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I split off an entry applications of (higher) category theory from the entry nPOV.
Hopefully we find the energy to further improve this entry in various ways. For the moment I just added a 1-line intro. And a quote, which I think hits the nail of this entry on the head.
Edit: I’ve now tweaked the sentences I found somewhat troubling.
Why don’t you be more specific, Eric? Which parts do you find offensive? (Much of it I recall came from the article on nPOV.)
Constructive criticism works best when it is focused and to the point.
Edit: I wrote that before seeing your edits. I can kind of see your point.
Constructive criticism works best when it is focused and to the point.
Yep yep. I generally try to adhere to the Azimuth Golden Rule
I agree Zoran. Thanks. I removed my objections now that the tweaks have been made.
I originally stated them hoping that you guys try harder to see things from other people’s perspectives.There is a reason for the image some people have of category theory as witnessed from a quick glance at MO.
I’ve also noticed a tendency to belittle the work of others because of some unification provided by category theory. It is probably not intentional, but it is there. Since it is there, I think more effort can be made to show that “category theory is your friend”. It sometimes sounds like, “Hah! Differential equations?! Nothing be $\infty$-Lie algebroids. Pffft!” “What? Cohomology theory? You loser. That is just hom spaces. I eat those for breakfast because I’m an n-category theorist.” I’m just saying what I feel when I read some of these things and, as I said, I love you guys. Imagine how someone who DOESN’T love you guys would read it.
Thanks for tweaking. Looks better now. I think I was getting a bit tired towards the end of today’s working day and my formulation skills declined.
I’ve also noticed a tendency to belittle the work of others
I think the problem is as much on the other side: the statement that something has been unified is received as a belittling, while it shouldn’t.
This is a very widespread misunderstanding. It is at the heart of it really a little weird, if you think about it. But I suppose it is human nature. Much of the opposition against category theory is not so much the category theorists saying something in the wrong manner, but the statements themselves. Some people feel better the more obscure their field of study looks.
If one person actively dislikes another person, that’s not necessarily the other person’s fault.
And I think the statements I had, even before the tweaking, were entirely true: there are certainly fields of study in math that you have no chance to even formulate if you don’t use higher category theory. If that fact makes anyone feel bad that’s sad, but not really my problem.
Similarly with the general dislike of category theory: I don’t see it as a task to campaign for the goods of category theory. If people don’t like it, so be it. There are many good ideas in the world that are feverishly attacked by some people. That’s not the fault of the good ideas.
@Eric I also have that feeling sometimes. We feel a ’joy’ that some hard analogy / problem / link /question suddenly feels simpler because we have reformulated it in the nPOV, perhaps it is even completely solved, and so are very enthusiastic about that advance. Others see what they think of as a too-abstract set of concepts and ideas that someone is saying is ’the best thing since sliced bread’. (I don’t like sliced bread, by the way! It is usually cut too thinly.)
There are some Ideas sections that go too quickly (and possibly too enthusiastically) without really getting an idea of where the idea comes from.
I gave a talk on crossed complexes, crossed n-cubes etc at Aberdeen and somehow managed to show to certain traditionalists (no names mentioned) algebraic topologists how that theory reflected concerns in their view of homotopy theory. I even managed to point out that the crossed module machinery was ‘traditional’ coming from Whitehead’s Combinatorial Homotopy 1 if not further back. Whether I convinced them enough to change their prejudices is another matter.
We do need to make the entries on key areas user friendly. I have noticed some comments on MO and felt a bit disappointed by the anti-category theory view. (I said in a comment in MO that Cat theory was not a Hob Goblin. I don’t think the comment was understood! I meant that it is a natural way of looking at things and not some fearful beast that lurks behind a commutative diagram trying to trick you into a state of confusion…. or something like that.)
Cohomology theory? You loser. That is just hom spaces. I eat those for breakfast because I’m an n-category theorist.
The thing is: everyone can have them for breakfast, once higher category theory has thus reduced the concept to something edible.
Saying something is an hom-space in a higher category is conceptually much simpler than most of the explicit definitions of cohomology theories. It is not “because I’m an n-category theorist” that I can think about hom-spaces. Every child can. It is because I took higher category theory seriously that I observe that it is that simple. For everybody.
I often run into people who when they hear that concept xyz is a special case of general abstract concept abc say: “Ah, but we don’t have to say it in such a difficult way. It is much easer to say just xyz.” Which is the wrong way round. The abstract statements are usualy way more simple. You can teach them to kids. That’s the point of the abstract theory: that it clarifies things and makes them simpler.
The problem is that some people develop emotional barriers if they realize that there is a simpler way of conceiving their subject. And I think that’s the main problem. And the only way to solve this problem is to teach the next young generation the good description from the very beginning.
