In reply to André : What seems reasonable to do is analysis, namely what is behind the success of category theory and how is this success related to the progress of mathematics. Which implies asking questions of mathematics, some of which have been aired in this discussion list. In this way, it should be possible to avoid seeming partisan, but to ask serious questions, which should help to steer directions, or suggest new ones. Of course lots of great maths does not arise in this way, but by following one's nose, but that does not mean that such analysis of direction is unhelpful. I know some argue that this excursion into what might be called the theory of knowledge, or into methodology, seems unnecessary to some. In reply I sometimes point to remarks of Einstein on my web site www.bangor.ac.uk/r.brown/einst.html or more mundanely retort that normal activities normally require some meta discussion: if you want to go on a holiday, you do some planning, you don't just rush to the station and buy some tickets. I develop this theme in relation to the teaching of mathematics in an article What should be the output of mathematical education? on my popularisation and teaching page. I gave a talk to school children on `How mathematics gets into knots' in the 1980s, and a teacher came up to me afterwards and said: `That is the first time in my mathematical career that anyone has used the word `analogy' in relation to mathematics.' Yet abstraction is about analogy, and very powerful it is too. This was part of the motivation behind the article 146. (with T. Porter) `Category Theory: an abstract setting for analogy and comparison', In: What is Category Theory? Advanced Studies in Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274. pdf There is also interest in the question of how category theory comes to be successful, and more successful than, say, the theory of monoids. This seems connected with the underlying geometric structure being a directed graph, i.e. allowing a `geography of interaction'. A category is also a partial algebraic structure, with domain of definition of the operation defined by a geometric condition. Is this enough to explain the success? It is worth noting that the article Atiyah, Michael, Mathematics in the 20th century, Bull. London Math. Soc., {34}, {2002}, 1--15, suggests that important trends in the 20th century were: higher dimensions, commutative to non commutative, local-to-global, and the unification of mathematics, but does not include the words `category' or `groupoid', let alone `higher dimensional algebra'! This kind of analysis needs to be presented to other scientists, and to the public, not only to mathematicians. There is a hunger for knowing what mathematics is really up to, in common language as far as possible, what new concepts, ideas, etc., and not just `we have solved Fermat's last theorem'. If your analysis of what category theory should do suggests some gaps, then that is an opportunity for work! Good luck Ronnie Brown Joyal wrote: Category theory is a powerful mathematical language. It is extremely good for organising, unifying and suggesting new directions of research. It is probably the most important mathematical developpement of the 20th century. But we cant say that publically. André Joyal [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
This discussion has been very interesting. I have a couple of comments and a request, but first a little background. I've only recently truly encountered category theory - I describe myself as a differential topologist and as yet see no reason to change that description, but I've increasingly needed to use at least the language of category theory to express some of the things that I come across in algebraic and differential topology and this has led to me learning some category theory at last. However, I sometimes feel as though I've stumbled into a party by mistake and can't find the way out. I'm quite enjoying being at the party, I ought to say, but every now and then I sit down in a corner and wonder how I got in, and also suspect that I missed the Big Announcement at the beginning that said what the party was for. All this discussion about a "well kept secret" has gone a bit over my head. I'm not sure what the secret is! My forays into the categorical landscape have been two-fold: understanding operations in cohomology theories and understanding smooth spaces. The first, paradoxically, relates to trying to un-categorify something ("decategorify" now has a mathematical meaning and I don't intend that); namely, the previous description of what we wanted to understand was extremely categorical and we wanted a much more "hands on" description, but that actually just led us from one categorical description to another (our own journey was quite tortuous, I should say). The second foray wouldn't have happened if those I'd been talking to hadn't already been speaking in categories - I had to learn the language just to join the conversation. So when you all talk of a "well kept secret" and something that "went wrong in the 60s" (didn't everything?), please remember that some of us weren't even born in the 60s, let alone thinking about mathematics, so haven't a clue what's going on. And, as I've tried to say above, I'm an outsider but one with a favourable view of category theory so if it's hard for me to figure out what the fuss is about, I'm not surprised that it's hard for anyone further out. Let me make these remarks a little more concrete with a request (or a challenge if you prefer). In my department, the colloquium is called "Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; I've been writing too many scripts lately!). I'm giving this talk in January. My original plan was to say something nice and differential, with lots of fun pictures of manifolds deforming or knots unknotting, or something like that. However, the discussion here has set me to thinking about saying something instead about category theory. It is a pearl of mathematics, it does have a certain beauty, there's certainly a lot that can be said, even to a fairly applied audience as we tend to have here (it is the Norwegian university of Science and Technology, after all), even without talking about programming (about which I know nothing). But for such a talk, I need a story. I don't mean a historical one (I'm not much of a mathematical historian anyway), I mean a mathematical one. I want some simple problem that category theory solves in an elegant fashion. It would be nice if there was one that used category theory in a surprising way; beyond the idea that categories are places in which things happen (so perhaps about small categories rather than large ones). I'm not trying to get anyone to write my talk for me! It's just that as someone who only recently engaged with category theory then I'm aware - painfully aware - that I often miss the point. But to counter that, then as someone who only recently engaged with category theory then I can still remember fairly vividly why I like it and what convinced me that it was worth thinking about (and learning about), which will hopefully give the talk a little more omph. Thanks in advance for your suggestions, Andrew Stacey PS I just remembered something else I was going to mention. Someone else mentioned MathOverflow. Well, there was a question about what was missing from undergraduate mathematics. I said "category theory". It currently lies 5th in the list (out of 28, my other suggestion "how to write with chalk so it doesn't squeak" is 12th). More interesting than it's placing is the vast number of comments that followed, mostly saying that too much "abstract nonsense" would be off-putting to students. You can read it at: http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergradua... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Andrew Stacey wrote:
All this discussion about a "well kept secret" has gone a bit over my head. I'm not sure what the secret is!
