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Alexander Prähauser (Jul 24 2026 at 14:30):Hello everyone! Very few of you might know me, I'm Alex (Alexander Prähauser), a Ph.D. student of Tobias Fritz. Tobias told me it was time to get someone else's reflection on my work and recommended I should put it here to see if any of you can take a look at it before uploading it to the Arxiv or submitting it to some publication.
I'm unsure as for how to preface this other than what I have written in the work, but I want to say one thing: this is self-consciously a long and somewhat heavy work, and it's fine to glance at it, but that cannot be the basis for a full evaluation. If I sound defensive, it is because I am: I've taken up some risk and a lot of hard work to create this text, and in a sense I've had to fight for it. But I believe in the worth of it, and so decided against Tobias's advice not to split it further, because I think that as it is now it constitutes a unit in a way where what is introduced in the beginning is paid off throughout the text, and that further splitting would hurt the overall quality. So I am asking something from the reader but I hope that I provide more in return.
For the same reason I hope the reader understands that I was faced with several competing and contradictory constraints and had to balance them in a way that is inherently subjective, and there are a variety of criticisms that I could make myself. For instance I noted the varying difficulty in the preface and tried to inform the reader about it. I also hope that some understanding of the amount of work that flowed into this text will leave the reader sympathetic even if some errors were found, even though I did my very best to eradicate them completely. On that note, I've had an LLM write me a Rocq-module, which contains many of the proofs but not all of them (yet), and in particular not that of the last theorem, which is by far the hardest. I've still included it in case it is of interest to someone. Running the (preliminary) build script would require adjusting the load paths though.
Finally I hope I'm forgiven for not checking in on this thread over the weekend because I know that any more than superficial engagement with my text takes some time and, not to be dramatic, but my nerves wouldn't take the constant background tension well. Let us rather have a relaxed weekend and do math if we want to but resist the drive to be "always on" that causes so much neurotic behavior!
And with that I thank any sympathetic readers for their time and attention and hope they will find some of the value in my work that I have!
main.pdf
rocq.tar.gz
Matteo Capucci (he/him) (Jul 24 2026 at 15:29):FYI it's unlikely you'll get feedback if you drop a big PDF without telling us what's interesting about it
David Egolf (he/him) (Jul 24 2026 at 16:51):Early on, the attached pdf introduces the following definition, which I find interesting:
Given an upward-bounded preorder , an element is small if the only such that is .
(An "upward-bounded" preorder is a preorder with a largest element .)
One can then contemplate the poset of subobjects of some object, and ask which subobjects are small.
Kevin Carlson (Jul 24 2026 at 17:12):I agree with David that the notion of smallness is intriguing, but I would note that it is not original.
fosco (Jul 24 2026 at 17:20):the notion is well-known in module theory, but I believe some literature in order theory (e.g. Grätzer monograph on lattices) has a similar definition somewhere https://en.wikipedia.org/wiki/Essential_extension
Kevin Carlson (Jul 24 2026 at 17:44):Well, I've looked at this as closely as I'm likely to at this point. To give an idea of what I could get out of this quickly: you note that every subobject lattice in every category allows for the notion of small (or superfluous) subobject one which has no complementary proper subobjects, in the sense of one whose only common upper bound with is In more structured categories, you also have the dual notion of "dense" subobject, and you give a number of results on how functors, regular functors, coherent functors, etc interact with these notions. You also give relative notions of smallness and density, and you give a universal property of the category UBP of posets with a top element in which subobject lattices naturally lie. You spend an exorbitant amount of space outlining many, many examples of these notions from many fields.
The work is, as you mention, over 100 pages long, and contains only two theorems of your own, both related to the universal property of UBP which, as you note in the introduction, seem to be incommunicable to the reader without considerable careful buildup. You also allow that you've used LLMs rather generously, to my mind, here, not only in checking your efforts but also in writing out examples and proofs, LLM output which you've considered it so unnecessary to refine that you expect the reader to see its "seams" as "readily...visible." That is, you've appeared here asking in rather strong terms for a careful and considered reading from experts of a long work, whose point you claim can't really be condensed (a serious "smell" in its own right), and which you couldn't even be bothered to write yourself! I don't think you're going to find many takers for that proposition.
It is good that you described your use of LLMs clearly, but you are far beyond the level of usage that's I'd consider at all appropriate for a Ph.D. student. For myself, I'm not at all convinced I'd accept LLM-generated text of proofs or examples as appropriate in any mathematical publication, and how much less from someone without the experience of producing an acceptable paper in the traditional way? I understand it's a strange time to be doing a math Ph.D, and it makes sense that you're experimenting with these powerful technologies to ideate, but you must in the end be fully intellectually responsible for anything you propose to publish; to my mind it's unlikely to be clear that you've cleared this bar unless you typed all of the text of the paper yourself without even continuous consulting of a robot, let alone copy-pasting from it.
The paper is also full of elements that set off one's crank-hackles, besides the overuse of LLMs: the rhetorical level is much too high at places, and I do not think most mathematicians will be impressed by the Hegel quotes, let alone those from Evanescence. A mathematician has to earn such idiosyncrasies on the basis of indubitably powerful technical work, a basis which you haven't yet established.
My recommendation is to listen to your advisor. I think it's highly unlikely that your work, though it clearly contains some good ideas that ought to see the light of day, will have anything like the impact you clearly hope for in anything like this form.
David Michael Roberts (Jul 24 2026 at 19:43):so decided against Tobias's advice not to split it further
A PhD student's adviser really does know better in many, most, if not all non-mathematical decisions like this, just on the basis of pure experience.
Kevin Carlson (Jul 24 2026 at 20:44):Yes, to amplify probably my strongest piece of advice:
Kevin Carlson said:
My recommendation is to listen to your advisor.
Morgan Rogers (he/him) (Jul 25 2026 at 11:31):I looked at the conclusion, which sent me to Section 3 (definition of smallness mentioned above), then Section 6, which opens with "essential subobject smallness". I can tell from this passage alone that you have not spoken to enough category theorists about this work: this condition is also called "well-poweredness", and features as a criterion in one of the famous adjoint functor theorems, but you do not acknowledge this; did you know it? More generally, you have plenty of concrete examples of subobject posets in the wild, but your work is extremely light on order-theoretic literature -- you don't acknowledge how much work has been done on the subobject functor and it's unclear whether you are aware of any of it.
This is a fairly typical result of working heavily with LLMs in my experience.
Alexander Prähauser (Jul 27 2026 at 06:50):Thank you everyone for your feedback so far, I will take it into consideration.
