* Re: A well kept secret?
@ 2009-12-17 23:30 peasthope
2009-12-18 4:09 ` John Baez
` (5 more replies)
0 siblings, 6 replies; 47+ messages in thread
From: peasthope @ 2009-12-17 23:30 UTC (permalink / raw)
To: categories
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/ ]
^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-17 23:30 A well kept secret? peasthope @ 2009-12-18 4:09 ` John Baez 2009-12-18 22:25 ` Ellis D. Cooper ` (5 more replies) 2009-12-18 10:48 ` A well kept secret? KCHM ` (4 subsequent siblings) 5 siblings, 6 replies; 47+ messages in thread From: John Baez @ 2009-12-18 4:09 UTC (permalink / raw) To: categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* A well kept secret? 2009-12-18 4:09 ` John Baez @ 2009-12-18 22:25 ` Ellis D. Cooper 2009-12-19 17:45 ` Ronnie Brown ` (4 more replies) 2009-12-20 17:50 ` Joyal, André ` (4 subsequent siblings) 5 siblings, 5 replies; 47+ messages in thread From: Ellis D. Cooper @ 2009-12-18 22:25 UTC (permalink / raw) To: categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-18 22:25 ` Ellis D. Cooper @ 2009-12-19 17:45 ` Ronnie Brown 2009-12-19 22:16 ` John Baez ` (3 subsequent siblings) 4 siblings, 0 replies; 47+ messages in thread From: Ronnie Brown @ 2009-12-19 17:45 UTC (permalink / raw) To: Ellis D. Cooper, categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-18 22:25 ` Ellis D. Cooper 2009-12-19 17:45 ` Ronnie Brown @ 2009-12-19 22:16 ` John Baez 2009-12-20 22:52 ` Greg Meredith ` (3 more replies) 2009-12-20 21:50 ` A well kept secret? jim stasheff ` (2 subsequent siblings) 4 siblings, 4 replies; 47+ messages in thread From: John Baez @ 2009-12-19 22:16 UTC (permalink / raw) To: Ellis D. Cooper, categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-19 22:16 ` John Baez @ 2009-12-20 22:52 ` Greg Meredith 2009-12-21 15:46 ` Zinovy Diskin ` (2 subsequent siblings) 3 siblings, 0 replies; 47+ messages in thread From: Greg Meredith @ 2009-12-20 22:52 UTC (permalink / raw) To: John Baez, categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-19 22:16 ` John Baez 2009-12-20 22:52 ` Greg Meredith @ 2009-12-21 15:46 ` Zinovy Diskin 2009-12-22 16:59 ` zoran skoda 2009-12-23 1:53 ` Tom Leinster 2009-12-23 19:10 ` CatLab Joyal, André 3 siblings, 1 reply; 47+ messages in thread From: Zinovy Diskin @ 2009-12-21 15:46 UTC (permalink / raw) To: John Baez, categories >> 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-21 15:46 ` Zinovy Diskin @ 2009-12-22 16:59 ` zoran skoda 0 siblings, 0 replies; 47+ messages in thread From: zoran skoda @ 2009-12-22 16:59 UTC (permalink / raw) To: Zinovy Diskin 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-19 22:16 ` John Baez 2009-12-20 22:52 ` Greg Meredith 2009-12-21 15:46 ` Zinovy Diskin @ 2009-12-23 1:53 ` Tom Leinster 2009-12-23 14:15 ` Colin McLarty 2009-12-23 19:10 ` CatLab Joyal, André 3 siblings, 1 reply; 47+ messages in thread From: Tom Leinster @ 2009-12-23 1:53 UTC (permalink / raw) To: categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-23 1:53 ` Tom Leinster @ 2009-12-23 14:15 ` Colin McLarty 0 siblings, 0 replies; 47+ messages in thread From: Colin McLarty @ 2009-12-23 14:15 UTC (permalink / raw) To: categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* CatLab 2009-12-19 22:16 ` John Baez ` (2 preceding siblings ...) 2009-12-23 1:53 ` Tom Leinster @ 2009-12-23 19:10 ` Joyal, André 3 siblings, 0 replies; 47+ messages in thread From: Joyal, André @ 2009-12-23 19:10 UTC (permalink / raw) To: categories, urs.schreiber 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-18 22:25 ` Ellis D. Cooper 2009-12-19 17:45 ` Ronnie Brown 2009-12-19 22:16 ` John Baez @ 2009-12-20 21:50 ` jim stasheff [not found] ` <d4da910b0912220859q3858b68am4e58749f21ce839d@mail.gmail.com> [not found] ` <4B322ACA.50202@btinternet.com> 4 siblings, 0 replies; 47+ messages in thread From: jim stasheff @ 2009-12-20 21:50 UTC (permalink / raw) To: Ellis D. Cooper, categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* Re: A well kept secret? [not found] ` <d4da910b0912220859q3858b68am4e58749f21ce839d@mail.gmail.com> @ 2009-12-23 4:31 ` Zinovy Diskin 2009-12-23 14:35 ` Ronnie Brown 0 siblings, 1 reply; 47+ messages in thread From: Zinovy Diskin @ 2009-12-23 4:31 UTC (permalink / raw) To: zoran skoda 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. I'm not going to defend the market metaphor -- it's doubtful anyway. Still, I'd like to clarify a couple of points. 1) I think that both sources of mathematical development, the internal one based on aesthetic criteria and consistency, and the external one based on applications, are equally important and mutually beneficial. Apart from posing interesting problems, applications provide novel interpretations of formal constructs, which is always fruitful. After all, effectively applicable mathematics turns out to be aesthetically appealing as well (Eugene Vigner wrote a famous paper about this "On inaccessible effectiveness of mathematics in natural sciences") 2) The problem of "category theory vs. mathematics" is beyond mathematics as such. A lot of problems could be avoided if a taste for categorical thinking were cultivated in high school, and the basics of category theory were taught to mathematicians, scientists, engineers at the undergrad level. But education is one of the most conservative social institutes with a huge inertia. Turning mathematical education towards category theory needs financial and administrative support, and an external demand. Applications of category theory to engineering problems would be beneficial in this respect too. Z. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-23 4:31 ` Zinovy Diskin @ 2009-12-23 14:35 ` Ronnie Brown 0 siblings, 0 replies; 47+ messages in thread From: Ronnie Brown @ 2009-12-23 14:35 UTC (permalink / raw) To: Zinovy Diskin 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* Re: A well kept secret? [not found] ` <4B322ACA.50202@btinternet.com> @ 2009-12-25 20:06 ` Zinovy Diskin 0 siblings, 0 replies; 47+ messages in thread From: Zinovy Diskin @ 2009-12-25 20:06 UTC (permalink / raw) To: Ronnie Brown On Wed, Dec 23, 2009 at 9:35 AM, Ronnie Brown <ronnie.profbrown@btinternet.com> wrote: > 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: > ------------------------------------------------------------------------- yes, I meant this paper, I apologize for the wrong reference. Actually, I literally translated the Russian translation of the title (that I remembered) back into English. The result turned out to be not an identity, and even not an isomorphism > ... 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. > Here's an absolutely rational explanation. Suppose that once upon a time there were two classes of people, say, A and B, with different logics and aesthetics and criteria of elegance. Correspondingly, they had developed different mathematics, MA and MB. It so happened that A-aesthetics and thinking based on it turned out to be inadequate for the reality, and people A were eaten by saber-toothed tigers. Mathematics MA was forgotten and its traces can now be found in ancient archives only. I'm afraid that the A-destiny is awaiting the opponents of cat theory and their non-categorical math. :) [Do not take it seriously, I understand that category theory is just a good mathematics, not a different mathematics]. Happy Holidays! Z. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-18 4:09 ` John Baez 2009-12-18 22:25 ` Ellis D. Cooper @ 2009-12-20 17:50 ` Joyal, André [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E2159B6AA@CAHIER.gst.uqam.ca> 2009-12-21 19:20 ` additions Michael Barr ` (3 subsequent siblings) 5 siblings, 1 reply; 47+ messages in thread From: Joyal, André @ 2009-12-20 17:50 UTC (permalink / raw) To: John Baez, categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* additions [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E2159B6AA@CAHIER.gst.uqam.ca> @ 2009-12-21 8:43 ` Joyal, André 2009-12-21 14:16 ` additions Bob Coecke ` (2 more replies) 0 siblings, 3 replies; 47+ messages in thread From: Joyal, André @ 2009-12-21 8:43 UTC (permalink / raw) To: categories In my message to John Baez, I wrote: >I can 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 The list is too restrictive. I would like to expand it further: 1) Algebraic topology and homological algebra 2) Abelian categories 3) Algebraic geometry and topos theory 4) General cartesian algebra 5) Categorical logic 6) Homotopical algebra 7) Elementary topos theory and set theory 8) Monoidal categories and enriched category theory 9) General tensor algebra and coalgebra 10) Category theory and computer science 11) Quantum field theory 12) Higher categories and homotopy theory Algebraic theories and limit sketches are included in (4). Multicategories, operads are included in (9). I have included Quillen homotopical algebra in (6). Best, André -------- Message d'origine-------- De: categories@mta.ca de la part de Joyal, André Date: dim. 20/12/2009 12:50 À: John Baez; categories@mta.ca Objet : categories: Re: A well kept secret? 