We could tell you... ... but then it wouldn't be a secret, now, would it? Seriously, I think the so-called "secret" is the power and glory of category theory. And I think some of the older category theorists on this mailing list have a different attitude than youngsters like you and me. They fought to convince the world that category theory was worthwhile. Some feel they lost that fight. We came along later and are a bit puzzled by that attitude: if you look around at the landscape of mathematics today, categories are everywhere! From Grothendieck to Voevodsky to Lurie, etc., much of the most exciting mathematics of our era would be inconceivable without categories. I don't think I'll try to tell you the old war stories: others are better qualified. But I hope the veterans of those wars take heed of your comments and realize many young mathematicians naturally find categories interesting, exciting, and/or useful. Certainly there is much about categories that these youngsters don't understand. But they can learn it if you explain it.` Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 15/12/2009, at 5:41 AM, Andrew Stacey wrote:
In my department, the colloquium is called "Mathematical Pearls" I'm giving this talk in January.
But for such a talk, I need a story.
Dear Andrew Back in the early 90s Todd Trimble gave a beautiful colloquium talk to our Mathematics Department at Macquarie. It was based on a question in a book by Halmos which involved finding some group (topological I think) doing something or other. It was not a categorical problem as such. Todd spoke about groups in a category with finite products. The only categorical theorem he needed was that finite product preserving functors take groups to groups. I believe he took the definition of category as known but defined functor, product and internal group. My vague memory is that he found a group solving the analogous problem in some fairly combinatorial (presheaf?) category, then found a product preserving functor to topological spaces to obtain the desired group. I hope Todd is reading this and I have jogged his memory enough to write in more detail. It takes work and ingenuity to design such pearls. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Date: Mon, 14 Dec 2009 21:12:53 -0800 John Baez wrote,
... older category theorists ... fought to convince the world that category theory was worthwhile. Some feel they lost that fight.
They won it ... but how prevalent is the subject in undergraduate programs? Vector algebra and analysis wasn't taught to engineers until what, 1900 or later. Now it is ubiquitous. Absolutely no offense to existing books but what about an energetic mathematician or two writing a _Schaum's Outline of Category Theory_? I'd expect it to sell off the shelves initially. Regards, ... Peter E. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Peter Easthope wrote: They won it ... but how prevalent is the
subject in undergraduate programs? Vector algebra and analysis wasn't taught to engineers until what, 1900 or later. Now it is ubiquitous.
Interestingly, in the late 1800s there was a period where quaternions were a mandatory examination topic in Dublin - and in some American universities they were the only advanced mathematics taught. Gibbs, who chopped the quaternion into its scalar and vector part and introduced the notation we use today, was the first person to get an engineering PhD in the United States, back in 1863. Absolutely no offense to existing books but what
about an energetic mathematician or two writing a _Schaum's Outline of Category Theory_? I'd expect it to sell off the shelves initially.
Great idea! I think it's premature to introduce category theory in the undergrad curriculum. Why? Merely because there aren't enough professors who'd see how to teach the subject at that level. It's bound to happen eventually - but right now we need category theory to become a standard course at the graduate level. Whenever they get a good taste of category theory, math grad students are eager to take a course on it. They think it's exciting, and they see it as a way to learn other subjects more efficiently. But right now it's usually taught as part of algebra, without enough detail, and without enough attention to its applications outside algebra. So, sometimes students start their own seminars on category theory! Once most math grad students take a class on category theory, we'll get professors who can conceive of teaching it at the undergrad level. The only real question is whether our current civilization, based on burning carbon, tearing up forests, and destroying oceans, lasts long enough to see this change. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
At 11:09 PM 12/17/2009, John Baez wrote:
I think it's premature to introduce category theory in the undergrad curriculum.
