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John Baez (May 29 2026 at 18:13):For decades, Cambridge has been a center of category theory. Olivia Caramello, Eugenia Cheng, Tom Leinster, Richard Garner, Christina Vasilakopoulou, Stephen Lack, Paul Taylor and many others learned it there.
It's still going strong in the department of computer science, but not within mathematics. Now the only person there teaching the usual course is Peter Johnstone - and they may not let him anymore:
P.T. Johnstone via Categories <categories-list@categories.org.au>
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
Btw, he almost died of COVID.
Nathanael Arkor (May 29 2026 at 19:06):It would be a big ask, but I wonder whether any of the lecturers in the Computer Science department at Cambridge might be willing to take on the Category Theory course in the Mathematics Tripos? (That said, I suspect this would still not be entirely satisfactory to Peter, as I imagine he wants to consider lecturing it himself.)
CC @Jon Sterling, who may be able to share some insights as one such member of the CS department.
Jon Sterling (May 29 2026 at 19:15):I don't think this is likely to happen, unfortunately. The Theory Group (where we category theorists live) is extremely load-bearing as far as computer science modules are concerned, and I think the likelihood of us having bandwidth to take on lecturing in the Mathematical Tripos is relatively low... In fact, we are struggling to cover (our own version of) Category Theory as well as other Theory modules at the moment because three of us are going on Sabbatical at the same time next year.
John Baez (May 29 2026 at 20:31):Are mathematics students allowed to take category theory at the computer science department? If so, do they?
(It's clear having Johnstone teach it is not a sustainable solution in the long run.)
Jon Sterling (May 29 2026 at 20:36):We certainly have no problem with them auditing category theory with us — as to whether they can be examined by us, I am not sure what the regulations are.
But I don't think this is going to be a solution either, because our category theory module is totally different from Peter's (being more lambda-calculus focused), and it covers a (very small) fraction of the material that Peter covers due to, shall we say, a different level of preparedness among the students of the respective triposes.
Jon Sterling (May 29 2026 at 20:37):To be honest, I have been advocating more of a "sink or swim" approach to our category theory module that might bring us a bit closer to parity with the Mathematics version, but I think I might be in the minority on that debate ;-)
John Baez (May 29 2026 at 20:45):Yes, the lambda calculus focus will not be what most math students need or yearn for.
Jon Sterling (May 29 2026 at 20:50):Personally I don't love our level of focus on the typed lambda calculus. The problem, in my opinion (with all due respect to Andy who wrote this module), is that once you are enlightened, "typed lambda calculus == CCC" is obvious and doesn't feel very profound. So, the result for this module is:
1) The students whom we manage to enlighten end up being let down, because the punchline (post-enlightenment) is some kind of change of notation rather than a useful theorem of computer science.
2) The students whom we do not manage to enlighten will be enthralled by the mystical connection of :place_of_worship: typed lambda calculus :place_of_worship: and :place_of_worship: CCCs :place_of_worship:, and I have never felt that we needed to send out even more poorly informed Curry–Howard–Lambek missionaries out into the world, who do not know what a colimit is.
I would greatly prefer to teach category theory indirectly through some mathematics that is relevant to computer science, such as domain theory. But failing that, if we must teach a _pure_ category theory course, I think it is important for students to come out with more breadth within the topic than we are currently giving them.
John Baez (May 29 2026 at 20:59):Do they come out knowing about adjoint functors?
Nathanael Arkor (May 29 2026 at 21:07):There was a year Marcelo Fiore taught the category theory course in the CS department that was much less focused on the STLC, but that material seems to have been forgotten once Andy returned from sabbatical.
Nathanael Arkor (May 29 2026 at 21:08):The students typeset lecture notes that year; I may be able to find them if there's interest.
Jon Sterling (May 29 2026 at 21:11):This is the current syllabus: https://www.cl.cam.ac.uk/teaching/2526/CAT/. They do learn adjoint functors.
Jon Sterling (May 29 2026 at 21:12):There is some good non-lambda material after the CCC stuff is finished. The presheaves and monads are very useful for computer science, of course.
John Baez (May 29 2026 at 21:12):Okay, that's good. Yes, limits and colimits seem like the first important thing to be left out. But there will always be something important left out.
Morgan Rogers (he/him) (May 30 2026 at 16:20):Heh... I applied for a position in the maths dept in Cambridge two years(?) ago, pointing out this possibility in my application, but I didn't make it to interview. I don't know what the precise factors were in the progressive decline in the population of category theorists in the maths department, but it's been a long time coming.
John Baez (May 31 2026 at 08:39):I know that at one point DPMMS (the Department of Pure Mathematics and Mathematical Statistics) at Cambridge made an offer to Charles Rezk. I believe Martin Hyland was head at this time, and hoped that Rezk's appointment would continue the prominence of category theory there. But Rezk wound up not going there.
Oscar Cunningham (Jun 02 2026 at 08:43):There's been an update from Johnstone with good news.
David Corfield (Jun 02 2026 at 08:46):Good news, but probably still worth using the next couple of years to prepare for a successor.
Nathanael Arkor (Jun 02 2026 at 08:49):How, given that there are no category theorists with hiring power in the mathematics department at Cambridge?
John Baez (Jun 02 2026 at 13:00):There might still be a possibility, but it feels like it's coming a few decades late. Hyland was head of DPMMS; that was the time to hire another category theorist.
Joe Moeller (Jun 03 2026 at 21:59):In general, if we want category theory to be a lively and prosperous field, we must always be investing in the new generation by teaching it to a broad audience, supporting the work of young researchers, and hiring them.