I am aware of the strict policy about offtopic and opinion-based questions here on MO.
I am ok with being redirected towards academia, or another stackexchange, but please, notice that my question is likely to be properly answered by mathematicians only: I kindly ask you at least the time to collect a few answers. In case you feel like writing me privately, you can reach me at email@example.com
This said, here’s my post.
I’m currently a young, not-so-young mathematician, finishing its second postdoc. I developed an interest for rather different topics in the last years but constantly, slowly converged towards something that has to do with (but at this point I’m quite unsure is) category theory and its applications. What motivated me in the study of mathematics back in the days was the desire to understand the mechanisms ruling algebraic topology; then the word “functor” came in, and I fell into the rabbit-hole.
At this point most of you would expect I’m not unsatisfied with the shape category theory has nowadays: isn’t a blurry mixture of homotopy theory and category theory precisely what I’m tackling?
I’m instead profoundly disappointed by the drift that categorical thinking has taken in the last ten years or so. And this is because the more I dwell into what “higher category theory” and “formal homotopy theory” became, the less I like both of them (I will somewhat refer as both with the portmanteau term “HTT”; I hereby stress that this acronym has no particular meaning whatsoever):
It is still absolutely unclear what good is HTT for category theorists. To my eye, it is certainly a masterpiece of applied mathematics (“spectral algebraic geometry” is applied mathematics), but it doesn’t seem to add a single grain of sand to the sea of category theory; instead, it re-does all the things you need to know to behave “as if” your homotopy-things were things, or to compactly bookkeep an infinite amount of data into a finite amount of space. These are honest practical motivations, addressed in a way I’m unable to judge; what I am able to judge, is the impact this impressive amount of material is having on category theory intended not as a part of mathematics, but as a way to look at mathematics from the outside. I feel this impact is near to zero. Not to mention that to my eye you do category theory only the australian way; everyone else is applying category theory towards the solution of a specific mathematical problem. And yet, I couldn’t think of two more distant languages than Australian CT and HTT; what’s wrong with me? What’s wrong with the community? Sure there have been attempts to circumvent this; I feel this is a beginning, and somehow the first example of HTT done by truly categorical means. But in the end, you open and read these papers, only to find that you still need to know simplicial sets and homotopy theory and the brainfucking lingo of topologists. This is not what I’m after.
When you use HTT, you are not providing a foundation for (higher) category theory; instead, you are relying quite heavily on the structure of a single category (simplicial sets), and on its quite complicated combinatorics. I am orrified by the naivety of people that believe HTT can serve as a foundation for higher category theory; I am frightened by the fact that these people are satisfied by what they have.
Struggling with the things I have been at the reach of my hand, I haven’t been able to find a single convincing word about neither of these terms (foundation, category, theory). Again, it seems that HTT is a framework to perform computations (be them in stable homotopy theory or intersection theory or something else), instead than a language explaining the profound reason why you already know what things intimately are (this is what category theory does, to me). It is also sadistically schizophrenic that HTT exhibits the double nature of a device taking (almost all) category theory for granted, and at the same time it wants to rebuild it from scratch. But rebuild it on what, and using which language, and why this, why not another?
There’s a rather deep asymmetry between category theory and homotopy theory: these two fields, although intimately linked, live different planets when it comes to outreach and learning. By its very nature, categorical thinking is trivial; there are few things to prove, and all of them are done with the same toolset, and instead there’s an extreme effort in carving deep definitions that can turn into milestones of thought (I take “elementary topos” as an example of such a definition). On the contrary, homotopy theory is a scattered set of results, fragmented in a cloud of subfields, speaking different dialects; every proof is technically a mess, uses ad-hoc ideas, complicated constructions, forces to re-learn things from scratch… in a few words, there is no Bourbaki for algebraic topology, and there’s no hope to see one coming. This double nature entails that there’s no way to learn HTT if you (like me) are not so acquainted with the use of concrete and painful arguments; in a few words, if you are not a good enough mathematician. The complexity of techniques you are requested to master is daunting and leaves outside some beginners, as well as some people caught at the wrong time in their formation process. Sure, the situation is changing; but it’s doing it slowly, too slow to perceive a real change in the pace, or in the sensibility, or in the sense of priority of the community. Until now, every single attempt I made to enter the field failed in the most painful way. I feel there’s no way I can understand fragmented, uncanny arguments like those. The few I can follow, I’d be absolutely unable to repeat, or reshape to prove something I need: they simply lie outside the language I’m comfortable with. Everytime I have to check whether something is true, I have absolutely no clue how to operate, apart from pretending that what I do happens in/for a 1-category. And this disability is not conceptual, it is utterly practical and seemingly unsolvable.