Alexander Prähauser (Jul 27 2026 at 06:53):Kevin Carlson said:
Well, I've looked at this as closely as I'm likely to at this point. To give an idea of what I could get out of this quickly: you note that every subobject lattice in every category allows for the notion of small (or superfluous) subobject one which has no complementary proper subobjects, in the sense of one whose only common upper bound with is In more structured categories, you also have the dual notion of "dense" subobject, and you give a number of results on how functors, regular functors, coherent functors, etc interact with these notions. You also give relative notions of smallness and density, and you give a universal property of the category UBP of posets with a top element in which subobject lattices naturally lie. You spend an exorbitant amount of space outlining many, many examples of these notions from many fields.
The work is, as you mention, over 100 pages long, and contains only two theorems of your own, both related to the universal property of UBP which, as you note in the introduction, seem to be incommunicable to the reader without considerable careful buildup. You also allow that you've used LLMs rather generously, to my mind, here, not only in checking your efforts but also in writing out examples and proofs, LLM output which you've considered it so unnecessary to refine that you expect the reader to see its "seams" as "readily...visible." That is, you've appeared here asking in rather strong terms for a careful and considered reading from experts of a long work, whose point you claim can't really be condensed (a serious "smell" in its own right), and which you couldn't even be bothered to write yourself! I don't think you're going to find many takers for that proposition.
It is good that you described your use of LLMs clearly, but you are far beyond the level of usage that's I'd consider at all appropriate for a Ph.D. student. For myself, I'm not at all convinced I'd accept LLM-generated text of proofs or examples as appropriate in any mathematical publication, and how much less from someone without the experience of producing an acceptable paper in the traditional way? I understand it's a strange time to be doing a math Ph.D, and it makes sense that you're experimenting with these powerful technologies to ideate, but you must in the end be fully intellectually responsible for anything you propose to publish; to my mind it's unlikely to be clear that you've cleared this bar unless you typed all of the text of the paper yourself without even continuous consulting of a robot, let alone copy-pasting from it.
The paper is also full of elements that set off one's crank-hackles, besides the overuse of LLMs: the rhetorical level is much too high at places, and I do not think most mathematicians will be impressed by the Hegel quotes, let alone those from Evanescence. A mathematician has to earn such idiosyncrasies on the basis of indubitably powerful technical work, a basis which you haven't yet established.
My recommendation is to listen to your advisor. I think it's highly unlikely that your work, though it clearly contains some good ideas that ought to see the light of day, will have anything like the impact you clearly hope for in anything like this form.
Just a a little bit of a defense, I was introduced LLM usage for mathematics by Tobias and he never raised this as an issue, to the contrary, I got the strong impression that he thought is was good I was getting as proficient with it as I could because they would grow so important.
Alexander Prähauser (Jul 27 2026 at 07:00):Morgan Rogers (he/him) said:
I looked at the conclusion, which sent me to Section 3 (definition of smallness mentioned above), then Section 6, which opens with "essential subobject smallness". I can tell from this passage alone that you have not spoken to enough category theorists about this work: this condition is also called "well-poweredness", and features as a criterion in one of the famous adjoint functor theorems, but you do not acknowledge this; did you know it? More generally, you have plenty of concrete examples of subobject posets in the wild, but your work is extremely light on order-theoretic literature -- you don't acknowledge how much work has been done on the subobject functor and it's unclear whether you are aware of any of it.
This is a fairly typical result of working heavily with LLMs in my experience.
I knew well-poweredness as a condition for the adjoint functor theorem, I had forgotten the concrete content. Can you point me towards some work that treats the subobject functor specifically? I indeed didn't find much existing literature on that.
Alexander Prähauser (Jul 27 2026 at 07:17):fosco said:
the notion is well-known in module theory, but I believe some literature in order theory (e.g. Grätzer monograph on lattices) has a similar definition somewhere https://en.wikipedia.org/wiki/Essential_extension
I mentioned the module-theoretic notion, I didn't find anything about it in general pre/partial orders. I will look up that monograph though.
Alexander Prähauser (Jul 27 2026 at 07:30):David Michael Roberts said:
so decided against Tobias's advice not to split it further
A PhD student's adviser really does know better in many, most, if not all non-mathematical decisions like this, just on the basis of pure experience.
I should not have mentioned that detail at all, and I got angry at myself for doing so. If it's necessary to split the work, I'll do so.
Alexander Prähauser (Jul 27 2026 at 07:43):I almost forgot: I found an error in one example and some typing issues in the last proof. Both were small and I corrected them, here is the overworked version.
main.pdf
Morgan Rogers (he/him) (Jul 27 2026 at 10:37):Alexander Prähauser said:
Can you point me towards some work that treats the subobject functor specifically? I indeed didn't find much existing literature on that.
It's easiest to me to give references within the topos theory and categorical logic literature.
The notion of (hyper)doctrine is an abstraction of the subobject functor. In particular, the subobject functor defines a doctrine, and the nicer the structure of the subobjects in the category, the nicer the doctrine, leading up to the notion of tripos.
The subobject functor is representable in a topos (this is one definition of the subobject classifier), and more generally the functor providing the strong subobjects is representable in a quasitopos (Sketches of an Elephant section A2.6).
In Freyd and Scedrov, Categories, Allegories, a bunch of work on subobject lattices is developed starting at regular categories (the minimum setting where composition of relations behaves nicely). In particular I think you should compare some the work you did in the regular case to that book (not that it's the easiest to read...)
Alexander Prähauser (Jul 27 2026 at 10:47):Morgan Rogers (he/him) said:
Alexander Prähauser said:
Can you point me towards some work that treats the subobject functor specifically? I indeed didn't find much existing literature on that.
It's easiest to me to give references within the topos theory and categorical logic literature.
The notion of (hyper)doctrine is an abstraction of the subobject functor. In particular, the subobject functor defines a doctrine, and the nicer the structure of the subobjects in the category, the nicer the doctrine, leading up to the notion of tripos.
The subobject functor is representable in a topos (this is one definition of the subobject classifier), and more generally the functor providing the strong subobjects is representable in a quasitopos (Sketches of an Elephant section A2.6).
In Freyd and Scedrov, Categories, Allegories, a bunch of work on subobject lattices is developed starting at regular categories (the minimum setting where composition of relations behaves nicely). In particular I think you should compare some the work you did in the regular case to that book (not that it's the easiest to read...)
Thank you! I've known about hyperdoctrines and mentioned them later in the text, but I should underline the connection early on. I'll look at the Sketches section and Categories, Allegories and if the connection to my work is decently strong, I'll work it in.
Tobias Fritz (Jul 27 2026 at 17:33):Thanks to everyone for the pertinent comments, and in particular to @Kevin Carlson and @Morgan Rogers (he/him) for the more detailed suggestions. This is a learning experience for me as a supervisor as well: I realize that I arguably should have been more assertive with my feedback.