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-21 8:43 ` additions Joyal, André @ 2009-12-21 14:16 ` Bob Coecke 2009-12-22 2:24 ` additions Joyal, André [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E370F5626@CAHIER.gst.uqam.ca> 2009-12-22 0:39 ` additions Mike Stay [not found] ` <Pine.LNX.4.64.0912211413340.15997@msr03.math.mcgill.ca> 2 siblings, 2 replies; 47+ messages in thread From: Bob Coecke @ 2009-12-21 14:16 UTC (permalink / raw) To: categories Dear all, May I point at another current activity: > 1) Algebraic topology and homological algebra > 2) Abelian categories > 3) Algebraic geometry and topos theory > 4) General cartesian algebra > 5) Categorical logic > 6) Homotopical algebra > 7) Elementary topos theory and set theory > 8) Monoidal categories and enriched category theory > 9) General tensor algebra and coalgebra > 10) Category theory and computer science > 11) Quantum field theory > 12) Higher categories and homotopy theory 13) Category theory in quantum information & quantum foundations While only a quite recent activity, in funding terms this may currently be the 2nd most funded category-theory related area after Category Theory in Computer Science, with, for example, dedicated large EU grants, US Office of Naval Research support, and many others. Also, this summer two researchers in the area obtained permanent positions in France, and here in Oxford we hired a 3rd faculty member in the area. While money and science quality are obviously not always that related to each other, funding is an essential component for sustaining a research activity, enabeling jobs for postdocs in the area, etc. Best wishes, Bob. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-21 14:16 ` additions Bob Coecke @ 2009-12-22 2:24 ` Joyal, André 2009-12-23 20:51 ` additions Thorsten Altenkirch ` (2 more replies) [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E370F5626@CAHIER.gst.uqam.ca> 1 sibling, 3 replies; 47+ messages in thread From: Joyal, André @ 2009-12-22 2:24 UTC (permalink / raw) To: Bob Coecke, categories Dear Bob, The subjects in my list were not chosen according to the size of research grants or the number of peoples hired in recent years by such and such institutions. I know first hand every subject in the list, at least at the basic level. They make an essential use of category theory, and they inspire new developements in the field. I confess that I am quite ignorant about quantum computing. You wrote: >in funding terms >this may currently be the 2nd most funded category-theory >related area after Category Theory in Computer Science, >with, for example, dedicated large EU grants, US Office >of Naval Research support, and many others. Also, this >summer two researchers in the area obtained permanent >positions in France, and here in Oxford we hired a >3rd faculty member in the area. I do not question the importance of the subject. But presently, I am not convinced that quantum computing can contribute significantly to category theory. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-22 2:24 ` additions Joyal, André @ 2009-12-23 20:51 ` Thorsten Altenkirch 2009-12-24 23:55 ` additions Dusko Pavlovic 2009-12-26 2:14 ` additions Peter Selinger 2 siblings, 0 replies; 47+ messages in thread From: Thorsten Altenkirch @ 2009-12-23 20:51 UTC (permalink / raw) To: André On 22 Dec 2009, at 02:24, Joyal, André wrote: > do not question the importance of the subject. > But presently, I am not convinced that quantum computing can > contribute significantly to category theory. I have to admit that I am quite ignorant about many of the areas mentioned in the previous email. On the other hand developments in Computer Science I know about don't seem to feature. Maybe they are too mundane for Mathematicians. As far as quantum computing goes, or maybe more general quantum theory from a computer science point of view, it seems to me that there are interesting interactions with category theory in the recent work by Samson Abramsky, Bob Coecke, Peter Selinger and many others. E.g. the completeness of the graph theoretical calculus for dagger compact closed categories modelling finite dimensional Hilbert spaces. Cheers, Thorsten [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-22 2:24 ` additions Joyal, André 2009-12-23 20:51 ` additions Thorsten Altenkirch @ 2009-12-24 23:55 ` Dusko Pavlovic 2009-12-26 2:14 ` additions Peter Selinger 2 siblings, 0 replies; 47+ messages in thread From: Dusko Pavlovic @ 2009-12-24 23:55 UTC (permalink / raw) To: categories bob coecke proposed to add quantum computing to andre joyal's list of important directions of categorical research, but andre rejected it. i cannot overstate my respect for andre's work and judgement. but this left me pondering. like andre, i must confess that i am quite ignorant about quantum computing. (unlike andre, i am also ignorant about many other categorical topics on his list.) but we probably all know the following. most results in quantum computing are theorems about hilbert spaces. quantum computing is a *tensor calculus*. but it is a tensor calculus of a special kind: it attempts to describe a wildly unintuitive world. even the greatest contributors, like von neumann and feynman, deplored the gap between the quantum world, imposed on us in the lab, and the intuitions imposed on us in everyday life. now category theory often helps where the common intuitions fail. many of its applications demonstrate this. so quantum computation might be an opportunity for an effective application of *geometry of tensor calculus*. is it really wise to reject an attempt to develop this, as objectionable as it might be in any details? physicists like string diagrams, category theorists like string diagrams. most communities would actively reach out... is it just my impression, or are category theorists a little more sceptical about the value of applications than most mathematical communities? they seem to seek a recognition that categories are useful across mathematics, but then hesitate to recognize the depth and value of the applications in the other areas. --- can it be that we suffer from a superiority complex of some sort? the questions raised in the *well kept secret* thread were: 1) why are the achievements of category theory not recognized publicly? 2) what have we done to deserve the opprobium? 3) how can we convince the sceptics? please allow me to add one more: 4) how can the achievements of category theory be used to expand its future developments and applications, and not to constrain them? with best wishes, -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-22 2:24 ` additions Joyal, André 2009-12-23 20:51 ` additions Thorsten Altenkirch 2009-12-24 23:55 ` additions Dusko Pavlovic @ 2009-12-26 2:14 ` Peter Selinger 2 siblings, 0 replies; 47+ messages in thread From: Peter Selinger @ 2009-12-26 2:14 UTC (permalink / raw) To: joyal.andre In defense of Andre's list, the explanation he gave for his original list was that subjects in category theory become hot from time to time in response to factors such as new developments outside category theory. The list was supposed to be a list of categorical subjects, not a list of the respective developments that have inspired their use and advancement. The current use of category theory in quantum foundations is clearly an interesting development, and has inspired new work in category theory. But I would still be comfortable, for the time being, in classifying this new work as falling within the existing subjects of "monoidal categories" and "category theory and computer science" on Andre's growing list. Recently also "topos theory" due to the work of Andreas Doering, Klaas Landsman, and others on topos models for basic physics. As for the completeness result that Thorsten mentioned, the reference is: M. Hasegawa, M. Hofmann and G. Plotkin, "Finite dimensional vector spaces are complete for traced symmetric monoidal categories". In Pillars of Computer Science: Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday, Springer LNCS 4800, pages 367-385, February 2008. The result is that an equation holds in all traced symmetric monoidal categories if and only if it holds in finite dimensional vector spaces. An immediate corollary is that the analogous result holds for compact closed categories. A