If that fact makes anyone feel bad that’s sad, but not really my problem.
At some point, like or not, you will need to get an actual “job”. When/if you get an actual “job”, you may have to actually start caring about what other people think.
Thanks for the advice Eric, and for putting job in quotation marks. But I do think you are missing my point.
If I think Earth is a ball and somebody else thinks its flat, I won’t invest energy into convincing that somebody, and I won’t apply for a job with him.
There is something to be said for the Bohemian lifestyle I suppose :)
The problem is that some people develop emotional barriers if they realize that there is a simpler way of conceiving their subject. And I think that’s the main problem.
I think that is precisely it, Urs. Although I’m not sure it always reaches the level of conscious realization; there is a kind of unconscious fear that there is a conceptually simpler way of seeing their subject. People become angry and defensive about their hard-won points of view.
And if the realization begins to break into consciousness, there may be something like the Kübler-Ross stages of dealing with it, like people insisting that these realizations are only superficial (denial), and so on.
There is something to be said for the Bohemian lifestyle I suppose :)
We can chat about questions of lifestyle and career strategies on another forum. Here the question is really a scientific one:
if scientist X knows that statement A is true, and hords of his colleagues deny A with good reason, then X will be a good scientist only if he sticks to claiming A even in the face of personal attacks and ridicule.
If we ran a wiki on biology, say, instead of math, we should advocate the idea that nothing in biology makes sense without the notion of evolution. This is the ePOV. There are hords of people who disagree with this statement. But a good biology wiki would not bend to their desires.
As someone who has both (a) been irritated by overreach in the past, and (b) appreciates categorical methods and regards them as indispensible, let me explain how and why it has happened to me.
The setting: I'm talking about some point about lambda-calculus and proof theory. A local category theorist listening in interjects "Isn't lambda calculus just cartesian closed categories?" I get irritated. The reason I got irritated was because typed lambda calculi often form CCCs, but they aren't just CCCs. Lambda calculi have a lot more structure to them which is erased/lost when you interpret them with a CCC.
This is not an unconscious fear that there is a simpler way of seeing my subject. Instead, I'm annoyed that a valuable perspective (traditional categorical proof theory) has somehow been turned into a set of blinders which keep another mathematician from understanding my subject. Concretely, CCCs equate beta-eta equal terms. This is indeed often a useful thing to do, but not always --- for example, if you want to prove normalization theorems, this perspective offers basically no help at all.
This is not to say that categorical machinery is unhelpful --- in fact, it's extremely useful! But you need to apply it in a sophisticated way in order to accurately reflect the concerns of proof theory. For example, one way of proving normalization theorems is to start by viewing rewriting as inducing a preorder on terms, turn this preorder into a site by giving families of covers corresponding to some technical conditions from rewriting, and then proceeding to interpret types as sheaves.
This is the essence of Girard's method of reducibility candidates, which is one of the coolest proof techniques in all of structural proof theory. And thanks to categorical methods, this proof technique is no longer only bravura Gallic wizardry: it is also revealed as a beautiful application of topological ideas to logic. But we could not have been reached this viewpoint if people had stopped at "lambda calculus is a special case of CCCs".
The barrier to entry into n-category theory-land is actually quite high, at least for those of us who prefer to understand all of the results that make it a usable theory.
Mastery of quasicategories, for instance, must be preceded by a mastery of simplicial homotopy theory, model category theory, the theory of simplicial categories, etc.
I also think that the amateur psychoanalyses performed above look quite foolish and should be redacted. It’s never good to try to act like an oppressed minority.
@Guest #15:
I understand why you might have been irritated – as academics (and I include postgrads like me in that) we all have a personal attachment of some sort to our subject area, simply because we couldn’t do the work we do without one. But even though the incident you describe could feel like belittlement of your field (or your approach to it), it’s also a perfect opportunity to explain what you’re doing:
Category theorist:
Isn’t lambda calculus just cartesian closed categories?
You:
No! And here’s why… rewriting… locally ordered bicategories… Seely… lax 2-adjunctions… Gallier… sheaves on hom-orders…
In my (admittedly limited) experience of mathematicians and computer scientists, I’ve found category theorists to be a lot more open than most to alternative mathematical points of view. Chances are your local CT person would be delighted to find out that sheaf theory can yield insights into normalization proofs – if not, and if he/she refuses to listen, then that’s just small-mindedness, and not your problem any more.
