The "derived drift" is pretty unsatisfying and dangerous to category theory (or at least, to me) [closed]

Question

I'm currently a young, not-so-young mathematician, finishing its second postdoc. I developed an interest for rather different topics in the last few 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):

1. It is still absolutely unclear what good is HTT for category theorists. To my eye, it is certainly a masterpiece of applied mathematics (in the sense that its tasks rest on the use of conceptualization as a tool, not as a target), 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 lingo of topologists. This is not what I'm after.

2. 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 perplexed 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 seems to be satisfied by what they have. So should I? Or shall I look for more? And where? Struggling with the books I had, 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 quite 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. Do I have to already know this stuff, to learn this stuff?

3. 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 [edit: now I know there is one, but it's evidently insufficient].

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. Every time 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.

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 people seem to avoid tackling foundations to do geometry and topology. Which is fine, but not my cup of tea.

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:

1. Settle down, learn my lesson, and fake to be a real mathematician, even though I know barely anything about the above mentioned 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 maintain 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.

2. 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.

3. 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 the definiendum. 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?

4. The questions I raised at point 3 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).

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One of the answers below advises me to "give HTT another try".

This is what to do. I've no clue about how, and this is why I'm looking for mathematical help. I can't find a way out of this cul-de-sac: doing new, unpolished mathematics is a social event, but I've lived the years of my PhD isolated and without a precise guidance aside from myself.

Answer 1

Higher category theory is, roughly speaking, where category theory meets homotopy coherent mathematics. It is hence relevant to those problems in which categorical structures and homotopy coherent phenomena play a significant role. Many areas of algebraic topology and algebraic geometry have this property. There are also many such areas who don't. From what I understood from your question, you like category theory, but not so much homotopy coherent mathematics. So far I would say you don't actually have any problem, since ordinary category theory itself is not, at least in my opinion, a domain in which homotopy coherent mathematics is crucially needed. This is mainly because the coherence issues that arise in category theory are very low-dimensional, to the extent that it is more cost effective to do them by hand (or simply neglect them), then to use fancy machinery. This leads me to the first possible solution to your problem:

Do category theory.

It has really not been my impression that this field is anywhere close to finished. This is especially true if you consider 2-category theory as an acceptable extension (here the coherence is again usually simple enough to do by hand). It also has many interactions with domains such as logic, set theory and foundations of mathematics. You will find many interesting discussions of all these topics, as well as links to state-of-the-art research, in the n-category café. It will also not surprise me if you will find people there who share your mathematical taste.

If you still maintain, for whatever reason, that it is imperative that you do things related to higher category theory, I can tell you that there are many domains in this topic which are very 1-categorical in flavor. For example, you can

Do model category theory.

This notion, one of many brilliant ideas of Quillen, allows one to magically reduce homotopy coherence issues into a 1-categorical framework. Model categories also share many of the aesthetic features of ordinary category theory, in the sense that everything seems to fit together very nicely, while still being extremely useful for real world homotopy coherent mathematics. A bit less known, but also very categorical in flavor are derivators. You might also look into triangulated categories.

Finally, as many of the comments above suggest, it's possible that the things that you don't like in homotopy coherent mathematics are actually not essential properties of the field, but rather of its young age. You may hence consider to

Give HTT another try.

In doing so, you may want to take into account the following: I strongly believe that no one has ever written a technical simplex-by-simplex combinatorial proof of an HTT-type result without knowing in advance that what they want to prove is true, and moreover why it is true. This is because, despite the technicality of some proofs, higher categories do behave according to fundamental principles. Sometimes these principles are the same as the 1-categorical case, but sometimes they're different. As a result, it may take a bit of time to acquire a guiding intuition for what should be true and when. It is, nonetheless, certainly doable. I would then suggest that, before reading a given proof, you try to think first why the announced result should be true. In addition, think how you would prove, say, the 1-categorical case, and then try to extend the proof to higher categories dimension by dimension, and see where this leads you. Then read the simplex-by-simplex argument. It may suddenly look very clear.

Answer 2

I'm coming a bit late to this party, but I'll put in my two cents anyway because they are rather different from everything else I've heard so far. In a nutshell, my response is:

1. Yes, I agree that this is a problem (though I do think you would have done better to post only the question and not the rant), and

2. What you can do is be part of the solution.

For a long time I resisted "homotopical higher category theory" too, for reasons that I think are not unrelated to yours. I even wrote a somewhat whiny blog post about it. What eventually "brought me on board" was not the applications to algebraic geometry or what-have-you (which is not, of course, to denigrate those applications), but the truly category-theoretic conceptual insights arising from what you call HTT. Examples include:

• Colimits in a 1-category cannot be as well-behaved as we would like them to be, and the reason is because a 1-category doesn't have enough "room"; an $(\infty,1)$-category fixes this. For instance, Giraud's axioms for a 1-topos assert "descent" only for coproducts and quotients of equivalence relations; the analogous axioms for an $(\infty,1)$-topos assert descent for all colimits.