simplified proof, and extension to dagger compact closed categories (w.r.t. finite dimensional Hilbert spaces), can be found here: P. Selinger, "Finite dimensional Hilbert spaces are complete for dagger compact closed categories". Extended abstract, to appear in Proceedings of the 5th International Workshop on Quantum Physics and Logic (QPL 2008), Reykjavik, 2010. Merry Christmas to all, -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* quantum information and foundation [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E370F5638@CAHIER.gst.uqam.ca> @ 2009-12-28 17:54 ` Joyal, André 2009-12-29 12:13 ` Urs Schreiber 2009-12-29 15:55 ` zoran skoda 0 siblings, 2 replies; 47+ messages in thread From: Joyal, André @ 2009-12-28 17:54 UTC (permalink / raw) To: categories Dear Bob, Thorsten and Dusko, I thank you for expressing frankly your position. Thorsten wrote: >I have to admit that I am quite ignorant about many of the areas >mentioned in the previous email. On the other hand developments in >Computer Science I know about don't seem to feature. Maybe they are >too mundane for Mathematicians. Dusko wrote: >is it just my impression, or are category theorists a little more >sceptical about the value of applications than most mathematical >communities? they seem to seek a recognition that categories are useful >across mathematics, but then hesitate to recognize the depth and value of >the applications in the other areas. --- can it be that we suffer from a >superiority complex of some sort? Bob wrote: >In this context there is the highly unfortunate fact that there are >certain quite prominent people in the category theory community who think >that any deviation from treating category as a branch of pure mathematics >and pure mathematics only is a bad thing! You seems to suggest that this is a debate between pure and applied category theorists. I disagree to the extend that "quantum foundation" and "quantum information" are very speculative subjects. The "Foundational Question Institute", http://www.fqxi.org/ which is known to support speculative research projects exclusively, is funding a project on Quantum Foundation by Bob Coecke: http://www.fqxi.org/grants/large/awardees/view/__details/2008/coecke It has funded a project called "Topos Quantum Theory" by Christopher Isham http://www.fqxi.org/grants/large/awardees/view/__details/2006/isham It is funding a project "Categorifying Fundamental Physics" by John Baez: http://www.fqxi.org/grants/large/awardees/view/__details/2008/baez Physics is in bad shape today according to Lee Smolin: http://www.amazon.ca/Trouble-Physics-String-Theory-Science/dp/061891868X/ His main critic is that string theory has lost contact with experiments. It has become an academically driven discipline. Maybe we should stop calling it physics. Of course, it can be interesting mathematically. 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. The notion of dagger compact closed category is interesting and purely mathematical, like the notion of quantum group. Quantum information science is also quite speculative: http://en.wikipedia.org/wiki/Quantum_Information_Science http://en.wikipedia.org/wiki/Topological_quantum_computer Again, there is nothing wrong with highly speculative research. So the present debate is not about real applications of category theory. Bob wrote: >What the quantum information `hype' has done is injected some new blood in >foundations of quantum mechanics research, an area which for several >reasons had been suffocated by by the end of the previous century, despite >the universal discomfort of the physics community with quantum mechanics. >(a typical slogan which reflects this is: ``don't ask questions just >compute'') One would expect that this surge of quantum foundations, which >meanwhile has led to many novel ideas, approaches, and radically different >manners to think about physics in general, will ultimately lead to new >mathematics. Moreover, the natural guise of many of these new ideas is >within category theory, a message that some including myself have been >trying to pass on within the foundations of physics communitee, with >moderate success. Feynam introduced his diagram as a method for computing the solutions of QED field equations. It is essentially a technique for enumerating the terms arising in perturbation theory. The method was extended to all physical fields and Penrose understood the connection between the diagrams and tensor calculus. The geometry of tensor calculus is just an abstraction of this connection. 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. Of course, a good formalism can stimulate new developements. But it should not be presented as radically new if it is not. Too much hype may backfire. It is not good for the reputation of category theory. Happy New Year to all! Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: quantum information and foundation 2009-12-28 17:54 ` quantum information and foundation Joyal, André @ 2009-12-29 12:13 ` Urs Schreiber 2009-12-29 15:55 ` zoran skoda 1 sibling, 0 replies; 47+ messages in thread From: Urs Schreiber @ 2009-12-29 12:13 UTC (permalink / raw) To: Joyal, André On Mon, Dec 28, 2009 at 6:54 PM, Joyal, André <joyal.andre@uqam.ca> wrote: > Physics is in bad shape today according to Lee Smolin: > > http://www.amazon.ca/Trouble-Physics-String-Theory-Science/dp/061891868X/ > > His main critic is that string theory has lost contact with experiments. > It has become an academically driven discipline. > Maybe we should stop calling it physics. > Of course, it can be interesting mathematically. I would like to expand on this remark, and point out an application of (higher) category theory that might deserve more attention from mathematicians. First a remark concerning the detachment of string theory from experiment: much of theoretical physics, not just string theory, is far remote from experiment, but -- in principle -- for a good reason: if experiment shows that a certain incarnation of mathematical structure X is relevant for the description of the physical world, then for understanding it well we ought to study also all other incarnations of structure X, even if they are not (yet) known to be relevant for the description of the world themselves. As a simple example: not all solutions of Einstein's equations describe anything in the real world. But we want theoretical physicists to understand as many as possible of them: while some particular cosmological model (say one with closed timelike geodesics) may look utterly irrelevant for the description of the real world (given the present state of experimental knowledge!), it is the understanding of the collection of all such models and their interrelation that helps with understanding the particular one that does describe the real world. This idea, that we may study a theory in terms of the collection of its models, should resonate with category theorists. >From that perspective string theory strongly deserves to be studied by theoretical physicsists, even in the absence of experimental evidence: the string perturbation series is a conceptually compelling variation of Feynman's celebrated sum over correlators of a 1d QFT. Every theoretical physicist worth his or her money should feel an itch to explore the analogous sums over correlators of 2d QFTs. And that's what (perturbative) string theory is. http://ncatlab.org/nlab/show/string+theory And indeed, the above idea that for understanding one model it helps to understand all its variations, is at work here, too: studying the string perturbation series has led to a better understanding of Feynman's perturbation series, since a few years quite spectacularly resulting in a previously undreamed of understanding of the higher loop Feynman terms in supergravity theories. The fact that the discovery of many other suggestive aspects of the string perturbation series made a whole community become so excited about it that they threw some care and scientific discipline in the wind is a problem, but one of the sociology of science, not a fault of the topic. The reason why I feel saying all this is worthwhile on a mailing list devoted to category theory, is that a closer look shows that the mathematical structures involved in string theory are not only an impressive source of examples of applications of higher category theory, but in some cases even their archetypical motivational examples. The cobordism hypothesis/theorem http://ncatlab.org/nlab/show/cobordism+hypothesis is arguably comparatively pivotal for higher category theory as, say, the Yoneda lemma is for ordinary category theory. (I really think it is.) With that in mind, it should not be forgotten that both its