director of the “Vietnam Proposal” was at the old Waltham Watch Factory, which had been taken over as a subsidiary of the California Think Tank. Naturally, my report met with utter disapproval. I took a bus from Waltham to NYC in order to defend my thesis at Hamilton Hall, in front of Eilenberg, Kadison, and Morgenbesser. Now I could complete my application for a teaching job at Reed College.
A few years later the New York Times reported on the failure of a large system that differed only in detail from the one I had analyzed. The supporters of the proposal had taken the plan and defected to another Think Tank.
Matt Earnshaw (Jul 23 2025 at 09:01):Jonas Frey said:
Matt Earnshaw maybe these could be added to the github repository you maintain?
In some sense the new archive (builds on and) supersedes the repository, but Danilo has encouraged mirroring so I am accruing the new stuff, to incorporate it eventually
Jonas Frey (Jul 23 2025 at 09:17):John Baez said:
Which of the papers says that?
This one. It's explained further down in the introduction.
Paolo Perrone (Jul 23 2025 at 09:20):This is wonderful.
By the way, has anyone seen the terminology "Markov category" used in the USSR (or elsewhere) in the '60s-'70s? If it's written or recorded anywhere, I would be extremely interested.
(Not to be confused with the related, but much more modern, Markov categories.)
David Corfield (Jul 23 2025 at 10:35):It was after 1962 that Godement’s notion of standard construction became developed by Kleisli, Huber, Eilenberg & Moore, and Beck, into the theory of algebras for a Monad. Once that theory is made explicit, an extremely compact description of the basic construction can be given, namely probabilistic mappings are just the morphisms in the Kleisli category of the probability monad. In fact, there are several reasons for considering instead the larger Eilenberg-Moore category of the same monad, because it is a symmetric monoidal closed category whose unit object is terminal; that permits numerous constructions involved in inference, et cetera to be expressed explicitly in terms of Kan extensions.
Is this under consideration today as a reason to look at the EM category? Last time I asked, @Paolo Perrone mentioned expectation values as his reason.
Tobias Fritz (Jul 23 2025 at 11:15):Concerning EM vs Kleisli for probability monads, there's also a MathOverflow discussion on this question: Why Kleisli Markov categories and not the Eilenberg-Moore categories of the associated monads
David Corfield (Jul 23 2025 at 13:31):Thanks. I was interested in what Lawvere's pointing to with
permits numerous constructions involved in inference, et cetera to be expressed explicitly in terms of Kan extensions
There's plenty of interest in relating machine learning algorithms to Kan extensions, such as Learning is a Kan Extension and Kan Extensions in Data Science and Machine Learning. But I don't think they invoke probabilities. Does anyone know what Lawvere means?
Tobias Fritz (Jul 23 2025 at 14:29):I believe that this allusion to Kan extensions refers to the ideas outlined in the long paragraph starting at the bottom of p.4, where he describes decision making and inference as dual problems that look like enriched Kan extensions.
Paolo Perrone (Jul 23 2025 at 14:35):It could be, but this is my very personal opinion, that Lawvere suspected that the analogy between Kan extensions and conditional expectations could be made precise in terms of enrichment.
Or maybe he even managed to establish the connection.
Paolo Perrone (Jul 23 2025 at 14:38):(This is one more instance of the problem I've mentioned at CT, where probability seems to have 2-cells, but they also seem impossible to define. I wonder if Lawvere managed to solve this. Maybe it's in the secret part of the document. But once again, this is just my personal, probably very biased, interpretation.)
John Baez (Jul 23 2025 at 14:43):The secret stuff would be connected to nuclear war and Vietnam, not 2-cells.
Paolo Perrone (Jul 23 2025 at 14:48):Yes, true.
David Corfield (Jul 23 2025 at 15:36):Tobias Fritz said:
I believe that this allusion to Kan extensions refers to the ideas outlined in the long paragraph starting at the bottom of p.4, where he describes decision making and inference as dual problems that look like enriched Kan extensions.
Thanks. I guess we can see this as providing an example to the passage on p. 2
That Hom is almost never free is one of the important reasons why it is necessary to consider the whole category of P-algebras (= “convex sets”), not just the Kleisli category; the vast machinery of enriched categories [2] can then be applied to construct functor categories, Kan extensions, etc. in order to analyze, design, and construct natural stochastic processes and decision procedures of all sorts.