I agree, by the way, that Gallier’s local-site idea is beautiful, and I’ve been wondering for a long time if there’s some connection with the higher-categorical stuff that Urs et al. have been describing on the nLab. Incidentally, there’s an unwritten rule here: you’re not allowed to mention a piece of mathematics here on the nForum without then writing something about it on nLab. You’ve just volunteered :)
if scientist X knows that statement A is true, and hords of his colleagues deny A with good reason, then X will be a good scientist only if he sticks to claiming A even in the face of personal attacks and ridicule.
But that is not the situation here. You make a claim. In most cases (not all), I don’t think anyone is saying the claim is false or denying it. At most, I said that you should consider more politically correct ways to state the claim. No one is suggesting that you not make claims.
I think your tone has a tendency to come across as arrogant. Have a look at the changes I made and compare to what was there before.
There is a plethora of applications that become transparent only from the nPOV. There are whole fields which cannot even be conceived without the language of higher category theory. This page lists and discusses examples.
“only from the nPOV”? Really? I’ll go out on a limb and say there are experts in each example field given on that page that are NOT interested in category theory and they probably think they have a fairly decent and transparent idea of what is going on. The first sentence is already quite arrogant.
The second sentence is worse. Every field listed on that page was born and has a happy existence completely independent of category theory. Category theory may have helped to illuminate and unify some things, but to suggests that one cannot fully understand any one of those fields without category theory is degrading. Those are not the words written, but that is the message that gets across. “You can’t seriously call yourself a mathematician if your field is cohomology, yet you do not view it as a hom space in $(\infty,1)$-topoi.” I mean how long have $(\infty,1)$-topoi even been around? Give people some time to let it soak in.
Closer to home, my doctoral research was in computational electromagnetics. From this higher perspective, you could say, “Bah! That is just U(1) gauge field theory. The simplest of the simplest of gauge fields. Nothing interesting could possibly come from that.” Ahem! I’ll tell you right now that no level of higher category theory would help you with some of the large scale scientific computing challenges one comes across when solving realistic systems such as computing the radar signature of a stealth aircraft.
Statements like the ones you are making are broad and can be interpreted as insulting.
This construction is probably literally unthinkable without adopting the n-point of view when approaching it. Using this point of view, the general strategy for approaching it however becomes naturally evident.
What does “probably literally unthinkable” even mean? And why say something like that when there are more reasonable ways to express the same thing?
We can chat about questions of lifestyle and career strategies on another forum.
We could, but I think this goes to the one aspect of the “unpopularity of category theory”. At this moment, how many higher category theorists are in a position that they could offer you a job today even if they wanted to? How many mathematicians in general are confident they will remain employed if the financial crisis worsens (as I expect it to) and funding for higher education evaporates? How long should one intend to be a post doc?
At some point, making friends with non-category theorists will become important. Starting an academic career in science has always been challenging, but as the world financial situation deteriorates, things will get more and more difficult.
What does “probably literally unthinkable” even mean?
It means that without the right language tool, you are not able to see it.
There are many example for this in the history of math.
To give a simple example: if you insist, you can do all of differential geometry with just calculus of matrces valued in smooth functions. That’s how it started out at the end of the 19th century. I claim that most of the results in differential geometry nowadays are unthinkable if you stick to that language. In the sense of: you’d never be able to think them up.
Guest: that’s a good illustration; I would indeed agree the local category theorist was behaving insensitively (both conceptually and emotionally), at least if he/she had just left it at that.
I have to agree with Eric that we’re not doing ourselves any favors by coming across as arrogant, as history should have shown us by now. It is something to keep in mind.
Nobody disagrees about arrogance. The two sentences that Eric is picking at I wrote last night after many hours of intense work on nLab entries when in the last minute I decided, after discussion with Zoran, to split off an “application” page from the nPOV page. So I had to quickly think of something to put as a lead in. What i ended up typing was ill-formulated, as Eric noticed and as I have already agreed on above. It has been improved, now let’s move on.
Okay, good, let’s move on. Just a few words to Harry: I agree it’s probably better not to behave in public like an oppressed minority, but that negative feelings towards category theory and category theorists are frequently expressed is undeniable (Tim Porter gave some examples recently), and it is natural to inquire into the causes of that. Amateur psychologists we may be, but I don’t understand how that sneer is actually an argument against anything anybody has said.
I just think that if we were discussing this in private somewhere, it would be fine to say things like this. However, outsiders to the category theory community do read this board, and some of the posts above make the whole community look a bit foolish. I think that if this discussion is going to be had in private, make it actually private. If not, it should be on the café so the other side can respond and clarify.
Basically, I don’t like the quasi-privacy of the nForum as a vehicle for this discussion.
I was going to react but I think it is probably best to move on to more constructive topics. My one parting shot is that we do need to be careful about pedagogic issues in the more accessible topics of the Lab, as it is being quoted in e-mails etc quite frequently, so we need to make it look good, even if we know there will always be improvements and changes to be made and there is a lot pf work in progress.