Kevin Carlson (Jul 27 2026 at 18:03):Alexander Prähauser said:
Just a a little bit of a defense, I was introduced LLM usage for mathematics by Tobias and he never raised this as an issue, to the contrary, I got the strong impression that he thought is was good I was getting as proficient with it as I could because they would grow so important.
Yes, I should say that there's really no consensus about the right amount of LLM use for a current Ph.D. student. The problem is that we know how to train Ph.D. students to write math papers and then get a job teaching and researching math, but nobody has any verifiably correct idea how to train Ph.D. students for whatever math jobs might exist in the 2030s, which might be dramatically different from what they've been since the 1890s given the disruption of LLMs.
So my advice is basically around how to become good at the things I actually know how to do; it's very plausible that it would be a better idea to focus more on becoming good at whatever will be available to do in the 2030's, which again might be quite different. But again I can say pretty confidently that I don't think the thing mathematicians will be wanted for in the 2030's, if anything, will be to produce artifacts that look a lot like 1890's-2020's research papers, but augmented with a whole lot of LLM-generated text. It's just not at all clear why I would want to read such a thing; for instance if I once saw your ideas about small and dense subobjects, why shouldn't I just prompt my own LLM to explain examples customized to my tastes?
I think that, though it's very early days with serious assistance for LLMs in math, so far the most compelling cases (as in the unit distance conjecture and the very fresh Jacobian conjecture) have involved an LLM solving a problem, and then many human mathematicians working hard to explain the solution to many other human mathematicians, setting it in the context of the literature, clarifying how one might have gotten the right intuition, and so on. This is a very different kind of activity than the one you're engaged in here: it suggests that your most important job as a human mathematician (at least for the next few years) is as an explainer of ideas which you may have relied on an LLM to rigorously establish. But this idea circles back to most of my advice from my first message: it remains unclear to me what the plausible benefit of posting a paper including a whole bunch of LLM output is, to anybody.
Alexander Prähauser (Jul 27 2026 at 22:52):Tobias Fritz said:
Thanks to everyone for the pertinent comments, and in particular to Kevin Carlson and Morgan Rogers (he/him) for the more detailed suggestions. This is a learning experience for me as a supervisor as well: I realize that I arguably should have been more assertive with my feedback.
And thank you Tobias.
Alexander Prähauser (Jul 27 2026 at 23:08):Kevin Carlson said:
Alexander Prähauser said:
Just a a little bit of a defense, I was introduced LLM usage for mathematics by Tobias and he never raised this as an issue, to the contrary, I got the strong impression that he thought is was good I was getting as proficient with it as I could because they would grow so important.
Yes, I should say that there's really no consensus about the right amount of LLM use for a current Ph.D. student. The problem is that we know how to train Ph.D. students to write math papers and then get a job teaching and researching math, but nobody has any verifiably correct idea how to train Ph.D. students for whatever math jobs might exist in the 2030s, which might be dramatically different from what they've been since the 1890s given the disruption of LLMs.
So my advice is basically around how to become good at the things I actually know how to do; it's very plausible that it would be a better idea to focus more on becoming good at whatever will be available to do in the 2030's, which again might be quite different. But again I can say pretty confidently that I don't think the thing mathematicians will be wanted for in the 2030's, if anything, will be to produce artifacts that look a lot like 1890's-2020's research papers, but augmented with a whole lot of LLM-generated text. It's just not at all clear why I would want to read such a thing; for instance if I once saw your ideas about small and dense subobjects, why shouldn't I just prompt my own LLM to explain examples customized to my tastes?
I think that, though it's very early days with serious assistance for LLMs in math, so far the most compelling cases (as in the unit distance conjecture and the very fresh Jacobian conjecture) have involved an LLM solving a problem, and then many human mathematicians working hard to explain the solution to many other human mathematicians, setting it in the context of the literature, clarifying how one might have gotten the right intuition, and so on. This is a very different kind of activity than the one you're engaged in here: it suggests that your most important job as a human mathematician (at least for the next few years) is as an explainer of ideas which you may have relied on an LLM to rigorously establish. But this idea circles back to most of my advice from my first message: it remains unclear to me what the plausible benefit of posting a paper including a whole bunch of LLM output is, to anybody.
I have to get this off my chest: I did not ask anyone read my paper. Not that I did not want people to, but what I was afraid of was exactly what I cautioned against in my preceeding remarks: that people would glance over it and toss it away for superficial reasons. I would rather not people read it in that case because, honestly, receiving such a treatment in public is even more hurtful than a dismissive mail or meeting. Yes, I was using LLMs, as will soon become the norm, and if you want to call my paper "crankish" because I was quoting from areas outside mathematics then so be it, but the one complaint I cannot accept is that it is bereft of spirit! And if the spirit flowed partially through me and partially through a machine, then what, fundamentally is the difference? We are both just vessels. That is the spirit I wrote this paper in, one of unselfish dedication, and, as I said, I think the fruits bore out. Your summary left out most of what makes this paper special: yes, smallness and density existed before, but reducedness, satedness, tablets, scales and Heyting spaces are all innovations. And if that is not enough, if only proofs are to be considered, because our most significant result is not a Lemma, then what about the result that density forms a coverage (in a regular category with strict initial objects)? That smallness stratifies each small preorder? I conscously used the name "Theorem" for effect, following Serre, who used them very sparsely to have them shine brighter. I put great thought into this matter, in no way whatsoever was this a purely mechanic process, and if this — a mathematics of spirit through the machine — is not the mathematics of the future, what hope is there for mathematics?
Kevin Carlson (Jul 27 2026 at 23:22):Perhaps it is! But even if you've seen what mathematics is going to be in the future, you still, at least for now, have to figure out how to communicate the results to people. If you would prefer that people not look at your very large paper unless they have spontaneously decided to read it closely on the basis of a quite minimal effort on your part to convince them to do so, well, there are simpler ways to ensure that nobody looks at your paper. You would be better off focusing on getting across more effectively in your Zulip post, in your abstract, in your introduction, and in talks you give on your work why somebody ought to dig in to all you've written, and you would also be better off thinking very carefully about why you don't think it's possible to write a 20-page paper explaining smallness, density, reducedness, satedness, tablets, scales, Heyting spaces, that smallness stratifies each small preorder, etc. Indeed I didn't mention these notions after looking at your paper for half an hour, which is perhaps more a reflection on how clear the paper was at getting across what you find most novel than it is on my reading comprehension--though certainly it may be some of both, you have no control over the latter.
