Category Theory under threat in Cambridge
As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid! I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year. I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so. Peter Johnstone
Hi Peter, I'm sad to hear of the decline of category theory in the Maths Faculty. All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason? To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory. All the best, Steve. ________________________________ From: P.T. Johnstone via Categories <categories-list@categories.org.au> Sent: Friday, May 29, 2026 10:11 AM To: Marino Gran via Categories <categories-list@categories.org.au> Cc: P.T. Johnstone <ptj1000@cam.ac.uk> Subject: [categories] Category Theory under threat in Cambridge CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe. As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid! I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year. I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so. Peter Johnstone
Why do we have this development? Decades ago the famous categorists were still young or at least younger and still working. Today we have a new generation of categorists teaching category theory and sharing memories of times when now the oldies were younger, and when everybody felt self-confident and proud about being part of the CT community. Young students now, unfortunately for CT, have more options to feel self-confident and proud. Category theory is useful, always was. Bernays and Gödel never saw that, but never mind (until the PS). CT is useful in math. CT is useful in computer science. CT is useful in real applications, but that was never acknowledged. Are we ourselves to blaim for not having transformed CT to something that lives and survives through times. Has it been too introvert? Just wondering. --- Is there a silverlining? What can we do? CT in CS has often been seen as a good example where CT comes into play. And it does, still today. But I cannot see it has improved, so CT in CS may decline similarly in a decade or two. I may be wrong, I hope I'm wrong, but I say so since I was recently at the Types 2026 conference in Gothenburg. I haven't been at those for a while and now I wanted to see if there is something new in CT in CS. I didn't see anything, I should to say, but thanks to a plenary (by Emily Riehl) there was at least a plea for CT (teaching) to remain in CS (teaching). Of course, there are CS teachers who can teach CT, but many of them are also not interested in real applications at all. Very few if any can explain why we need CT. --- "Real applications", what's that? I was nominated to professor in 1995, and from start I recognized my three tasks, i.e., tasks we recognize as our three tasks in universities. One, we teach. Two we do research. And it is in that order! Three, we cooperate with the surrounding society. We are obliged to do one and two, and CT teachers often do not have the luxury and time of doing two. Not even CT professors. Take CT people and take math people in general. Is there a difference between how they at average engage in the third task (ignoring the situation about how much time they have for such engagements)? Am I right or am I right? If I'm right, what happens before a meeting between a categorist professor and, say, a medical professor. Will the medical professor be curious to know something about CT "before" the meeting? No. Will the CT professor be curious to know something about the medical domain to be discussed "before" the meeting? Should be, but in most cases will not. So what happens after the meeting? The medical professor go backs to office and continues research in, say, cardiology. The CT professor goes back to office and continues to remember the good old days when we didn't even think about talking to professors in other disciplines. Am I wrong or am I wrong? If I'm wrong, then there is hope, otherwise not. And whenever we might go "real", do remember that we must "FIRST find the problem, THEN design a solution", not the other way around ("FIRST design a solution, THEN find a problem where the solution comes into play"). I rest my case. --- Best, Patrik Eklund PS CT became CT some less than a century ago. Suppose CT had become CT some decades before, say, even before Hilbert wrote his foundations of geometry. In that case, CT had been around by the time Bernays was recruited to Göttingen, and I bet Gödel's ideas would never have reached publication in 1931. Hilbert's and Bernays' two volumes on the foundation of mathematics might even have been preceded by another two volumes on the foundation of the foundation of mathematics. Who knows. I bet Mac Lane never thought about it that way. Otherwise he would have called Bernays and said "Hey, we really need to talk!". But I don't see that in Bernays correspondence, so here we are, 75 years later, still not talking to each other. On 2026-05-31 16:29, Steven Vickers via Categories wrote:
Hi Peter,
I'm sad to hear of the decline of category theory in the Maths Faculty.
All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason?
To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory.
All the best,
Steve.
-------------------------
From: P.T. Johnstone via Categories <categories-list@categories.org.au> Sent: Friday, May 29, 2026 10:11 AM To: Marino Gran via Categories <categories-list@categories.org.au> Cc: P.T. Johnstone <ptj1000@cam.ac.uk> Subject: [categories] Category Theory under threat in Cambridge
CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.
As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid!
I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year.
I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so.
Peter Johnstone _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
This is very depressing. I've always been on the geometrical, conceptual side of mathematics (and I've never really been any good at numerical calculation). And the geometrical side of mathematics goes very far back, to ancient Greece at least. Could we not try to emphasize this? Graham White London On Mon, 1 Jun 2026, 08:20 peklund via Categories, < categories-list@categories.org.au> wrote:
Why do we have this development?
Decades ago the famous categorists were still young or at least younger and still working. Today we have a new generation of categorists teaching category theory and sharing memories of times when now the oldies were younger, and when everybody felt self-confident and proud about being part of the CT community.
Young students now, unfortunately for CT, have more options to feel self-confident and proud.
Category theory is useful, always was. Bernays and Gödel never saw that, but never mind (until the PS). CT is useful in math. CT is useful in computer science. CT is useful in real applications, but that was never acknowledged.
Are we ourselves to blaim for not having transformed CT to something that lives and survives through times. Has it been too introvert?
Just wondering.
---
Is there a silverlining? What can we do?
CT in CS has often been seen as a good example where CT comes into play. And it does, still today. But I cannot see it has improved, so CT in CS may decline similarly in a decade or two. I may be wrong, I hope I'm wrong, but I say so since I was recently at the Types 2026 conference in Gothenburg. I haven't been at those for a while and now I wanted to see if there is something new in CT in CS. I didn't see anything, I should to say, but thanks to a plenary (by Emily Riehl) there was at least a plea for CT (teaching) to remain in CS (teaching).
Of course, there are CS teachers who can teach CT, but many of them are also not interested in real applications at all. Very few if any can explain why we need CT.
---
"Real applications", what's that?
I was nominated to professor in 1995, and from start I recognized my three tasks, i.e., tasks we recognize as our three tasks in universities. One, we teach. Two we do research. And it is in that order! Three, we cooperate with the surrounding society. We are obliged to do one and two, and CT teachers often do not have the luxury and time of doing two. Not even CT professors.
Take CT people and take math people in general. Is there a difference between how they at average engage in the third task (ignoring the situation about how much time they have for such engagements)?
Am I right or am I right?
If I'm right, what happens before a meeting between a categorist professor and, say, a medical professor. Will the medical professor be curious to know something about CT "before" the meeting? No. Will the CT professor be curious to know something about the medical domain to be discussed "before" the meeting? Should be, but in most cases will not. So what happens after the meeting? The medical professor go backs to office and continues research in, say, cardiology. The CT professor goes back to office and continues to remember the good old days when we didn't even think about talking to professors in other disciplines.
Am I wrong or am I wrong?
If I'm wrong, then there is hope, otherwise not.
And whenever we might go "real", do remember that we must "FIRST find the problem, THEN design a solution", not the other way around ("FIRST design a solution, THEN find a problem where the solution comes into play").
I rest my case.
---
Best,
Patrik Eklund
PS CT became CT some less than a century ago. Suppose CT had become CT some decades before, say, even before Hilbert wrote his foundations of geometry. In that case, CT had been around by the time Bernays was recruited to Göttingen, and I bet Gödel's ideas would never have reached publication in 1931. Hilbert's and Bernays' two volumes on the foundation of mathematics might even have been preceded by another two volumes on the foundation of the foundation of mathematics. Who knows. I bet Mac Lane never thought about it that way. Otherwise he would have called Bernays and said "Hey, we really need to talk!". But I don't see that in Bernays correspondence, so here we are, 75 years later, still not talking to each other.
On 2026-05-31 16:29, Steven Vickers via Categories wrote:
Hi Peter,
I'm sad to hear of the decline of category theory in the Maths Faculty.
All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason?
To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory.
All the best,
Steve.
------------------------------ *From:* P.T. Johnstone via Categories <categories-list@categories.org.au> *Sent:* Friday, May 29, 2026 10:11 AM *To:* Marino Gran via Categories <categories-list@categories.org.au> *Cc:* P.T. Johnstone <ptj1000@cam.ac.uk> *Subject:* [categories] Category Theory under threat in Cambridge
*CAUTION:* This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.
As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid!
I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year.
I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so.