To move on:
What expertise on the various areas of applicability of n-cats etc., can we offer other that the obvious ones in mathematical physics and algebraic topology/geometry? I have been thinking of some of the ideas of higher dimensional automata and higher dimensional transition systems. This would also link with directed homotopy theory. It might give some ideas on applications in CS. Any other ideas so that when I (or anyone else) has time to start pages on those topics it is simpler to create a fairly coherent picture from the start.
@Tim: What happened at Bangor had to do with category theory as well?
Let us say that there was a research assessment exercise done UK-wide. I do not think that the research group at Bangor had a nationally high profile because category theory was never flavour of the month in UK. It was regarded internationally within category theory circles, but that was not enough, of course, so we got a poor grade in that exercise. There was a period in which we got virtually no research grants in the area of our kind of categorical algebraic homotopy. We put in a large number of grant applications and each time received good reports from all but one referee! That was enough to kill the application. As the UK system became more and more focussed on research income as a measure of what is called ’research excellence’, the bigger universities cornered more and more of the grants and so there was less for the minnows. The situation was thus very complex and it is not a good idea to try to go into too much detail, but the fact that category theory is not generally highly regarded, even when it is being applied, probably played a role in the closure.
@Harry #23: you’re right. It’s not the right forum for this type of discussion.
@Urs #21
Nobody disagrees about arrogance. The two sentences that Eric is picking at I wrote last night after many hours of intense work on nLab entries when in the last minute I decided, after discussion with Zoran, to split off an “application” page from the nPOV page. So I had to quickly think of something to put as a lead in. What i ended up typing was ill-formulated, as Eric noticed and as I have already agreed on above. It has been improved, now let’s move on.
Yep yep. Back in comment #7, after I modified the page, you said
Thanks for tweaking. Looks better now. I think I was getting a bit tired towards the end of today’s working day and my formulation skills declined.
That is understood (we’ve all been there) and if it had stopped there then I would have happily let it rest, but you then continued in the same comment:
I think the problem is as much on the other side: the statement that something has been unified is received as a belittling, while it shouldn’t.
There are likely many reasons for this (to the extent that it is true). Based on what little I have seen, I think one reason could be the “manner” in which you confront people with the fact. From my observation, you tend to present it as child’s play. Your words were, “You can teach them to kids.” If that is true, please teach it to me. I have been trying (whenever I can find time) for years to understand this stuff and still don’t. Once I do understand, I may also think it is child’s play (everything is easy after you understand it), but the road to that point is not easy.
Back to my computational EM example, if you did come to me one day and told me that category theory, in a single stroke, DID tell me how to model the physics, develop a numerical solution method, find a fast algorithm to solve it, and write the code efficiently, then my initial reaction would be, “No way! You just simplified 6 years of my life into a meta process that could be followed by a child!” I’d have a hard time to believe it. I may even deny it initial. In fact, I would probably even go through the (Kübler-Ross) five stages of grief Todd mentioned. Note: Only once you reach stage 5 do you get “Acceptance”.
This is a very widespread misunderstanding. It is at the heart of it really a little weird, if you think about it. But I suppose it is human nature.
Yes! Understanding and appreciating this human aspect would help.
Much of the opposition against category theory is not so much the category theorists saying something in the wrong manner, but the statements themselves.
The former impacts the latter. If you present a statement in a way that immediately puts me on the defensive, it is natural for me to react negatively as well.
Some people feel better the more obscure their field of study looks.
If this is true, it is a symptom, not the disease. I spent 15 years immersed in physics and engineering. 8 of those years were focused in a very specialized subject. I feel like I am an expert in that field. Maybe even a leading scientist. If you came into my office one day and told me, “Did you know that your field is nothing more than hom spaces in an $(\infty,1)$-topoi? Did you also know I can teach that to children?”, I would “kindly” show you the door.
My message here is that of an admirer. I am telling you the way I see things because I fear that people that might not be admirers could see things even more negatively than I do. I hope the tone can change for everyone’s sake.
And I think the statements I had, even before the tweaking, were entirely true: there are certainly fields of study in math that you have no chance to even formulate if you don’t use higher category theory. If that fact makes anyone feel bad that’s sad, but not really my problem.
Yes. Before I tweaked them, in my original comment, I said they were true, but presented less than optimally. The last sentence above made me throw up my hands. I hope it is not useless trying to help, but there you go.
Similarly with the general dislike of category theory: I don’t see it as a task to campaign for the goods of category theory. If people don’t like it, so be it. There are many good ideas in the world that are feverishly attacked by some people. That’s not the fault of the good ideas.
