TITLE: The "derived drift" is pretty unsatisfying and dangerous to category theory (or at least, to me)
QUESTION [49 upvotes]: 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):

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.
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?
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:

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.
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 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?
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).



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.

REPLY [68 votes]: 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:

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
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.