• Passing "all the way to $\infty$" has a "stabilizing" effect that enables $(\infty,1)$-categories and $(\infty,1)$-category theory to "describe itself" in ways that 1-category theory can only approximate. For instance, the 1-category of 1-categories does not include enough information to characterize the "correct" notion of "sameness" for 1-categories, namely equivalence (at least, not unless you hack it with something like a Quillen model structure); for that you need the 2-category of 1-categories, or at least the $(2,1)$-category of 1-categories. But the $(\infty,1)$-category of $(\infty,1)$-categories does characterize them up to the correct notion of equivalence. (Although for many purposes one still needs the $(\infty,2)$-category of $(\infty,1)$-categories, pointing towards the still largely-unexplored territory of $(\infty,\infty)$-categories.) Similarly, a 1-topos can only have a subobject classifier, classifying those objects that are "internally $(-1)$-categories", i.e. truth values; but an $(\infty,1)$-topos can have an object classifier that classifies all objects (up to size limitations).

• Various mysterious phenomena in 1-topos theory are explained as shadows of $(\infty,1)$-topos-theoretic phenomena. For instance, the analogy between open geometric morphisms and locally connected ones is explained by seeing them as the steps $k=-1$ and $k=0$ of a ladder of locally $k$-connected $(\infty,1)$-geometric morphisms, and similarly for proper and tidy geometric morphisms. Moreover, various apparently ad hoc notions of the "homotopy theory of toposes", such as cohomology, fundamental groups, shape theory, and so on, are explained as manifestations of the $(\infty,1)$-topos-theoretic "shape", which is characterized by a simple universal property.

• Perhaps most importantly, the fundamental idea that the basic objects of mathematics are not just sets, but $\infty$-groupoids. Thus, for instance, the really good notion of "ring" should be an $\infty$-groupoid with a coherent multiplication and addition structure (i.e. a ring spectrum), including the set-based notion of "ring" as simply a special case. And so on.

Note that none of these ideas depends on any concrete model for $(\infty,1)$-categories, and most of them have nothing to do with homotopy theory; they are purely category-theoretic ideas. So I think even a category theorist who cares nothing about homotopy theory ought to be interested in a kind of "category theory" where these are true.

That said, I think a good category theorist should care at least somewhat about homotopy theory, if for no other reason then for the same reason that a good category theorist should care about other applications of category theory. Like all fields of mathematics, category theory is supported and invigorated by its connections to other fields of mathematics, and the close tie between higher category theory and homotopy theory has great potential to stimulate both subjects. That this potential has been realized more fully on the homotopy-theoretic side is, I think, largely an accident of history and personality.

Why is $(\infty,1)$-category theory not usually done "Australian-style"? I believe it is just because people doing $(\infty,1)$-category theory don't know, or at least don't appreciate, Australian-style 1- and 2-category theory, while many Australian-style category theorists don't know or appreciate $(\infty,1)$-category theory. This creates a tremendous opportunity for anyone who is willing to put in the effort to be a bridge, teaching category theorists how to think about $(\infty,1)$-categories "category-theoretically" and teaching $(\infty,1)$-category theorists the benefits of "really thinking like a category theorist".

One way to be such a bridge is to learn the simplicial technology that's currently used for $(\infty,1)$-category theory and "do them Australian-style". For instance, as far as I know there is still no $(\infty,2)$-monad theory with the power and flexibility of 2-monad theory; someone should do it. Enriched $(\infty,1)$-categories are only starting to be investigated. The $(\infty,2)$-category of $(\infty,1)$-profunctors has been used for some applications, but its category-theoretic potential is largely unexplored. As far as I know, no one has even defined $\infty$-double-categories yet. (Edit: They've been defined, but apparently not systematically studied; see comments.) What about generalized $\infty$-multicategories? Etc. etc.

While a worthy endeavor, I suspect that this is not what you want to do. In particular, it sounds like you don't feel able to spend the time to really understand simplicial technology. I can sympathize with that; it's difficult enough for me, and I was already exposed to lots of simplicial stuff as a graduate student since my advisor was an algebraic topologist. So I generally avoid using simplicial technology as much as possible. One way to do this, which I have pursued myself, is to study $(\infty,1)$-categories using 1- and 2-categorical machinery, including Quillen model categories (which, by the way, have an algebraic version that is rather more pleasing to a category theorist's heart) but also homotopy-level structures such as derivators, homotopy 2-categories, and homotopy proarrow equipments.