How would Einstein react? Would he scoff at the imbeciles who deny his ideas, or would he battle the deniers with the beauty of his theories and through the elegance of his exposition? I think that the utility of category theory is undeniable (in time), so please please (everyone) make more of an effort to do battle using the beauty of exposition as your primary weapon. That would do a lot more for category theory than the correctness of the statements made. And although it may sound silly, please do keep the five stages in mind.
I think I’ve said what I want to say as clearly as I can say it, so I’ll now try to go back to my role as passive student and active admirer of the work you guys do.
PS: For what it is worth, to the degree many in mathematics have been shielded by recent world events, I fear that this will change in the coming years. Budget cuts have been deep and could get a lot deeper. It is not too early to begin thinking of survival. This is my opinion as a professional risk manager.
How would Einstein react?
Probably by plagiarizing more of Hilbert’s work =p
@Eric I think believe there is an old Chinese proverb: One showing is worth a thousand tellings. That is true, but only looks at the visual as against the verbal form of communication. I sometimes wonder if the form of communication we are using is not tending to lie somewhere between the two. The results are therefore unpredictable. As I said earlier it is very important to reflect on the pedagogic / communication aspect of what we are doing. It is becoming non negligible since we have been managing to do some things quite well (:-)) so people (and not just mathematicians) are looking in the Lab to see what we have available.
Perhaps we need some pedagogic discussion here (this thread is almost that but not quite). There seems to me two tasks that need doing from that aspect. The first is to improve the ideas parts of entries. Sometimes these go too deep too quickly, so can put off someone who is not that au fait with the stuff. That is linked to the second task, learning how best to use the multitrack multilayer nature of the lab to best advantage for a ’popularising’ / ’ teaching’ role. If people are wanting to look at things that we are doing, the Lab becomes a show case where we can show that it is ’fun’, and (initially) not that frightening, a ’shop window’ for a set of approaches that we find useful, natural etc. That is why I have been adding in some knots stuff as one can get to quite deep ideas using geometric ideas and strangely as Lou Kaufmann shows the whole time, knots and links help you to understand other ideas, since, perhaps they are simple and yet have a ’simple complexity’.
On that point, I started to write some pages on colourability of knots where you use a set of group elements to label the arcs of a knot diagram. This gives a neat way to prove that knots exist. (I have used this as a way into explaining various algebraic and categorical aspects of the theory in the past.) I suddenly started thinking about the problem from loads of other directions, e.g. seeing if other algebraic gadgets could give interesting labelling systems (anyone know?). (I will probably give a list of questions that I do not know the answers of when I get towards the end of what i plan to write.) That area also gives a way into state sum type approaches to TQFTs etc. so … . Hopefully other directions will become apparent as I progress and following the idea will lead to the naturality of certain of the concepts that are there. This process should be helped by the multilayer nature of the Lab, and should be a challenge to write well.
@Tim That sounds good to me. I think you just volunteered :)
By the way, one of my favorite nLab pages that is in the spirit you’re talking about is
THAT is an awesome page.
Maybe instead of expanding the “Idea” section of each page, which would significantly increase the size, we should start a series of “Motivation for…” pages written like this one.
I also tried to start an “Understanding…” series here
Guest: that's a good illustration; I would indeed agree the local category theorist was behaving insensitively (both conceptually and emotionally), at least if he/she had just left it at that.
This story didn't have a happy ending, but it didn't have a sad one either. At the time, I didn't know about Gallier's work on the reducibility candidates method, but I was able to explain why I didn't want to equate all the proofs CCCs did. (I think I explained my problem in terms of the distinction between comma and conjunction in sequent calculus.) The local category theorist was a smart guy, and ended up understanding what I wanted to do, but not why I wanted it.
I was left with the impression was that he found categorical proof theory sufficiently pretty that problems not easily expressible in its language became unquestions.[*] Since then I've become very sensitized to this effect, which can occur when any two areas touch. To really understand a field, you have to understand why its problems are interesting, but if we had a good mathematical characterization of that then they wouldn't be problems any more. So it's hard to quickly explain why certain problems are interesting or not.
[*] The converse also happens; as you doubtless know, models of dependent types as lcccs are troublesome due to needing to make a choice of pullbacks, but proof theorists tend to find it weird that substitution is problematic, but not apparently for any of the usual reasons.