I think there are enough very interesting simple examples of categories that the language and diagrams could be introduced to high school students. For example, lists are terrific examples for discussion of the free monoid functor, its unit, and counit, but they don't have to be called by their official names. And tables give a 2-dimensional version of that discussion, with an exchange law that is simple but interesting. Kinship trees or the trees used in high school probability class can be used to talk about partially ordered sets, but they don't have to be called that. The idea would be to get diagrams into the student consciousness, so they learn about connecting the dots. Advanced high school students know about multiplication of matrices, so they could learn something about arrows standing for linear transformations, and composition of arrows corresponding to matrix multiplication. The slogan is, "algebra is the geometry of notation," and high school students can learn to look at and play with diagrams. I bet some kind of school-yard game could be based on diagram chasing. As I see it the greater problem is that high school mathematics teachers need more education. Therefore, I am preparing a book with no calculus beyond the AP level, but using Robinson infinitesimals, Kolmogorov probability spaces, and Eilenberg-Mac Lane categories wherever these things come up simply and naturally in a certain context to do with biology. It has been announced for pre-order at amazon.com by World Scientific and should be available in April, 2010. The work-in-progress is available for examination (and feedback to me!) upon request. Ellis D. Cooper [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
From teaching first year analysis I saw that we need rules for constructing continuous (and then differentiable) functions (as the texts do, of course). I guess this led me later to emphasise constructions of continuous functions in topology, and this leads naturally in many cases to universal properties. (Are categorical methods relevant to functions of bounded variation?) This is the conceptual revolution in which of course a particular construction is defined by its relation to all other objects of the `category of discourse'. This can be related to programming; given any input of the required type, the output is a function or morphism. It also emphasises properties rather than mode of construction. So categorical methods can be used without explicitly saying at the first instance that one is doing `category theory'; I was also an advocate of set notation in calculus, for example to name the domains of functions defined by formulae, without introducing `set theory' as a `big deal'. Ronnie Brown Ellis D. Cooper wrote:
At 11:09 PM 12/17/2009, John Baez wrote:
I think it's premature to introduce category theory in the undergrad curriculum.
I think there are enough very interesting simple examples of categories that the language and diagrams could be introduced to high school students. For example, lists are terrific examples for discussion of the free monoid functor, its unit, and counit, but they don't have to be called by their official names. And tables give a 2-dimensional version of that discussion, with an exchange law that is simple but interesting. Kinship trees or the trees used in high school probability class can be used to talk about partially ordered sets, but they don't have to be called that. The idea would be to get diagrams into the student consciousness, so they learn about connecting the dots. Advanced high school students know about multiplication of matrices, so they could learn something about arrows standing for linear transformations, and composition of arrows corresponding to matrix multiplication. The slogan is, "algebra is the geometry of notation," and high school students can learn to look at and play with diagrams. I bet some kind of school-yard game could be based on diagram chasing.
As I see it the greater problem is that high school mathematics teachers need more education. Therefore, I am preparing a book with no calculus beyond the AP level, but using Robinson infinitesimals, Kolmogorov probability spaces, and Eilenberg-Mac Lane categories wherever these things come up simply and naturally in a certain context to do with biology. It has been announced for pre-order at amazon.com by World Scientific and should be available in April, 2010. The work-in-progress is available for examination (and feedback to me!) upon request.
Ellis D. Cooper
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear categorists - At 11:09 PM 12/17/2009, John Baez wrote:
I think it's premature to introduce category theory in the undergrad curriculum.
On Fri, Dec 18, 2009 at 2:25 PM, Ellis D. Cooper <xtalv1@netropolis.net>wrote:
I think there are enough very interesting simple examples of categories that the language and diagrams could be introduced to high school students.