roots in the ideas of Witten, Atiyah and Segal, as well as its present rather impressive applications in the work of Freed-Hopkins-Lurie-Teleman http://arxiv.org/abs/0905.0731 are situated in the conceptual framework that was opened by the step from the Feynman perturbation series to string theory: as John Baez mentioned in a previous message, cobordism representations are being speculated to encode quantized general relativity, but that speculation should not make us forget that what made theoretical physicists eventually pass from the study of quantum field theories defined on Minkowski space or similar, to "full" quantum field theories defined on all possible cobordisms was the idea that the Feynman perturbation series ought to have a generalization from a sum over graphs to a sum over cobordisms of higher dimension: conformal field theory used to be studied on R^2 for years until string theory opened the perspective that a CFT ought to be defined on general surfaces. Today the classification of such full 2dCFT -- the representation theory of 2-dimensional conformal cobordism categories -- is an impressive result in the theory of modular tensor categories. http://golem.ph.utexas.edu/string/archives/000813.html Indeed, it seems to me that the most substantial conceptual progress on the grand perspective exhibited by the passage to the string perturbation series has recently come not out of the physics departments (which seem to be curiously stuck with throwing insufficient formal tools at their grand targets), but out of the math departments, those math departsments where higher category theory has an influence in one way or other. In order to proliferate this observation, with AMS publishing we are currently preparing a book volume that is devoted to exhibiting aspects of the full story behind this claim. http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory The text at that link may provide more details on the point that I am trying to make here. I can summarize this point maybe as follows: pure mathematicians and especially category theorists and higher category theorists should not be tricked by complaints such as voiced in Smolin's book into thinking that it is ill-advized to have a closer look at the mathematical structures to be found in string theory, well hidden under physicist's nonsense as they may be. On the contrary: much of what makes the present practice of string theory so tiresome is that the lively activity of the 1980s of mathematically inclined researchers looking into the mathematical structures of the theory has largely vanished, at least in the physics departments. The theory is much more interesting than the average talk of its current practicioners. And much deeper. One of the foremost powers of category theory is its ability to unravel hidden structures and make them become mathematically active. String theory is a vast reservoir of crucial (higher) categorical structures that is, while recently beginning to be investigated as such, largely like a huge bag of disjoint LEGO pieces which physicist dream of putting together to a grand edifice, but which is waiting for the higher category theorist to actually assemble it. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: quantum information and foundation 2009-12-28 17:54 ` quantum information and foundation Joyal, André 2009-12-29 12:13 ` Urs Schreiber @ 2009-12-29 15:55 ` zoran skoda 1 sibling, 0 replies; 47+ messages in thread From: zoran skoda @ 2009-12-29 15:55 UTC (permalink / raw) To: Joyal, André Dear Prof. Joyal, 1. I agree with you that the hype about combinatorics of Feynman diagrams is, while important for constructing good practial theories and calculational methods, not appropriate target for understanding and changing the very foundations of quantum theory. 2. I disagree with you that quantum groups have no applications to real quantum physics. Surely, they do not change the very foundations of quantum theory, but do have numerous and significant applications to concrete models in quantum physics. Most of the significant applications are limited to the quantum groups at root of unity. They appear as symmetries of numerous integrable models, e.g. quantum spin chain models, and hidden symmetries of some conformal field theories to name the most well-understood applications. Harmonic analysis on quantum groups is important to calculate analytic expressions for correlation functions in some of the models, and the representation thoery at root of unity has a Kazhdan-Lusztig type correspondence in some cases to vertex operator algebra representations. This involves not a superficial but a very intricate picture. As a physicist I despise when people come with quantum and string terminology when not at least vaguely and indirectly appropriate, revelations by mathematician that they found the true meaning of some physical concepts and alike. A typical claim is of many mathematicians that vertex operator algebras are THE SAME as conformal field theories, while they feature just a part of the true story. I witnessed a talk by a young hot mathematician who gave an introduction that CFT as a discipline is a SUBSET of string theory. When I told him that CFT originated and is fruitful outside of string theory too (e.g. in study of critical phenomena in condensed matter physics), and thus should not be DEFINED subordinated to its particular hot and popular application, he started substantiating his claim waving hands that somebody has proved that "this and this is the same as that and that" (I am not paraphrasing but citing!!! what kind of psychology drives these young postdocs from Princeton-level hype places snowing the audience with misterious claims and referal to untouchable authorities whom they seen somewhere and half-understood ??). 3. As far as quantum computation and quantum information, the engineering boundaries of the field are not natural place of subject within physics and math. If one looks at the textbooks on quantum information more than half of the books are just standard material on quantum physics, not a different area. Topological quantum computation on the other hand, is more of topology, monoidal categories and QFT-type in its technology so it is already included in divisions listed. Various measures of coherence on the other hand in the literature are rather nonrigorous and somehow trivial variations are publiashable. I have been a referree 2 times and witnessed extremely content-free papers building the merit on 2-3 elementary and obvious observations which were claimed to have connections to algebaric geometry etc. while the authors were not being able to say anything nontrivial other than fancying about formal similarity in a polynomial describing some quantity. The other referree, from optical engineering has suggested the papers for publications as "significant" in J. Phys. A which accepted it against my recommendations. Baisng publications on hype and superficial remarks other than substantial content is a sign of an unhealthy standpoint of the community. I agree with John Baez that there is a healthy potential in quantum computing, but do not think that the area is well-defined, not subsumed to already listed areas of applications (like QFT), and would remark that it is overfunded for the present extent of true significant research. 4. It is not very important how we subdivide the applications of categories, but it is more important that we educate each other with aspects and overview of the subjects some of us are not specialized in but others can help. Awareness of possible applications amy help to bridge the gap between special areas and main focuses of current pure research. Thus while the lists like the one compiled in this discussion may be fun to mobilize a bit of cross-disciplinary discussion, more educative efforts and true discussions would do more. Zoran [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-21 8:43 ` additions Joyal, André 2009-12-21 14:16 ` additions Bob Coecke @ 2009-12-22 0:39 ` Mike Stay 2009-12-23 11:19 ` additions Steve Vickers [not found] ` <Pine.LNX.4.64.0912211413340.15997@msr03.math.mcgill.ca> 2 siblings, 1 reply; 47+ messages in thread From: Mike Stay @ 2009-12-22 0:39 UTC (permalink / raw) To: Joyal, André On Mon, Dec 21, 2009 at 12:43 AM, Joyal, André <joyal.andre@uqam.ca> wrote: > In my message to John Baez, I wrote: > >>I can distinguish approximatly 6 major currents: > >>5) Category theory and computer science I'm trying to expose my fellow programmers to the joys of category theory, but none of them have a math or physics background (or even a funcitonal programming background), which is where most of my experience with CT has been. What have been the major applications of category theory to computer science that have affected programmers? Are there new algorithms? Are there really nice ways of solving certain problems? The fact that data types with equivalence classes of lambda terms between them form a cartesian closed category doesn't seem to inspire them very much. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-22 0:39 ` additions Mike Stay @ 2009-12-23 11:19 ` Steve Vickers 2009-12-23 18:06 ` additions Mike Stay 2009-12-23 19:06 ` additions Thorsten Altenkirch 0 siblings, 2 replies; 47+ messages in thread From: Steve Vickers @ 2009-12-23 11:19 UTC (permalink / raw) To: Mike Stay Dear Mike, Of course, in functional programming the applications of categories go far beyond lambda terms. (1) At a fairly elementary level, the treatment of list types in functional programming languages gives a good lead in to universal properties (e.g. list type = free monoid or free (empty, cons)- algebra). Things you do with the universal properties are present as well known tools in functional programming with lists. The universal properties can then be used to motivate the abstract structure of categories: they describe data types by their external interfaces with the rest of the world rather than by their concrete internal structure, and the morphisms play the role of saying what the external interactions are. Expositionally, for Mac Lane universal properties were an important example where the working mathematician had been doing category theory all along without knowing it. (2) More advanced, Haskell has made important use of monads as a programming technique for bringing side-effects, I/O etc into functional programming in an elegant way. (The way this came about is that it has long been more or less self-evident that categories are just what you need for describing the semantics, and the categorical experience of the semanticists led to the practical application of monads.) So it could be that the best way forward is to teach them Haskell first. (I gave a short introduction to categories at Imperial, in the Compujting Department, and I exploited heavily the fact that they had all done Miranda, a predecessor of Haskell.) Regards, Steve Vickers. On 22 Dec 2009, at 00:39, Mike Stay wrote: > On Mon, Dec 21, 2009 at 12:43 AM, Joyal, André > <joyal.andre@uqam.ca> wrote: >> In my message to John Baez, I wrote: >> >>> I can distinguish approximatly 6 major currents: >> >>> 5) Category theory and computer science > > I'm trying to expose my fellow programmers to the joys of category > theory, but none of them have a math or physics background (or even a > funcitonal programming background), which is where most of my > experience with CT has been. > > What have been the major applications of category theory to computer > science that have affected programmers? Are there new algorithms? > Are there really nice ways of solving certain problems? The fact that > data types with equivalence classes of lambda terms between them form > a cartesian closed category doesn't seem to inspire them very much. > -- > Mike Stay - metaweta@gmail.com > http://math.ucr.edu/~mike > http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-23 11:19 ` additions Steve Vickers @ 2009-12-23 18:06 ` Mike Stay 2009-12-24 13:12 ` additions Carsten Führmann 2009-12-24 19:23 ` additions Dusko Pavlovic 2009-12-23 19:06 ` additions Thorsten Altenkirch 1 sibling, 2 replies; 47+ messages in thread From: Mike Stay @ 2009-12-23 18:06 UTC (permalink / raw) To: Steve Vickers On Wed, Dec 23, 2009 at 3:19 AM, Steve Vickers <s.j.vickers@cs.bham.ac.uk> wrote: > Dear Mike, > > Of course, in functional programming the applications of categories go far > beyond lambda terms. > > (1) At a fairly elementary level, the treatment of list types in functional > programming languages... > (2) More advanced, Haskell has made important use of monads as a programming > technique... > So it could be that the best way forward is to teach them Haskell first... Thanks, everyone for your replies! Many of you suggested the same approach as Steve, functional programming and monads. At Google, however, we use Java, C++ and Python (collectively "JCP") for programs that run on our servers and JavaScript for programs that run in our webpages. So there's not a lot of call for learning a functional programming language either. On the wikipedia page for monads in functional programming, I see these examples: * I/O--JCP aren't functional; side effects are easy. * Maybe--This one's really exception handling, built into JCP. * Identity--so trivial it's never used. * Lists--built into JCP as arrays, together with the function "map"; the rest of the data structures are in the standard libraries, too. Monads and catamorphisms certainly give a more unified picture, but still not enough benefit to a programmer to justify the investment in learning category theory. * State & environment--as before, JCP aren't functional. State and environment are easy to come by. * Continuation passing style transformation--useful for functional language compiler writers for turning recursive programs into iterative ones that don't consume the stack. Very small audience. Continuations are also useful for coroutines and "threadless actors", but malicious code can consume all the resources--in this model, there's no preemptive multitasking; it's all voluntary. JCP are eagerly evaluated. One monad that isn't on the wiki page is for making a program be lazily evaluated. That can come in handy sometimes, but still doesn't justify learning category theory. I suppose the strongest argument I've heard for learning functional programming (and thereby justifying learning category theory) is that functional programs are much easier to test: there's no inaccessible state to worry about setting up properly. Monads allow the functional programmer to do easily all these things he's used to from imperative programming, while gaining the benefits of easy testability. Other reasons I've received involve solving problems in specialized domains. I think if I have a long enough list of these, I could probably convince my friends of category theory's utility. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-23 18:06 ` additions Mike Stay @ 2009-12-24 13:12 ` Carsten Führmann 2009-12-24 19:23 ` additions Dusko Pavlovic 1 sibling, 0 replies; 47+ messages in thread From: Carsten Führmann @ 2009-12-24 13:12 UTC (permalink / raw) To: Mike Stay Dear Mike, > Thanks, everyone for your replies! Many of you suggested the same > approach as Steve, functional programming and monads. At Google, > however, we use Java, C++ and Python (collectively "JCP") for programs > that run on our servers and JavaScript for programs that run in our > webpages. So there's not a lot of call for learning a functional > programming language either. It might be worth noting that JavaScript is a functional language. (It has a lambda operator ("function"), closures, and can pass functions as parameters and return values.) However, because it has eager evaluation, the whole monad business does not apply, at least not in the way it applies to Haskell. In fact, JavaScript is probably the most widely used functional language on the planet. But there are two strange phenomena: - Functional-programming experts keep on overlooking JavaScript (probably because it is so ugly from a theorists point of view) - Most professional JavaScript programmers fail to see the enormous functional potential of JavaScript. It is a very strange situation: the whole world uses a functional language and almost nobody is aware of it. Anyway, even though I am very category-prone, I must admit that category theory might be a very tough sell for the JavaScript crowd :) Best, Carsten [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-23 18:06 ` additions Mike Stay 2009-12-24 13:12 ` additions Carsten Führmann @ 2009-12-24 19:23 ` Dusko Pavlovic 1 sibling, 0 replies; 47+ messages in thread From: Dusko Pavlovic @ 2009-12-24 19:23 UTC (permalink / raw) To: Mike Stay Mike Stay wrote: > At Google, > however, we use Java, C++ and Python (collectively "JCP") for programs at one point, some googlers were interested in the conceptual views of ranking. FWIW, a categorical view of the PagRank and some extensions is in http://arxiv.org/abs/0802.1306 eg, the comma construction qpplies naturally... -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-23 11:19 ` additions Steve Vickers 2009-12-23 18:06 ` additions Mike Stay @ 2009-12-23 19:06 ` Thorsten Altenkirch 1 sibling, 0 replies; 47+ messages in thread From: Thorsten Altenkirch @ 2009-12-23 19:06 UTC (permalink / raw) To: Steve Vickers Hi