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Ben Sprott (Jan 29 2026 at 22:15):Hello everyone,
I’m posting to invite high-level mathematical feedback on a short paper I’m sharing (PDF attached). The paper is a category-theoretic attempt to make precise a claim that has been coming up more and more in practice: tools and procedures in the lab (and the ways we refine them) are part of the evolving structure of what can be known.
A small contextual note: Terence Tao recently remarked—in both a recent write-up and an interview—that using ChatGPT as a “research assistant” saved him hours on a concrete mathematical task (in particular, exploratory computation / debugging and checking details). I take that as an encouraging example of a broader theme: the modern research workflow increasingly involves structured interaction with instruments (software, pipelines, automated checking, computational probes) whose capabilities evolve with our access to information.
My paper tries to formalize that theme in a way that is faithful to actual laboratory practice:
Model an instrument (the lab interface: configurations, allowable adjustments, readouts) as a small category (C).
Model an experiment not as a process (A \to B) acting on a presumed input “system,” but as a presheaf (F \in [C^{op},\mathbf{Set}]) encoding structured observational assignments over configurations.
Model causal accessibility / approximation via ideal completion on posets (domain-theoretic style), and extend this to (\mathbf{Pos})-enriched categories by completing hom-posets.
Combine these into a unified completion (\mathbb{G}(C)) (schematically: presheaves over a causally completed instrument), with observers as coalgebras after an appropriate 2-categorical dualization.
What I’d love from experts here (even very brief replies are valuable):
Foundational sanity check: Are the variance choices (2-monad vs 2-comonad, dualization) and the ambient 2-categorical setting presented coherently? What’s the cleanest “home” for the construction?
Prior art / naming: Does this “causal completion + free cocompletion” (or its interaction/distributive law) already exist under a standard name in enriched category theory / domain theory / topos theory?
Technical pressure points: What assumptions are actually needed for the interaction between ideal completion and presheaves to behave well (smallness, algebraicity/continuity, accessibility, enriched limits/colimits, etc.)?
Better formalization choices: If you were rewriting the paper, what would you swap in (dcpo-enrichment, quantaloid enrichment, profunctor semantics, sheaves vs presheaves, probabilistic enrichment, etc.)?
If you’re willing, I’m especially grateful for responses of the form “this is already X,” “this fails because Y,” or “state this as Z instead.”
Thanks for taking a look.
Benjamin Sprott
Full_Theory_2.pdf
Kevin Carlson (Jan 29 2026 at 22:28):Ben, I am not interested in reading a paper co-authored with ChatGPT. You can see elsewhere on this server than we are inundated with fake papers on the category theory ArXiv these days, exactly none of which has contained any progress whatsoever. Yours, too, does not contain any actual progress: you have the idea that perhaps a scientific instrument can be modelled as a category of its states, with a presheaf corresponding to the measurements possible in each state, which is a pretty reasonable idea, but then rather than giving any actual examples, which would inevitably lead you to notice that your definitions don't actually work to do anything you want to do, you (or, perhaps, your robotic friend) run off into the stratosphere of a stack of iterated generalizations which lead nowhere. There is some discussion these days about ways in which AIs can assist researchers, and they are filling more and more substantial gaps in even quite advanced research work, but the idea that they can take a quarter-baked idea and turn it into a real paper, when the human author lacks experience producing strong papers independently, is to date a fantasy. It cannot work. I personally do not expect it to ever work. You would do much better to spend your time learning some math (Spivak's work on polynomial functors is a well-developed categorical theory of devices and their interfaces that you might build on, rather than starting from scratch) than trying to do solo research at this point. I only put this so harshly because I have seen you making posts about your research efforts for over a decade, here and at MO, and that is much longer than it takes to simply do a Ph.D.! It's really wonderful that you are interested in category theory, but this is just not how it is done. I'm sorry I don't have anything more supportive to say. Best of luck.
John Baez (Feb 01 2026 at 18:40):By the way, the example of Terence Tao using AI may fool nonexperts into thinking "if a bigshot like Tao is using AI, maybe I could do better writing math papers with help from AI!" But one should keep a couple things in mind here:
Terence Tao won the Fields medal before using AI, and his work using AI is at a much lower level of quality. He's like someone who won the Tour de France bicycle race testing out a unicycle.
Furthermore, he's applying his intense intelligence and practiced competence to evaluate and critique the results produced by AI.
I'm sure that he's thinking: "if AI ever gets good, I don't want to be left behind - I want to be very good at using it." But so far the problems he's solving with AI are much less mathematically interesting than the problems he works on without it.