I agree! Just to be clear: by "premature" I wasn't trying to say that undergraduates or even high school students are too young to learn and profit from category theory. I meant that there aren't enough high school teachers who understand category theory well enough to teach it - except for isolated experiments here and there. Math trickles down. Right now we need more category theory taught at the graduate level, so someday enough professors will understand it well enough to teach it at the undergrad level, so that eventually enough high school teachers will know enough to teach it at the high school level. If this seems overly optimistic, it's worth thinking about calculus, which in Newton's day was regarded as comprehensible only by a few experts. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear All, As one of MacLane's working mathematicians who follows the catlist, I would like to add some thoughts about perceptions of category theory (CT). I earned a Bachelor's in Physics at Berkeley in 1960 and went to grad school in Mathematics at the University of Oregon the following fall. In Frank Anderson's graduate algebra course I was first exposed to CT and hated it. My background and ability in algebra were marginal anyway and to have my first definition of tensor product be in terms of commuting diagrams was disastrous. Fortunately, I got a summer job at the Jet Propulsion Lab and one of my coworkers, Gus Solomon, gave me the classical constructive definition of tensor products. Sammy Eilenberg came by Eugene and gave a lecture on CT which did nothing to change my opinion of it. When I heard of Serge Langs's characterization of CT as "abstract nonsense" it reinforced what I already thought (See however, http://en.wikipedia.org/wiki/Abstract_nonsense which does not mention Serge Lang in the body of the article). My fascination with, and love of, CT was ignited in 1966 when I was a postdoc with Gian-Carlo Rota at the Rockefeller University. Ron Graham and I were collaborating on a conjecture of Rota; that the lattice of partitions of an n-set has the same property that Erwin Sperner had demonstrated for the lattice of subsets of an n-set (the largest antichain is the largest rank). We had some partial results on Rota's conjecture and in the course of writing them up I realized that they implicitly involved a notion of morphism for the Ford-Fulkerson maxflow problem. I thought this was a promising insight and incorporated it with the other material. I gave the finished paper to Ron for approval but when it came back to me it had been rewritten and all mention of flowmorphisms eliminated. I took this as a challenge to show that flowmorphisms could lead to further insight into Sperner problems. The result, which I called The Product Theorem, was natural conditions under which the product of Sperner posets must also be Sperner. The key was to realize that the concept of normalized flow introduced in Graham-Harper (which is stronger than the Sperner property) is equivalent to a flowmorphism from the given poset to a chain (total order). If two posets, P,Q, have normalized flows then product, being a bifunctor, will induce a flowmorphism from their product to the product of their chains. All I had to do then was to find natural conditions under which the product of (weighted) chains has a normalized flow. The Product Theorem generalized known theorems of Sperner, deBruijn et al & Erdos. and has since been applied to prove at least 3 new conjectures. Having such a success with flowmorphisms motivated me to dig more deeply. I showed that flowmorphisms have pushouts and was asking about pullbacks (though I did not use those terms because I did not know them) when (about 1971) my office mate at JPL, Dennis Johnson, introduced me to Saunders MacLane's classic, Categories for the Working Mathematician. This was, of course, a revelation and changed my mathematical universe. I joined the faculty of the University of California at Riverside in the fall of 1970. In alternate years I taught a 2-quarter graduate course on combinatorics. Over the next 36 years it evolved into two independent courses having a common thesis. One was on maximum flows in networks and Sperner problems, the other on minimum paths in networks and combinatorial isoperimetric problems. I believe it is no accident that maximum flow and minimum path (aka dynamic programming) problems are central to algorithmic analysis and that they both have nice notions of morphism. The common thesis of the two courses is that morphisms can be effective in solving hard problems. In 2004 the notes for one course were published under the title Global Methods for Combinatorial Isoperimetric Problems. If I live long enough its companion volume on Sperner problems will appear. It will show how several steps in the eventual resolution of the Rota Conjecture were illuminated by flowmorphisms. It has been a personal goal, since the early 1970s, to demonstrate the existence and usefulness of morphisms for combinatorial problems. This often comes down to questions of 1) How to use symmetry to systematically simplify the problem? 2) How to pass to a continuous limit? I like to call this endeavor the relativity theory of combinatorics. Albert Einstein asked "What are the symmetries of the universe and what do they tell us about it?" To show the depth and subtlety of such questions, consider that two of the leading mathematicians of his age, Henri Poincare and Hendrick Lorentz, studied Lorentz transformations five years before Einstein. However they both missed the epoch-making relation E = mc^2 that is easily deduced from Lorentz transformations. In studying a problem through its morphisms we need all the help we can get. CT is invaluable as the road map to morphism country! Regards, Larry Harper Professor Emeritus of Mathematics University of California, Riverside [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
2009/12/19 Larry Harper <harper@math.ucr.edu>: mentions
When I heard of Serge Langs's characterization of CT as "abstract nonsense" it reinforced what I already thought (See however,
http://en.wikipedia.org/wiki/Abstract_nonsense
which does not mention Serge Lang in the body of the article).
Lang's Algebra uses "abstract nonsense" to describe his favorite kind of one line proof that some given construction (such as tensor product) defines its results uniquely up to isomorphism. It is discussed on the discussion page of the wikipedia article. best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
This seems like an excellent advertisement of `thinking categorically' and not necessarily writing in that dialect. jim Larry Harper wrote:
Dear All,
As one of MacLane's working mathematicians who follows the catlist, I would like to add some thoughts about perceptions of category theory (CT). I earned a Bachelor's in Physics at Berkeley in 1960 and went to grad school in Mathematics at the University of Oregon the following fall. In Frank Anderson's graduate algebra course I was first exposed to CT and hated it. My background and ability in algebra were marginal anyway and to have my first definition of tensor product be in terms of commuting diagrams was disastrous. Fortunately, I got a summer job at the Jet Propulsion Lab and one of my coworkers, Gus Solomon, gave me the classical constructive definition of tensor products. Sammy Eilenberg came by Eugene and gave a lecture on CT which did nothing to change my opinion of it. When I heard of Serge Langs's characterization of CT as "abstract nonsense" it reinforced what I already thought (See however,
http://en.wikipedia.org/wiki/Abstract_nonsense
which does not mention Serge Lang in the body of the article).