A few more words of explanation are in order, to understand the real nature of my claims. As I said, I began with a real, concrete interest in mathematics. Years, choices, and a difficult character, carved my interest into something different. Learning HTT requires to abandon categorical thinking from time to time; you are forced to show that something is true in a specific model, using a rather specific and particular technique, without relying on completely formal arguments. It is an unsatisfying, poor language from the point of view of a category theorist, and yet money is injected in (ab)using this non-language. Even though the theory is in its absolute infancy, many people seem to be more interested in eluding foundations to do geometry and topology. I disagree with this philosophy and (see below) I don’t feel I master enough the technicalities to participate; in the long run, this made me lose some interest in trying to collaborate.
It is at this point extremely likely that, by lack of ability, or simply because I can’t recognize myself in (the absence of) their philosophy, I won’t be part of the crew of people that will be remembered for their contributions to higher category theory. What shall I do then? The echo-chamber where I live in seems to suggest a “love or leave it” approach, without any space for people that couldn’t care less about chromatic homotopy theory, algebraic geometry, differential geometry, deformation theory…
So, what shall I do? I can list a few answers, all equally frightening:
Settle down, learn my lecture, and fake to be a real mathematician, even though I know barely anything about the abovementioned homotopy theory, algebraic geometry, differential geometry, deformation theory? To a certain extent, it is working: my thesis received surprisingly positive reports, I happen to be able to mantain a position, even though scattered and temporary. But I’m also full of discomfort; I fear that my nature is preventing me from becoming a good mathematician; I am unsatisfied and I feel I’m denying my true self. What’s worse: I feel I have to deny it, posting this rant with a throwaway account, because the ideas I proposed here are unpopular and could cost my academic life.
Shall I quit mathematics, since at this point there’s no time to learn something new (I have to employ my time writing to avoid death)? I have to do mathematics with what I have; I feel what I have, what I know at the deep level I want, is barely nothing. And I can’t use things I don’t know, that’s the rule.
Shall I face the fact that I’ve been defeated in my deepest desire, becoming exactly the kind of mathematician (and human being) I’ve always hated, the one who uses a theorem like a black box and makes guesses about things he ignores the true meaning of? But mathematics works this way: there is no point in knowing that something is true, until you ignore why it’s true. Following a quite common idea among category theorists, I would like to go further, knowing why something is trivial. I don’t want to know a definition, I want to know why that definition is the only possible way to speak about something. And if it’s not, I want to be aware of the totality of such ways: does this totality carry a structure? The presence/absence of it have a meaning? Is there a totality of totalities, and how it behaves? When I first approached HTT I thought that answering these very questions was its main task. You can see how deeply I’m disappointed. And you can see the source of my sense of defeat: I feel stupid, way more limited, distracted from learning technicalities, way more than people that do not tackle this search for an absolute meaning. Younger than me, many colleagues began studying HTT, rapidly reaching a certain command of the basic words and subsequently began producing mathematics out of this command. To them, category theory is just another piece of mathematics, not different from another (maybe more beautiful); you do your exercises, learn to prove theorems, that’s it. To me, category theory is the only satisfying way to think. Am I burdened by this belief to the point that it’s preventing me from being a good mathematician?
The questions I raised at point 4 do not pertain mathematics; I should do something else. In fact, the only reason why I tried to become a mathematician was that I felt that mathematics is the only correct meaning of the word “philosophy”, and the only correct way to pursue it. But turning to philosophy would be, if possible, even a more unfortunate choice: philosophers tend to be silly, ignorant people who claim to be able to explain ethics (=a complicated and elusive task) ignoring linear algebra (=something that shall be the common core of knowledge of every learned person).
I am this guy. I feel alone, worried, dumb, and hopeless. Which is not something nice to say on Christmas, but it’s true. I’m looking for help.
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