You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
JR Learnstomath (Apr 10 2026 at 12:24):Help! Idk whether and how one would do some work, and then say, "There. I have applied CT to X."
My first working informal definition was that if I think of a few ideas in X, say x, y, and z, I would be able to read some set of CT papers A and specify a map X -> A that sends x, y, and z to one or more a_i's in A. Hmm, writing this, now I'm wondering if I have that backwards.
This is a burning question for me, because I have spent a long time looking at Tae-Danae Bradley's "What is applied category theory?" and lots of textbooks on ACT/CT, and even logic, but have not found an explicit ACT definition.
What is a good, simple one (like mapping concepts to CT structures or vice-versa), and where can I find it please?
fosco (Apr 10 2026 at 13:37):uh, not much time to write, but I will throw some raw ideas
Many things happened since 2018, the community inflated/deflated the importance of some research lines...
...community that I do not feel I represent, so don't take me seriously. I think I lean towards the "hasn't category theory always been applied?" gang, although I think I give a very peculiar meaning to this rethorical question.
From the outside, I see that the community is looking for an identity, or to reach a critical mass so that sub-communities will split (the biggest: probability theorists); many things will happen, and I believe an especially liminal moment will be the next 2-3 ACT conferences.
Lawvere did a great deal of "applied category theory" in a sense that I find difficult to debate; but he did also something equally important, i.e. clarify the importance of categorical constructions per se, given their ubiquities in applications. Probably an hot take: "Diagonal arguments and cartesian closed categories" and "Quantifiers as adjoints" are papers on "applied category theory", because the focus is to use CT to do something. But they also clarified the mathematics of adjointness, and the extreme ubiquity of fixpoint theorems regarded with a categorical lens. These two results are purely mathematical.
Sometime ago there has been a quite heated discussion on "what are some real-world applications of category"; after some time, André Joyal replied in the most elegant way, saying that "category theory's best application is to mathematics. Mathematics is a part of the real world." I don't think there exists a better answer.
fosco (Apr 10 2026 at 13:47):(...Joyal who coined an entire theory, rooted in category theory, to explain the meaning of combinatorial identities. That was applied category theory too.)
You should not listen to me, as I am a pure mathematician, educated by ultra-Bourbakist mathematicians who actively despised any attempt to descend from the Empyrean tagging it as a sign of weakness, but if you really want my take, it is that "applied" category theory is the use of category theory to do mathematics that does not pertain to category theory.
fosco (Apr 10 2026 at 13:47):In such a sense, it is almost unavoidable to do it, at some point in one's life.
fosco (Apr 10 2026 at 13:57):(incidentally, I think it's impossible to have a formal definition for ACT, as much as it is impossible to have a formal definition of what was Queneau's OuLiPo project. Animated by a certain philosophical take, some people made statements and built artifacts to backup those claims. You can say more, but not substantially more than that)
Joe Moeller (Apr 10 2026 at 19:30):My take is that Applied Category Theory is a community, not a topic. So people have been applying categories to a wide variety of topics for a long time, and where the line is for "applied" and "not applied" isn't really relevant in this lens. Whoever is organizing, speaking, publishing in our venues is a member of the community. It sorta feels like categorical quantum computation is becoming less ACT in that I see them less at the conference, and opposite for cellular sheaves. I don't have hard numbers for these, just my gut.
John Baez (Apr 12 2026 at 01:03):JR Learnstomath said:
Help! Idk whether and how one would do some work, and then say, "There. I have applied CT to X."
You can say it whenever you want; it's just a question of how many people will believe you. :wink:
In other words, there's a continuum of "application", where at the beginning you take a concept from some field and say something about it using category theory...
and then you develop a collection of ideas around this concept...
and then you prove some theorems about these ideas...
and then you write some software that takes advantage of these theorems...
and then you use the software to do something useful in the field that motivated you initially...
and so on.
For example, my student Brendan Fong wrote his thesis on applied category theory:
and it's very worth reading. Later my student Kenny Courser simplified some of these ideas, making it easier to develop them further:
Then @Evan Patterson and @James Fairbanks and @Sophie Libkind and many other people developed a software framework called AlgebraicJulia based on these ideas and many other ideas from category theory:
You'll see many specific software packages on this page.
Then my friend Nathaniel Osgood and started developing software for epidemiology based on AlgebraicJulia. This is a good introduction I think:
However I still don't feel that the process of application is complete, because we haven't really used this software to fight disease or even write a research paper on epidemiology!
I often feel that application is only "done" when you've helped feed someone, or kept someone from getting sick, or prevented a forest from getting cut down, or something equally concrete.