Peter Johnstone
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
Not having Peter Johnstone's lectures in category theory would be bad for Cambridge students. This is a very serious problem, and I don't understand why has this discussion moved somewhere else. George Janelidze ________________________________ From: Graham White via Categories <categories-list@categories.org.au> Sent: Monday, June 1, 2026 10:22 To: peklund via Categories <categories-list@categories.org.au> Cc: Steven Vickers <s.j.vickers.1@bham.ac.uk>; ptj1000@cam.ac.uk <ptj1000@cam.ac.uk>; peklund@cs.umu.se <peklund@cs.umu.se>; Graham White <g.graham.white@gmail.com> Subject: [categories] Re: Category Theory under threat in Cambridge CAUTION: This email originated outside the UCT network. Do not click any links or open attachments unless you know and trust the source. This is very depressing. I've always been on the geometrical, conceptual side of mathematics (and I've never really been any good at numerical calculation). And the geometrical side of mathematics goes very far back, to ancient Greece at least. Could we not try to emphasize this? Graham White London On Mon, 1 Jun 2026, 08:20 peklund via Categories, <categories-list@categories.org.au<mailto:categories-list@categories.org.au>> wrote: Why do we have this development? Decades ago the famous categorists were still young or at least younger and still working. Today we have a new generation of categorists teaching category theory and sharing memories of times when now the oldies were younger, and when everybody felt self-confident and proud about being part of the CT community. Young students now, unfortunately for CT, have more options to feel self-confident and proud. Category theory is useful, always was. Bernays and Gödel never saw that, but never mind (until the PS). CT is useful in math. CT is useful in computer science. CT is useful in real applications, but that was never acknowledged. Are we ourselves to blaim for not having transformed CT to something that lives and survives through times. Has it been too introvert? Just wondering. --- Is there a silverlining? What can we do? CT in CS has often been seen as a good example where CT comes into play. And it does, still today. But I cannot see it has improved, so CT in CS may decline similarly in a decade or two. I may be wrong, I hope I'm wrong, but I say so since I was recently at the Types 2026 conference in Gothenburg. I haven't been at those for a while and now I wanted to see if there is something new in CT in CS. I didn't see anything, I should to say, but thanks to a plenary (by Emily Riehl) there was at least a plea for CT (teaching) to remain in CS (teaching). Of course, there are CS teachers who can teach CT, but many of them are also not interested in real applications at all. Very few if any can explain why we need CT. --- "Real applications", what's that? I was nominated to professor in 1995, and from start I recognized my three tasks, i.e., tasks we recognize as our three tasks in universities. One, we teach. Two we do research. And it is in that order! Three, we cooperate with the surrounding society. We are obliged to do one and two, and CT teachers often do not have the luxury and time of doing two. Not even CT professors. Take CT people and take math people in general. Is there a difference between how they at average engage in the third task (ignoring the situation about how much time they have for such engagements)? Am I right or am I right? If I'm right, what happens before a meeting between a categorist professor and, say, a medical professor. Will the medical professor be curious to know something about CT "before" the meeting? No. Will the CT professor be curious to know something about the medical domain to be discussed "before" the meeting? Should be, but in most cases will not. So what happens after the meeting? The medical professor go backs to office and continues research in, say, cardiology. The CT professor goes back to office and continues to remember the good old days when we didn't even think about talking to professors in other disciplines. Am I wrong or am I wrong? If I'm wrong, then there is hope, otherwise not. And whenever we might go "real", do remember that we must "FIRST find the problem, THEN design a solution", not the other way around ("FIRST design a solution, THEN find a problem where the solution comes into play"). I rest my case. --- Best, Patrik Eklund PS CT became CT some less than a century ago. Suppose CT had become CT some decades before, say, even before Hilbert wrote his foundations of geometry. In that case, CT had been around by the time Bernays was recruited to Göttingen, and I bet Gödel's ideas would never have reached publication in 1931. Hilbert's and Bernays' two volumes on the foundation of mathematics might even have been preceded by another two volumes on the foundation of the foundation of mathematics. Who knows. I bet Mac Lane never thought about it that way. Otherwise he would have called Bernays and said "Hey, we really need to talk!". But I don't see that in Bernays correspondence, so here we are, 75 years later, still not talking to each other. On 2026-05-31 16:29, Steven Vickers via Categories wrote: Hi Peter, I'm sad to hear of the decline of category theory in the Maths Faculty. All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason? To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory. All the best, Steve. ________________________________ From: P.T. Johnstone via Categories <categories-list@categories.org.au<mailto:categories-list@categories.org.au>> Sent: Friday, May 29, 2026 10:11 AM To: Marino Gran via Categories <categories-list@categories.org.au<mailto:categories-list@categories.org.au>> Cc: P.T. Johnstone <ptj1000@cam.ac.uk<mailto:ptj1000@cam.ac.uk>> Subject: [categories] Category Theory under threat in Cambridge CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe. As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid! I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year. I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk<mailto:is200@cam.ac.uk>) and tell him so. Peter Johnstone _______________________________________________ Categories mailing list -- categories-list@categories.org.au<mailto:categories-list@categories.org.au> To unsubscribe send an email to categories-list-leave@categories.org.au<mailto:categories-list-leave@categories.org.au> _______________________________________________ Categories mailing list -- categories-list@categories.org.au<mailto:categories-list@categories.org.au> To unsubscribe send an email to categories-list-leave@categories.org.au<mailto:categories-list-leave@categories.org.au> Disclaimer - University of Cape Town This email is subject to UCT policies and email disclaimer published on our website at https://www.uct.ac.za/main/email-disclaimer or obtainable from +27 21 650 9111. If this email is not related to the business of UCT, it is sent by the sender in an individual capacity. Please report security incidents or abuse via https://csirt.uct.ac.za/report-incident
Dear All, The threat to category theory in particular and mathematics in general is a recurring attack on sensiblity and reason dating back at least to late Professor Ronnie Brown's letter to this very CatList. I don't know of one math department that's been disbanded / merged with IT / AI, or one math faculty forced to resign / retire, or one category theory course deleted from the course catalog reinstated, as a result of (the requested) email campaigns (none would be more happier than me if I were factually incorrect in this regard). As one born in a nation (India) subjected to genocidal Islamic invasions, unspeakable crimes-against-humanity, including killing of teachers, burning univerties, and libraries for centuries, creating an educational void, was possible not because of the barbarism of Turks, but by what predated it all: elitistism of educators. Educating the gifted; education ecosystem metamorphing into an exclusionary system. There's no caste called Brahmin, but none can say no if one chooses to name himself Jesus; so is the case with the self-baptized brahmins. Thus reified, they had their brief moment of glory. Fast-forward, now, the scientific community is the new priesthood demanding FAITH (shameless British Nature publishing anti-scientific editorials for pennies: https://www.nature.com/articles/529437a). Not surprisingly, noone trusts science / scientists in the USA, for good reasons (the idiocracy that's pre-poned: https://youtu.be/Fje510WMHPQ?si=FgZB7N0H86_W-xrf; the first thought, after noticing she's indian, that came to my mind is: somebody plz kill me ;) and thanks to eminences reciting received scripts, including our fellow grandmaster of mystification ( https://open.substack.com/pub/conceptualmathematics/p/mystification-aka-categorification?utm_source=share&utm_medium=android&r=1vwbhg) John Baez (for openly calling all those European professors protesting unscientific COVID restrictions: born-stupid in his zulipchat). There's always noise, pollution, and crooks; in spite of it all we continue to work to advance science. We have a solution: Madam Fatima Lawvere; follow in her footsteps (or, speaking of which, I have nothing to add, especially upon recollecting the Editor-in-Chief of Notices of American Mathematical Society, in response to my email with an overview of Madam Fatima Lawvere's great service to the promotion and practice of mathematics in imaginging the unimaginable and dedicating her entire life ensuring the materialization of her mind-made Conceptual Mathematics textbook, asked me: what do you want me to do, which for me was the end card). In closing, we don't have to think for ourselves ( https://youtube.com/shorts/7Flovu26nxQ?lc=UgzQj9D63zZi7XAbOkl4AaABAg&si=emI42Nm--ME8XJEU), it's all spelled out in readily comprehensible simple words: Injustice anywhere is a threat to justice everywhere (MLK, whose national holiday is celebrated by going to work in the USA ;) Let's not go down the road of I am not communist, when they come after communists; I'm not a socialist, when they roundup socialists; sooner or later they will come after you, not for some grand conflict of ideologies, but something as schzoid as not sending your granddaughter to their madrasa. The threat is NOT new; it's not against category theory per se, but against each and every one that tries to be sensible-and-reasonable. Thanking you, Yours respectfully, posina P.S. Mathematicians, especially the current breed of Field's Medalists, when, in a conference: The Future of Mathematics, find themselves confronted with: "What is Mathematics?" should at least go borrow a couple of neurons and synapses to address / ignore / delete my answer to their question, instead of hiding behind burkhas: https://www.youtube.com/watch?v=tN4hsT5t0nw