julesh

Applied Compositional Thinking

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Last Notes

2026-10-31 22:04:09 GMT

Something just clicked for me, while down a random youtube rabbit hole watching a guy trying to make an ancient bulldozer run
Engineers do not work on engines (except maybe on ships, but ships also have their own dialect words for "left" and "right", so I think it's reasonable to ignore them). In English, people who work on engines are called mechanics. Software engineering is exactly like being a mechanic: you don't have any foundations, you don't need to know calculus, you just have a bunch of black box systems connected together that don't work and you have to use a bunch of tricks and trial and error to figure out what's wrong. Software is like an engine, in that sometimes the thing that fixes it is injecting some extra flammable gas and whacking it with a big hammer

2026-08-03 18:12:17 GMT

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@nprofile…qwda I'm also running Tuxedo OS, and I installed gnome on day 1 but I am possibly having regrets because gnome is not much like it was 15 years ago

2026-08-03 13:15:30 GMT

Alright nerds I need linux advice, I think I have probably come to the right place
This laptop ships with a fairly heavily modified fork of Ubuntu that removes a lot of the Canonical stuff and replaces Unity with KDE
Last time I used linux was about 15 years ago. Back then I used Ubuntu with Gnome, and I personally disliked the feel of KDE a lot. How have things changed since then, and in particular in the shitshow that has been the last 6 months? If the first thing I do to this machine is replace KDE with Gnome, am I going to have regrets?

2026-08-03 12:06:53 GMT

I think it is not unreasonable to say that companies that force you to use American tech are literally treasonous

2026-06-28 20:42:54 GMT

In this paper we improve the bound on the value of Dylan's constant 𝐷. This value, first defined by Dylan [1963], is the number of roads that a man must walk down.
We briefly recap the history of Dylan's constant. Although it is not a priori obvious that a man must walk down a finite number of roads, a relatively elementary nonconstructive proof that 𝐷 is finite, using the pigeonhole principle, was quickly given by Marrow, Stookey and Travers [1963]. The first explicit bound, a stack of exponentials of height 5, was found by Wonder [1966]. This bound was significantly improved by Baez [1976] to 𝐷 ≤ 420 using methods of higher category theory. Although this bound remains the best known currently, Adams [1980] conjectured that 𝐷 = 42 based on extensive computer calculations.

2026-06-27 18:06:35 GMT

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In theory a dot matrix printer could print 1 page in the time a laser printer could print like 20 pages, but in reality a dot matrix printer could print 1 page in the time that a laser printer can tell you that it's out of cyan

2026-06-27 18:02:48 GMT

Why are there open source 3d printers but no open source 2d printers? Surely a dot matrix printer wouldn't be *that* hard to make and it would be able to handle 99% of things most people need

2026-03-31 10:00:37 GMT

More than 6 years after noticing it I have *finally* understood the Bayesian chain rule (*) intuitively
If you have conditional distributions P(y | x) and P(z | y), the Bayesian chain rule really just says
P(x | z) = ∫dy P(y | z) P(x | y)
Only if you expand P(z | y) using Bayes' law you get the term P(y), which is the pushforward of the prior. It's exactly like writing the differential chain rule in Leibniz notation,
dz/dx = dy/dx dz/dy
which similarly notationally suppresses the pushforward compared to the Lagrange/Jacobi notation
(g.f)'(x) = f'(x) g'(f(x))
* https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.24

2026-02-21 14:04:23 GMT

Downside of LLM culture that I have not seen mentioned anywhere before: The best way to understand something is to explain it to someone else, and people don't ask people to explain things anymore

2026-02-19 14:00:36 GMT

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@nprofile…97ud There is a better resolution using type-in-type: an omnipotent God can create a stone so heavy that they can't lift it, and then they can lift it

2026-02-19 12:33:17 GMT

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@nprofile…sqfw What is this heresy? Category theory is not superior to the Son or the Holy Ghost, but equal to them

2026-02-16 21:22:17 GMT

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@nprofile…4hp5 I wonder if there's some model theoretic universality result that all finite (or not too big) commutative rings embed into one specific commutative ring that involves numbers, like the ring of rationals or something like that?

2026-02-16 20:56:15 GMT

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@nprofile…4hp5 I wonder what this looks like for rings that have very little to do with numbers, like boolean rings or rings of functions

2026-02-13 12:08:17 GMT

It appears that I do need bunched contexts. I now have a bunch of problems.