Steve & Mike, I completely agree with Steve, I'd like to add that instead of functional programming we could have said "mathematically structured programming". But then indeed the two terms are very closely related. Category Theory helps us to structure abstractions. In Computer Science and in other areas (e.g. Physics). Some people seem to think that abstractions don't buy your anything concrete. E.g. they don't deliver faster algorithms or new physical theories. These people often overlook that everything they do relies essentially on abstractions which have been established a while ago. Hence, while it is hard to measure the impact of abstractions exactly, IMHO it is almost impossible to underestimate their value. Cheers, Thorsten On 23 Dec 2009, at 11:19, Steve Vickers wrote: > Dear Mike, > > Of course, in functional programming the applications of categories > go far beyond lambda terms. > > (1) At a fairly elementary level, the treatment of list types in > functional programming languages gives a good lead in to universal > properties (e.g. list type = free monoid or free (empty, cons)- > algebra). Things you do with the universal properties are present as > well known tools in functional programming with lists. The universal > properties can then be used to motivate the abstract structure of > categories: they describe data types by their external interfaces > with the rest of the world rather than by their concrete internal > structure, and the morphisms play the role of saying what the > external interactions are. Expositionally, for Mac Lane universal > properties were an important example where the working mathematician > had been doing category theory all along without knowing it. > > (2) More advanced, Haskell has made important use of monads as a > programming technique for bringing side-effects, I/O etc into > functional programming in an elegant way. (The way this came about > is that it has long been more or less self-evident that categories > are just what you need for describing the semantics, and the > categorical experience of the semanticists led to the practical > application of monads.) > > So it could be that the best way forward is to teach them Haskell > first. (I gave a short introduction to categories at Imperial, in > the Compujting Department, and I exploited heavily the fact that > they had all done Miranda, a predecessor of Haskell.) > > Regards, > > Steve Vickers. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* RE : categories: additions [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E2159B6B3@CAHIER.gst.uqam.ca> @ 2009-12-23 17:08 ` Joyal, André 0 siblings, 0 replies; 47+ messages in thread From: Joyal, André @ 2009-12-23 17:08 UTC (permalink / raw) To: Michael Barr, categories Michael Barr wrote: >I would add something between 2 and 3 about Triples (allright, >monads) and Equational theories. I agree. Let me expand the list further: 1) Algebraic topology and homological algebra 2) Abelian categories 3) Adjoint functors and monads 4) Algebraic geometry and topos theory 5) General universal algebra 6) Limit sketches and locally presentable categories 7) Categorical logic 8) Categorical model theory 9) Homotopical algebra 10) Elementary toposes theory and set theory 11) Monoidal categories and enriched category theory 12) General tensor algebras and coalgebras 13) Category theory and computer science 14) Quantum field theory 15) Higher categories and homotopy theory Best, André -------- Message d'origine-------- De: Michael Barr [mailto:barr@math.mcgill.ca] Date: lun. 21/12/2009 14:20 À: Joyal, André Cc: categories@mta.ca Objet : Re: categories: additions I would add something between 2 and 3 about Triples (allright, monads) and Equational theories. Here is an example of the sort of thing we are up against. A colleague called me this morning because a student had taken a set of notes (in French) on his course and was interested in publishing it. My colleague had an objection because in describing conformal isomorphism from the complex plane (or maybe sphere) to itself, the student had used the word "towards" (vers) instead of "on". His objection was that a conformal isomorphism was something between two spaces, not from one to the other. My answer was a specific such map was a map from one to the other. His reply essentially was, "Oh, it's category theory language. Well, I won't allow any of that in MY notes. No analyst would use that language." Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-18 4:09 ` John Baez 2009-12-18 22:25 ` Ellis D. Cooper 2009-12-20 17:50 ` Joyal, André @ 2009-12-21 19:20 ` Michael Barr 2009-12-27 23:14 ` quantum information and foundation Dusko Pavlovic ` (2 subsequent siblings) 5 siblings, 0 replies; 47+ messages in thread From: Michael Barr @ 2009-12-21 19:20 UTC (permalink / raw) To: Joyal, André, categories I would add something between 2 and 3 about Triples (allright, monads) and Equational theories. Here is an example of the sort of thing we are up against. A colleague called me this morning because a student had taken a set of notes (in French) on his course and was interested in publishing it. My colleague had an objection because in describing conformal isomorphism from the complex plane (or maybe sphere) to itself, the student had used the word "towards" (vers) instead of "on". His objection was that a conformal isomorphism was something between two spaces, not from one to the other. My answer was a specific such map was a map from one to the other. His reply essentially was, "Oh, it's category theory language. Well, I won't allow any of that in MY notes. No analyst would use that language." Michael On Mon, 21 Dec 2009, Joyal, André wrote: > In my message to John Baez, I wrote: > >> I can 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 > > The list is too restrictive. I would like to expand it further: > > 1) Algebraic topology and homological algebra > 2) Abelian categories > 3) Algebraic geometry and topos theory > 4) General cartesian algebra > 5) Categorical logic > 6) Homotopical algebra > 7) Elementary topos theory and set theory > 8) Monoidal categories and enriched category theory > 9) General tensor algebra and coalgebra > 10) Category theory and computer science > 11) Quantum field theory > 12) Higher categories and homotopy theory > > Algebraic theories and limit sketches are included in (4). > Multicategories, operads are included in (9). > > I have included Quillen homotopical algebra in (6). > > Best, > André > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: quantum information and foundation 2009-12-18 4:09 ` John Baez ` (2 preceding siblings ...) 2009-12-21 19:20 ` additions Michael Barr @ 2009-12-27 23:14 ` Dusko Pavlovic [not found] ` <Pine.GSO.4.64.0912272037140.28761@merc3.comlab> [not found] ` <Pine.GSO.4.64.0912281630040.29390@merc4.comlab> 5 siblings, 0 replies; 47+ messages in thread From: Dusko Pavlovic @ 2009-12-27 23:14 UTC (permalink / raw) To: joyal.andre 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* Re: quantum information and foundation [not found] ` <Pine.GSO.4.64.0912272037140.28761@merc3.comlab> @ 2009-12-28 16:38 ` Bob Coecke 0 siblings, 0 replies; 47+ messages in thread From: Bob Coecke @ 2009-12-28 16:38 UTC (permalink / raw) To: Dusko Pavlovic Dear Andre, Thorsten, Dusko, and others, Andre Joyal wrote: > Quantum information science is also quite speculative: > > http://en.wikipedia.org/wiki/Quantum_Information_Science > http://en.wikipedia.org/wiki/Topological_quantum_computer It depends whether one is talking about: (1) having a quantum computer in the shops (2) theoretical discovery and experimental verification of new physical phenomena inspired by approaching nature in information-theoretic terms. While the first is indeed pure speculation, the second is a fact, with many recently discovered physical phenomena, some of which are embodied in terms of computational models, having effectively been established in the lab. Well-known examples are quantum teleportation and quantum key exchange. To mention one example of a phenomenon embodied in terms of computational model: the ability to universally alter the state of quantum systems by only relying on observations (= the measurement-based quantum computational model). Actually, certain guises of quantum information technology are effectively available for purchase at: ID quantique: http://www.idquantique.com/ MagiQ: http://www.magiqtech.com/MagiQ/Home.html Smart Quantum: http://www.smartquantum.com/SmartQuantum.html These three companies are not at all controversial, as opposed to for example D-Wave. There must be well over 1000 researchers active in the area which has its `own