This works quite well for surprisingly many things, and doesn't require you to learn any simplicial technology. However, it does often depend on the fact that someone has proven something using simplicial technology in order to "get into the world" where you're working. Moreover, you've also expressed some skepticism about the very idea of simplicial technology and concrete models. I think it'd be good if you can get over this to a degree — mathematics has to move forward with what we have, even if it's not perfect, and later on someone can make it better — but I do also sympathize with it, because for instance of the last conceptual insight I mentioned above:

• The basic objects of mathematics are not just sets, but $\infty$-groupoids.

How can this be, if an $\infty$-groupoid is defined in terms of sets (e.g. as a Kan complex)?

Well... there is now a way to study $\infty$-groupoids directly, without defining them in terms of sets: it's called homotopy type theory (HoTT). HoTT is (among other things) a foundational theory, on roughly the same ontological level as ZFC, whose basic objects can be regarded as $\infty$-groupoids; I wrote a philosophical introduction to it from this perspective. (There's also work on an analogous theory whose basic objects are $(\infty,1)$-categories.) Thus, HoTT offers the promise of an approach to homotopy theory and higher category theory that's almost completely free of simplicial technology, and incorporates the conceptual insights of $(\infty,1)$-category theory "from the ground up", allowing us to build intuition for, and work directly with, higher-categorical and higher-homotopical structures without having to construct them explicitly out of sets. When I read or write a proof in $(\infty,1)$-topos-theoretic language, I'm never quite sure whether I've dotted enough "i"s to make all the coherence come out right; but when I instead write it in HoTT then I am, not only because with HoTT I understand the profound reason why you already know what things intimately are (as you put it), but because a HoTT proof can be formalized and verified with a computer proof assistant. There are already some graduate students who have "grown up" with HoTT and can "think in it" in ways that surpass those of us who "came to it late".

Now, this "promise" of HoTT is not yet fully realized. Many coherent higher-categorical structures can be represented simply and conceptually in HoTT; but many others we don't know how to deal with yet. So here's another way you can be part of the solution: improve the ability of HoTT to represent higher category theory, so that eventually it becomes powerful enough that even the "applied" $(\infty,1)$-category theorists can do away with simplices. This is, in large part, what I am now working on myself.

Answer 3

This is too long for a comment, but doesn't exactly answer the question. However, I've had enough eggnog this Christmas that I'm going to post it anyway (despite knowing almost nothing about category theory).

Reading the question and skimming over the comments, I see a lot of romantic descriptions of the practice of mathematics that bear little relationship to how it is actually practiced. It is truly wonderful when a single elegant idea can completely illuminate and render transparent some part of the subject. However, these ideas are usually the end product of a long development that starts with a hacked together, complicated mess of arguments. And they are discovered by people who are deeply immeshed in the subject.

To put it another way, while it is great to have a strong philosophical take on what mathematics is and how it should be practiced, if that philosophy is not informed by the actual practice of mathematics, then it is unlikely to lead anywhere. Philosophical clarity comes at the end and not at the beginning.

To be successful at research, you have to be willing to get your hands dirty. If you don't enjoy the ordinary craft of doing mathematics, then it is unlikely that you will be happy as a research mathematician. But it is a craft. I strongly disagree with various comments that make it sound like you have to be some kind of crazy romantic hero taking superhuman risks or something. I certainly am not like that, but I have been able to make a career out this.

Now, it is impossible for us to give you personal advice on what you should do with your life or what direction your research should take. We don't know you. But I can say that everyone goes through periods of doubt and frustration. What I always do in those situations is to take a brief break from the front lines of research and go back to the sources that drew me to mathematics in the first place. Read some great mathematics, be refreshed, and then get back at it.

Answer 4

Questions about aesthetics are of course inherently subjective. In your third point you seem to be expressing a strong aesthetic preference for mathematics which starts from a smallish set of axioms and builds a large, unified theory, constructed with maximal generality. I can think of other areas of mathematics than Australian-style category theory that fit this description, like universal algebra, point-set topology, finite group theory, and several areas of set theory and logic. I believe a young mathematician in any of those areas could have posted a similar complaint bemoaning the general drift away from the kind of mathematics they care about.

