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To avoid filling up the transgression topic, I've moved the discussion here.
If people feel it's too strong, please remove it.
I found the statement to be a very good analog of the original statement in biology. And yes, I certainly think of ordinary category theory and 0-category theory as being included in higher category theory.
I feel like we could call everything category theory, but that's not going to change whether or not categorical methods are useful. For example, category theory really has no place in Differential equations (and to be honest, it doesn't seem like it ever could be). Similar statements are true of analytic number theory and a whole host of other subjects.
For example, category theory really has no place in Differential equations (and to be honest, it doesn’t seem like it ever could be).
Lawvere felt that category theory had to say something about differential equations. See differential equation.
Today people think of the theory of differential equations often as being a part of the theory of D-modules. That places the topic firmly into the realm of higher category theory.
There is more along these lines: way back Cartan taught that differential equations are to be thought of as exterior differential systems. As described at that entry, these may be understood naturally as sub -Lie algebroids of a tangent Lie algebroid.
I mean the kinds of PDEs that a person who does PDEs would do. The fact is that you may be able to build a wonderful theory of PDEs, but to actually solve PDEs, you have to get your hands dirty.
added to nPOV a little bit of material on the following points
but to actually solve PDEs, you have to get your hands dirty.
So? Back to square 19.
and now also a little bit at nPOV under Physics gauge theory supergravity.
Somebody should put in something on Langlands and twisted Yang-Mills theory and derived algebraic geometry…
Harry, do you know much about PDEs?
Neither do I :-) but I still feel like subjecting your statement
For example, category theory really has no place in Differential equations (and to be honest, it doesn’t seem like it ever could be).
to scrutiny. From what I gathered in my graduate student days, categorical methods do enter PDE theory. Much of topological vector space theory, e.g., the theory of distributions, nuclear spaces, etc. has its origins in PDE theory and is intensely conceptual (categorical) in spirit. My teacher, Francois Treves, had plenty of experience in this area; he would qualify as “a person who does PDEs”.
Regarding “to actually solve PDEs, you have to get your hands dirty” – see, what actually counts as a solution is part of what we’re talking about here. Again, from what I gather, it is routine these days to accept distributional solutions, but it wasn’t always so, and it was the efficacy of the abstract TVS theory which changed people’s minds.
I may have an inkling of what you’re trying to say, but none of us has a crystal ball, so it’s premature to say just how far category theory will penetrate the theory of PDEs in the future.
one more thing at nPOV, a sentence on 3d-TFT and 2d CFT. But now I am really running out of steam.
Oh, and about analytic number theory. I guess it might depend on what you think analytic number theory comprises, but I would certainly consider the Riemann hypothesis (complex zeroes of zeta functions of number fields) as firmly within analytic number theory. You’re telling me that categorical methods will never have anything to say about that?
I do recall that Andreas Blass once wrote something along the lines of what you said about analytic number theory, but that was a long time ago. I always wondered about that, and I wonder how strongly he still believes it.
I would certainly consider the Riemann hypothesis (complex zeroes of zeta functions of number fields) as firmly within analytic number theory. You’re telling me that categorical methods will never have anything to say about that?
Right, for instance -technology is supposed to help, right? And that’s “low-dimensional higher category theory”.
Urs #10: thanks. I’m glad you put that in.
Urs #12: yes, that’s the sort of thing I had in mind, although all that stuff is extremely conjectural. I also have in mind people like Connes, who has thought deeply about RH and is not afraid of using a category on occasion. :-)
However, in fairness to both sides of the discussion, there are fields where there has apparently been little interaction between it and category theory. For example, graph theorists and category theorists don’t seem to interact that deeply (to my knowledge). This sometimes puzzles me. Also, David mentioned Ramsey theory.
My general (and hopefully not too banal) POV though is that mathematics is terrifically fluid; compartmentalization of mathematics is sometimes sociologically convenient (for bureaucracy, for turf-defenders, and much else) but it doesn’t have much to do with mathematical reality. I imagine no one here would argue much against that! It follows that category theory, which by its very nature is a globalizing discipline, is bound to interpenetrate many, perhaps most of those areas of mathematics which are “alive” in some sense.