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
John Baez wrote:
They fought to convince the world that category theory was worthwhile. Some feel they lost that fight. We came along later and are a bit puzzled by that attitude: if you look around at the landscape of mathematics today, categories are everywhere! From Grothendieck to Voevodsky to Lurie, etc., much of the most exciting mathematics of our era would be inconceivable without categories.
Like most fields of mathematics, category theory keeps growing and evolving. It may be hard to identify the mechanism of this evolution but fashion must be playing a role. But why are certain subjects becoming hot at a given time? Probably because they resonate with new developments outside category theory. When a trend becomes hot, it gives rise to a permanent current. I was able to distinguish approximatly 6 major currents: 1) Algebraic topology and homological algebra 2) Abelian categories 3) Algebraic Geometry and topos theory 4) Logic and elementary topos theory 5) Category theory and computer science 6) Higher categories with homotopy theory Here is an example of a recent applications of category theory to geometry: "Associahedral categories, particles and Morse functor" by Jean-Yves Welschinger http://arxiv.org/abs/0906.4712 The n-category caffé is an extraordinary experiment in research collaboration and dissimination of knowledge. It maybe the way of the future. But an old mathematicians like me find it difficult to adapt to this new form of collaboration.
The only real question is whether our current civilization, based on burning carbon, tearing up forests, and destroying oceans, lasts long enough to see this change.
Yep! And we should not remain passive. Best, AJ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear John, et al, If this seems overly optimistic, it's worth thinking about calculus, which
in Newton's day was regarded as comprehensible only by a few experts.
i totally agree! Back when i was pushing the process algebras into the corporate software sector i would regularly "shame" exec/engineers who claimed the formalism too complex by demonstrating that i could teach the π-calculus to 13 year-old's and they could use it, fruitfully. There are branches of mathematics that really require steady application to a steep learning curve for an extended period of time, but there are many -- computation and category theory being among them -- where there is a core that really is accessible to anyone with a certain penchant for abstraction. Engaged and engaging teachers and practitioners are a key ingredient -- without which many go hungry at the table of mathematics. Best wishes, --greg On Sat, Dec 19, 2009 at 2:16 PM, John Baez <john.c.baez@gmail.com> wrote:
Dear categorists -
At 11:09 PM 12/17/2009, John Baez wrote:
I think it's premature to introduce category theory in the undergrad curriculum.
On Fri, Dec 18, 2009 at 2:25 PM, Ellis D. Cooper <xtalv1@netropolis.net
wrote:
I think there are enough very interesting simple examples of categories that the language and diagrams could be introduced to high school students.
I agree! Just to be clear: by "premature" I wasn't trying to say that undergraduates or even high school students are too young to learn and profit from category theory. I meant that there aren't enough high school teachers who understand category theory well enough to teach it - except for isolated experiments here and there.
Math trickles down. Right now we need more category theory taught at the graduate level, so someday enough professors will understand it well enough to teach it at the undergrad level, so that eventually enough high school teachers will know enough to teach it at the high school level.
If this seems overly optimistic, it's worth thinking about calculus, which in Newton's day was regarded as comprehensible only by a few experts.
Best, jb
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I think there are enough very interesting simple examples of categories that the language and diagrams could be introduced to high school students.
I've heard that Piaget experimented, successfully, with teaching category theory to 12-year-old children (but I do not have any references).
Math trickles down. Right now we need more category theory taught at the graduate level, so someday enough professors will understand it well enough to teach it at the undergrad level, so that eventually enough high school teachers will know enough to teach it at the high school level.
If this seems overly optimistic, it's worth thinking about calculus, which in Newton's day was regarded as comprehensible only by a few experts.
For calculus, the transformation of an esoteric into a basic discipline was largely driven by engineering applications. After mathematicians demonstrated that calculus could be applied to practical engineering problems, and developed a methodology for such applications, engineers recognized that calculus should be taught at the then-undergrad level. This created a demand in professors capable of teaching calculus to engineers, and further along the chain, as John described. This mechanism should work for category theory as well: software engineering is saturated with problems to which categories have something essential to offer. The situation is even more favorable because software engineers themselves reinvent categorical constructs (more accurately, their inventions can be seen as a reinvention of categorical constructs). I believe that software engineering is ready (theoretically :) to accept categorical methods. Of course, much needs to be done to adapt category theory as a basic mathematical discipline for software engineering but it would not be a waste of time and effort. This work should be profitable for categories in two ways: 1) Public appreciation, funding etc. 2) Engineering applications are a source of interesting problems and interpretations that may be mathematically fruitful. Focusing on engineering allows treating "the opprobrium issue" in a different way (my apologies if it is too vulgar). Category theory provides methods and tools, and there are other tools on the market. At least part of the attempts to sell category theory to a general mathematical public is like selling it to a competing vendor, and hence doomed to fail from the very beginning. It's more fruitful to sell (whatever that means) to prospective users/customers. Working mathematics, physics, computer science are such users, and they do appreciate category theory. However, these groups of customers are not particularly numerous. A very promising prospective user is software engineering: it's massive, dynamic, and eager (as any other engineering) to adapt any widget helpful to do the job, be it calculus, vector algebra or abstract nonsense. Having such a customer would dramatically change the market situation for categories similarly to the case of mechanical engineering-calculus. Z. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Zinovy, I can say that I dislike your selling/marketing despair and do not share excitement in the existence of an easy niche market you propose. Having such a customer
would dramatically change the market situation for categories similarly to the case of mechanical engineering-calculus.