wikipedia': http://www.quantiki.org/wiki/index.php/Main_Page The general expectation would be that it are the quantum communication protocols which will be the first transitions to mainstream technology, and these may become components within some hybrid information processing device. Andre Joyal wrote: > But the mystery of quantum physics lies elsewere: the extraction of a > probability distribution from the complex values of a wave function. Thorsten Altenkirch wrote: > I agree that the big question in quantum theory is the "measurement > problem". The measurement-based quantum computational model is interesting in that it considers what for a long time was the most controversial ingredient of quantum theory, as the main processing resource: von Neumann's projection postulate which describes how the state changes under observation. These changes under observations of typically highly entangled states can be conveniently modeled by certain interacting Frobenius algebras in monoidal categories: http://arxiv.org/abs/0906.4725 http://arxiv.org/abs/0902.0500 I don't see any speculation here, just a convenient manner of representing physical phenomena which effectively have been observed in the lab, by using structures which are considered as category-theoretic. A software package to help with this is also under development: http://web.comlab.ox.ac.uk/people/Aleks.Kissinger/projects.html http://dream.inf.ed.ac.uk/projects/quantomatic/ In this context, recently Ross Duncan and Simon Perdrix solved an open problem in the area of measurement-based quantum computing, which has to do with guarantying a deterministic answer for certain sequences of measurements, and the formulation of the answer crucially relies on the Frobenius algebras. (their paper is forthcoming) Dusko Pavlovic wrote: > most physicists would probably say that they are happy with hilbert > spaces. but many of them (albeit mostly theoreticians) ar enot. In fact, it are the experimentalists which tend to get quite excited about the use of graphical languages to describe quantum phenomena since these are more `operational' than the usual Hilbert space treatments. Theoretcians have a harder time to denounce the things to which they are used, except when you are called John von Neumann and you crafted the Hilbert space quantum mechanical formalism a few years earlier. Best wishes for the new year, Bob. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* Re: quantum information and foundation [not found] ` <Pine.GSO.4.64.0912281630040.29390@merc4.comlab> @ 2009-12-28 18:17 ` Bob Coecke 0 siblings, 0 replies; 47+ messages in thread From: Bob Coecke @ 2009-12-28 18:17 UTC (permalink / raw) To: Dusko Pavlovic Dear Andre and others, Andre Joyal wrote: > I disagree to the extend that "quantum foundation" and "quantum > information" are very speculative subjects. The "Foundational Question > Institute", > > http://www.fqxi.org/ > > which is known to support speculative research projects exclusively, > is funding a project on Quantum Foundation by Bob Coecke: > > http://www.fqxi.org/grants/large/awardees/view/__details/2008/coecke I addressed "quantum information" and "speculation" in a separate posting. Since the above may slightly misrepresent the activity within our group allow me to providea short description of what we do. While my FQXi grant (which meanwhile ended) indeed addresses the more speculative end of physics research, it is only a very small fraction of the research portfolio within our group here at Oxford University Computing Laboratory led my Abramsky and myself, which meanwhile has close to 30 members: http://web.comlab.ox.ac.uk/activities/quantum/ (good knowledge in category theory is part of the entrance fee and most of the research is on categorical quantum mechanics and related things, which stretches as far as computational linguistics) The three major contributing agencies are the Future and Emerging Technologies scheme of the European Union, the Information Technology panel of the British Engineering and Physical Sciences Research Council (EPSRC), and the US Office of Naval Research (ONR). For each of these the application process has very strong requirements on the potential for transition to society of the funded research. As mentioned in my other posting, software development based on categorical structures is one of them: http://web.comlab.ox.ac.uk/people/Aleks.Kissinger/projects.html http://dream.inf.ed.ac.uk/projects/quantomatic/ There is obviously nothing speculative here since this is a tool which (semi-)automates reasoning about quantum systems by exploiting a discrete (ie no complex field etc) representation of a fragment of quantum theory. This software relies on results in pure category theory such as Steve Lack's work on PROPs, on which my student Andrei Akhvlediani (formerly Walter Tholen's MSc student) is currently elaborating. Jamie Vicary who has a strong interest in higher-dimensional category theory (eg http://arxiv.org/abs/0805.0432) is hired on a software development-related ONR grant. The FQXi grant was important for our group since it acknowledges that while we are based in a computer science department we have an important activity at the more speculative end of fundamental physics. My personal philosophy on all of this is to try to span the whole spectrum, from no-nonsense straight computer science research, which provides stability, to the speculative end of the physics spectrum, where there is a desperate need for something radical to happen, which brings us to the following: Andre Joyal wrote: > Physics is in bad shape today according to Lee Smolin: > > http://www.amazon.ca/Trouble-Physics-String-Theory-Science/dp/061891868X/ > > His main critic is that string theory has lost contact > with experience. It has become an academically driven discipline. > Maybe we should stop calling it physics. The main problem is that string theory has suffocated many other approaches to foundational physics. Lee Smolin recently mentioned to me that he sees great promise in the work which some people in the quantum foundations community are doing. For example, he participated in this "Reconstructing Quantum Theory" workshop at the Perimeter Institute for Theoretical Physics: http://pirsa.org/C09016 Perimeter Institute for Theoretical Physics is an institute which aims to compensate for the lack of funding in foundational areas of physics, and is highly regarded in the physics community. Initial funding came from the Blackberry-RIM boss, and Lee Smolin was the first academic to be hired by them. Besides a talk by myself, there are also talks by the two faculty members in quantum foundations of the Perimeter Institute, Lucien Hardy and Rob Spekkens, who actually both have meanwhile been infected by some category theory: http://web.comlab.ox.ac.uk/publications/publication3026-abstract.html (draft! many typos etc, ...) http://arxiv.org/abs/0912.4740 (see the related work section) Lucien Hardy, Andreas Doering and myself also organized a conference on category theory and physics at the Perimeter Institute entitled Categories, Quanta, Concept: http://pirsa.org/C09008 Best wishes, Bob. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-17 23:30 A well kept secret? peasthope 2009-12-18 4:09 ` John Baez @ 2009-12-18 10:48 ` KCHM 2009-12-19 20:55 ` Vaughan Pratt 2009-12-22 12:21 ` additions Mark Weber ` (3 subsequent siblings) 5 siblings, 1 reply; 47+ messages in thread From: KCHM @ 2009-12-18 10:48 UTC (permalink / raw) To: categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: A well kept secret? 