And, it goes both ways: other fields have something to say about category theory. I recall some somewhat subtle combinatorial arguments (borrowed from and extending Girard’s linear logic) needed in my thesis on coherence problems. I wish I knew more model theory – now there’s an area that I intuitively feel could have a lot to say to category theory, and yet there aren’t extremely many category theorists who are conversant with model theory at a deep level.
people like Connes, who has thought deeply about RH and is not afraid of using a category on occasion. :-)
That’s true, he is not afraid of it. But it seemed to me that at least for a long time he was rather sceptical about the merit of abstract category theory. But maybe that has changed by now.
If you can get categories working in Hungarian-style combinatorics (Gowers, Green, Tao), I'll be impressed.
For PDE, the best overview I ever read was PDE as a Unified Subject. Tao is surely onto something when he says
Any given system of PDE tends to have a combination of ingredients interacting with each other, such as dispersion, dissipation, ellipticity, nonlinearity, transport, surface tension, incompressibility, etc. Each one of these phenomena has a very different character. Often the main goal in analysing such a PDE is to see which of the phenomena “dominates”, as this tends to determine the qualitative behaviour (e.g. blowup versus regularity, stability versus instability, integrability versus chaos, etc.) But the sheer number of ways one could combine all these different phenomena together seems to preclude any simple theory to describe it all. This is in contrast with the more rigid structures one sees in the more algebraic sides of mathematics, where there is so much symmetry in place that the range of possible behaviour is much more limited. (The theory of completely integrable systems is perhaps the one place where something analogous occurs in PDE, but the completely integrable systems are a very, very small subset of the set of all possible PDE.)
I just want to say:
hopefully Harry doesn’t feel discouraged (that’s hard to image, anyway) and tries to list further mathematical disciplines that seem to look to him like “category theory has no place” in them. That seems like a very productive conversation for the nPOV!
Could still be right about Connes there. I think he and most very bright mathematicians are pragmatists at heart and would agree with the spirit of Freyd’s remark about “trivially trivial”. So it’s how sympathetic he is to the “Grothendieckian spirit” that’s in question.
David #16: very true. Nolo contendere that point about Hungarian combinatorics.
Great quote by Tao!!
Oh, and about analytic number theory. I guess it might depend on what you think analytic number theory comprises, but I would certainly consider the Riemann hypothesis (complex zeroes of zeta functions of number fields) as firmly within analytic number theory. You're telling me that categorical methods will never have anything to say about that?
I mean, when I'm talking about analytic number theory, it's misleading to start talking about the Riemann hypothesis, which has algebraic analogues and then talking about a way to possibly lift the algebraic case to the analytic case. I mean analytic number theory with log log log log logarithmic integral log log log log sigma log log = O(f) finding bounds and stuff. I mean, there are a lot of really cool things about special functions and algebra that are conjectured from Langlands, but as far as doing any work on the analytic side, category theory doesn't really have too much to day about it.
Well, maybe that’s the type of thing Blass had in mind, but as far as the future goes that’s just guesswork IMO. Never underestimate the power of conceptual methods, even for hard analysis! (“Hard analysis.” So many analysts crow about “hard analysis”, as opposed to wimpy soft analysis I suppose. (Flaming.) As if real men, real red-blooded types, don’t deal in that wussy categorical stuff (flame, flame). I admit this arouses some emotion in me.)
Actually, speaking of Tao, he also had a great article on conceptual methods in “hard analysis” that center on ideas from nonstandard analysis, where infinitesimal language makes some things much easier to say. Perhaps it’s not exactly category theory, but you could still call it conceptual mathematics.
However, speaking about the present day, I think you Harry have a valid point, and there are definitely plenty of areas where category theory hasn’t interacted deeply. I don’t know, perhaps it would be productive to discuss possible reasons for such cases, including some that might seem (superficially) surprising. David gave a great quote from Tao; I’d like to hear more along these lines, of similar depth.