Unlike your search for an easy "customer", I am CONVINCED that really interaction with central parts of mathematics which you find CONCURRENT in some odious market sense, is the main and natural one I could intellectually, artisticially and purposefully like to find in interaction with category theory. It can not satisfy me knowing that some PARTs of category theory can be easily sold to your proposed customer, if I know that some natural parts are INTRINSICALLY interwoven with many other subjects and this is ignored. I do not think that the category theorists should approach other mathematicians/computerists/others just to find SOME company to share what they know, but rather because they find a natural need to do so. I do not talk to person A because I have nobody else to talk to, but because I am interested in what specifically person A can offer in the communication. This said I can not replace person A with different person B who has other values to enhance me. This is not to diminish software engineering as a valuable field of interaction, but having software engineers accept category theory, does not solve the problem that some who should accept category theory do not. Many category theorists themselvses are of the same ignorant kind by not accepting higher category theory and homotopy theory which are naturally form a whole with classical category theory. Zoran [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Status: RO
Those depressed about the social status of category theory might be cheered up by a look at Math Overflow, http://mathoverflow.net . This is a website where you can ask and answer questions about any part of mathematics. You might expect that for a site with this problem-solving format, category theory wouldn't be much in evidence. But according to the site's own statistics, it's the 3rd most popular topic for questions. The site's been up only a few months but has been enormously successful, with some extremely clever and knowledgeable people contributing regularly. It really doesn't seem like a pro-category niche group. But I've seen little or no anti-category sniping there. The kind of cynicism that most of us have experienced just isn't in evidence. A particular reason to find this cheering is that the demographic of the contributors is skewed towards the young, the American, and the algebraic geometers. (Young American algebraic geometers are a definite minority, though - there's quite a wide spread.) Why might that be particularly cheering? "Young" because it suggests a bright future, "American" because, as I gather, the NSF has historically been loath to support category theory, and "algebraic geometers" because it suggests that the anti-category theory backlash in that influential subject may be nearing an end. Best wishes, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Status: RO
2009/12/22 Tom Leinster <tl@maths.gla.ac.uk>: substantially understates
[as to] "algebraic geometers" because it suggests that the anti-category theory backlash in that influential subject may be nearing an end.
the book about Wiles's proof of Fermat's Last Theorem @BOOK{ModForms, editor = "Cornell, Gary and Silverman, Joseph and Stevens, Glenn", TITLE = "Modular Forms and {F}ermat's {L}ast {T}heorem", PUBLISHER = "Springer-Verlag", YEAR = "1997", } takes no explicit stand but is inevitably full of Grothendieck's categorical tools. More recently, from the beginning graduate level to research we have books explicitly explaining or building on Grothendieck's methods: @BOOK{SzamuelyGal, AUTHOR = "Szamuely, Tam{\'a)s", TITLE = "Galois Groups and Fundamental Groups", PUBLISHER = "Cambridge University Press", YEAR = "2009", } @BOOK{FGAexplained, AUTHOR = {Fantechi, Barbara and Angelo Vistoli, and Lothar Gottsche, and Steven L. Kleiman, and Luc Illusie, and Nitin Nitsure}, TITLE = {Fundamental Algebraic Geometry: {G}rothendieck's {FGA} Explained}, PUBLISHER = {American Mathematical Society}, YEAR = {2005}, } @BOOK{LurieHigher, AUTHOR = {Lurie, Jacob}, TITLE = {Higher Topos Theory}, PUBLISHER = {Princeton University Press}, YEAR = {2009}, } and a series of printed or web-published works by Voevodsky. There will be anti-Grothendieck backlashers always. Life is like that. But for decades algebraic geometry at the top schools has been impossible without Grothendieck tools and now that is being mainstreamed. best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Status: RO
First a slight correction: The paper referred to was I think Wigner, E.P., The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Comm. in Pure Appl. Math. (1960), reprinted in Symmetries and reflections: scientific essays of Eugene P. Wigner, Bloomington Indiana University Press (1967). Here are some quotations from this article: ------------------------------------------------------------------------- ... that the enormous usefulness of mathematics in the physical sciences is something bordering on the mysterious, and that there is no rational explanation for it. Mathematics is the science of skilful operations with concepts and rules invented just for this purpose. [this purpose being the skilful operation ....] The principal emphasis is on the invention of concepts. The depth of thought which goes into the formation of mathematical concepts is later justified by the skill with