Everybody works hard on their math papers. Everybody has at least one paper full of wonderful ideas that nobody has ever paid any attention to (ask me how I know.) Many people have only such papers. What you want is to work hard not only on writing the paper but on making it easy for somebody to understand why your paper is great, what its key ideas are, and to reasonably quickly read it to reasonable depth. It is also a good idea to think hard about how you want to spend "weirdness" points. Even if you really like some idiosyncrasy, the paper is not, in the end, primarily for you. It seems very clear to me that you could do a lot more on all these points to make your paper shine, as it were, more brightly.
Alexander Prähauser (Jul 27 2026 at 23:32):Kevin Carlson said:
Perhaps it is! But even if you've seen what mathematics is going to be in the future, you still, at least for now, have to figure out how to communicate the results to people. If you would prefer that people not look at your very large paper unless they have spontaneously decided to read it closely on the basis of a quite minimal effort on your part to convince them to do so, well, there are simpler ways to ensure that nobody looks at your paper. You would be better off focusing on getting across more effectively in your Zulip post, in your abstract, in your introduction, and in talks you give on your work why somebody ought to dig in to all you've written, and you would also be better off thinking very carefully about why you don't think it's possible to write a 20-page paper explaining smallness, density, reducedness, satedness, tablets, scales, Heyting spaces, that smallness stratifies each small preorder, etc. Indeed I didn't mention these notions after looking at your paper for half an hour, which is perhaps more a reflection on how clear the paper was at getting across what you find most novel than it is on my reading comprehension--though certainly it may be some of both, you have no control over the latter.
Everybody works hard on their math papers. Everybody has at least one paper full of wonderful ideas that nobody has ever paid any attention to (ask me how I know.) Many people have only such papers. What you want is to work hard not only on writing the paper but on making it easy for somebody to understand why your paper is great, what its key ideas are, and to reasonably quickly read it to reasonable depth. It is also a good idea to think hard about how you want to spend "weirdness" points. Even if you really like some idiosyncrasy, the paper is not, in the end, primarily for you. It seems very clear to me that you could do a lot more on all these points to make your paper shine, as it were, more brightly.
You are right about some of that and I will still improve. Moreover, I'm not sure why exactly you assume that I think these ideas are not summarizable, I actually have a summary I'm writing on. I just didn't do a further summary here because I thought the abstract and introduction brought most things out decently. Apparently I was wrong, and/or I should still have done a further summary. But what more disconcerns me is this format: as you say, my time would be much better spent working on my paper, but I can't. I'm restless and constantly circling back to this thread in my thoughts. I'm not sure if this is just me, but I feel like this is just not right for me. I should be sleeping! I've mostly kept from social media, and when I didn't, it was mostly a waste of my time. I feel like this is terrible for my productivity and I don't think I can engage like this on a regular basis. Maybe it's just me, but I do feel like this entire format is very stressful.
Kevin Carlson (Jul 28 2026 at 01:41):Yes, that's perfectly reasonable, and I'm sorry it's been so stressful. I try to be very direct on here because I hope that will be helpful compared to silence or vague commentary. But social media has a lot of problems. It's probably safer to use a platform like this for more narrower, more well-defined questions, where you're getting a big benefit from a number of sets of eyes on the problem, or else for questions of broad general interest. For a question like this, which was more or less "what do you think of this distillation of everything I've been thinking about for years", it's not obviously the best fit. But still perhaps worth something if you don't have many other places beyond Tobias to reach out directly for feedback.
Todd Trimble (Jul 28 2026 at 03:15):Maybe it's just me, but I do feel like this entire format is very stressful.
What format? The format of speaking with human mathematicians in your subject area?
I'm very worried about the new generation of mathematics learners and what looks to me like very heavy and increasing reliance on AI and less independence of critical thought, less experience with just thinking hard by one's self and scribbling on paper and talking with others, the way it had always been done until very recently. My own radical piece of advice to any mathematician would be to close the clam shell at regular intervals and do just that. Like working out with weights, you get stronger that way, and in a better position to tell when the AI is full of crap or is off on some tangent that isn't useful.
To what degree have you done the following? Write out what you understand on a pad of paper (or with a LaTex editor) without consulting an LLM, and bring your honest understanding of what you understand to Tobias. It could be an interesting experience anyway, to close the clam shell and test your baseline level of intellectual independence away from the machine, which you will absolutely need to have before you give seminar talks and defend your thesis.
but the one complaint I cannot accept is that it is bereft of spirit!
I didn't see anyone say anything about spirit. Let's not talk about spirit. The object of the discussion should be the mathematics. Since you're here and you've been working on this, it should be possible to at least outline what it is you're doing without making people read the pdf in order to find out. And it could be good practice for you as well.
Morgan Rogers (he/him) (Jul 28 2026 at 11:33):For context, here are the definitions of tablet and scale from Alexander's work.
Since you said you think these things are important, I took a closer look. Normally I would fixate on the choice of names, but I'm going to resist that here. Instead, I'd like to discuss the following remark:
I notice that even though "morphism of upward-bounded preorders" is ambiguous (in the morphisms are not required to preserve the top element), if no element is mapped to by then the notion of smallness under becomes vacuous, so it might as well be -preserving. I also notice that if multiple elements are mapped to then the notion of smallness again becomes vacuous. I notice that the notion of "reduced under " depends only on the image of . As such, the dual of the remark above also applies to tablets: why not instead define a tablet to be a sub-bounded-preorder?
But I'll go further: it seems to me that the notions of "small" and "reduced under" are not affected if we upward-close the image of . That is, we can take to be an up-set... but then it looks like the notions of smallness for tablets and (the corresponding reduction to up-sets of) scales coincide.
Curiously, the notions of "reduced under" a tablet and "reduced over" a scale don't obviously coincide, since the latter definition uses more of the structure of . In fact, the latter notion subsumes the former: given an up-set , let , where the latter is the preorder with the same elements as but with iff this is true in or . Then being reduced under the inclusion of into is the same as being reduced over the quotient map.
With all of that said: what do you (@Alexander Prähauser ) think is gained from considering these two definitions separately? Do you think your examples justify the extra generality that scales provide over tablets?
Alexander Prähauser (Jul 28 2026 at 20:13):Morgan Rogers (he/him) said:
For context, here are the definitions of tablet and scale from Alexander's work.
Since you said you think these things are important, I took a closer look. Normally I would fixate on the choice of names, but I'm going to resist that here. Instead, I'd like to discuss the following remark:
I notice that even though "morphism of upward-bounded preorders" is ambiguous (in the morphisms are not required to preserve the top element), if no element is mapped to by then the notion of smallness under becomes vacuous, so it might as well be -preserving. I also notice that if multiple elements are mapped to then the notion of smallness again becomes vacuous. I notice that the notion of "reduced under " depends only on the image of . As such, the dual of the remark above also applies to tablets: why not instead define a tablet to be a sub-bounded-preorder?