Nonetheless, most people would call all the papers I listed applied category theory, so you can look at them if you're wanting an example of ACT.
There are plenty of other examples; this is just the one I know best.
You're not going to get a "formal definition" of applied category theory, and you shouldn't want one.
JR Learnstomath (Apr 13 2026 at 10:45):fosco said:
(...Joyal who coined an entire theory, rooted in category theory, to explain the meaning of combinatorial identities. That was applied category theory too.)
You should not listen to me, as I am a pure mathematician, educated by ultra-Bourbakist mathematicians who actively despised any attempt to descend from the Empyrean tagging it as a sign of weakness, but if you really want my take, it is that "applied" category theory is the use of category theory to do mathematics that does not pertain to category theory.
All of your raw thoughts made me think, "Ah! Se cambia todo!!!!" This gives me a lot to reflect on. Anyway, I want to become a pure mathematician in future, so I'm furiously taking notes.
JR Learnstomath (Apr 13 2026 at 10:52):This is really helpful also, and also changes my previous way of thinking completely.
Joe Moeller said:
My take is that Applied Category Theory is a community, not a topic. ... Whoever is organizing, speaking, publishing in our venues is a member of the community.
By participating in Zulip chats and asking questions, does that make me a member of the community?
I don't know when I'd be able to call myself an "applied CT-ist"... I'm definitely an aspiring "(A)CT-ist" (if the difference between (A)CT is not a stark one)! In my head, I'd have to have some sort of paper or give a talk that "uses CT to do non-CT-math or to talk about non-math stuff" where a current upstanding community member says, "ah -- that looks like an (A)CT-ist to me".
JR Learnstomath (Apr 13 2026 at 10:56):Your example above is great, @John Baez , thank you -- another facet to all my views changing. It also makes me think of those itineraries from the medieval times. Like, one could make a syllabus out of this example, and maybe cap it off with a "Choose your own Adventure" like continuation of this story.
Makes me think of @Joe Moeller your work with engineers.
It'll take a little while to rewire my brain away from searching for a formal "ACT" definition, but it's DEFINITELY better for my health to do so, thank you all!
fosco (Apr 13 2026 at 11:43):Just to avoid misunderstandings: I disagree with my masters, yet I consider myself a pure mathematician. Said masters believed topos theory was important, but ignored and despised logic, considering it a pastime activity for mathematicians who failed at doing "real stuff" (they were Pavlov'ed into thinking so, by another old master who had a similar attitude towards logic). I believe I'm a bit better than that...
I have been terminally online since, I think, the end of high school, and the people who know me since that time remember that I used for very long the nickname "killing_buddha", I have thus always been happy to slay my masters when time comes.
fosco (Apr 13 2026 at 11:46):JR Learnstomath said:
By participating in Zulip chats and asking questions, does that make me a member of the community?
It's a good start, but I would say you can safely consider yourself part of the community of X-ers when you give your first talk about your work, at a gathering of X-ers. The bar is, of course, movable a little bit up or a little bit down.. the impostor syndrome can make one feel that even after 20 years they are still not "true X-ers", but one learns to dismiss that voice, with experience...
JR Learnstomath (Apr 13 2026 at 13:49):fosco said:
JR Learnstomath said:
By participating in Zulip chats and asking questions, does that make me a member of the community?
when you give your first talk about your work, at a gathering of X-ers.
This gives a concrete goal to work towards, and with one of the links in the chain that John mentioned above as a topic, then, as well as perhaps volunteering to help organize a gathering. Planning for my future path is one of my development areas and this conversation is filling in a very important piece of that...
John Baez (Apr 13 2026 at 18:42):I'd say you're a member of the community when some people in the community think you're in the community and not too many people violently disagree. (This might be true of many communities?)
Kevin Carlson (Apr 13 2026 at 18:59):This is a recursive definition, which somewhat conceals the possibility of schisms.
John Baez (Apr 13 2026 at 19:22):Indeed! And the schisms don't need to be violent. As Joe mentioned, the "categorical quantum computing" community - or whatever you want to call it - is now fairly separate from the "applied category theory" community. At the first ACT conference, namely ACT2018, there were a lot more folks working on quantum logic / quantum computing. But now quantum computing gets so much $$$ and is so focused on its own concerns that it's hard to imagine it as a strongly integrated part of the ACT community.