P.P.S. Here's my answer to dem Fields Medalists and their minions' question: What is mathematics? @ZerothLawOfMotion Thank you for sharing your fascinating take on the future of mathematics. If I may, ~08:00:00, there's this question: What is mathematics? Much of what follows is based on my [mis?]understanding of the work of Professor F. William Lawvere. Mathematics, in its ancient sense, means learnable-AND-teachable. Having gotten initiated into what is now called Artificial Intelligence (AI) thanks to Caianiello's Thinking Machines, my thinking maybe outdated and most likely unpalatable to most of you. With that full disclosure, here's my unvarnished critique. First, soon after the success of AI in proving theorems, Minsky, upon recognizing that theorems are but statements in a story, so to speak, put forward the agenda of abstracting theories (of which theorems proved by AI are but tiny fragments), which is now conveniently forgotten, which, in turn, would be understandable if we didn't have any methods to abstract theories. Unfortunately (for those aficionados of AI predicting math's future (forgot Yogi Bohr ;) here's how you abstract theories from a given category of particulars: with Set-valued measurements of the properties of a category of particulars (e.g., functions, dynamical systems, groups, directed graphs, reflexive graphs) construed as representable functors, the corresponding representing objects along with their morphisms constitute the theory (of, say, graphs), which happens to be a subcategory of the category of graphs. Models of the thus abstracted theory of graphs are contravariant functors from the theory category to the discrete / constant subcategory within the category of graphs. Where does formalism fit in all this? There is formal-conceptual adjointness that's distinct from the aforementioned theory-models adjointness. Presentations (in terms of generators and relations / symbols and equations) are required for calculations, just as words and sentences (symbolic languages) are required to communicate our thoughts. Concepts are presented in words and represented as percepts, in the lingo of CogSci. There is no direct path from the (syntax of) presentations to the (semantics of) representations; there's is a factorization via concepts. Simply put, words-sans-concepts are meaningless. Once you have a theory (abstracted with respect to a doctrine; cf. screwing the doctrine up and down reveals various phenomena, as Maxwell recognized aeons ago; in our context, for example, in a finite-product doctrine, only product-preserving functors are models). Reminiscent of how the success of AI in proving theorems relegated the very idea of advancing AI by way of abstracting theories to an invisible background, the logical successes in going from one proposition to another, occluded the fact that propositions are made up of concepts, and that further advances in logic require finding the laws of rational passage between concepts. Along these lines, the objective logic intrinsic to a category / topos (functions, graphs, et al) is determined by the theory subcategory, beginning with the calculation of truth value object / subobject classifier, and all logical operations can then be characterized in terms the thus calculated truth value object. So is the case with number theory, the familiar arithmetic of natural numbers is that of the category of sets. One (hopefully familiar) illustration of objective number theory: in the category of pointed sets, we find: 1 + 1 = 1 (cf. with timeline as a set of timepoints, take origin as the distinguished point). Once again our applaudable proficiency in counting points (discrete / constant) abstracted from cohesively extended objects of everyday physics, impoverished our mental faculties so much so that we rarely think about the problem of counting extended objects. Mathematics is about qualities, with quantities providing, at best, a first approximation. As though success, in contrast to the psyche characteristic of the USA, is capable of incapacitating, a few instances of which we alluded to earlier, the success of calculus in solving all sorts of physics problems involving change, ended up with physicists oblivious to the integration problems, notwithstanding the fact that none other than Newton pointed out the significance of compounding after resolving. We may ignore, but reality has a way of making itself palpable: the structure-respecting maps / natural transformations / Becoming consistent with Being / Unity-respecting Change as the Zeroth Law of Change (that failed to hit Newton when that proverbial apple fell on his head) is impossible to pretend unawares. Now, let's get to the elephant in the room: it's sensible-and-reasonable to begin, before going artificial or whatever, with a definition of 'intelligence', notwithstanding the patently ridiculous take of Turing (definitions, as with weddings in California, can, if ain't working, be divorced ;) Using the most advanced mathematical method of defining, which is in terms of "good for" / universal mapping property, which is a refinement of the yesteryuga functional definitions (the flaws of which, for instance, Gould explicated in the context of evolution), we define intelligence in terms of 'what intelligence is good for', or equivalently, 'what is that which wouldn't be but for intelligence?' But for human intelligence, there wouldn't be science: science understood as ever-proper alignment of reason with experience. Those of us who are into neuro-namedropping, how about contemplating the profound insight of, once again, not a no-name neuroscientist, Sherrington: Neuron (with its generic operation of dot product) is a model of the integrative actions of the brain (hint: generalize from quantities to qualities; to ease your life, both intensive (color) and extensive (shape) qualities are defined as part of the axiomatization of cohesion). To those neuro / cogsci / philpeople who are whiling away betting in bars about one or other baptized hard problems (e.g., qualities, binding) with which they and only they, in their printed thinking, are chosen to solve, it may be of some use, upon sobering up, to look at how mathematicians are addressing those very problems including numerous problems pertinent to neuroscience, cognitive science, and AI such as: compounding epistemology and ontology into which reality is resolved. One unsolicited suggestion to those who are serious about AI, looking up to neuroscience / cognitive science / philosophy as a method of advancing AI is like holding on to a drowning dog's tail as a way to get across raging river Godavari. Most important of all: teaching is learning-with-students (not shock-and-awe with useless lecturing ad nauseam about the mathematical nuisance that is paradoxes and inconsistencies). The future of mathematics is in recognizing a category (of objects) as an objectification of the essence in which every object of the category partakes and as such every transformation between objects of a category preserves the essence that's characteristic of the category. For example, space is an object of a category of spaces, all of which partake in the essence of cohesion / stick-together. Geometry is self-founded, with the algebra of measures as its subcategory; i.e., math doesn't need any symbolic language as its foundations, leave alone the justification of the symbolic non-foundation, which is wasting mathematical muscle, although professionally rewarding for those engaged in the non-foundation delulu, which has been brought into figural salience for all to see by way of not figuring in everyday practice of mathematics. Continuing with the mind-made future of math includes, just to name one, advancing our understanding of the relation between space (characterized by the quality: stick-together) and time (characterized by the quality: urge to move) beyond that given in quantitative velocity. Quantities are quantities of things (thinking of a motion of a thing as a thing is one of the major milestones in the advancement of science) with qualities. Of course, it's one's birthright to be oblivious to all of the above and fixate on data / particulars (and persist on popularizing novel terminology that even comedians (e.g., Bill Maher) don't want to know anything about), but the problem with particular is that it is unlimited (cf. Fodor's dog), and that's exactly the reason abstracting ([finite / limited] generals / theories from particulars) is the method of science. In closing, truth is not a statistical notion. Pondering the contrast---statistical vs. mathematical---may be of some help in getting to answer, just in case none of the above didn't: What is mathematics? ------------------------------------------ On Mon, 1 Jun, 2026, 3:58 pm George Janelidze via Categories, < categories-list@categories.org.au> wrote:
Not having Peter Johnstone's lectures in category theory would be bad for Cambridge students. This is a very serious problem, and I don't understand why has this discussion moved somewhere else.
George Janelidze ------------------------------ *From:* Graham White via Categories <categories-list@categories.org.au> *Sent:* Monday, June 1, 2026 10:22 *To:* peklund via Categories <categories-list@categories.org.au> *Cc:* Steven Vickers <s.j.vickers.1@bham.ac.uk>; ptj1000@cam.ac.uk < ptj1000@cam.ac.uk>; peklund@cs.umu.se <peklund@cs.umu.se>; Graham White < g.graham.white@gmail.com> *Subject:* [categories] Re: Category Theory under threat in Cambridge
CAUTION: This email originated outside the UCT network. Do not click any links or open attachments unless you know and trust the source.
This is very depressing. I've always been on the geometrical, conceptual side of mathematics (and I've never really been any good at numerical calculation). And the geometrical side of mathematics goes very far back, to ancient Greece at least. Could we not try to emphasize this?
Graham White London
On Mon, 1 Jun 2026, 08:20 peklund via Categories, < categories-list@categories.org.au> wrote:
Why do we have this development?
Decades ago the famous categorists were still young or at least younger and still working. Today we have a new generation of categorists teaching category theory and sharing memories of times when now the oldies were younger, and when everybody felt self-confident and proud about being part of the CT community.
Young students now, unfortunately for CT, have more options to feel self-confident and proud.
Category theory is useful, always was. Bernays and Gödel never saw that, but never mind (until the PS). CT is useful in math. CT is useful in computer science. CT is useful in real applications, but that was never acknowledged.
Are we ourselves to blaim for not having transformed CT to something that lives and survives through times. Has it been too introvert?