2026-02-10 13:21:55 GMT

I begrudgingly agree with everything Macron said. I for one welcome our new French centrist overlords
https://www.bbc.com/news/articles/ce8n1zdnpd3o

2026-02-03 20:27:18 GMT

Higher category theory for synth players:
globular = sine wave
cubical = square wave
simplicial = sawtooth

2026-01-28 13:56:42 GMT

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@nprofile…97ud Leading to a hilarious moment of "how did *that* end up falling in a black hole?"

2026-01-27 14:37:43 GMT

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@nprofile…tfwq I'm preparing to move my stuff off Godaddy and OVH are my leading contender, do you still recommend them?

2026-01-24 00:22:20 GMT

Today I learned that during the ice age there was a really big river in Doggerland that included the Rhine, the Seine and the Thames as tributaries. Or I guess you could say, the Rhine continued and gained the Seine and Thames as tributaries
https://en.wikipedia.org/wiki/Channel_River

2026-01-23 18:08:59 GMT

For sale: Linear variable, never used

2026-01-20 18:52:07 GMT

It doesn't seem to be widely reported that the EU will be imposing some chonky tariffs on the US starting in a couple of weeks (source: BBC news)
https://media.mathstodon.xyz/media_attachments/files/115/928/922/547/121/982/original/b1bd65c735a48465.png
https://media.mathstodon.xyz/media_attachments/files/115/928/924/490/812/728/original/e13031c518506250.png

2026-01-18 14:22:02 GMT

"Machine learning is linear algebra" factoid statistically inaccurate. Machine learning is statistics

2026-01-17 11:19:41 GMT

I used to think the "used future" aesthetic was cool but a bit silly, like in Star Wars where canonically technology hasn't really advanced in thousands of years, but then I realised that technology hasn't really advanced in about 15 years and it feels suddenly plausible

2026-01-14 14:30:50 GMT

The hardest problem in computer science is naming things, therefore Nicolaas de Bruijn solved the hardest problem in computer science

2026-01-14 12:26:21 GMT

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@nprofile…amew Well yes but everything is that, that doesn't distinguish what is special about transformers

2026-01-14 11:15:46 GMT

It's weirdly difficult to find good sources that explain how transformers and attention actually work, considering how important they are. Everybody just seems to repeat a variant of this diagram from the original paper, as though it actually explains anything
https://media.mathstodon.xyz/media_attachments/files/115/893/159/408/201/698/original/cdb664f51e5d8838.png

2026-01-06 20:23:55 GMT

Assembling furniture without instructions feels like solving a well designed puzzle, right down to that moment when you have finally understood the puzzle deeply enough to figure out what the real puzzle is

2025-12-31 13:16:05 GMT

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@nprofile…4x5t I've still got a good 30 years to go until retirement and I have enough funding runway right now to spend time reskilling as a lumberjack or a circus juggler or a vtuber or whatever, so.... bet you my career that it doesn't?

2025-12-31 11:01:57 GMT

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@nprofile…4x5t I don't doubt that, but it doesn't change what I said

2025-12-30 19:23:04 GMT

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@nprofile…4x5t There was me thinking that (human) chess was currently going through its greatest golden age since the cold war

2025-12-26 21:42:42 GMT

I sat down next to the fire with my laptop but then a lap cat pulled rank so my laptop is now a chairarmtop
https://media.mathstodon.xyz/media_attachments/files/115/788/040/685/647/847/original/64b0ef912691f480.jpg

2025-12-26 13:07:44 GMT

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@nprofile…8eua This is definitely true for keyboard instruments and most ensembles, but most non-keyboard instruments have a "natural" key (eg. G for violin, which causes the lowest string to resonate on the fundamental and the next string to resonant on the 5th), and I reckon some very pedantic people playing solo instrumental music must tune a few cents off the standard tuning to match that

2025-12-26 11:06:23 GMT

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@nprofile…8eua A discrete manifold equipped with an approximate scalar field? That sounds like it could be useful for some other stuff too

2025-12-25 13:32:19 GMT

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@nprofile…8eua Today is the first day of Christmas so you should only expect a(1) = 1 gifts

2025-12-25 10:39:36 GMT

The number of gifts sent to you by your true love up to and including the 𝑛th day of Christmas is
𝑛(𝑛+1)(𝑛+2)/6
https://oeis.org/A000292