But I'll go further: it seems to me that the notions of "small" and "reduced under" are not affected if we upward-close the image of . That is, we can take to be an up-set... but then it looks like the notions of smallness for tablets and (the corresponding reduction to up-sets of) scales coincide.
Curiously, the notions of "reduced under" a tablet and "reduced over" a scale don't obviously coincide, since the latter definition uses more of the structure of . In fact, the latter notion subsumes the former: given an up-set , let , where the latter is the preorder with the same elements as but with iff this is true in or . Then being reduced under the inclusion of into is the same as being reduced over the quotient map.
With all of that said: what do you (Alexander Prähauser ) think is gained from considering these two definitions separately? Do you think your examples justify the extra generality that scales provide over tablets?
I am grateful about such deeper engagement.
"As such, the dual of the remark above also
applies to tablets: why not instead define a tablet to be a sub-bounded-preorder?"
This is correct: I was singling out tablets for educational purposes because I thought it was more obvious. But the same caveats apply as I introduced for scales: a pair of tablets and scales can define both smallness and density for the same object, and that information cannot be reduced to an ideal and an upward-bounded sub-preorder, it takes an additional filter and a downward-bounded sup-preorder. Moreover, the preservation and reflection statements in the next section I find are just plain simpler expressed with a morphism-based calculus. Finally, when I lift the calculus of tablets and scales to a categorical level in the Applications, morphisms again just seem like the right choice, particularly with scales, where more sophisticated requirements such as additivity are often introduced.
"But I'll go further: it seems to me that the notions of "small" and "reduced under" are not affected if
we upward-close the image of . That is, we can take to be an up-set... but then it
looks like the notions of smallness for tablets and (the corresponding reduction to up-sets of) scales
coincide."
I'm not exactly sure how you arrive at that conclusion. The generic case is that the image of the top under is equal to , e.g. that is a top-preserving preorder embedding that we can even assume is full, as I note in the text. This is e.g. the case for the embedding of closeds into a topological space : the largest closed set is itself. But the dual, that is not always the case, which matters, e.g. for the density analysis for subsets of modules over a ring (Example 4.4), where the meet of all tablets is that over all submodules and thus contains , but is usually the powerset, which contains the empty set.
For instance, if is again a topological space and the inclusion of all closed subsets, then a (not necessarily closed) subset is small with respect to if and only if the only closed subset of that joins with to , i.e. whose common union is , is itself. Since we have negations we can reformulate this with the set-theoretic complement of , which of course is the smallest such that (where I'm using the -symbol to suggest the set-theoretic union): so if the closure of is , then is small. So if a scale were to exist that subsumes this notion of smallness, then that would mean that each nowhere dense topological subset the set-theoretic complement of a subset of all but the image of .
If I follow your suggestion for the equivalence of tablets and scales as best I can, the most natural way to implement it would be to define a scale which is the identity on all small elements and just maps all other elements to . And for topological subsets this kind of works, although we only arrived at such a scale by considering the tablets, i.e. closed subsets, which are the actually natural notion.
But the topological case is very well-behaved and I can give you an elementary example where this does not work: let be the upward-bounded preorder on elements with relations and, of course, above everything. Then and . Let moreover be the embedding of . Then both and are -small but the join of and is , so that they are not small under any scale that trivializes and by mapping them to but preserves and by mapping them below . The underlying reason for this is that, unlike in the absolute case, where the tablets are where smallness is stable under joins, this is not always the case for non-trivial tablets. So in particular -small elements are can be non-small against each other.
So I'm hoping I've clarified the relation a little bit. Some of how smallness and density play out depends on the particularities of , which I went into deeper at the beginning of the next section. I'm now considering placing that part a little prior, perhaps that gives the reader more proficiency with the notions.
Alexander Prähauser (Jul 28 2026 at 20:45):Todd Trimble said:
Maybe it's just me, but I do feel like this entire format is very stressful.
What format? The format of speaking with human mathematicians in your subject area?
I'm very worried about the new generation of mathematics learners and what looks to me like very heavy and increasing reliance on AI and less independence of critical thought, less experience with just thinking hard by one's self and scribbling on paper and talking with others, the way it had always been done until very recently. My own radical piece of advice to any mathematician would be to close the clam shell at regular intervals and do just that. Like working out with weights, you get stronger that way, and in a better position to tell when the AI is full of crap or is off on some tangent that isn't useful.
To what degree have you done the following? Write out what you understand on a pad of paper (or with a LaTex editor) without consulting an LLM, and bring your honest understanding of what you understand to Tobias. It could be an interesting experience anyway, to close the clam shell and test your baseline level of intellectual independence away from the machine, which you will absolutely need to have before you give seminar talks and defend your thesis.
but the one complaint I cannot accept is that it is bereft of spirit!
I didn't see anyone say anything about spirit. Let's not talk about spirit. The object of the discussion should be the mathematics. Since you're here and you've been working on this, it should be possible to at least outline what it is you're doing without making people read the pdf in order to find out. And it could be good practice for you as well.
But we are not speaking. I cannot see your face, you cannot see mine, I and you have to type and wait for each other, in a way that is attention-grabbing, incentivising short-term responses (unlike a letter or even email) and alienating because, again, emotional expression is far restricted. In some ways it is good to type rather than talk, it allows for more reflection, but this is a very specific way of interaction, where, not to mention, all is preserved and people can sift tens of years later on through everything you said because some norm changed and this can be used against you. I'm sorry, but I have long lost the naïveté to think that I do not have to discipline myself online. I am far too undisciplined, if anything.
Of course I talked about Tobias about this matter several times, and each time I left with the impression that he agreed with me about the principal value of my content, even if I might place the overall value slightly higher. I can't say whether my estimation is due to vanity or not, but unless our communication suffered a total breakdown, which is manifestly not the case, he agreed with me that there is some there there. I also have to say that I was writing mathematics long before I used AI and while my rigour back then was more subpar than today (whatever that is worth), I did write a lot, including the core of this text. And when Tobias spoke up here, though perhaps what he wrote can be read either way, when he asked me whether he should write something, as he felt was right, I very much took it as his support, saying that this is not solely my fault but both of ours, and not merely in a "I should have been more strict" way, but in very specific ways we both talked about beforehand and which were in part my suggestion. I also want to say, regarding deep thought, that, as much as LLMs helped me, and they did, all the most crucial parts of my paper were derived by me lying down to a point where I almost forgot my body, and thinking, visualizing and going again and again through whatever the point was, often for multiple sessions. I never got into the habit of note-taking, perhaps to my detriment, but, perhaps because of that, this has always been my modus operandi. And, to be honest, even if I wanted them to, machines could not replace this right now: they could take much from me, and they helped a lot in telling me what I did wrong but when I was really stuck, I had to think for myself.