Well, maybe that's the type of thing Blass had in mind, but as far as the future goes that's just guesswork IMO. Never underestimate the power of conceptual methods, even for hard analysis! ("Hard analysis." So many analysts crow about "hard analysis", as opposed to wimpy soft analysis I suppose. (Flaming.) As if real men, real red-blooded types, don't deal in that wussy categorical stuff (flame, flame). I admit this arouses some emotion in me.)
I will readily admit that anyone who is good enough to do hard analysis/analytic number theory professionally is more of a man than I am (even the women). That is the kind of math for which I have absolutely no talent.
Deriving results even from a hundred years ago is not a trivial task (try to prove the Chebyshev type bound or the PNT without a hint!)
I will readily admit that anyone who is good enough to do hard analysis/analytic number theory professionally is more of a man than I am
Ha! Don’t believe that. Actually, those macho hard analysts find algebra hard, and secretly admire/envy you. :-)
(And we should envy them, because they get more funding. :-( )
Todd #21: I'm gathering together material here on the idea of there being a difference between parts of math. Tao on open, closed and hybrid conditions seems to be onto something. On the other hand, if nonstandard analysis can change a hybrid condition to a closed condition, maybe the framework in which a theory is written makes a big difference.
There's some very good material on the hard/soft analysis distinction by Tao here.
Harry wrote:
I mean the kinds of PDEs that a person who does PDEs would do.
Todd wrote:
... see , what actually counts as a solution is part of what we ' re talking about here.
My interpretation: One does not understand (classical) electrodynamics if one is able to prove that the Maxwell equations admit solutions (classical, weak, whatever). You have to study different (physical) settings, their solutions and their physical interpretation. The expectation that category theory will change anything about this fact is unwarranted.
Todd wrote:
For example, graph theorists and category theorists don't seem to interact that deeply (to my knowledge). This sometimes puzzles me.
Is there a way to explain your puzzlement? There are many graphical languages that use points and arrows, I don't expect that every field using one of those could profit from category theory.
The expectation that category theory will change anything about this fact is unwarranted.
What’s unwarranted, Tim, is the charge that I expected it to.
Is there a way to explain your puzzlement? There are many graphical languages that use points and arrows, I don’t expect that every field using one of those could profit from category theory.
Maybe later, Tim.
Todd wrote:
What's unwarranted, Tim, is the charge that I expected it to.
Sorry, my statement wasn't directed at anyone - that's why I chose a passive construction.
Thank you very much, David – that’s a very interesting collection on your page. I’ll have to read and reread the quotes from Tao.
Tim #27, okay (and thanks), but that expectation doesn’t reside with anyone that I know of.
Back to #25: it’s not that I expect every field or subfield that uses arrow notation to interact with category theory in every instance. But graph theory is a pretty big and multifarious subject, and so is category theory, and at least superficially, one might expect at least some nontrivial interaction. For example, just by looking at the types of structure involved: directed graphs are endospans internal to , and categories are monad spans internal to . Also, while it may be true that graph theory is (today) more attractive to clever problem-solvers than to theory builders, it’s much more than a bag of clever tricks: there is some real theoretical depth involved (as in the graph minor theorem).
I can think of a few instances of interactions between the two fields, but to my knowledge there’s not a lot going on, and without having thought about it deeply, I find it mildly surprising. Is there some satisfying explanation for that, I wonder? Is there much more potential for interaction? I’m really not sure.
(Even more surprising to me is that there isn’t more communication between category theorists and model theorists. I stand in open-mouthed admiration at the obviously very deep achievements of model theorists, but I find it difficult to establish a deep personal connection to that field. I wish it were otherwise.)
@Todd #30: So do something about it :)
@Todd #30. I see, I know graph theory from the problem solver perspective only, that is as "algorithmic graph theory". The "graph minor theorem" I do not know, that's a buzzword I will look up. As a problem solver I would consider graphs in graph theory as objects in a suited category, by which transition all superficial similarities are lost.
(I wish I would be able to comment on model theory, but that's far beyond my expertise).