2025-12-22 13:12:42 GMT

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@nprofile…8eua It didn't occur to me before to switch to HAL, I'll actually think about that now

2025-12-21 21:01:43 GMT

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@nprofile…8eua I'm not sure I have that much to say, especially not that you haven't already said. To be honest, I feel like doing interdisciplinary research is so difficult that if you're in a position where advice from a blog would help you, you've probably already failed

2025-12-21 14:53:35 GMT

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@nprofile…8eua I had to scan back through my blog to see if I'd forgotten anything, but no, I never wrote about it. Well, pretty sure I wrote a few twitter threads about it, but it would be easier to write it again than to track those down

2025-12-21 14:07:34 GMT

There is no AI bubble. There is an LLMs bubble, and among all the other problems it causes it's also blocking progress in AI far more effectively than all the AGI doomer bros could ever hope to achieve

2025-12-21 13:58:21 GMT

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@nprofile…8eua I submit {me, Philipp Zahn} (with several other people joining at various points) as a successful ACT collaboration. My experience broadly matches yours, this collaboration is more than a decade long and required very much meeting each other half way

2025-12-20 17:32:37 GMT

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@nprofile…4x5t I'm not the right person to answer that (I happen to be an Idris programmer); my vague feeling is that Agda is the main language used as a testbed for new ideas (ie. the same role that Haskell used to have for polymorphic type theory), so it will probably continue to evolve, provided they have the courage to periodically break everybody's stuff like ghc always has. Perhaps Lean will do that too, but I would guess they probably face more social pressure to not break the big libraries

2025-12-20 16:47:31 GMT

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Science advances one funeral at a time, and a bunch of mathematicians who are currently young and betting their career on Lean are eventually going to be dinosaurs sitting on hiring committees and insisting to their grave that it is good actually that their favourite proof assistant falls over if you hit it with a corner case of dependent pattern matching on something defined with coinduction-corecursion or whatever

2025-12-20 16:44:23 GMT

While it's cool and good that a bunch of mathematicians are experimenting with proof assistants, I'm very confident that one of the biggest consequences is that 20 years in the future, it will finally be sufficiently understood how to write a dependent typechecker, and a bunch of mathematicians are going to learn the hard way the meaning of the term "technical debt"

2025-12-16 12:07:57 GMT

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@nprofile…ztfm I always thought about it like this: "any measure that becomes a target ceases to be a good measure" is such a widely known principle that whenever anybody proposed any measure whatsoever as a target I immediately knew that they were totally incompetent, and so I could safely disregard everything else they say too

2025-12-15 12:24:00 GMT

What the LaTeX feature of Mathstodon is supposed to be used for: Writing formulas that appear beautifully typeset (except to people on any other server and anyone using any client except the web client)
What I actually use it for: quick access to the → unicode symbol for talking about my train journeys

2025-12-15 12:17:56 GMT

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@nprofile…u6hy I was wondering which cities I've been to between 5 and 10 times. My first thought was Amsterdam

2025-12-14 16:38:14 GMT

"Boymoding enbies on trains" doesn't have quite the same ring to it, but oh well
https://media.mathstodon.xyz/media_attachments/files/115/718/900/049/912/602/original/90df5c547cfeefa3.jpg

2025-12-09 17:30:15 GMT

If software engineering is replaced by vibe coding, we should also replace electrical engineering with vibe wiring, civil engineering with vibe bridging, aerospace engineering with vibe flying etc

2025-12-09 13:28:37 GMT

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@nprofile…ztfm uBlock Origin has no problem at all with native adverts on other sites, so I guess it's a non issue

2025-12-08 12:42:28 GMT

This is fine and entirely reasonable, but I kinda wish the arXiv had generated this error during the submission process instead of immediately afterwards
https://media.mathstodon.xyz/media_attachments/files/115/683/997/610/455/664/original/ab57cc75f6a55f76.png

2025-12-07 15:48:11 GMT

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@nprofile…ral2 I find theorem proving (even for non open problems) impressive for the same reason that AlphaGo still blows my mind: algorithmic search for such high dimensional problems with no known metric seems to be obviously impossible

2025-12-03 23:15:50 GMT

Catching the subway home at 11.15 and I haven't missed the last one, finally I'm in a real city

2025-12-02 16:18:10 GMT

First I lived in Oxford when Lewis was still be regularly filmed, and later a chunk of a Transformers film was also filmed there. Then I moved to Leipzig where a police procedural for local tv was filmed in the same building I worked in. Then I moved to Glasgow where many Hollywood films are filmed (in the 5 years I was there there was a bunch of Batman, a bunch of the Catwoman that was infamously deleted for tax reasons, and a scene of Indiana Jones). Now I live in London and *something* is being filmed outside our window, although I haven't figured out what yet.