@Eric I think the idea of Motivation pages is a good one. Too long an ideas section means that someone who knows what the idea is but wants to check on things and links etc, will have to use the pagedown arrow for a long time. A short ideas section together with a more leisurely motivation or even understanding entry would be a good thing. (Actually some grad students could write some pretty good ’Understanding’ entries as they start not understanding. Magnus Forrester Barker did that as an exercise (without being asked to) as he needed to understand crossed modules. The result is quoted as being one of the clearest introductions to the concept. )
@Guest: #15, now you would have the chance to rebut “Isn’t lambda calculus just cartesian closed categories?” with “No, it’s (at least) cartesian closed 2-categories”, as here.
I have been looking at the paper: Eric Goubault and Samuel Mimram, Formal Relationships Between Geometrical and Classical Models for Concurrency
It looks very interesting and provides a wealth of connections between cubical sets and applications in CS. One of these is with Mazurkiewicz trace theory and the trace monoid. This in turn seems to be related to what is called the history monoid. (see Wikipedia). Winskel wrote a paper looking at relations between this area and quantum stuff and the histories approach there.
I am wondering if anyone has explored this as I am finding if hard to come up with on-line stuff that gives a clear view of the trace stuff. (There is a nice paper on trace semantics as coinduction but it assumes knowledge that I do not have in detail. )
My reasons for doing this is partially to learn more about it but also eventually to try to draw together material that can be useful in giving ’applications’ and interpretations of the 1-directed homotopy hypothesis, if and when someone among us has a detailed attack on it. I feel there is a very close link here as the canonical structure that Goubault and Mimram use is a symmetric cubical set.
The discussion has gone on from this, but …
Mastery of quasicategories, for instance, must be preceded by a mastery of simplicial homotopy theory, model category theory, the theory of simplicial categories, etc.
Quasicategories seem much simpler to me than any of those other things. Why would I want to master them first?
Proving nontrivial things about quasicategories requires a deep understanding at least of simplicial homotopy theory, which is the theory of (oo,0)-categories forming the base of the (oo,1)-theory. In the course of dealing with quasicategories (say to describe the quasicategory of small quasicategories), you have many forced encounters with these much more complex subjects.
I’d say that a deep understanding of simplicial homotopy theory is necessary, and that one should have an intuitive feeling for model categories (and simplicially enriched categories are the least important of the three).
I’d be delighted to see evidence to the contrary, but in my experience, this is true.
As an ’old hand’ in Simplicial Homotopy and an early worker on quasicategories (in the 1980s), let me say that to work with quasicategories it is probably necessary to have seen the way that simplicial homotopy theory built up (historically, I mean) so skimming through Curtis’s famous early survey article was what I recommended as a starting point, with explanation of the links to PL topology (where some of the motivation came from, and which was a source of ideas for proof methods). (Look at Lurie’s notes on Microbundles, they are very good indeed.) Simplicial groups plus a smattering of Q model cats in some form, then gets you where you can start, (provided you know lots of category theory. If I remember the reasons for the comment earlier, it was for beginning students in quasicats. My problem would be to decide what were the useful questions to ask.
I frequently observe this in pure math students: the idea that in order to proceed beyond one definition, one needs to have “full mastery” of all the theory going along with the definitions that it depends on.
I see that some pure math students are at high risk of getting themselves stuck this way. They open a textbook and as soon as the author mentiones a classical theorem without proof, they will not rest until they recursed through all the cited literature, dug out the original references for that proof and confirmed it themselves.
This is a very good trait in a mathematician. But if it becomes compulsory there is the risk that everything will grind to a halt.
And it also goes against the spirit of math in this way: the whole point of a theorem in math (as opposed to statements in other sciences) is that once it has been proven, it can be trusted to be true. So I can just use it as a black box. I don’t need to have full mastery over the subjects that the autors of the theorem had to derive it, I just need to know how and where it applies.
To come back to the question at hand: of course to fully master quasicategory you need to also fully master simplicial homotopy theory and lots of other things. But to just understand something about quasicategories (particularly that something that you may need in your own research) often much less will do.
My main point is: don’t scare yourself away from complex math, just because it is complex. In general, don’t be scared of math. Math is fundamentally easy: everything proceeds one trivial step at a time. Just go and take these steps. Quasicategories are no exception.
There is a story that when Grothendieck was at the IHES a visitor complained that the library was not that good, and AG replied ’ we write mathematics we do not read it!’. A bit tough, but it is the bottom line. He was also asked ’how much do you need to know to do algebraic geometry?’ and his reply: everything or nothing. Or so the story goes!
everything or nothing.
A fun anecdote, but taken literally it makes my point: even Grothendieck doesn’t and didn’t know everything. Not by far. Math is vast. So then the only conclusion is: we all quit and do macrame instead. That can’t be it. Instead: go and start somewhere. Don’t be intimidated!
it makes my point
Exactly why I mentioned it. :-)
Exactly why I mentioned it. :-)
Oh, I see. Well, it seemed to me that the anecdote was supposed to be saying hat Grothendieck thought that to really understand geometry, one needs to understand the whole edifice of math. It seems at least plausible that he would have suggested something like this, because that is really what his main thrust has been: show that algebraic geometry is but a special case of a general abstract theory of geometry.