By contrast let me try to articulate how I perceive the dominant aesthetic preference of the mathematical community, without necessarily endorsing it myself. One imagines that there is a "core" of mathematics, consisting perhaps of geometry, arithmetic and analysis. Questions in these areas are inherently interesting and deserve to be studied. Other parts of mathematics deserve to be studied only inasmuch that they can be applied to those core areas. From this perspective HTT is "superior" to Australian style category theory only because it has been more useful in algebraic geometry, K-theory and topology, which are all close to that "core".

A more blunt take is that this is memetic evolution in action. In a toy model, all mathematicians start out being interested only in the specific problem assigned to them by their advisor. During their career they'll become interested in more things, determined by randomness, personal preferences, but above all self-interest: "oh, it seems I can maybe prove something about X (which I care about) if I only learn a bit of Y (which I have only heard of)"... Rightly or wrongly, such a process will "reward" areas exclusively on the basis of whether they are useful to other areas, and "punish" those which aren't.

So what can you do if you work in an area which is out of vogue? Of course you can advertise your work and your point of view, but you can't force people to care about the theorems you prove. So you have two extreme options: (a) Keep working on what you care about, no matter what others think. This is not easy, but it can be done - you'll just have fewer job opportunities, fewer funding options, etc. Or: (b) Switch fields. This is not easy either, but it can also be done. This is of course easier if you know someone you can collaborate with and learn from, and if you can switch to something reasonably close to your own interests. Most people will maybe do something inbetween - try to nudge their own interests in the direction of things that can be applied to what other people are working on. Good luck.

Answer 5

I think the answer is obvious. If you want to stay in mathematics, quit HTT and go back to the basics of what got you interested in math in the first place. It's easy for young people to be seduced into highly abstract areas just because there is a community of senior mathematicians around them doing it, and I've seen it many times.

Your assumption that you don't have enough time to learn something new is incorrect. It does not take long to learn something different and even publish in that field if your heart is in it. Do another postdoc and take time to explore your own interests, publish and find a new community.

Answer 6

This is also merely a long comment.

'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.'

History shows that the 'australian way' (whatever this might mean) is not the way to go; and that the correct approach is to regard category theory as an auxiliary branch of mathematics. Basically anything useful in category theory was developped to help solving problems and formalising definitions in algebraic topology, algebraic geometry, etc. Grothendieck himself (the patron of everything abstract) regarded algebra in this way: as a means, not as an end. I know that some people prefer to write 2000 pages of purely formal arguments. Because that is way easier than solving some real problems that have been around for dozens of years.

'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).'

Maybe it is your task to develop a category-theoretic, linear-algebraic approach to ethics, and to enlighten those 'ignorant people'? I don't know whether you are joking (I certainly am), but if you really think that liner algebra has meaningful applications in ethics, you should go back to the beginning: enrol again as an undergrad and make a new start.

Answer 7

Let me address something which is not explicitly mentioned in other answers. Job issues exist, and they exist not only for category-theorists or the like, even much more ''working'' (in the sense of MacLane) pure mathematicians are struggling. In my opinion, the reasons (and possibly the solutions) for this are socio-political in character, we can discuss that elsewhere.

To return to math, I have not seen all the comments, but. First, using HTT to prove something, understanding the proofs of HTT, and doing HTT-level of papers are three different levels of difficulty. It is very instructive to have a problem from another field for which you would need higher categories, then it becomes much easier to navigate in the literature to find what you need, or ask precise questions to the experts. Later steps seem to follow as you get sucked in. On the go, you will perhaps start to value the aesthetics of the state-of-the-art in higher category theory, happy days. Even if your pure interest is category theory or something of that form, it helps. You learn more about (higher) category theory by applying it.

And, we have to remember that it has been only 20 years since the active development of the field. Various papers which followed since the HTT book show that more thinking leads to more transparency. You mention that there is no Bourbaki for homotopy theory; certain people believe that it is yet to come, and working towards it represents a big prospect in mathematical foundations.

Answer 8

This is merely a long comment.

You say that "[category theory] role is to trivialize [..] something", and I suppose part of the problem is that to trivialise in a new weird way something already well understood appears, by itself, of little interest.

I wonder if you would consider the following observations as examples of category theory trivializing something: one may note that a number of elementary properties are defined using iterated orthogonals of a single morphism, e.g. compactness "trivialises" as ${(\{ \{a\}\longrightarrow\{a\rightarrow b\} \}^r_{<5})}^{lr}$, and that the definitions of a topological and uniform space "trivialises" as a simplicial object in the category of filters. However, it appears unclear what theorems should these observations lead to. And by themselves, they are no theorems, as everything is very standard and trivial to check.