Just wondering.
---
Is there a silverlining? What can we do?
CT in CS has often been seen as a good example where CT comes into play. And it does, still today. But I cannot see it has improved, so CT in CS may decline similarly in a decade or two. I may be wrong, I hope I'm wrong, but I say so since I was recently at the Types 2026 conference in Gothenburg. I haven't been at those for a while and now I wanted to see if there is something new in CT in CS. I didn't see anything, I should to say, but thanks to a plenary (by Emily Riehl) there was at least a plea for CT (teaching) to remain in CS (teaching).
Of course, there are CS teachers who can teach CT, but many of them are also not interested in real applications at all. Very few if any can explain why we need CT.
---
"Real applications", what's that?
I was nominated to professor in 1995, and from start I recognized my three tasks, i.e., tasks we recognize as our three tasks in universities. One, we teach. Two we do research. And it is in that order! Three, we cooperate with the surrounding society. We are obliged to do one and two, and CT teachers often do not have the luxury and time of doing two. Not even CT professors.
Take CT people and take math people in general. Is there a difference between how they at average engage in the third task (ignoring the situation about how much time they have for such engagements)?
Am I right or am I right?
If I'm right, what happens before a meeting between a categorist professor and, say, a medical professor. Will the medical professor be curious to know something about CT "before" the meeting? No. Will the CT professor be curious to know something about the medical domain to be discussed "before" the meeting? Should be, but in most cases will not. So what happens after the meeting? The medical professor go backs to office and continues research in, say, cardiology. The CT professor goes back to office and continues to remember the good old days when we didn't even think about talking to professors in other disciplines.
Am I wrong or am I wrong?
If I'm wrong, then there is hope, otherwise not.
And whenever we might go "real", do remember that we must "FIRST find the problem, THEN design a solution", not the other way around ("FIRST design a solution, THEN find a problem where the solution comes into play").
I rest my case.
---
Best,
Patrik Eklund
PS CT became CT some less than a century ago. Suppose CT had become CT some decades before, say, even before Hilbert wrote his foundations of geometry. In that case, CT had been around by the time Bernays was recruited to Göttingen, and I bet Gödel's ideas would never have reached publication in 1931. Hilbert's and Bernays' two volumes on the foundation of mathematics might even have been preceded by another two volumes on the foundation of the foundation of mathematics. Who knows. I bet Mac Lane never thought about it that way. Otherwise he would have called Bernays and said "Hey, we really need to talk!". But I don't see that in Bernays correspondence, so here we are, 75 years later, still not talking to each other.
On 2026-05-31 16:29, Steven Vickers via Categories wrote:
Hi Peter,
I'm sad to hear of the decline of category theory in the Maths Faculty.
All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason?
To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory.
All the best,
Steve.
------------------------------ *From:* P.T. Johnstone via Categories <categories-list@categories.org.au> *Sent:* Friday, May 29, 2026 10:11 AM *To:* Marino Gran via Categories <categories-list@categories.org.au> *Cc:* P.T. Johnstone <ptj1000@cam.ac.uk> *Subject:* [categories] Category Theory under threat in Cambridge
*CAUTION:* This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.
As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid!
I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year.
I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so.
Peter Johnstone
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
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Disclaimer - University of Cape Town This email is subject to UCT policies and email disclaimer published on our website at https://www.uct.ac.za/main/email-disclaimer or obtainable from +27 21 650 9111. If this email is not related to the business of UCT, it is sent by the sender in an individual capacity. Please report security incidents or abuse via https://csirt.uct.ac.za/report-incident _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
Dear All,
The threat to category theory in particular and mathematics in general is a recurring attack on sensiblity and reason dating back at least to late Professor Ronnie Brown's letter to this very CatList. I don't know of one math department that's been disbanded / merged with IT / AI, or one math faculty forced to resign / retire, or one category theory course deleted from the course catalog reinstated, as a result of (the requested) email campaigns (none would be more happier than me if I were factually incorrect in this regard).
As one born in a nation (India) subjected to genocidal Islamic invasions, unspeakable crimes-against-humanity, including killing of teachers, burning univerties, and libraries for centuries, creating an educational void, was possible not because of the barbarism of Turks, but by what predated it all: elitistism of educators. Educating the gifted; education ecosystem metamorphing into an exclusionary system. There's no caste called Brahmin, but none can say no if one chooses to name himself Jesus; so is the case with the self-baptized brahmins. Thus reified, they had their brief moment of glory.
Fast-forward, now, the scientific community is the new priesthood demanding FAITH (shameless British Nature publishing anti-scientific editorials for pennies: https://www.nature.com/articles/529437a).
Not surprisingly, noone trusts science / scientists in the USA, for good reasons (the idiocracy that's pre-poned: https://youtu.be/Fje510WMHPQ?si=FgZB7N0H86_W-xrf; the first thought, after noticing she's indian, that came to my mind is: somebody plz kill me ;) and thanks to eminences reciting received scripts, including our fellow grandmaster of mystification (https://open.substack.com/pub/conceptualmathematics/p/mystification-aka-categorification?utm_source=share&utm_medium=android&r=1vwbhg [1]) John Baez (for openly calling all those European professors protesting unscientific COVID restrictions: born-stupid in his zulipchat).
There's always noise, pollution, and crooks; in spite of it all we continue to work to advance science.
We have a solution: Madam Fatima Lawvere; follow in her footsteps (or, speaking of which, I have nothing to add, especially upon recollecting the Editor-in-Chief of Notices of American Mathematical Society, in response to my email with an overview of Madam Fatima Lawvere's great service to the promotion and practice of mathematics in imaginging the unimaginable and dedicating her entire life ensuring the materialization of her mind-made Conceptual Mathematics textbook, asked me: what do you want me to do, which for me was the end card).
In closing, we don't have to think for ourselves (https://youtube.com/shorts/7Flovu26nxQ?lc=UgzQj9D63zZi7XAbOkl4AaABAg&si=emI42Nm--ME8XJEU [2]), it's all spelled out in readily comprehensible simple words:
Injustice anywhere is a threat to justice everywhere (MLK, whose national holiday is celebrated by going to work in the USA ;)
Let's not go down the road of I am not communist, when they come after communists; I'm not a socialist, when they roundup socialists; sooner or later they will come after you, not for some grand conflict of ideologies, but something as schzoid as not sending your granddaughter to their madrasa.
The threat is NOT new; it's not against category theory per se, but against each and every one that tries to be sensible-and-reasonable.
Thanking you, Yours respectfully, posina
P.S. Mathematicians, especially the current breed of Field's Medalists, when, in a conference: The Future of Mathematics, find themselves confronted with: "What is Mathematics?" should at least go borrow a couple of neurons and synapses to address / ignore / delete my answer to their question, instead of hiding behind burkhas:
https://www.youtube.com/watch?v=tN4hsT5t0nw
P.P.S. Here's my answer to dem Fields Medalists and their minions' question:
What is mathematics?