which these concepts are used. The statement that the laws of nature are written in the language of mathematics was properly made three hundred years ago; [it is attributed to Gallileo] it is now more true than ever before. The observation which comes closest to an explanation for the mathematical concepts cropping up in physics which I know is Einstein's statement that the only physical theories which we are willing to accept are the beautiful ones. It stands to argue that the concepts of mathematics, which invite the exercise of so much wit, have the quality of beauty. --------------------------------------------------------------------------------- There is also a question of what is expected from a mathematical area. At a conference in Baku in 1987 I was asked `what are the big theorems in category theory? People sometimes want to know:`What are the big problems in category theory?' That these `big' things may not exist (comments?) does say something about the nature of category theory, and also of mathematical progress, and what this is conceived of by various groups of mathematicians.. Part of Grothendieck's success was his aims for maximum generality and for making things tautological. So some simple things (to category theorists) like `left adjoints preserve colimits' are very useful in a variety of fields, and make tautological some apparently difficult procedures. And also allow analogies between different fields. Hence my paper with Tim Porter: `Category theory: an abstract basis for analogy and comparison'. (Just one aspect, of course.) Ronnie Brown Zinovy Diskin wrote:
Dear Zoran,
You misunderstood my posting, or I phrased it badly, because
On Tue, Dec 22, 2009 at 11:59 AM, zoran skoda <zskoda@gmail.com> wrote:
Dear Zinovy,
I can say that I dislike your selling/marketing despair and do not share excitement in the existence of an easy niche market you propose.
in the list "despair-excitement-easy niche", only the second term is true. Building mathematical models for engineering problems is a hard business, and the suggestion to view it as a fruitful area for categorical applications stems from optimism about the power of category theory, rather than from despair.
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Status: RO
Dear Urs, The nLab is a very nice thing! http://ncatlab.org/nlab/show/HomePage You wrote:
It is a wiki-lab for collaborative work on Mathematics, Physics and Philosophy especially from the n-point of view: insofar as these subjects touch on higher algebraic structures.
The nLab is devoting a lot of space to category theory. It would be nice to have a CatLab devoted to category theory per se. Is this something that can be created ? My knowledge of wiki-technology is null. Maybe you could create a wiki-lab for homotopy theory too (a HoLab?) Maybe all these labs could be connected. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Ellis D. Cooper wrote:
At 11:09 PM 12/17/2009, John Baez wrote:
I think it's premature to introduce category theory in the undergrad curriculum.
I think there are enough very interesting simple examples of categories that the language and diagrams could be introduced to high school students. agreed diagrams yes vocabulary can wait until the ideas have grown
jim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
dear andre, first of all, i would like to thank you again for this invigorating thread. it seems that we are getting to some points that seem to be of general interest, so i'll add some comments.
Feynman diagrams are very useful in physics and mathematics. But the mystery of quantum physics lies elsewere: the extraction of a probability distribution from the complex values of a wave function. I dont think that a categorical formalism based on Feynman diagrams is very different from what the physicists are currently doing. This maybe why your formalism is having a moderate success among the physicists.
physicists and category theorists are certainly drawing very similar string diagrams. the *meaning* of these diagrams is, however, not completely identical. for physicists, string diagrams are a convenient shorthand for some constructions with hilbert spaces and operators. for category theorists, string diagrams represent constructions available in any category with enough structure. how important is this difference? more precisely, how important is it to go beyond hilbert spaces and look for some ***nonstandard models***? most physicists would probably say that they are happy with hilbert spaces. but many of them (albeit mostly theoreticians) ar enot. von neumann was very unhappy, and worked a lot to provide alternatives. and failed. but many people are thinking hard about "toy models" these days, capturing certain quantum phenomena and not other, generating some independence results, axiomatics etc. maybe category theory can help with this. (eg, bob coecke et al's recent work, as well as some bits that i have worked on, show that some crucial quantum phenomena, even entire quantum algorithms, can be represented using funny constructions with relations.) of course, my view of this may be biased, and nonstandard models of quantum mechanics may be irrelevant. but this is just one direction, showing a general way in which popping up from concrete sense into abstract nonsense may be a good thing.