There is clearly from the conversation and worry, which I understand, about the increasing adoption of AI among the young, and, reading everything as sympathetically as I reasonably can, this worry was applied to me in a way that suggested "ill gains", and only original in very restricted ways, i.e. the final two theorems — a lack of spirit concealed by a reliance on the machine — this is what I protested against. And given that I spoke the word already let me also say this: people have used the word "spirit" from the dawn of civilization to now in almost all contexts, certainly with regard to mathematics since Pythagoras at the least. Truthfully, it is a very normal word. My worry is not so much about the problems of the next generation in its approach to the new technology but the widening gulf between the mathematical community and society, a gulf the mathematical community seems to me to be a little too eager to accept without an understanding of the deep pathologies it not is caused by but causes, not least within the mathematical community itself. I have long known and accepted this, but I have to say: I can drop the mention of any spirit, which I so understand myself beholden to, or I can talk as if this were a normal conversation to me — I cannot do both.
Alexander Prähauser (Jul 28 2026 at 23:16):Todd Trimble said:
Maybe it's just me, but I do feel like this entire format is very stressful.
What format? The format of speaking with human mathematicians in your subject area?
I'm very worried about the new generation of mathematics learners and what looks to me like very heavy and increasing reliance on AI and less independence of critical thought, less experience with just thinking hard by one's self and scribbling on paper and talking with others, the way it had always been done until very recently. My own radical piece of advice to any mathematician would be to close the clam shell at regular intervals and do just that. Like working out with weights, you get stronger that way, and in a better position to tell when the AI is full of crap or is off on some tangent that isn't useful.
To what degree have you done the following? Write out what you understand on a pad of paper (or with a LaTex editor) without consulting an LLM, and bring your honest understanding of what you understand to Tobias. It could be an interesting experience anyway, to close the clam shell and test your baseline level of intellectual independence away from the machine, which you will absolutely need to have before you give seminar talks and defend your thesis.
but the one complaint I cannot accept is that it is bereft of spirit!
I didn't see anyone say anything about spirit. Let's not talk about spirit. The object of the discussion should be the mathematics. Since you're here and you've been working on this, it should be possible to at least outline what it is you're doing without making people read the pdf in order to find out. And it could be good practice for you as well.
To describe what I am doing in the paper in a fairly quick way: I am using a notion of smallness, which is a joint generalization of small/superfluous modules and nowhere dense subsets, and its abstract dual, density, which generalizes topological density. These notions are applicable in any upward-bounded preorder (for smallness) or downward-bounded preorder (for density). Since the subobject preorder of every object in every category is upward-bounded, smallness can be applied in any category and, since the subobject preorder of any object in a category with an initial object is also downward-bounded. I also introduce two derived notions, reducedness and satedness, which generalize e.g. radicals of ideals, giving an overall calculus of four notions that can be very widely applied and specializes to interesting, often already known, notions. I also generalize how topological properties are infered on all subsets of a topological space from its open and closed subsets, and how negligibility is infered from the morphism of a measure in a joint tablets and scales calculus that makes the smallness/density/reducedness/satedness calculus applicable relative to such inference procedures.
Then I trace out how this calculus unfolds in categories as I add assumptions. This is where I derive the most results, such as that in a regular category with a strict initial object, dense subobjects are pullback-stable and thus form a coverage, and I give as an example for sheaves with respect to such a coverage the rational morphisms of varieties. I follow a specific thread: from
plain categories,
to categories with pullbacks,
to ones regular ones,
to coherent ones.
I think this is a fairly natural path of development but I'm also setting this up for future work.
Finally I provide two applications: the first is a general theory of dualities of opens and closeds, not just in the topology but also in, e.g. algebraic geometry. In particular I introduce what I call a Heyting Space, a generalization of topological spaces where we essentially replace unbounded unions of opens with a Heyting implication binary unions. The core point of a Heyting space is in having a space in which information about arbitrary subsets is infered like with the opens and closeds of a topological space while replacing an unbounded operation, unions of opens, with a binary one, the Heyting implication, so that all operations involved have a finite signature. This is in particular important for an abstraction of semialgebraic topology, since semialgebraic sets (open or otherwise) are not closed under arbitrary unions. The second (admittedly shorter) application is a generalization of measures that can be applied "under", i.e. directly on the subobjects of each particular category. I also show that these two notions interact well: we can use such a measure on the opens (or closeds) of a Heyting space to obtain a measure on their constructible closure.
Finally I close with two theorems that describe the subobject 2-functor and the 2-category of upward-bounded preorders through universal properties. Basically, this universal property shows that reflects monic categories (i.e. categories with only monics) the same reflects all (small) categories through taking slice categories.
I hope this was more helpful than my introduction. None was written by an LLM, of course. Neither was the example I give in my other comment.
Todd Trimble (Jul 29 2026 at 02:59):Referring back to this.
Well, okay, I suggested that we not go on about "spirit". It is an ordinary word, yes, but going on about it as you are here is -- I would again suggest -- not really a topic well-adapted to this forum. That's why I suggested we not talk about "spirit" and focus instead on the mathematics. I appreciate your responding on a more mathematical level as you did in your later comment. As time and interest allow, I may return with more questions, and track the conversation you are having with Morgan. But I would need more time to study what you have written.
Are there specific problems are gripping you that gave rise to your topic, e.g., yes/no questions about some phenomena for which you really want to know the facts of the matter? Are they questions that other people have already expressed interest in?
I also want to say, regarding deep thought, that, as much as LLMs helped me, and they did, all the most crucial parts of my paper were derived by me lying down to a point where I almost forgot my body, and thinking, visualizing and going again and again through whatever the point was, often for multiple sessions.
That sort of description does resonate with me and how I myself behave when I'm actually trying to "sink down" into mathematics.
I never got into the habit of note-taking, perhaps to my detriment, but, perhaps because of that, this has always been my modus operandi.
I appreciate the honesty. Most of my "note-taking" as such still consists of scribbles, but scribbles that can be converted readily by me into statements and proofs, or at the very least would be a basis for that.
My worry is not so much about the problems of the next generation in its approach to the new technology but the widening gulf between the mathematical community and society, a gulf the mathematical community seems to me to be a little too eager to accept without an understanding of the deep pathologies it not is caused by but causes, not least within the mathematical community itself. I have long known and accepted this, but I have to say: I can drop the mention of any spirit, which I so understand myself beholden to, or I can talk as if this were a normal conversation to me — I cannot do both.