Eric, I’ll drop everything and get right on it. :-) Seriously, I wish I had a clone who could spend a year or two doing nothing but digging into model theory; it’s the material surrounding Classification and Stability Theory, largely the invention of the great genius Saharon Shelah, that I wish I knew. I am told that model theorists like to think of definable subsets of a model as akin to algebraic subvarieties, and Model Theory itself as akin to Algebraic Geometry, and the applications to Algebraic Geometry itself are, from everything I gather, mind-blowingly powerful.
and forcing is just like adding an indeterminate to a ring.
Or at least it would be, if set theorists would stop messing around with countable models and generic filters because they want all models to “live inside the real universe of sets” or some such. (-:
This discussion inspired me to create the page forcing with some remarks about this analogy.
Or at least it would be, if set theorists would stop messing around with countable models and generic filters because they want all models to “live inside the real universe of sets” or some such. (-:
Hear, hear.
I’m really glad you wrote that article, Mike, because I never understood this funny business about countable models before – why the need to make that move in the traditional accounts. At least I perceive that there’s a real technical point there, but the sheer technicality of it makes it look clunky by comparison with the Scott-Solovay Boolean-valued model approach.
The whole countable thing has to do with things being computable, somehow. I think it's got stuff to do with Goedel.
By the way, model theory and algebraic geometry are not at all alike.
I'm sorry, but I disagree that forcing has anything at all to do with the nPOV. Anyway, you guys should ask François or Joel from MO if they would write a forcing article here, because they're the only two experts on forcing that I know.
Regarding the relationship between model theory and category theory, there might be something worth extracting from this cafe discussion.
What do people reckon about Awodey's claims?
Sheaves and forcing. Early research on topos theory and set theory [41, 11, 13, 14, 10, 36] clearly displayed the sheaf-theoretic aspect of forcing, but it suffered from the inherent diffificulty of interpreting set theory in the resulting sheaf toposes. AST provides a framework that is more amenable to sheaf-theoretic forcing by providing a proper interpretation of elementary set theory, without sacrificing the "structural" character that permits its preservation under formation of sheaves. The first examples of models of AST in [25] included sheaf models, and the main result of the ambitious work [33] was to demonstrate closure under the formation of sheaf categories for a predicative form of AST. Some current research is devoted to providing a systematic sheaf-theoretic treatment of forcing (subsuming also permutation models): the case of presheaves was recently treated in [45]; constructions of sheaf models for certain special cases have also recently been given in [43, 18]. And research continues into this promising application of AST, unifying two profound ideas from farflung branches of mathematics: Grothendieck's theory of sheaves and Cohen's method of forcing. A brief introduction to algebraic set theory
@Todd #37: there is definitely a technical point there, but my current understanding is that it’s only technical and not conceptual, and really just because you want your generic set to exist as a “real” set in the ambient universe you were given, instead of passing to the internal logic of some new universe. It seems that for some people, the resulting technicality is preferable to having to step outside of “the” universe of sets. (But I’d love to have a discussion about this with a set theorist – maybe I should as a provocative question on MO and see if Francois or Joel steps up.)
@Harry: I don’t know that forcing has much to do with higher categories, but there is definitely a valuable category-theoretic POV on it, namely that it is basically another way of talking about sheaves. And I would be happy to have set theorists contributing to the nLab about anything! But of course one doesn’t have to be an expert set theorist to write a bit about forcing, especially from a category-theoretic point of view.
@David: I agree entirely with this:
Early research on topos theory and set theory [41, 11, 13, 14, 10, 36] clearly displayed the sheaf-theoretic aspect of forcing, but it suffered from the inherent diffificulty of interpreting set theory in the resulting sheaf toposes.
And AST is indeed one way of resolving those difficulties. But another way, which I think more directly “solves” the original problem rather than changing it into a different problem which the original tools are adequate to solve, is to use the stack semantics of the sheaf topos instead of its internal logic.
The whole countable thing has to do with things being computable, somehow. I think it’s got stuff to do with Goedel.
