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Pierre R (Oct 03 2025 at 20:56):Hello Guys,
I approached the only Category Theory professor in my department to collaborate in applied category theory in AI as a general guiding research topic for part of my master thesis. She was very kind but asked me a very specific question that I wasn’t ready to answer with my intro-level CT:
Her question, summarized:
In computational/AI contexts, is Category Theory mainly being used just as a language to describe constructions more succinctly, or are there cases where theorems from particular categorical structures are essential to prove non-trivial results in neural networks, algorithms, or other AI subfield — results that could not be established (or would be much harder) without CT?
She compared this to Hopf algebras, where CT is necessary to reach classification theorems (I may not correct citing her here), and asked whether similar things exist in AI/ML.
Ask: I would appreciate references (titles only are fine) to recent papers (2023–present preferred) of this type — either in applied CT for AI, or in applied CT in other computational areas that may be relevant.
Thanks!
Ryan Wisnesky (Oct 04 2025 at 03:32):my usual rejoinder to this kind of question is that, when you have a compositional system, category theory just applies, same as how when you have a symmetric system group theory just applies; whether or not you must use some underlying aspect of a system such as symmetry or compositionally is often 'no' simply because of human cleverness. Anyway, when database theorists try to figure out if various classes of logic programming language are closed under composition, they don't think of themselves as doing category theory, even though they are directly probing the compositionality of a given system, and this was a major community goal - 20 years ago. Others will have to speak for the more modern kind of AI (database theory being 'symbolic' AI / deductive logic).
James Deikun (Oct 04 2025 at 08:34):From what I can tell, one of the major problems of AI as a field is that there are hardly any theoretical results that apply to the kind of AI/ML that people use in practice, and all this engineering of tremendous civilizational importance and which huge numbers of people and projects are directly exposed to is being done purely by vibe-guided experimentation, with a very short path to deployment. So probably there aren't any results of importance out there that essentially use category theory, but that says very little about whether category theory could be essential to important results.
John Baez (Oct 04 2025 at 13:29):Since the person asking the question
are there cases where theorems from particular categorical structures are essential to prove non-trivial results in neural networks, algorithms, or other AI subfield
was not an AI expert but "the only category theory professor" in @Pierre R's department, this was probably not the usual question we all hear, where someone is doubting the usefulness of category theory to their field, and challenging us to change their mind. ("See if you can change my mind! I have it firmly set not to change!")
Instead, it could be a case of a category theorist wondering how their field could be useful to some other subject!
But personally, I find that the usefulness of categories in "applied category theory" is often less about theorems and more about using them as part of a software environment. The theorems exist, but often they merely show that the software makes sense.
Pierre R (Oct 04 2025 at 15:48):Hello John,
You are completely right in your reading of the question. She is not doubting Category Theory—she works in theoretical CT—and neither am I.
I think what she is really asking me to do is look for AI papers where CT concepts are essential to reaching the main conclusions of the work.
As a beginner, I tend to think that any paper using CT in a meaningful way already makes CT essential. But from her point of view, simply reframing things in terms of Categories, Functors, or Natural Transformations is not enough to count as strong theoretical CT.
That’s probably what prompted my response, since I’m not sure whether my understanding is too basic to even classify work in that way.
Morgan Rogers (he/him) (Oct 05 2025 at 11:36):Some relevant work appeared at ACT this year, I think. You could check there for slides.