I'm not sure I know (or could even plausibly guess) what you're talking about. ("Pathologies" caused from within the mathematics community.) But judging from past such very general conversations at this Zulip and how they often go -- how they become so wide-ranging and splinter off and also get emotional -- I again think it'd be better to stay focused in this thread on your specific mathematical issues. The conversation with Morgan looks like a decent start.
Morgan Rogers (he/him) (Jul 29 2026 at 09:42):@Alexander Prähauser you can select and quote just part of a message, by the way!
Morgan Rogers (he/him) (Jul 29 2026 at 11:02):In my message yesterday, I said
if no element is mapped to by then the notion of smallness under becomes vacuous, so it might as well be -preserving.
I don't know exactly how I arrived at that (partly wishful thinking?). If the image of doesn't contain the top element then an element is small over when (or is undefined) for all , which is the case for (if this exists) but not the case for , so it's clearly not vacuous. This undermines most of the rest of what I said, especially about scales subsuming tablets, which is unfortunate. At least my questions, which boil down to "why did you present things in this way?" still make sense..!
As such, the dual of the remark above also applies to tablets: why not instead define a tablet to be a sub-bounded-preorder?
This is correct: I was singling out tablets for educational purposes because I thought it was more obvious.
I think it would have been safe to use the substructure as the definition of tablet and then point out that any morphism induces a tablet. After all, a lot of this work is about subobjects!
But the same caveats apply as I introduced for scales: a pair of tablets and scales can define both smallness and density for the same object, and that information cannot be reduced to an ideal and an upward-bounded sub-preorder, it takes an additional filter and a downward-bounded sup-preorder.
This doesn't really satisfy me: the upward-bounded sub-preorder and downward-bounded sub-preorder are in fact the same thing if you're using the same tablet to define smallness and density; meanwhile, for the scale you're using dual information for the dual properties: a scale determines both an ideal and a filter, but those are independently used to define density and smallness, so there's no need to lump them together.
Moreover, the preservation and reflection statements in the next section I find are just plain simpler expressed with a morphism-based calculus.
I disagree. It would be a lot clearer to consider how smallness interacts with the ordering on sub-preorders (tablets) and up-sets (scales) and then to argue about how these interact with order-homomorphisms, rather than varying all three at the same time and requiring convoluted auxiliary definitions to constrain the maps involved. The inverse image of an up-set is an up-set, so the scale version of smallness should be reflected along homomorphisms in a natural way, while I expect the tablet version of smallness would be readily preserved.
Finally, when I lift the calculus of tablets and scales to a categorical level in the Applications, morphisms again just seem like the right choice, particularly with scales, where more sophisticated requirements such as additivity are often introduced.
Can you be more specific? I can see places where additivity made it easier to argue about which sets lie in the filter, but can you point to a place where the fact a scale is a morphism was leveraged?
But I'll go further: it seems to me that the notions of "small" and "reduced under" are not affected if
we upward-close the image of . That is, we can take to be an up-set... but then it
looks like the notions of smallness for tablets and (the corresponding reduction to up-sets of) scales
coincide.I'm not exactly sure how you arrive at that conclusion.
I'm not sure either :face_in_clouds: . The definition of reduced superficially only cares about the minimal elements of the image of , but adding things above these would indeed change which elements are small... Please forgive my misunderstanding here (and thanks for providing a counterexample!)
Morgan Rogers (he/him) (Jul 29 2026 at 11:23):One thing I think it's important to do early on in the tablets and scales section, however they shake out, is the trivial/absolute case: you mentioned above that the original definition of smallness corresponds to the identity tablet on ; I gather the absolute case also corresponds to the identity map as a scale. I don't know if you do this somewhere else and I missed it.
Something further, since I was shown to be wrong above: you should explain clearly that neither extension of the notion of smallness subsumes the other, perhaps by using the counterexample you gave above, but also on a more conceptual level. That is, reflect on why these extensions of the notions of smallness still fulfil one's expectations about what it should mean to be small. Your Aristotle quote motivating the absolute notion of smallness in the introduction is great, but the motivation for these relative versions is missing. In my opinion showing that Proposition 3.7 (small objects form an ideal*) extend to these relative versions would be an essential component of this argument, since one motivation for the dual definition of filter is as an axiomatization of which elements (usually subsets of some kind of space) are "large".
I suppose that would then beg the question of which ideals are realised as ideals of small objects with respect to the different definitions, since that would also address the differences between the notions of smallness (and better motivate the hybrid definition, perhaps).
*you should explicitly say in that proposition 'any joins that exist', since joins of small objects are not assumed to exist there
Morgan Rogers (he/him) (Jul 29 2026 at 11:49):I'll read the topos theory aspects at some other point.
In the OP you said your reasoning for not splitting this into two parts was:
I think that as it is now it constitutes a unit in a way where what is introduced in the beginning is paid off throughout the text, and that further splitting would hurt the overall quality.
I don't really understand your reasoning here. Presented as a 100 page document, the effect is simply that the back half won't be read by most readers (I've experienced this with my own papers that ran over 50 pages, and you're way beyond that). Moreover, this work really does split into parts: there is the order-theoretic aspect with lots of examples, including the interaction with homomorphisms, and then there is the analysis of the subobject functor. It would be relatively straightforward to concisely reintroduce the definitions of smallness etc. in a paper devoted to the second subject, and you could just quote the necessary results from a paper on the first half without having to reprove them.
Alexander Prähauser (Jul 29 2026 at 13:34):Todd Trimble said:
Referring back to this.
Well, okay, I suggested that we not go on about "spirit". It is an ordinary word, yes, but going on about it as you are here is -- I would again suggest -- not really a topic well-adapted to this forum. That's why I suggested we not talk about "spirit" and focus instead on the mathematics. I appreciate your responding on a more mathematical level as you did in your later comment. As time and interest allow, I may return with more questions, and track the conversation you are having with Morgan. But I would need more time to study what you have written.
Are there specific problems are gripping you that gave rise to your topic, e.g., yes/no questions about some phenomena for which you really want to know the facts of the matter? Are they questions that other people have already expressed interest in?
I also want to say, regarding deep thought, that, as much as LLMs helped me, and they did, all the most crucial parts of my paper were derived by me lying down to a point where I almost forgot my body, and thinking, visualizing and going again and again through whatever the point was, often for multiple sessions.
That sort of description does resonate with me and how I myself behave when I'm actually trying to "sink down" into mathematics.
I never got into the habit of note-taking, perhaps to my detriment, but, perhaps because of that, this has always been my modus operandi.
I appreciate the honesty. Most of my "note-taking" as such still consists of scribbles, but scribbles that can be converted readily by me into statements and proofs, or at the very least would be a basis for that.