@ZerothLawOfMotion
Thank you for sharing your fascinating take on the future of mathematics. If I may, ~08:00:00, there's this question: What is mathematics? Much of what follows is based on my [mis?]understanding of the work of Professor F. William Lawvere. Mathematics, in its ancient sense, means learnable-AND-teachable. Having gotten initiated into what is now called Artificial Intelligence (AI) thanks to Caianiello's Thinking Machines, my thinking maybe outdated and most likely unpalatable to most of you. With that full disclosure, here's my unvarnished critique. First, soon after the success of AI in proving theorems, Minsky, upon recognizing that theorems are but statements in a story, so to speak, put forward the agenda of abstracting theories (of which theorems proved by AI are but tiny fragments), which is now conveniently forgotten, which, in turn, would be understandable if we didn't have any methods to abstract theories. Unfortunately (for those aficionados of AI predicting math's future (forgot Yogi Bohr ;) here's how you abstract theories from a given category of particulars: with Set-valued measurements of the properties of a category of particulars (e.g., functions, dynamical systems, groups, directed graphs, reflexive graphs) construed as representable functors, the corresponding representing objects along with their morphisms constitute the theory (of, say, graphs), which happens to be a subcategory of the category of graphs. Models of the thus abstracted theory of graphs are contravariant functors from the theory category to the discrete / constant subcategory within the category of graphs. Where does formalism fit in all this? There is formal-conceptual adjointness that's distinct from the aforementioned theory-models adjointness. Presentations (in terms of generators and relations / symbols and equations) are required for calculations, just as words and sentences (symbolic languages) are required to communicate our thoughts. Concepts are presented in words and represented as percepts, in the lingo of CogSci. There is no direct path from the (syntax of) presentations to the (semantics of) representations; there's is a factorization via concepts. Simply put, words-sans-concepts are meaningless. Once you have a theory (abstracted with respect to a doctrine; cf. screwing the doctrine up and down reveals various phenomena, as Maxwell recognized aeons ago; in our context, for example, in a finite-product doctrine, only product-preserving functors are models). Reminiscent of how the success of AI in proving theorems relegated the very idea of advancing AI by way of abstracting theories to an invisible background, the logical successes in going from one proposition to another, occluded the fact that propositions are made up of concepts, and that further advances in logic require finding the laws of rational passage between concepts. Along these lines, the objective logic intrinsic to a category / topos (functions, graphs, et al) is determined by the theory subcategory, beginning with the calculation of truth value object / subobject classifier, and all logical operations can then be characterized in terms the thus calculated truth value object. So is the case with number theory, the familiar arithmetic of natural numbers is that of the category of sets. One (hopefully familiar) illustration of objective number theory: in the category of pointed sets, we find: 1 + 1 = 1 (cf. with timeline as a set of timepoints, take origin as the distinguished point). Once again our applaudable proficiency in counting points (discrete / constant) abstracted from cohesively extended objects of everyday physics, impoverished our mental faculties so much so that we rarely think about the problem of counting extended objects. Mathematics is about qualities, with quantities providing, at best, a first approximation. As though success, in contrast to the psyche characteristic of the USA, is capable of incapacitating, a few instances of which we alluded to earlier, the success of calculus in solving all sorts of physics problems involving change, ended up with physicists oblivious to the integration problems, notwithstanding the fact that none other than Newton pointed out the significance of compounding after resolving. We may ignore, but reality has a way of making itself palpable: the structure-respecting maps / natural transformations / Becoming consistent with Being / Unity-respecting Change as the Zeroth Law of Change (that failed to hit Newton when that proverbial apple fell on his head) is impossible to pretend unawares. Now, let's get to the elephant in the room: it's sensible-and-reasonable to begin, before going artificial or whatever, with a definition of 'intelligence', notwithstanding the patently ridiculous take of Turing (definitions, as with weddings in California, can, if ain't working, be divorced ;) Using the most advanced mathematical method of defining, which is in terms of "good for" / universal mapping property, which is a refinement of the yesteryuga functional definitions (the flaws of which, for instance, Gould explicated in the context of evolution), we define intelligence in terms of 'what intelligence is good for', or equivalently, 'what is that which wouldn't be but for intelligence?' But for human intelligence, there wouldn't be science: science understood as ever-proper alignment of reason with experience. Those of us who are into neuro-namedropping, how about contemplating the profound insight of, once again, not a no-name neuroscientist, Sherrington: Neuron (with its generic operation of dot product) is a model of the integrative actions of the brain (hint: generalize from quantities to qualities; to ease your life, both intensive (color) and extensive (shape) qualities are defined as part of the axiomatization of cohesion). To those neuro / cogsci / philpeople who are whiling away betting in bars about one or other baptized hard problems (e.g., qualities, binding) with which they and only they, in their printed thinking, are chosen to solve, it may be of some use, upon sobering up, to look at how mathematicians are addressing those very problems including numerous problems pertinent to neuroscience, cognitive science, and AI such as: compounding epistemology and ontology into which reality is resolved. One unsolicited suggestion to those who are serious about AI, looking up to neuroscience / cognitive science / philosophy as a method of advancing AI is like holding on to a drowning dog's tail as a way to get across raging river Godavari. Most important of all: teaching is learning-with-students (not shock-and-awe with useless lecturing ad nauseam about the mathematical nuisance that is paradoxes and inconsistencies). The future of mathematics is in recognizing a category (of objects) as an objectification of the essence in which every object of the category partakes and as such every transformation between objects of a category preserves the essence that's characteristic of the category. For example, space is an object of a category of spaces, all of which partake in the essence of cohesion / stick-together. Geometry is self-founded, with the algebra of measures as its subcategory; i.e., math doesn't need any symbolic language as its foundations, leave alone the justification of the symbolic non-foundation, which is wasting mathematical muscle, although professionally rewarding for those engaged in the non-foundation delulu, which has been brought into figural salience for all to see by way of not figuring in everyday practice of mathematics. Continuing with the mind-made future of math includes, just to name one, advancing our understanding of the relation between space (characterized by the quality: stick-together) and time (characterized by the quality: urge to move) beyond that given in quantitative velocity. Quantities are quantities of things (thinking of a motion of a thing as a thing is one of the major milestones in the advancement of science) with qualities. Of course, it's one's birthright to be oblivious to all of the above and fixate on data / particulars (and persist on popularizing novel terminology that even comedians (e.g., Bill Maher) don't want to know anything about), but the problem with particular is that it is unlimited (cf. Fodor's dog), and that's exactly the reason abstracting ([finite / limited] generals / theories from particulars) is the method of science. In closing, truth is not a statistical notion. Pondering the contrast---statistical vs. mathematical---may be of some help in getting to answer, just in case none of the above didn't:
What is mathematics?
------------------------------------------
On Mon, 1 Jun, 2026, 3:58 pm George Janelidze via Categories, <categories-list@categories.org.au> wrote:
Not having Peter Johnstone's lectures in category theory would be bad for Cambridge students. This is a very serious problem, and I don't understand why has this discussion moved somewhere else.
George Janelidze
-------------------------
From: Graham White via Categories <categories-list@categories.org.au> Sent: Monday, June 1, 2026 10:22 To: peklund via Categories <categories-list@categories.org.au> Cc: Steven Vickers <s.j.vickers.1@bham.ac.uk>; ptj1000@cam.ac.uk <ptj1000@cam.ac.uk>; peklund@cs.umu.se <peklund@cs.umu.se>; Graham White <g.graham.white@gmail.com> Subject: [categories] Re: Category Theory under threat in Cambridge
CAUTION: This email originated outside the UCT network. Do not click any links or open attachments unless you know and trust the source.
This is very depressing. I've always been on the geometrical, conceptual side of mathematics (and I've never really been any good at numerical calculation). And the geometrical side of mathematics goes very far back, to ancient Greece at least. Could we not try to emphasize this?
Graham White London
On Mon, 1 Jun 2026, 08:20 peklund via Categories, <categories-list@categories.org.au> wrote:
Why do we have this development?
Decades ago the famous categorists were still young or at least younger and still working. Today we have a new generation of categorists teaching category theory and sharing memories of times when now the oldies were younger, and when everybody felt self-confident and proud about being part of the CT community.
Young students now, unfortunately for CT, have more options to feel self-confident and proud.
Category theory is useful, always was. Bernays and Gödel never saw that, but never mind (until the PS). CT is useful in math. CT is useful in computer science. CT is useful in real applications, but that was never acknowledged.
Are we ourselves to blaim for not having transformed CT to something that lives and survives through times. Has it been too introvert?
Just wondering.
---
Is there a silverlining? What can we do?
CT in CS has often been seen as a good example where CT comes into play. And it does, still today. But I cannot see it has improved, so CT in CS may decline similarly in a decade or two. I may be wrong, I hope I'm wrong, but I say so since I was recently at the Types 2026 conference in Gothenburg. I haven't been at those for a while and now I wanted to see if there is something new in CT in CS. I didn't see anything, I should to say, but thanks to a plenary (by Emily Riehl) there was at least a plea for CT (teaching) to remain in CS (teaching).
Of course, there are CS teachers who can teach CT, but many of them are also not interested in real applications at all. Very few if any can explain why we need CT.
---
"Real applications", what's that?
I was nominated to professor in 1995, and from start I recognized my three tasks, i.e., tasks we recognize as our three tasks in universities. One, we teach. Two we do research. And it is in that order! Three, we cooperate with the surrounding society. We are obliged to do one and two, and CT teachers often do not have the luxury and time of doing two. Not even CT professors.
Take CT people and take math people in general. Is there a difference between how they at average engage in the third task (ignoring the situation about how much time they have for such engagements)?
Am I right or am I right?
If I'm right, what happens before a meeting between a categorist professor and, say, a medical professor. Will the medical professor be curious to know something about CT "before" the meeting? No. Will the CT professor be curious to know something about the medical domain to be discussed "before" the meeting? Should be, but in most cases will not. So what happens after the meeting? The medical professor go backs to office and continues research in, say, cardiology. The CT professor goes back to office and continues to remember the good old days when we didn't even think about talking to professors in other disciplines.
Am I wrong or am I wrong?
If I'm wrong, then there is hope, otherwise not.
And whenever we might go "real", do remember that we must "FIRST find the problem, THEN design a solution", not the other way around ("FIRST design a solution, THEN find a problem where the solution comes into play").