Of course, a good formalism can stimulate new developements. But it should not be presented as radically new if it is not. To much hype might backfire, with bad consequence for the social image of category theory.
i cannot agree with this more. my first post in this thread was that maybe we should not advertise too much, but just make our tools available. ("nature will find the way" says the mathematician in jurassic park)
In mathematics, the word "quantum" is often used as a prefix to express some vague connection to quantum physics, like non-commutative algebras and Feynman diagrams. By itself it is no proof that the named notion is fitting something in the natural world. There are quantum groups, quantum algebras, quantum Grassmanians, quantum planes, quantum bundles, quantum Schubert cells, quantum cohomology theories, quantum fields, quantum Yan-Baxter operators, etc. The theory of quantum groups is mathematically very interesting but it has no applications that I know to real quantum physics:
http://en.wikipedia.org/wiki/Quantum_group I have a Phd student working on quantum quasi-shuffle algebras and he needs not to know about quantum physics because it is irrelevant.
oh but is that a bad thing? differential calculus was first physics, and then captured a lot of other things as well. and some of it did not reflect back into physics. as a computer scientist, i tend to think of quantum mechanics as a theory of a particular computational resource: **entanglement**. it seems to me that this concept raises fundamental worries for every computer scientist --- completely independent on its physical realisation. church's thesis said that computability was a very robust notion: whatever kind of a computer you take, you can compute the same. and for a while, it seemed that feasibility would be similar: there are various complexity classes, but they are all strictly subexponential with respect to each other. --- then came quantum algorithms with their "exponential beast", lurking from entanglement. now we know that computation happens in many models: on the internet, in a cell, distributed among the members of a mailing list. can some of them compute essenticall more than others? i don't know much about physics, but i cannot stay away from thinking about entanglement, and tensors, and string diagrams... with the very best wishes, -- dusko PS re **nonstandard models** again, i am wondering whether the hasegawa-hoffman-plotkin-selinger (HHPS) results, referred to by peter selinger, imply that there are no nonstandard models. the HHPS results say that a diagram commutes in a dagger-compact (resp compact, traced monoidal) category if and only if it holds in finitely dimensional hilbert (resp vector) spaces. so if a nonstandard model must be dagger compact, then anything validated in it must be validated in hilbert spaces? that would pretty much kill my nonstandard models, wouldn't it? i am not sure that i completely understand the HHPS results (so please correct me if i am wrong), but it does not seem to me that they provide anything like a representation theorem. a representation theorem, say for abelian categories, says something like: you give me a small abelian category AA, and i produce a ring R and an embedding AA--->Mod-R. in contrast, the HHPS theorems say: you give me a diagram D that commutes in a dagger-compact category, and i provide a field K such that that D also commutes in FHilb_K. so for every D, i need to construcat a new field K_D, right? well, this would provide an embedding of a dagger compact-category CC into FHilb_H for some field H only if there was a way to all fields K_D for all diagrams D that commute in CC into one big field H. how much hope is there for that? and even if i could do that, it would take some massage to embed FHilb_H into the standard model, consisting of *complex* hilbert spaces. so i still think that hilbert space may be a needlessly big place. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Thu, Dec 17, 2009 at 03:30:30PM -0800, peasthope@shaw.ca wrote:
... but how prevalent is the subject in undergraduate programs?
For the record, there was a course in category theory for undergraduates at Monash University (Melbourne) in the early 1970s. This was in the third year of what, for people interested in mathematics, was usually a four-year degree. It was taught by G B Preston, as in `algebraic theory of semigroups' using MacLane and Birkhoff (1967, not Birkhoff and MacLane). It partly took the line that category theory unified the basic algebraic and topological constructions and partly that it was a subject to study in its own right. Students then were simultaneously being taught the general Tichonoff theorem using ultrafilters, smooth manifolds and multilinear algebra (more universal constructions, as in Greub), and Hilbert space theory. This provided a strong context for category theory. Heady days. Kirill -- ===================================== http://kchmackenzie.staff.shef.ac.uk/ ===================================== [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
KCHM wrote:
For the record, there was a course in category theory for undergraduates at Monash University (Melbourne) in the early 1970s.
The (a?) counterpart of this at Berkeley at the start of the 1970s was Ed Spanier's algebraic topology course, whose first lecture would begin by exhibiting a functor between two categories, I forget which (I was not then at all into categories) but perhaps Top and Grp, and giving a two-line proof (of a representation?) to make the point that category theory could be a powerful tool when expertly deployed. I mention this because the experience at the UACT conference in 1993 at MSRI on the hill overlooking Berkeley rather created the impression that Berkeley would be the last place to welcome category theory, particularly when then-director of MSRI Bill Thurston welcomed us all at the opening of the meeting with his announcement that the very thought of the opposite of a category made him ill. Such an opening remark would be more appropriately made about CO2 at the currently running conference in Copenhagen. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (17)
-
Andrew Stacey -
Colin McLarty -
Dusko Pavlovic -
Ellis D. Cooper -
Greg Meredith -
jim stasheff -
John Baez -
Joyal, André -
KCHM -
Larry Harper -
peasthope＠shaw.ca -
Ronnie Brown -
Ross Street -
Tom Leinster -
Vaughan Pratt -
Zinovy Diskin -
zoran skoda