I think here is where higher category theory comes into the game: it is impossible for a human being to really understand all of math in its details . But with higher category theory used as an organizational tool, it is possible to understand all of math in its grand structure .
For me this is the central motivation for higher category theory: it alows us not to drown in math but see the grand organization of it all.
So in that sense I am all in favor of saying “we need to understand everything”.
And in that sense everybody can “master” $(\infty,1)$-categories (to come back to our example), by saying: “the $(\infty,2)$-category of $(\infty,1)$-categories is the well-pointed $(\infty,2)$-topos”.
The rest is details. Some of these details one may have time to absorb in a life-span. Others not.
@Zoran I think you are right on the details of the AG story.
@ Harry
I agree that one needs to know simplicial homotopy theory to understand quasicategories, but I disagree that one needs to know it first; one needs to know it eventually and can learn it in the course of studying quasicategories. In analogy, one needs to know groupoid theory to understand categories, but one doesn’t need to know it first; one usually learns groupoid theory in the course of studying categories. (That might not the best order, of course; groupoids are simpler than categories, and simplicial sets are simpler than quasicategories, so maybe one should learn about them first. It was wrong of me to include simplicial homotopy theory in the list of things that quasicategories are simpler than.)
But I don’t agree that one needs to know model category theory. Theoretically, one uses quasicategories to avoid model categories. In practice, one might very well use a model category to present a quasicategory, but you no more have to learn model categories first than you have to learn about group presentations before studying group theory.
I guess that complete mastery of quasicategories would require mastery of all those other things, but it would not have to preceded by them.
traditionally more extensive and to the point than those ridiculous seminars which often take place in the west where only 1-2 questions are allowed and that let’s clap our hands
There are two sides to that coin. At the U of C Drinfeld and Beilinson run a “Russian style” seminar which I attended (while I was there) a few times when the subject sounded interesting, but it always went on and on long past my point of exhaustion, because of the insistence on understanding everything the speaker was saying in detail, in real time, rather than allowing him/her to tell a more complete higher-level story in a reasonable amount of time by glossing over details which can be worked out in the privacy of one’s home or office, or read about in a paper or preprint or even a textbook. Perhaps this seminar was not representative, and/or perhaps this is just a difference in preferred style of doing mathematics, but I think my preferred style is just as valid. Ideally, a seminar speaker will be around for a while after the seminar, so that people interested in a deeper discussion can join him/her for it, while people who are not interested can leave after the seminar is officially over and still have gained a higher-level picture.
The hi-level you talk about is about research seminars. How about studying seminars where a student is expressing something what is important for the program of the group of people in the background ? I recall that to my undergraduate development it was extremely important that I had opportunity in Zagreb to present 8 seminars in differential geometry of 2 academic hours length each, where I went step by step on basic things like associated bundles and Ambrose-Singer theorem, and others from the audience, would ask questions or give alternative proofs of some statements, or hint at generalizations and so on. I benefited from those much more (and from similar ones by other people) then from most guest seminars from foreign hi guests.
Edit: in the same seminar we had weeks when the "foreign speakers" would come and also some sessions when an expositional talks would be given. The point is to mix. Seminars which prescribe level miss one dimension by making speakers uneasy of the plan. level and style they like to proceed in.
Far from true in non-ideal situations.
Hence why I used the word “ideal.” (-:
Of course, a student seminar is different than a research seminar; I was talking about research seminars. And the point about there not being many books in the past is an interesting historical explanation for how things came to be, but not an argument for why one style is better than another now. Cultural differences (e.g. acceptability of leaving in the middle of a seminar) and differences in the makeup of the audience may also make a difference – I was just saying that I think a blanket description of non-Russian seminars as “ridiculous” is unjustified.
a student seminar is different than a research seminar; I was talking about research seminars
As I explained, the point is in mixing the two. Once the communities separate into different time slot and different audience one does not have the advantage of having advanced and beginning listeners together.