David Michael Roberts (Apr 14 2026 at 00:30):I'm surprised no one mentioned Lawvere's early note from 1962 https://lawverearchives.com/wp-content/uploads/2025/07/1962.probmap.pdf Very much applied category theory to my mind, based on the provenance of the work, and the eventual contemporary descendants that we have, eg https://arxiv.org/abs/1301.6201 and subsequent work that lots of people have done in this space.
Joe Moeller (Apr 14 2026 at 00:32):If we do want to talk about the history of applying category theory outside of pure math, then Lawvere is certainly an example in many ways. His fundamental motivation was always understanding physics.
John Baez (Apr 14 2026 at 00:33):At the time, he seemed so far from actually having any impact on physics. I think that day is just beginning to dawn. One reason it was slow is that he was focused on continuum mechanics and didn't like quantum mechanics, while category theory first became visibly important in physics through topological quantum field theory.
(He really didn't like my work on categories in TQFT, and he wasn't shy about saying it.)
David Michael Roberts (Apr 14 2026 at 03:56):Yes, he very much disapproved when I sat next to him at dinner in 2015 and I told him I'd gotten into category theory from reading TWF etc.
John Baez (Apr 14 2026 at 05:49):Hmm I don't remember you mentioning that. I wear the badge of dishonor proudly - since what else can I do?
David Michael Roberts (Apr 14 2026 at 06:53):Lawvere disapproved, you weren't there! His daughter had to explain his disapproval: he was so disgusted it was conveyed in a non-obvious non-verbal way. It was very much also a complaint about people who worked on higher but especially -categories, saying they just repeated theorems writing in front of everything.
David Michael Roberts (Apr 14 2026 at 06:56):I'd brought them up to talk about Urs' work on cohesive higher toposes, but obviously didn't even get to the meat of that topic. (Now there's a way in which category theory is applied, in a way that's disjoint from ACT as colloquially understood!)
John Baez (Apr 14 2026 at 18:44):David Michael Roberts said:
Lawvere disapproved, you weren't there!
What I meant was: I don't think you've every told me that story before.
I wonder what was conveyed in a non-obvious non-verbal way.
David Michael Roberts (Apr 14 2026 at 23:15):In the words of Lawvere's daughter: "right religion, wrong guru" (the second half I remember distinctly, because she had to repeat it, the first half is approximate). It was a reaction so suppressed and puzzling she needed to explain. And, as you can see, had an explanation ready to hand!
I didn't think it polite to spread what seemed to me Lawvere's somewhat ignorant disdain of higher category theory while he was around, nor soon after his passing.
fosco (Apr 15 2026 at 06:45):David Michael Roberts said:
especially -categories, saying they just repeated theorems writing in front of everything.
he must have opened the nLab during the years 2010-2015... 
fosco (Apr 15 2026 at 06:45):(please take this as friendly banter, it is a joke impossible not to make)
Alex Kreitzberg (Apr 15 2026 at 06:59):My favorite sense of applied mathematics was inspired by an observation of Poincaré.
Henri Poincaré said:
For example, I seek to show that some property pertains to some object whose concept seems to me at first indefinable, because it is intuitive. At first I fail or must content myself with approximate proofs; finally I decide to give to my object a precise definition, and this enables me to establish this property in an irreproachable manner.
"And then," say the philosophers, "it still remains to show that the object which corresponds to this definition is indeed the same made known to you by intuition; or else that some real and concrete object whose conformity with your intuitive idea you believe you immediately recognize corresponds to your new definition. Only then could you affirm that it has the property in question. You have only displaced the difficulty."
That is not exactly so; the difficulty has not been displaced, it has been divided. The proposition to be established was in reality composed of two different truths, at first not distinguished. The first was a mathematical truth, and it is now rigorously established. The second was an experimental verity. Experience alone can teach us that some real and concrete object corresponds or does not correspond to some abstract definition. This second verity is not mathematically demonstrated, but neither can it be, no more than can the empirical laws of the physical and natural sciences. It would be unreasonable to ask more.
Emphasis is mine.
To me "Pure mathematics" is "everything that can be established rigorously in mathematics". Which is a useful activity because, if it's done well, then you know mistakes made while doing "Applied mathematics" are in the model, or the calculations, not the pure theory.
Most of the applied category theory I've read fits this pattern in my opinion. There's a mix of "necessarily empirical" applications, as opposed to provable applications, that are supported by "pure category theory".
David Michael Roberts (Apr 15 2026 at 11:04):fosco said:
David Michael Roberts said:
especially -categories, saying they just repeated theorems writing in front of everything.
he must have opened the nLab during the years 2010-2015...
He couldn't even bring himself to say "infinity", he merely gestured with his finger to make the symbol. Like it was something you shouldn't say else you give it power.