2025-12-01 10:29:28 GMT

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@nprofile…ral2 They made up a number and got mad at it

2025-11-28 22:15:58 GMT

This idiot left their laptop in a cafe so I'm forced to (gasp) read books
Just did the first half of Logicomix and now making a start on Piketty's Capital

2025-11-25 21:54:19 GMT

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@nprofile…w8t5 This is not a crazy idea but I'm pretty sure you did just make it up, I've never heard of an "etiquette" like that in my life
Mostly it sounds like you're trying to get a mob of humans to manually simulate tree threading, which would be done much better by sending mastodon a pull request for the feature

2025-11-24 12:05:03 GMT

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@nprofile…ztfm i saw Helfgott give a talk about this proof once. I understood basically none of the details (starting from the explanation of the classical Hardy-Littlewood proof that it starts from), but it was still one of the coolest talks I ever saw

2025-11-23 12:30:07 GMT

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@nprofile…4x5t I've been calling it a Tambara comodule, I wrote a tiny stub about it here, I noticed the connection to traces but didn't write about that part yet
https://julesh.com/forest/000D/index.xml
An equivalent but less slick definition plays a big role in an old paper of mine, "The game semantics of game theory", I called it something like a "context functor" back then

2025-11-18 11:17:57 GMT

It's funny how a bunch of my local mastodon discourse today is about how the "u" sound doesn't exist in American, that's a thing I've joked about for years. Waaaay back in the day my twitter display name was "Categories with Jules" and only after like 2 years I got rid of that because I suddenly discovered that anyone who learned USEn just didn't get the joke (because they would pronounce this nLab page as "categories with dools", which amuses me to this day)
https://ncatlab.org/nlab/show/category+with+duals

2025-11-18 11:13:07 GMT

A genre of twoot I just don't understand is screenshots of google dot com showing a hilariously and/or dangerously wrong answer. Like, you're using an LLM and trying to treat it like it's a search engine, literally what did you expect?

2025-11-14 13:31:07 GMT

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@nprofile…rrqe
https://julesh.com/forest/all/index.xml

2025-11-10 14:13:59 GMT

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@nprofile…4x5t @nprofile…0775 For me the interesting question is why every actegory gives you both, while general enrichments only give Para and only tensored ones give you Copara. Feels like a broken symmetry

2025-11-10 11:54:27 GMT

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@nprofile…4x5t As far as I can tell the only place it's mentioned in my blog is here, in passing
https://julesh.com/posts/2024-04-12-enriched-closed-lenses.html
I also recently started writing notes about it here
https://julesh.com/forest/0005/index.xml
but that's very much still half finished
https://media.mathstodon.xyz/media_attachments/files/115/525/262/141/552/431/original/acb829c0c95a1b26.png

2025-11-10 11:04:24 GMT

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@nprofile…4x5t Inside our group that was Dylan. I don't remember if it ever got written down anywhere though

2025-11-08 11:43:09 GMT

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@nprofile…u6hy Argh, the Android client I use (Moshidon) already supported quote-tweets...... but not in a compatible way 😭
https://media.mathstodon.xyz/media_attachments/files/115/513/887/862/020/438/original/50ce7f340c966124.jpg

2025-11-05 17:21:37 GMT

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@nprofile…ral2 Fun fact, since negation is strictly involutive (p and not-not-p are different names for the same proposition), proof of negation and proof by contradiction are actually different names for the same logical rule, they only differ by a relabelling

2025-11-02 13:03:20 GMT

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@nprofile…mk8f No, 100% organic artisan CSS

2025-11-02 12:28:12 GMT

https://julesh.com/ is finally starting to look close to how I would like, after changing the font to Open Sans 3
https://media.mathstodon.xyz/media_attachments/files/115/480/092/771/387/183/original/b698cecc81ada5ba.png