No, no, no. Please don’t pretend to knowledge you don’t have.
By the way, model theory and algebraic geometry are not at all alike.
I wasn’t just making stuff up; I was referring to how some actual model theorists think spiritually about their field. I could dig out some representative quotes, but it might take a while to track them down.
I just recently heard a talk by a bigshot on model theory as akin to algebraic geometry. The basic idea was very simple: you impose relations on a structure, which is like passing to the zeros of some polynomials. But I suppose there is more to it as one digs deeper. :-)
Harry said:
For example, category theory really has no place in Differential equations (and to be honest, it doesn’t seem like it ever could be)
There are different opinions: there is a whole school of Mikio Sato (read the interview with Sato, the link is at his nlab page!) who think different (algebraic analysis). There is also the Vinogradov’s diffiety ideology which emphasises on categorical aspects (you should see his translation of a book on deformations of PDEs,a nd his manifesto that the book is done wrong way as they did not understand the natural underlying category to deal with). Maybe the area is not sufficiently developed enough in that direction ? I do not see why solving algebraic equations would be for category theory, while its slight generalization to e.g. differential equations with algebraic coefficients to start not. I think the truth is in between: the differential equations are naturally typically related to categorically less structured mathematics, however part of the reason is in the tradition.
You see a typical branch of PDEs is classical field theory in physics; if you know the second quantization it is a formal procedure bringing instead of function solution, solutions which in addition can have multiparticle states. Hence the QFT is just a slight extension of a particular PDE setup, and QFT can be treated via solutions of PDEs (like Green functions and so on).
Do you then say that category theory is useless for QFT ? Should we deny your passport into nlab :) ?
the differential equations are naturally typically related to categorically less structured mathematics
I mean in comparison to algebraic equations, aka algebraic geometry.
Harry wrote:
...category theory really has no place in Differential equations...
When you take a look at the history of functional analysis, you will see that there is a shift of perspective somewhere between 1900 and 1950. In the beginning people saw equations that they wanted to solve (differential and integral equations). For example, the Fredholm alternative is a theorem about an integral equation and states when it has solutions.
Over the time this perspective changes, and people begin to see an integral operator instead, and the Fredholm alternative becomes a theorem about the eigenvalues resp. the resolvent of this operator. This shift of perspective from equations that have to be solved to maps of certain spaces is what I would call the categorification of functional analyis, which took place long before the term "category theory" was coined. I guess this is a little piece of the puzzle that Urs has in mind when he says "categories are everywhere".
I feel like the following point here still needs more emphasis:
For instance: there is hardly anything more general abstract category theoretic than the theory of groups. It is in effect a special case of entirely pure category theory. And yet, when you look at the models for this general abstract theory, then a whealth of concrete structure appears. Where was the existence of the Monster group, with all its intricacies, all its numerical peculiarities, encoded in the book of general abstract category theory? And still, it follows from it. And requires problem-solving to extract it.
Then: there is hardly anything more general abstract higher category theoretic than the theory of homotopy groups. This is nothing but part of the notion of equivalence in higher topos theory. What could be more general abstract? And still, when working out examples, a mind-boggling richness of phenomena emerges. Where were these tables of stable homotopy groups of spheres encoded when the world was created from abstract nonsense? This is concrete structure that just emanates out of its abstract source by itself.
Where is it encoded in the grand scheme of things that there are precisely four normed division algebras? Probably nowhere that you could point your finger to. What we have is the general abstract theory of algebra. From that springs the structure of normed division algebra by itself. It requires problem-solving to extract it. One just has to work it out.
And this is getting interesting: because – as opposed maybe to the homotopy groups of spheres – the four normed division algebras have a close relation to the physical reality that we see around us.
To me, this is the creation myth that category theory tells: there are general abstract structures out there that just exists in themselves in their own right and are manifestly evident and self-contained. Topos theory just exists and the more abstract the better, because that means no human intervention made it, it just is.
But then from that, by some miracle, a wealth of concrete structure emanates.