My worry is not so much about the problems of the next generation in its approach to the new technology but the widening gulf between the mathematical community and society, a gulf the mathematical community seems to me to be a little too eager to accept without an understanding of the deep pathologies it not is caused by but causes, not least within the mathematical community itself. I have long known and accepted this, but I have to say: I can drop the mention of any spirit, which I so understand myself beholden to, or I can talk as if this were a normal conversation to me — I cannot do both.
I'm not sure I know (or could even plausibly guess) what you're talking about. ("Pathologies" caused from within the mathematics community.) But judging from past such very general conversations at this Zulip and how they often go -- how they become so wide-ranging and splinter off and also get emotional -- I again think it'd be better to stay focused in this thread on your specific mathematical issues. The conversation with Morgan looks like a decent start.
I'm glad we seem to be finding more of a common basis for communication Todd, and I am fine with sticking to the mathematics, even if for me these things cannot be so clearly separated as might seem apparent. To be honest, and that is part of the problems with this format I was talking about, I never seriously expected anyone to form a decent understanding of what I've written over the weekend (I couldn't have!), this was just the interval of mental calm I afforded myself. I am in no rush. Moreover, I do have a better eye towards the expository flaws of the text now.
To your question: this work actually arose from my development of a "Closed Geometric Logic", a partial dualization of open geometric logic that does some things similarly to the standard open setting but also has to do some things differently to arrive at the right place, which I've talked about this in a short presentation I gave for last year's categorical Octoberfest. For this I needed these notions of smallness and reducedness particularly, and the way this text really got rolling was when Tobias noticed an error I had made that had me develop the calculus of tablets to really express what had gone wrong, and then I found that this material actually deserves its independence (and Tobias told me I should split the text more).
Alexander Prähauser (Jul 29 2026 at 13:48):Morgan Rogers (he/him) said:
Alexander Prähauser you can select and quote just part of a message, by the way!
Thank you Morgan for this feedback! I will go through it attentively and incorporate it into the text. To your question of where in the applications I'm using the morphism-based description in the applications, for instance in my definition of an open-closed duality as a particular family of natural transformations (which of course consist of morphisms, though even that you could theoretically describe through sub-preorders). When I describe my framework for categorical measure theory I also don't demand but work towards categorical measures that are additive, which cannot be expressed in terms of an ideal anymore.
It may or may not be unclear that when I was talking about Applications, I meant the "Applications" section of the text, i.e. Section 7, not the examples I provided. This was confusingly formulated by myself.
I will go through your feedback in the coming days, but I may me checking Zulip a little less. Truth be told, I have not eaten anything but the bare minimum throughout the last few days. This is obviously not due to anyone I have been in dialogue with, which has become both more welcoming and more useful after the initial cold reception, but I think almost exclusively due to the interaction of my psyche and this medium: I've obsesessed over what I have written here to a point where I could only walk up and down anymore, thinking through it again and again. I need to learn to distribute my mental resources better, and to gain a degree of self-control that allows me to approach these things much more calmly. But I also really need to regain my appetite.
David Michael Roberts (Jul 30 2026 at 07:05):There is no rush to respond, the asynchronous nature of the communication should be a benefit, not a down-side. As a PhD student I used to want to rush to respond as if I were having a live conversation, in blog comments in 2005-06, where my interlocutors were in Germany, the UK and California (and no internet at home, just at university). But these days I would tell myself not to be so hasty.
Take care of yourself.
Andrea S (Aug 10 2026 at 03:30):Hi @Alexander Prähauser, it might take me some time to get around to this (I need to finish writing my dissertation, move to a new continent, and start on my postdoc research in September) and it seems you've received a lot of feedback to chew on already, but I'm in a similar situation to you re:Needing more perspectives digging deep into my work (and perhaps with struggling to sufficiently attract the interest of readers to my work due to it being quite long, introducing new language, and lacking a compelling summary/intro :sweat_smile: ) I would be very happy to go over your work with a fine-toothed comb if you could do the same for me.
A particular struggle I've encountered is that most non-category theorists have been very excited and curious about my work, but category theorists I've presented my work to have had a more mixed reception, and have broadly seemed less impressed or interested in my work than I had hoped. I am still uncertain how to communicate my results to the categorical audience in a more compelling fashion. It also might be nice to discuss effective LLM use in the process—I've been extremely cautious to avoid using LLMs much in my work, but I worry about dinosaur extinction events.
If you'd like to take a glance at my work to see if you're interested, here's where I posted the latest draft (and you can see some of the earlier feedback): , but I hope to publish a new version with some further improvements soon so there is no rush.
(Lastly, if there is anybody else reading this in similar shoes to me or Alexander, I would love to include more folks in such a reading / criticism group!)
Alexander Prähauser (Aug 10 2026 at 08:30):Andrea S said:
Hi Alexander Prähauser, it might take me some time to get around to this (I need to finish writing my dissertation, move to a new continent, and start on my postdoc research in September) and it seems you've received a lot of feedback to chew on already, but I'm in a similar situation to you re:Needing more perspectives digging deep into my work (and perhaps with struggling to sufficiently attract the interest of readers to my work due to it being quite long, introducing new language, and lacking a compelling summary/intro :sweat_smile: ) I would be very happy to go over your work with a fine-toothed comb if you could do the same for me.
A particular struggle I've encountered is that most non-category theorists have been very excited and curious about my work, but category theorists I've presented my work to have had a more mixed reception, and have broadly seemed less impressed or interested in my work than I had hoped. I am still uncertain how to communicate my results to the categorical audience in a more compelling fashion. It also might be nice to discuss effective LLM use in the process—I've been extremely cautious to avoid using LLMs much in my work, but I worry about dinosaur extinction events.
If you'd like to take a glance at my work to see if you're interested, here's where I posted the latest draft (and you can see some of the earlier feedback): , but I hope to publish a new version with some further improvements soon so there is no rush.
(Lastly, if there is anybody else reading this in similar shoes to me or Alexander, I would love to include more folks in such a reading / criticism group!)
Hi @Andrea S! I'm also in the middle of many things, and in particular I've started rewriting the paper based on the comments I've received thus far. I've made similar experiences to you, and I'll gladly look at your work (can't also say how throughly yet because time)! I'm also hoping to have a reworked version done soon. Lastly, that idea of a reading / criticism group might be interesting. I've done Vienna Math Circles and also a digital reading group with non-mathematicians, but it would be nice to have something in-between that, with people who have some understanding of the general frameworks teaching each other online. A kind of informal, almost Boubarki-like seminar. I'm just spinning ideas here, but maybe you can see where I'm coming from, because I do feel like we should use the opportunities for self-organization we have some more, or maybe in a different way that I've never quite managed to pin down.