I think a blanket description of non-Russian seminars as "ridiculous" is unjustified
Come on, who said that ? I said that there is a ridiculous norma in many western seminars to allow just one-two short questions, no extension of talk by the speaker (like you already went 1 minute over time!), and repeating the phrase "let's thank the speaker again", plus looking bad at people who leave earlier and scorning people who complain if they do not get the answer. All that flexibility should be allowed, otherwise it has no point. If a lady came to Wisconsin, spent about 500 dollars of grant money for airplane and two nights of the hotel and dinners, then I find it criminal toward the taxpayers that she is not obliged to stay something like quarter of the hour talking to real audience, but rather runs for dinner with her old friends. There are many seminars in west whcih do rebel that ridiculous format, for example the famous algebraic geometer Abhyankar at Purdue leads a seminar for several decades where he will not let you go until you said things in a way which is precise and palatable. There was a case in which an incompetent speaker did not write anything on the board but let X be a Riemann surface, then he asked what do you mean, the discussion started and never finished (I heard of that one). In another there was a guy who was not sure if he works in projective or in affine setup, as he presented an algorithm which used blow ups what means projective, but he was often reducing to local argument what means affine. Abhyankar was insisting to reconcile that and tried to help, but the author did not succeed. At the end the talk was rescheduled to be redone in few weeks. My knowledge in classical algebraic geometry is not much and i learned quite a bit in few of the session of Abhyankar's seminar.
As I explained, the point is in mixing the two.
You also drew a distinction between them in #51.
who said that ?
I thought you did in #44. Apparently I over-interpreted; I apologize.
I think there is a balance to be struck, and the right balance depends on the goals of the seminar and the people in attendance. I could also cite examples of seminars where insistence on completely explaining basic concepts has prevented the speaker from getting to anything interesting or educational before the audience got exhausted – or at least I remember vividly sitting in such seminars, but I couldn’t tell you what they were about because they never got to the point! A seminar doesn’t have to be precise in order to be educational; in fact I think that often precision can be the enemy of clarity. On the other hand I definitely agree that a speaker should have time to talk with the audience. At Chicago there is a tradition of a “pre-seminar” which is theoretically aimed more at students, is more informal and feels more free to insist on explaining basic things, before the “seminar” where the speaker is traditionally allowed to finish their talk and convey the important new ideas for the purpose of which they were invited. Such a division may not work everywhere, but it has advantages.
Thinking about this some more, I agree also that it is more common in the West, and more detrimental, to err on the side of fewer questions rather than more. Also, with a good speaker who has prepared well and is on top of his/her subject and pitched it at an appropriate level to the audience, there are probably less likely to be lots of derailing questions — while if people become accustomed to and expect the audience to insist on understanding, it might (ideally?) incentivize them to give better talks in the first place.
I have added to applications of (higher) category theory
the rising sea quote
the pointer to homotopy type theory (I was struck looking back at the entry remembering that the piece about “$\infty$-logic” there was written before that term was common, or at least before it had an nLab entry)
This is prompted by this MO discussion.
I couldn’t resist wading in.
I like your comment, but I doubt the suggestion that those big historical names you mention were not also interested in solving problems.
The trouble is that there are “small problems” (such as “does $x^n + y^n = z^n$ have integer solutions?”) which the proverbial grandmother can understand, and then there are “large problems” (such as: “what is the deep principles of arithmetic geometry”?). Progress on the large problems is typically what solves the small ones in passing (as in the above case), but it seems these large problems may get too large for some people to still recognize them as problems. It’s the proverbial wood not recognized for all those trees.
The most striking current example: these days I hear people say “Lurie won’t get the Fields, because he doesn’t have a theorem yet”. That is like standing in front of a mountain and complaining the lack of a pebble stone.
Perhaps I wasn’t clear then. I didn’t say they weren’t interested in solving problems, but that they were not always constrained by problems that people of the time (equivalents of someone just learning about cohomology today, as he admits) would have seen as ’down-to-earth’.
I like your comment, but I doubt the suggestion that those big historical names you mention were not also interested in solving problems.
I liked it too, very much, but I didn’t hear that suggestion. Just looking at a different class of problems, as you go on to suggest yourself.
I am referring to David’s line
not always constrained by the need to resolve the down-to-earth problems
My point is that in developing theories these people may very well have been driven by what they realized (but others did not) as concrete problems that need to be addressed.
Okay, I see your point. I took David’s “down-to-earth” as referring to what those ’others’ think of as down-to-earth.
Yes, that’s the point. ’Down-to-earth’ is obviously a completely relative notion. The thing the MO poster wrote as concrete would have sounded out-of-this-world to people a century ago. The only function of ’Down-to-earth’ is to distinguish things you see the point of from those that you don’t understand, but which have been proposed as important. Really what he needs is the story of the rational development of mathematics from the point at which his current understanding has reached to the current best understanding of where it’s going. This doesn’t have to take the form of the construction of means to solve things that he can understand.
Given what he wrote, I wouldn’t envy you the task of so illuminating him.
It’s a bit like with environmental problems. The main problems are often beyond people’s grasp or interest.
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