2025-11-01 12:06:59 GMT

Announcement! Me and @nprofile…kkl9 are moving to London later this month
I will still be visiting Glasgow roughly once a month, for the foreseeable future

2025-10-31 23:09:13 GMT

I am unable to say the word "impredicative" without it turning into something from the "am I pregnet?" video

2025-10-29 19:58:00 GMT

I have a probably-basic analysis question
Nonnegative numbers satisfy the inequality
𝑥₁𝑦₁ + 𝑥₂𝑦₂ ≤ (𝑥₁ + 𝑥₂)(𝑦₁ + 𝑦₂)
Do nonnegative functions satisfy the corresponding inequality
∫ 𝑓𝑔 ≤ (∫ 𝑓)(∫ 𝑔) ?
(In case it helps I think I can also guarantee that the integrals are all finite)

2025-10-29 17:46:14 GMT

Oh, I've been spelling it "adaptors" for years, and only just realised it was meant to be "adapters"
https://www.cs.ox.ac.uk/people/jeremy.gibbons/publications/poptics.pdf
https://media.mathstodon.xyz/media_attachments/files/115/458/692/411/023/619/original/6b348afb95ea3efd.png

2025-10-29 12:44:08 GMT

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@nprofile…ral2 When ported all my blog posts recently I checked all the links, and noticed that a few that claimed to be to my twitter threads were dead

2025-10-28 22:40:44 GMT

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@nprofile…stct "Almost alarmingly feasible" is a fun phrase to see in a serious (?) physics paper
Hopefully there is a plan to maintain those "small course corrections" over the presumably millions of years this takes, otherwise it could make a bit of a mess

2025-10-26 11:12:58 GMT

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@nprofile…jahe I might do that if I get stuck, but I only ever use an llm if I want to do something without learning how to do it, and I do in fact want to learn css

2025-10-25 11:35:36 GMT

I really want to improve the visual style of https://julesh.com. Unfortunately this requires learning something about CSS. This is quite a dilemma.

2025-10-25 10:15:25 GMT

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@nprofile…ral2 I think the hype bubble was so ridiculous that nobody can be taken seriously for working on it even like 3 decades after it burst

2025-10-23 21:51:20 GMT

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@nprofile…ztfm @nprofile…8eua
This is the orange site, if you don't expect the worst from it then you haven't been around it long enough

2025-10-23 20:33:53 GMT

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@nprofile…ztfm @nprofile…8eua
https://news.ycombinator.com/item?id=45677397
(but nothing especially worth seeing there)

2025-10-22 16:58:36 GMT

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@nprofile…ztfm She does have a daemon, you just can't see or interact with it

2025-10-22 15:16:27 GMT

Liminal space in the National Museum of Scotland
https://media.mathstodon.xyz/media_attachments/files/115/418/475/795/015/861/original/9ec15f4304d4d8cd.jpg

2025-10-20 23:12:24 GMT

The proof is left as an exorcism for the reader

2025-10-18 19:53:07 GMT

Big shoutout to the community of people who could be developing better ballistic missiles or earning a fortune as quants but instead are saving frames in super mario 64 tool assisted speedruns
https://www.youtube.com/watch?v=_yx0eutBwII
https://media.mathstodon.xyz/media_attachments/files/115/396/904/348/980/713/original/a546437906d8786b.png
https://media.mathstodon.xyz/media_attachments/files/115/396/905/151/410/243/original/2c971f3473c99180.png
https://media.mathstodon.xyz/media_attachments/files/115/396/906/387/838/030/original/258420204396ade0.png
https://media.mathstodon.xyz/media_attachments/files/115/396/907/177/808/101/original/535ba346b61cf345.png

2025-10-18 10:36:51 GMT

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@nprofile…ral2 My pet theory is that the main purpose of sleep is to copy memories from the cortex to somewhere more stable, but that requires suspending normal operation of the cortex. Meanwhile anything running on digital hardware you can generally copy anything freely without disrupting anything

2025-10-17 11:13:19 GMT

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@nprofile…ral2 My gut feeling is some version of the first thing will actually be necessary, but not the second thing (whatever is the function of sleep, my gut feeling is that it is probably a quirk of the weird hardware the mind runs on)

2025-10-16 20:58:10 GMT

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@nprofile…w8t5 Academics gonna academic