I rest my case.
---
Best,
Patrik Eklund
PS CT became CT some less than a century ago. Suppose CT had become CT some decades before, say, even before Hilbert wrote his foundations of geometry. In that case, CT had been around by the time Bernays was recruited to Göttingen, and I bet Gödel's ideas would never have reached publication in 1931. Hilbert's and Bernays' two volumes on the foundation of mathematics might even have been preceded by another two volumes on the foundation of the foundation of mathematics. Who knows. I bet Mac Lane never thought about it that way. Otherwise he would have called Bernays and said "Hey, we really need to talk!". But I don't see that in Bernays correspondence, so here we are, 75 years later, still not talking to each other.
On 2026-05-31 16:29, Steven Vickers via Categories wrote:
Hi Peter,
I'm sad to hear of the decline of category theory in the Maths Faculty.
All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason?
To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory.
All the best,
Steve.
-------------------------
From: P.T. Johnstone via Categories <categories-list@categories.org.au> Sent: Friday, May 29, 2026 10:11 AM To: Marino Gran via Categories <categories-list@categories.org.au> Cc: P.T. Johnstone <ptj1000@cam.ac.uk> Subject: [categories] Category Theory under threat in Cambridge
CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.
As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid!
I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year.
I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so.
Peter Johnstone _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au Disclaimer - University of Cape Town This email is subject to UCT
Thanks, Posina. Yes, there is threat, and, as you also indicate by bringing up Madam Fatima Lawvere, there is opportunity. Like in any risk management, threat and opportunity are antinomies. Maybe there is a Galois connection to be described for this antinomy? Category theory saving itself? Best, Patrik On 2026-06-01 22:32, Posina Venkata Rayudu via Categories wrote: policies and email disclaimer published on our website at https://www.uct.ac.za/main/email-disclaimer or obtainable from +27 21 650 9111. If this email is not related to the business of UCT, it is sent by the sender in an individual capacity. Please report security incidents or abuse via https://csirt.uct.ac.za/report-incident _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au Links: ------ [1] https://open.substack.com/pub/conceptualmathematics/p/mystification-aka-categorification?utm_source=share&utm_medium=android&r=1vwbhg [2] https://youtube.com/shorts/7Flovu26nxQ?lc=UgzQj9D63zZi7XAbOkl4AaABAg&si=emI42Nm--ME8XJEU
Dear Professor Graham White, Reading your email made me happy in more than one respect, one of which is that which you suggest emphasizing, i.e., the conceptual and the geometrical side of mathematics dating back to ancient Greece (in the words of Professor F. William Lawvere): https://conceptualmathematics.wordpress.com/2012/09/23/comfortable-with-sheh... I just thought you might find it of some interest. Thanking you, Yours respectfully, posina P.S. Early this morning, I asked the shopkeeper, who gave me a bill of Rs. 6009, for a 1 rupee candy, so that I can pay a round figure of Rs. 7000 (happy to be in your good company ;) On Mon, Jun 1, 2026 at 2:31 PM Graham White via Categories <categories-list@categories.org.au> wrote:
This is very depressing. I've always been on the geometrical, conceptual side of mathematics (and I've never really been any good at numerical calculation). And the geometrical side of mathematics goes very far back, to ancient Greece at least. Could we not try to emphasize this?
Graham White London
On Mon, 1 Jun 2026, 08:20 peklund via Categories, <categories-list@categories.org.au> wrote:
Why do we have this development?
Decades ago the famous categorists were still young or at least younger and still working. Today we have a new generation of categorists teaching category theory and sharing memories of times when now the oldies were younger, and when everybody felt self-confident and proud about being part of the CT community.
Young students now, unfortunately for CT, have more options to feel self-confident and proud.
Category theory is useful, always was. Bernays and Gödel never saw that, but never mind (until the PS). CT is useful in math. CT is useful in computer science. CT is useful in real applications, but that was never acknowledged.
Are we ourselves to blaim for not having transformed CT to something that lives and survives through times. Has it been too introvert?
Just wondering.
---
Is there a silverlining? What can we do?
CT in CS has often been seen as a good example where CT comes into play. And it does, still today. But I cannot see it has improved, so CT in CS may decline similarly in a decade or two. I may be wrong, I hope I'm wrong, but I say so since I was recently at the Types 2026 conference in Gothenburg. I haven't been at those for a while and now I wanted to see if there is something new in CT in CS. I didn't see anything, I should to say, but thanks to a plenary (by Emily Riehl) there was at least a plea for CT (teaching) to remain in CS (teaching).
Of course, there are CS teachers who can teach CT, but many of them are also not interested in real applications at all. Very few if any can explain why we need CT.
---
"Real applications", what's that?
I was nominated to professor in 1995, and from start I recognized my three tasks, i.e., tasks we recognize as our three tasks in universities. One, we teach. Two we do research. And it is in that order! Three, we cooperate with the surrounding society. We are obliged to do one and two, and CT teachers often do not have the luxury and time of doing two. Not even CT professors.
Take CT people and take math people in general. Is there a difference between how they at average engage in the third task (ignoring the situation about how much time they have for such engagements)?
Am I right or am I right?
If I'm right, what happens before a meeting between a categorist professor and, say, a medical professor. Will the medical professor be curious to know something about CT "before" the meeting? No. Will the CT professor be curious to know something about the medical domain to be discussed "before" the meeting? Should be, but in most cases will not. So what happens after the meeting? The medical professor go backs to office and continues research in, say, cardiology. The CT professor goes back to office and continues to remember the good old days when we didn't even think about talking to professors in other disciplines.
Am I wrong or am I wrong?
If I'm wrong, then there is hope, otherwise not.
And whenever we might go "real", do remember that we must "FIRST find the problem, THEN design a solution", not the other way around ("FIRST design a solution, THEN find a problem where the solution comes into play").
I rest my case.
---
Best,
Patrik Eklund
PS CT became CT some less than a century ago. Suppose CT had become CT some decades before, say, even before Hilbert wrote his foundations of geometry. In that case, CT had been around by the time Bernays was recruited to Göttingen, and I bet Gödel's ideas would never have reached publication in 1931. Hilbert's and Bernays' two volumes on the foundation of mathematics might even have been preceded by another two volumes on the foundation of the foundation of mathematics. Who knows. I bet Mac Lane never thought about it that way. Otherwise he would have called Bernays and said "Hey, we really need to talk!". But I don't see that in Bernays correspondence, so here we are, 75 years later, still not talking to each other.
On 2026-05-31 16:29, Steven Vickers via Categories wrote:
Hi Peter,
I'm sad to hear of the decline of category theory in the Maths Faculty.
All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason?
To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory.
All the best,
Steve.
________________________________ From: P.T. Johnstone via Categories <categories-list@categories.org.au> Sent: Friday, May 29, 2026 10:11 AM To: Marino Gran via Categories <categories-list@categories.org.au> Cc: P.T. Johnstone <ptj1000@cam.ac.uk> Subject: [categories] Category Theory under threat in Cambridge
CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.
As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid!
I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year.
I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so.
Peter Johnstone
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
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Dear Posina, perhaps it might be of interest for you to know that when Bill referred to the notion of arithmos he relied on many previous conversations in Florence during that period, between Bil and John Mayberry (and also John Bell and me). John stayed in Florence for one year during which he wrote a book which is based on the Euclidean concept of “set” (Foundations of Mathematics in the Theory of Sets, 2000, and at the time Bill liked to spend at least one week each year in Florence to discuss with me about topics concerning issues of philosophy and logic. So, John Mayberry’s pioneering, and deep, investigation on the notion of arithmos was a recurrent topic of conversation. There were specific points on which Bill and I didn’agree with John. What you read in the mail you kindly allow us to access is a byproduct of Bill’s disagreement with John, who argued for the primacy of the notion of set over any other, so that - along similar lines to those of Kreisel and Feferman -, Bill’s categorical approach to foundations could not be accepted.As for my objections to John, they were implicit in a section of a paper also published in 2000 "An Essay on the Notion of Schema" but I preferred not to say that explicitly. Later, John had a brain ictus and our dialogue had to stop. John was a man of incredibly wide culture and had a “classic” mind. He didn’t accept Bill’s dialectical view and looked at my attempt to free Bill’s intuitions from Hegel’s frame and reframe them into a naturalistic view (while keeping some aspects of Kantian philosophy of knowledge) as an attempt by a "neo-Aristotelean", something I didn’t know to be and didn’t want to look, though mainly for reasons related to Aristotle’s perspective on metaphysics and epistemology rather than on his “naturalistic” reorientation of Plato’s views. Best, Alberto
Il giorno 5 giu 2026, alle ore 10:58, Posina Venkata Rayudu via Categories <categories-list@categories.org.au> ha scritto:
Dear Professor Graham White,
Reading your email made me happy in more than one respect, one of which is that which you suggest emphasizing, i.e., the conceptual and the geometrical side of mathematics dating back to ancient Greece (in the words of Professor F. William Lawvere):
I just thought you might find it of some interest.