I think in parts the talk about the “Two cultures in math” is misleading. It makes it sound as if there were two different strategies to understand the same things. I don’t think that’s true. It’s two optimized strategies to understand two very different things: one is:
The other is:
Of course both questions touch somewhere, where they nourish each other: the abstract theory builder can tell the problem-solver which problems look like they are worth being solved. The problem-solver can tell the theory-builder where there appears to be unkown theory still to be found.
But neither of the two aspects is to be expected to eventually replace the other.
@Todd #44 Angus MacIntyre is a big supporter of bringing Grothendieck into model theory - Model theory: Geometrical and set-theoretic aspects and prospects, The Bulletin of Symbolic Logic, Volume 9, 2003, pp. 197–212.
@Zoran #46 Vinogradov is certainly making some bold claims for his approach to PDE in Cohomological analysis of partial differential equations and secondary calculus.
This book deals with principles of a theory playing the same role for general systems of (nonlinear) partial differential equations as algebraic geometry does for algebraic equations.
Diffieties are, generally infinite-dimensional, manifolds carrying a structure that may be called the infinite-order contact structure, and are locally equivalent to infinite prolongations of differential equations. Due to this structure, a very special kind of differential calculus, called Secondary Calculus, can be developed on a diffiety.
...we can be sure that quantum physics and, above all, quantum field theory and its generalizations find their natural mathematical background in Secondary Calculus.
@David #51: being a former student of Vinogradov I know those claims very well and I’d like to add some comments. First of all I should say that he hasn’t published anything where the claims about the role of secondary calculus in QFT are really proven, and it’s also rather difficult do get a precise answer from him about these matters directly. You might want to read the semi philosophical article where he sketches one program of how to make this precise, but to the best of my knowledge that program has not been carried on since then.
Nevertheless there is some evidence that the whole theory might\should be of relevance in finding a mathematical framework for QFT. One general argument is simply: PDEs appear everywhere in classical physics (field theory), hence having a conceptual framework for PDEs seems like a natural first step before quantizing*. It seems to me that diffieties are such a framework (very closely related to D-modules\D-schemes and exterior differential systems which where mentioned here before. In my mind they are actually all the same as I’v started discussing with Zoran elsewhere). I hope to write more at the nLab about this eventually. In any case, even if the theory is not of relevance for QFT it seems to me that an nPOV is quite adequate for it (as you might even perceive from reading the introduction to the book you linked to). Let me repeat that: it seems to me that an nPOV is essential in nonlinear PDEs! (admittedly I’m still trying to figure that out by learning some higher categories on the other discussion I started)
*of course that’s an argument from a mathematician, some physicist might rightly argue that it’s nonsense.
Let me repeat that: it seems to me that an nPOV is essential in nonlinear PDEs!
You also said that commutative rings, schemes, algebraic spaces, and algebraic stacks were "wrong", so I'm a little bit wary to believe you here...
@Michael #52: Sounds interesting, I for my part discovered D-modules only recently, and stumbled upon the concept of holonomic quantum fields, do you happen to know this topic?
Did Vinogradov explain what the problems are in advancing his program?
I hope it's not the typical physics situation, where an author claims to have a new framework, publishes a paper applying it to a toy model, announces that more serious applications will follow and is never heard of again :-)
@Tim 54: I had heard about holonomic quantum fields but don’t know anything about it.
Did Vinogradov explain what the problems are in advancing his program?
Not really. As I said it was hard to get precise statements from him regarding his ideas on QFT. At least for me (:
@Harry 53:
You also said that commutative rings, schemes, algebraic spaces, and algebraic stacks were “wrong”
You might be confusing me with someone else. Or you can’t tell the difference between a question and a statement.
I didn't read your whole post. I read where Zoran quoted you, which made it seem like a statement rather than a question, whence came my (admittedly dissmissive) response. I apologize.
Finally somebody on MO asks explicitly for the nPOV! And then nobody mentions it. So I added a comment.
(That’s how it goes. First we spend all time with discussion of how to state it not to upset people, and then when they explicitly ask for it…)
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