2025-10-16 20:06:18 GMT

Personally I think it would be more appropriate if Birmingham New Street Station was renamed Crazy Train Station
https://media.mathstodon.xyz/media_attachments/files/115/385/639/167/805/207/original/40e162a1c34b6207.png

2025-10-16 13:54:54 GMT

New blog post!
Dependent optics II: Optics via forcing costates
https://julesh.com/posts/2025-10-16-dependent-optics-ii.html

2025-10-16 11:08:23 GMT

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@nprofile…w8t5 Hm, I find this a pretty interesting question, at first I strongly disagreed but I'm sort of coming round to your point of view. If the rules of form are only being checked externally, either by a machine or by other humans, are you working formally or informally? If you submit classical paintings over and over until the curator accepts you got the form right, were you following the laws of form? Is it possible that Bach intuited the formal rules of fugue so deeply that he was actually writing informally?

2025-10-14 20:51:10 GMT

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@nprofile…ral2 Use a static site generator, which compiles (mostly) markdown into html. The big one is Jekyll (we use that for the Cybercat blog), I'm using Hakyll which is written and configured in Haskell, some other more obscure options also exist. I considered using a paid web host (specifically Godaddy, who are also my domain registrar), but since I wanted to be open source anyway I'm going with Github Pages to actually serve the site (I wish there was a less shitty option that is free, but Codeberg Pages seems to be in an unstable state right now)
When you do the migration, take a whole day to focus on it and read all the guides on all of Wordpress, Github Pages (or whatever) and your domain registrar. For me the main steps of the process were:
1. Unlock the domain registration from Wordpress
2. Order the domain transfer from Godaddy
3. Transfer the DNS nameservers from Wordpress to Godaddy (at this point you're on the clock before your site goes offline)
4. Following instructions on Github Pages, add a DNS record to link your Github account
5. Add 8 more DNS records that make your domain point to the Github Pages site
A bunch of that can technically be skipped if you continue using Wordpress as the registrar while hosting externally, but I tried that last time and I don't recommend it. While Wordpress is a domain registrar, all of their domain management tools are built assuming that you use their in-house hosting

2025-10-14 18:29:30 GMT

Migrating https://julesh.com off https://wordpress.com has been something I've wanted to do since at least 2019. I'm so so happy to have finally done it.

2025-10-11 12:45:42 GMT

I have questions about the kleisli category of the continuation monad K_R on the category of sets
1. I think that the kleisli category is independent of the choice of R (up to equivalence) as long as R has cardinality at least 2, so there are many monads but they all share the same kleisli category. Is that correct?
2. The kleisli categories have coproducts (as all kleisli categories do) and it is coclosed. The cartesian product of sets fails to lift to a monoidal product, it's only premonoidal. Does the kleisli category have any monoidal structures other than the coproduct?

2025-10-09 19:22:56 GMT

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@nprofile…v2av I have good news, if you skip step 2 then you can actually drop the requirement of decidability

2025-10-07 16:22:35 GMT

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@nprofile…fa4k It could still work if you firmly think of them as open-ended projects with no defined endpoint

2025-10-07 15:08:21 GMT

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@nprofile…fa4k Is it feasible to say all of the above, all interleaved in a random order

2025-10-04 17:24:33 GMT

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@nprofile…hszg They are models of language and they are pretty large, calling it an LLM is both accurate and puts it appropriately in the context of NLP as a field

2025-09-29 22:20:31 GMT

For some reason I was 100% convinced Wikipedia was based in Sweden, I'm actually horrified to learn that it's entirely American
So...... rest-of-world alternatives to Wikipedia?

2025-09-11 09:51:06 GMT

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@nprofile…pmdx My understanding is that it's a takeover of Kent by the management of Greenwich... but it's a reasonable guess that it won't work out great for the staff at Greenwich either

2025-08-28 16:49:09 GMT

For a while I've been struggling with how "abstract syntax" means so many different things in a modern compiler pipeline, but I finally came up with a good naming scheme:
raw syntax: the character string read in from the file
rare syntax: after lexing and parsing, but no static indexing
medium rare syntax: statically well scoped
well done syntax: statically well typed
overdone syntax: the object code produced by the backend

2025-08-27 15:39:41 GMT

Me and @nprofile…xh84 are mostly done with implementing a noncommutative lambda calculus where types denote endofunctors, terms denote natural transformations, tensor denotes endofunctor composition and arrow denotes right kan extension