David Corfield (Jul 23 2025 at 15:43):The p.4 paragraph is very closely related to my association of inference with triangle-filling here. E.g.,
the following is a kind of abduction, hence his use of "causes":
An opposite sort of triangle results if two spaces are assumed to be equipped with given morphisms to (rather than from) a third space Y that is observable (rather than hidden) while consists of (names for) “causes” via . The sought-for then postdicts causes at least for the part of parameterized by .
Tobias Fritz (Jul 24 2025 at 00:58):David Corfield said:
The p.4 paragraph is very closely related to my association of inference with triangle-filling [here]
Interesting! That actually makes the distinction between deduction, induction and abduction nicely tangible to a category theorist. So it looks like induction is formally analogous to Bill's decision-making, while abduction to Bill's causal inference. Do you know if there's a way to make "decision-making = induction" and "causal inference = abduction" make intuitive sense?
David Corfield (Jul 24 2025 at 08:40):Well, "causal inference = abduction" is quite straightforward. Abduction is often described as providing an explanatory hypothesis for a phenomenon. (It is sometimes glossed as "inference to the best explanation", though there are reasons to think that's not quite what Peirce meant.) A good chunk of what's involved in explaining a phenomenon is giving a causal narrative that results in it. (Naturally, there's vast amounts of philosophical writing on this, including discussions on the nature of causality.)
I like thinking of an abductive lift in the literal terms of lifting one's gaze from a shadow on the ground to some object in the air casting the shadow. Why is there a dark circular patch on the ground? It's a shadow caused by that balloon blocking the sunlight.
In the notes, I ponder whether the story of these modes of inference makes better sense in certain settings. Certainly a category like seems better suited than one like to provide a proper directionality. I mused in those notes as follows:
But, of course, you yourself have provided a notion of causality in a Markov category. I should take a look.
Paolo Perrone (Jul 24 2025 at 08:51):Causality in a Markov category is not exactly the same as in the traditional sense, I believe.
At least the way I interpret it, it's along the lines of "if two quantities are almost surely equal, then they are still almost surely equal after we gather more knowledge".
(For examples and explanations, see this paper, section 2.4.)
David Corfield (Jul 24 2025 at 09:25):Thanks. I see from here there's another feature deemed 'causal':
David Corfield (Jul 24 2025 at 10:26):Tobias Fritz said:
Do you know if there's a way to make "decision-making = induction" and "causal inference = abduction" make intuitive sense?
Taking up now the first equality, there's something a little odd. Here is Lawvere:
Suppose is a space of parameters presumed to characterize a system of interest but not directly measurable, and suppose a morphism specifies what a correct decision would be if we knew the true value of . Suppose there is a morphism describing an experiment whose outcome can in fact be measured, thus the problem is to determine a morphism that makes a decision based on the reading of the experimental outcome. Yet even if both given morphisms are deterministic, there may be no such strictly commutative diagrams, so that a statistically best solution may be sought (and usually exists), namely, a point of the convex set whose experimental transform into is as near as possible to the given correct decision.
In my way of thinking about induction, a paradigmatic case is where the map is injective, so providing a sample from a population. We know the classification results for the sample, i.e., we have the map . Then the task is to extend this map to so that we then can classify the whole population.
This is not so close to Lawvere's description. However, I point out that sometimes we achieve this extension by constructing a retraction from the population back to the sample. This is the case with the nearest neighbor algorithm. We know how a certain number of points are classified, so to decide how to classify a fresh point, look to see the result on the nearest member of the sample. Then things look closer to what Lawvere is describing. We want to know how to classify points in . We can't access itself, but can project it to , where it's possible to find a decision function (or one that's as good as possible). So this is like choosing an extension (or one as near as possible) but along an epi rather than a mono.
Tobias Fritz (Jul 24 2025 at 12:54):Thanks very much @David Corfield, that is quite intriguing!
Tobias Fritz (Jul 24 2025 at 12:58):I don't think that abductive reasoning is directly related to a causality axiom, but more to causal inference in the sense of Bayesian networks. For the latter, there is a full account in terms of Markov categories up to the d-separation criterion in The d-separation criterion in Categorical Probability. What's needed for that is the existence of conditionals in the Markov cat under consideration, which implies that it satisfies the causality axiom.
Notification Bot (Jul 28 2025 at 12:42):Jonas Frey has marked this topic as resolved.
Notification Bot (Jul 28 2025 at 12:58):Jonas Frey has marked this topic as unresolved.
John Baez (Jul 29 2025 at 12:06):Check out Gavin Wraith's comments on how Lawvere's work for RAND radicalized him.