Thanking you, Yours respectfully, posina P.S. Early this morning, I asked the shopkeeper, who gave me a bill of Rs. 6009, for a 1 rupee candy, so that I can pay a round figure of Rs. 7000 (happy to be in your good company ;)
On Mon, Jun 1, 2026 at 2:31 PM Graham White via Categories <categories-list@categories.org.au> wrote:
This is very depressing. I've always been on the geometrical, conceptual side of mathematics (and I've never really been any good at numerical calculation). And the geometrical side of mathematics goes very far back, to ancient Greece at least. Could we not try to emphasize this?
Graham White London
On Mon, 1 Jun 2026, 08:20 peklund via Categories, <categories-list@categories.org.au> wrote:
Why do we have this development?
Decades ago the famous categorists were still young or at least younger and still working. Today we have a new generation of categorists teaching category theory and sharing memories of times when now the oldies were younger, and when everybody felt self-confident and proud about being part of the CT community.
Young students now, unfortunately for CT, have more options to feel self-confident and proud.
Category theory is useful, always was. Bernays and Gödel never saw that, but never mind (until the PS). CT is useful in math. CT is useful in computer science. CT is useful in real applications, but that was never acknowledged.
Are we ourselves to blaim for not having transformed CT to something that lives and survives through times. Has it been too introvert?
Just wondering.
---
Is there a silverlining? What can we do?
CT in CS has often been seen as a good example where CT comes into play. And it does, still today. But I cannot see it has improved, so CT in CS may decline similarly in a decade or two. I may be wrong, I hope I'm wrong, but I say so since I was recently at the Types 2026 conference in Gothenburg. I haven't been at those for a while and now I wanted to see if there is something new in CT in CS. I didn't see anything, I should to say, but thanks to a plenary (by Emily Riehl) there was at least a plea for CT (teaching) to remain in CS (teaching).
Of course, there are CS teachers who can teach CT, but many of them are also not interested in real applications at all. Very few if any can explain why we need CT.
---
"Real applications", what's that?
I was nominated to professor in 1995, and from start I recognized my three tasks, i.e., tasks we recognize as our three tasks in universities. One, we teach. Two we do research. And it is in that order! Three, we cooperate with the surrounding society. We are obliged to do one and two, and CT teachers often do not have the luxury and time of doing two. Not even CT professors.
Take CT people and take math people in general. Is there a difference between how they at average engage in the third task (ignoring the situation about how much time they have for such engagements)?
Am I right or am I right?
If I'm right, what happens before a meeting between a categorist professor and, say, a medical professor. Will the medical professor be curious to know something about CT "before" the meeting? No. Will the CT professor be curious to know something about the medical domain to be discussed "before" the meeting? Should be, but in most cases will not. So what happens after the meeting? The medical professor go backs to office and continues research in, say, cardiology. The CT professor goes back to office and continues to remember the good old days when we didn't even think about talking to professors in other disciplines.
Am I wrong or am I wrong?
If I'm wrong, then there is hope, otherwise not.
And whenever we might go "real", do remember that we must "FIRST find the problem, THEN design a solution", not the other way around ("FIRST design a solution, THEN find a problem where the solution comes into play").
I rest my case.
---
Best,
Patrik Eklund
PS CT became CT some less than a century ago. Suppose CT had become CT some decades before, say, even before Hilbert wrote his foundations of geometry. In that case, CT had been around by the time Bernays was recruited to Göttingen, and I bet Gödel's ideas would never have reached publication in 1931. Hilbert's and Bernays' two volumes on the foundation of mathematics might even have been preceded by another two volumes on the foundation of the foundation of mathematics. Who knows. I bet Mac Lane never thought about it that way. Otherwise he would have called Bernays and said "Hey, we really need to talk!". But I don't see that in Bernays correspondence, so here we are, 75 years later, still not talking to each other.
On 2026-05-31 16:29, Steven Vickers via Categories wrote:
Hi Peter,
I'm sad to hear of the decline of category theory in the Maths Faculty.
All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason?
To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory.
All the best,
Steve.
________________________________ From: P.T. Johnstone via Categories <categories-list@categories.org.au> Sent: Friday, May 29, 2026 10:11 AM To: Marino Gran via Categories <categories-list@categories.org.au> Cc: P.T. Johnstone <ptj1000@cam.ac.uk> Subject: [categories] Category Theory under threat in Cambridge
CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.
As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid!
I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year.
I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so.
Peter Johnstone
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
Dear Professor Alberto Peruzzi, Thanks so much for sharing what I recognize as a teaching-moment on and about how individual ideation is not individual's own: no mind is an island disconnected from the mainland of one's time et al., and more immediately / importantly: How our thinking is shaped and reshaped by the waves reaching the shores time and time again can't be left to Cognitive Science (a failed enterprise going by their own audit: https://drive.google.com/file/d/1iBIj5yLnnOIjzwRuBy0rlRQXTWBQj5a5/view?usp=d..., and then there's, of course, my curse for refusing to include Mathematics in their logo ;) As Professor F. William Lawvere highlighted: An explicit philosophy, dealing with the mutual transformation of individual thinking and collective thinking, could be very helpful (for example, when reforming the schools) and adequate mathematical models will be an essential basis of clarity ( https://lawverearchives.com/wp-content/uploads/2024/12/1994-tools-for-the-ad... p. 44). If I may, one point that puzzled me is the notion of the primacy of 'set'. Isn't a set a set of things, to begin with; but for elements, what is the Kardinalen a kardinalen of (cf. in speaking of, say, length of a line, is the primacy of line vis-à-vis its length open to debate)? One line of thinking, which seems to leave room for the primacy of sets appears to be that of Professor F. William Lawvere's category of sets, wherein elements of a set are points of the set, i.e., functions defined in terms of sets (there are some extremist set theorists, like my good friend: https://zenodo.org/records/4039859, who after all said-and-written arrived at our beloved category of graphs :). I'll write to you again soon after carefully studying your paper: An Essay on the Notion of Schema. Given that contrast is the basic building block on which rests our consciousness, thinking, sentience, and conscience (cf. Yarbus: stabilized images fade away; discounting the illuminant; as much as I love Cantor, had we all been his lauter Einsen, we wouldn't be ;) isn't it reasonable an approach to deploy arithmos-chaos in analysing given variation and cohesion with their contrasting constancy and discreetness ( https://conceptualmathematics.wordpress.com/wp-content/uploads/2013/02/sets-... pp. 245 - 246)? Last and yet the one that troubles me the most is how much of long-dead philosophy is healthy, especially given the mathematical brilliance of my role model: Professor Stephen H. Schanuel, who wasn't distracted by philosophy ( https://conceptualmathematics.wordpress.com/2023/05/08/mathematicians-mathem... )? My problem is compounded by the fact that I am drawn to "philosophy": beginning with the ancient Indian concept of Dharma, which, in the sense of its etymological roots, means: That which holds it all together, which is none other than Unity-respecting Change (e.g., plants growing into trees, water flowing down the gradient, birds flying high in the sky), or equivalently, Becoming consistent with Being ( https://conceptualmathematics.wordpress.com/wp-content/uploads/2025/02/conce... p. 152), which is all changes inundating us in the sense of: I don't remember turning into a caterpillar on my way to Malabar Cafe for coffee and returing home as a butterfly, all of which can be summed up as miracles-prohibited in our mathematical understanding of our everyday experience populated with varieties of categories of objects, all of which partake in the abstract essence(s) characteristic of their corresponding categories; as such, morphisms between objects of a category are necessarily structure-respecting maps: maps prohibited from tearing apart in mapping, say, a domain graph into a codomain graph (ibid., p. 210). It's this Unity-respecting Change / Becoming consistent with Being, or in a more familiar phrasing: Structure-respecting maps of any category of objects represented as Natural Transformations, appears to be the Zeroth Law of Change, the Dharma that holds it all together (Full Disclosure: Indian-by-birth). Please be kind enough to correct my mistakes (until I found Professor F. William Lawvere and Stephen H Schanuel, Conceptual Mathematics textbook / my Bible :) I used to think: math exams are deliberately designed to trick unsuspecting students into making mistakes, a' la Kahneman; https://philpapers.org/archive/POSFTS.pdf; Fodor provides a more-entertaing denuding of Kahneman's claim to fame: heuristics). Thanking you, Yours respectfully, posina P.S. Knowledge tends to inhibit the flight of imagination. Therefore, a certain naiveté, unburdened by conventional wisdom, can sometimes be a positive asset (Harish-Chandra). P.P.S. The singular Western contributions to the birthing, rearing, and development of modern science seems to have benefitted from not burdening themselves with the ancient intellectual inheritance of India (with its abundance of every concievable concept / thought / theory ;) P.P.P.S. One dangerous delusion that's gained a strong foothold in MathEd is that students' intellectual success / failure depends on that of their ancestors. This is eugenics: inheritance of intellect, or simply put, you need a Blockchain dad to invent blockchain, the rediculous nature of this anti-social paid-career of direspecting the disadvantaged amongst us is all around us since the dawn of sense and reason (http://disq.us/p/2ubeted). Notwithstanding my schooling, Kim, got promoted: from Quanta magazine to The Notices of AMS¿ On Thu, 11 Jun, 2026, 2:36 pm Alberto Peruzzi, <alberto.peruzzi@unifi.it> wrote:
Dear Posina,
perhaps it might be of interest for you to know that when Bill referred to the notion of *arithmos *he relied on many previous conversations in Florence during that period, between Bil and John Mayberry (and also John Bell and me). John stayed in Florence for one year during which he wrote a book which is based on the *Euclidean* concept of “set” (*Foundations of Mathematics in the Theory of Sets, *2000, and at the time Bill liked to spend at least one week each year in Florence to discuss with me about topics concerning issues of philosophy and logic. So, John Mayberry’s pioneering, and deep, investigation on the notion of *arithmos *was a recurrent topic of conversation. There were specific points on which Bill and I didn’agree with John. What you read in the mail you kindly allow us to access is a byproduct of Bill’s disagreement with John, who argued for the primacy of the notion of set over any other, so that - along similar lines to those of Kreisel and Feferman -, Bill’s categorical approach to foundations could not be accepted.As for my objections to John, they were implicit in a section of a paper also published in 2000 "An Essay on the Notion of Schema*"* but I preferred not to say that explicitly. Later, John had a brain ictus and our dialogue had to stop. John was a man of incredibly wide culture and had a “classic” mind. He didn’t accept Bill’s dialectical view and looked at my attempt to free Bill’s intuitions from Hegel’s frame and reframe them into a naturalistic view (while keeping some aspects of Kantian philosophy of knowledge) as an attempt by a "neo-Aristotelean", something I didn’t know to be and didn’t want to look, though mainly for reasons related to Aristotle’s perspective on metaphysics and epistemology rather than on his “naturalistic” reorientation of Plato’s views.
Best, Alberto
Il giorno 5 giu 2026, alle ore 10:58, Posina Venkata Rayudu via Categories <categories-list@categories.org.au> ha scritto:
Dear Professor Graham White,
Reading your email made me happy in more than one respect, one of which is that which you suggest emphasizing, i.e., the conceptual and the geometrical side of mathematics dating back to ancient Greece (in the words of Professor F. William Lawvere):
I just thought you might find it of some interest.
Thanking you, Yours respectfully, posina P.S. Early this morning, I asked the shopkeeper, who gave me a bill of Rs. 6009, for a 1 rupee candy, so that I can pay a round figure of Rs. 7000 (happy to be in your good company ;)
On Mon, Jun 1, 2026 at 2:31 PM Graham White via Categories <categories-list@categories.org.au> wrote:
This is very depressing. I've always been on the geometrical, conceptual side of mathematics (and I've never really been any good at numerical calculation). And the geometrical side of mathematics goes very far back, to ancient Greece at least. Could we not try to emphasize this?
Graham White London
On Mon, 1 Jun 2026, 08:20 peklund via Categories, < categories-list@categories.org.au> wrote:
Why do we have this development?
Decades ago the famous categorists were still young or at least younger and still working. Today we have a new generation of categorists teaching category theory and sharing memories of times when now the oldies were younger, and when everybody felt self-confident and proud about being part of the CT community.
Young students now, unfortunately for CT, have more options to feel self-confident and proud.
Category theory is useful, always was. Bernays and Gödel never saw that, but never mind (until the PS). CT is useful in math. CT is useful in computer science. CT is useful in real applications, but that was never acknowledged.
Are we ourselves to blaim for not having transformed CT to something that lives and survives through times. Has it been too introvert?
Just wondering.
---
Is there a silverlining? What can we do?
CT in CS has often been seen as a good example where CT comes into play. And it does, still today. But I cannot see it has improved, so CT in CS may decline similarly in a decade or two. I may be wrong, I hope I'm wrong, but I say so since I was recently at the Types 2026 conference in Gothenburg. I haven't been at those for a while and now I wanted to see if there is something new in CT in CS. I didn't see anything, I should to say, but thanks to a plenary (by Emily Riehl) there was at least a plea for CT (teaching) to remain in CS (teaching).
Of course, there are CS teachers who can teach CT, but many of them are also not interested in real applications at all. Very few if any can explain why we need CT.
---
"Real applications", what's that?
I was nominated to professor in 1995, and from start I recognized my three tasks, i.e., tasks we recognize as our three tasks in universities. One, we teach. Two we do research. And it is in that order! Three, we cooperate with the surrounding society. We are obliged to do one and two, and CT teachers often do not have the luxury and time of doing two. Not even CT professors.
Take CT people and take math people in general. Is there a difference between how they at average engage in the third task (ignoring the situation about how much time they have for such engagements)?
Am I right or am I right?
If I'm right, what happens before a meeting between a categorist professor and, say, a medical professor. Will the medical professor be curious to know something about CT "before" the meeting? No. Will the CT professor be curious to know something about the medical domain to be discussed "before" the meeting? Should be, but in most cases will not. So what happens after the meeting? The medical professor go backs to office and continues research in, say, cardiology. The CT professor goes back to office and continues to remember the good old days when we didn't even think about talking to professors in other disciplines.
Am I wrong or am I wrong?
If I'm wrong, then there is hope, otherwise not.
And whenever we might go "real", do remember that we must "FIRST find the problem, THEN design a solution", not the other way around ("FIRST design a solution, THEN find a problem where the solution comes into play").
I rest my case.
---
Best,
Patrik Eklund
PS CT became CT some less than a century ago. Suppose CT had become CT some decades before, say, even before Hilbert wrote his foundations of geometry. In that case, CT had been around by the time Bernays was recruited to Göttingen, and I bet Gödel's ideas would never have reached publication in 1931. Hilbert's and Bernays' two volumes on the foundation of mathematics might even have been preceded by another two volumes on the foundation of the foundation of mathematics. Who knows. I bet Mac Lane never thought about it that way. Otherwise he would have called Bernays and said "Hey, we really need to talk!". But I don't see that in Bernays correspondence, so here we are, 75 years later, still not talking to each other.
On 2026-05-31 16:29, Steven Vickers via Categories wrote:
Hi Peter,
I'm sad to hear of the decline of category theory in the Maths Faculty.
All the same, I know there are people in the Computer Lab who would be be qualified to teach it at pt III level, and they would be on the University payroll. Is that route not possible for some reason?
To me it's a sign of how category theory has developed into a piece of applied maths, just not in the DAMTP sense. At Birmingham too, nobody in the School of Maths really does category theory.
All the best,
Steve.
________________________________ From: P.T. Johnstone via Categories <categories-list@categories.org.au> Sent: Friday, May 29, 2026 10:11 AM To: Marino Gran via Categories <categories-list@categories.org.au> Cc: P.T. Johnstone <ptj1000@cam.ac.uk> Subject: [categories] Category Theory under threat in Cambridge
CAUTION: This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.
As many subscribers to this list will know, I have continued to lecture a course on Category Theory in Part III of the Mathematical Tripos in Cambridge every year since Martin Hyland and I retired (there being no-one on the current Faculty who is able to do it). I'm quite proud of the fact that I didn't even miss a year as a result of Covid!
I have now been informed that Cambridge University has a new policy of not allowing people who are not on the University payroll to teach examinable courses, which if implemented strictly means that Category Theory will cease to be an option in Part III from next year.
I know that there are many people on this list who first encountered categories through Martin's or my Part III lectures. If you feel, as I do, that the suppression of this option is a disgrace to Cambridge, I'd be grateful if you could write to the Head of DPMMS, Ivan Smith (is200@cam.ac.uk) and tell him so.
Peter Johnstone
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
_______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au
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participants (7)
-
Alberto Peruzzi -
George Janelidze -
Graham White -
P.T. Johnstone -
peklund＠cs.umu.se -
Posina Venkata Rayudu -
Steven Vickers