2009-12-18 10:48 ` A well kept secret? KCHM @ 2009-12-19 20:55 ` Vaughan Pratt 0 siblings, 0 replies; 47+ messages in thread From: Vaughan Pratt @ 2009-12-19 20:55 UTC (permalink / raw) To: categories 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-17 23:30 A well kept secret? peasthope 2009-12-18 4:09 ` John Baez 2009-12-18 10:48 ` A well kept secret? KCHM @ 2009-12-22 12:21 ` Mark Weber 2009-12-23 0:05 ` additions Scott Morrison [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E2159B6B8@CAHIER.gst.uqam.ca> ` (2 subsequent siblings) 5 siblings, 1 reply; 47+ messages in thread From: Mark Weber @ 2009-12-22 12:21 UTC (permalink / raw) To: Michael Barr, categories Dear Michael, 2009/12/21 Michael Barr <barr@math.mcgill.ca> > ... His reply essentially was, "Oh, it's category theory language. Well, > I won't allow any of that in MY notes. No analyst would use that language." > There's an easy reply to people infected with such silliness -- ask them if Terry Tao is an analyst, to which they'd probably reply "of course", and then tell them go to Tao's blog http://terrytao.wordpress.com/ and do a search for "category" (the search bar on Terry's page is on the left just below "recent comments"). The results will speak for themselves. Mark Weber [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-22 12:21 ` additions Mark Weber @ 2009-12-23 0:05 ` Scott Morrison 2009-12-23 14:13 ` additions Mark Weber 0 siblings, 1 reply; 47+ messages in thread From: Scott Morrison @ 2009-12-23 0:05 UTC (permalink / raw) To: Mark Weber Dear Mark, this is unfortunately a bad example. If you click through any of the results for "category" <http://terrytao.wordpress.com/?s=category> on Terry's page, you'll see that in nearly all cases, the only use of the word "category" is in "n-Category Cafe", which appears in the sidebar of every page, amongst the links to other blogs. best, Scott Morrison On Tue, Dec 22, 2009 at 06:21, Mark Weber <mark.weber.math@googlemail.com> wrote: > Dear Michael, > > 2009/12/21 Michael Barr <barr@math.mcgill.ca> > >> ... His reply essentially was, "Oh, it's category theory language. Well, >> I won't allow any of that in MY notes. No analyst would use that language." >> > > There's an easy reply to people infected with such silliness -- ask them if > Terry Tao is an analyst, to which they'd probably reply "of course", and > then tell them go to Tao's blog > > http://terrytao.wordpress.com/ > > and do a search for "category" (the search bar on Terry's page is on the > left just below "recent comments"). The results will speak for themselves. > > Mark Weber > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-23 0:05 ` additions Scott Morrison @ 2009-12-23 14:13 ` Mark Weber 0 siblings, 0 replies; 47+ messages in thread From: Mark Weber @ 2009-12-23 14:13 UTC (permalink / raw) To: Scott Morrison, Michael Barr, categories I wished to make the point that Tao uses categorical ideas and perspectives freely. It would've been better if I'd referred to the specific postings in which he does so ... http://terrytao.wordpress.com/2009/10/19/grothendiecks-definition-of-a-group/ http://terrytao.wordpress.com/2009/12/21/the-free-nilpotent-group/ These postings aren't about themselves about category theory, but in them he exhibits no inhibitions in using categorical language. Regards, Mark Weber On Wed, Dec 23, 2009 at 1:05 AM, Scott Morrison <scott@tqft.net> wrote: > Dear Mark, > > this is unfortunately a bad example. If you click through any of the > results for "category" <http://terrytao.wordpress.com/?s=category> on > Terry's page, you'll see that in nearly all cases, the only use of the > word "category" is in "n-Category Cafe", which appears in the sidebar > of every page, amongst the links to other blogs. > > best, > Scott Morrison [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* Re: CatLab [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E2159B6B8@CAHIER.gst.uqam.ca> @ 2009-12-23 21:04 ` Urs Schreiber 0 siblings, 0 replies; 47+ messages in thread From: Urs Schreiber @ 2009-12-23 21:04 UTC (permalink / raw) To: Joyal, André Dear André, thanks for your message. You write > The nLab is a very nice thing! I am pleased to hear that you think working on such wikis may be worthwhile! That's very encouraging to hear, from somebody like you. The nLab is just a first, tiny and tentative step, but eventually, with more work, this kind of activity could be rather useful, I believe. You write: > Maybe you could create a wiki-lab for homotopy theory too (a HoLab?) > Maybe all these labs could be connected. Before I enter the technicalities of setting up separate wiki-labs below, let me ask you one question: as you indicate, one useful aspect of wiki material is its interconnectedness, via hyperlinks. I am imagining that eventually, with plenty of time, effort and contributors, a large web of wiki-material will grow, covering many parts of math and exhibiting their connectedness. For instance pure abstract category theory entries may point to entries about physics where the concepts are applied, and vice versa. We currently think of the nLab as a wiki to encompass all that, in principle -- being well aware, of course, that its current state is at best just a puny approximation to what it could be, eventually. So my question is: could you tell me what it is that makes you feel that topics like category theory or homotopy theory need to be more physically separated out of the nLab as a whole (if that's what you feel)? Could't you see a useful way that they exist as clusters of related entries inside the whole structure? This is at least currently the idea that we have: when clusters of entries corresponding to a given topic begin to accumulate, we currently try to equip them with shared "floating tables of contents" that give them a kind of sub-web existence within the nLab, without losing the close contact to the rest of the web. Such sub-clusters do currently include, among various others, the topics category theory -- http://ncatlab.org/nlab/show/category+theory+-+contents higher category theory -- http://ncatlab.org/nlab/show/higher+category+theory+-+contents We don't have as yet a reasonable such cluster on homotopy theory, though we have for instance the beginning of one on "abstract homotopy theory", i.e. model category theory -- http://ncatlab.org/nlab/show/model+category+theory+-+contents As I said, none of this is meant to be anything close to perfect, it's just to indicate seeds of possible structure within the nLab. So my gut reaction to your question would be: let's all improve and expand on the existing seeds of topic sub-clusters within the nLab, such as notably those on category theory! Let's not spread these efforts over too many different software installations. The more they are connected by one single web of links, the better. All this said, here is a little non-exhaustive bit of information on technicalities of setting up wiki-webs: what is straightforward is to branch off separate "instiki webs", as they are technically called, from the nLab. All these webs work just as the nLab itself does, they look and feel the same, and all run on the same software. Currently we follow the practice of offering personal webs to contributors who feel that they want to put material into the wiki, but in a more personal or private way than on the main nLab web. You can see the list of all sub-webs that current exist here: http://ncatlab.org/web_list http://ncatlab.org/nlabmeta/show/directory+of+personal+webs It is very easy to create such a sub-web titled "category theory" or "homotopy theory" or the like. If you do tell me that you would be likely to start adding material on, say, category theory, to such a sub-web "category theory", while not as likely to add the same material within the nLab web as such, I will be sure to take care that such a sub-web is created soon. But in that case, maybe we could jointly try to think about what would be necessary requirements, from your point of view, to instead keep all the material within the nLab itself. You see, in practice the main difference between two entries within the same web and two entries on different webs is, apart from maybe the color scheme of the entry header, just that hyperlinks within one web are easily and conveniently created -- one just types "[[keyword]]", while hyperlinks across webs are more tedious, one has to type "[[webname:keyword]]". The user who views these pages may not even be able to tell the difference, though! Finally, it is of course also possible to set up entirely separate installations of the wiki software. We are very lucky to have Andrew Stacey among us, who has the expertise and energy to handle such things. He is our software administrator, if you wish. Originally Jacques Distler helped us with these matters, but he is rather busy with lots of other things. If you think you would want an entire separate software installation of the wiki software on some server, then you should ask Andrew Stacey about this. If I remember well, he is already running a second wiki installation of the nLab kind for some other purpose. With best regards, Urs On Wed, Dec 23, 2009 at 8:10 PM, Joyal, André <joyal.andre@uqam.ca> wrote: > 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/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* Re: additions [not found] ` <4B3368C1.3000800@bath.ac.uk> @ 2009-12-24 16:25 ` Mike Stay 2009-12-26 0:03 ` additions Toby Bartels [not found] ` <7f854b310912240825s39f195b2x2db16cc8f3a5cde7@mail.gmail.com> 1 sibling, 1 reply; 47+ messages in thread From: Mike Stay @ 2009-12-24 16:25 UTC (permalink / raw) To: Carsten Führmann 2009/12/24 Carsten Führmann <c.fuhrmann@bath.ac.uk>: > Dear Mike, > >> Thanks, everyone for your replies! Many of you suggested the same >> approach as Steve, functional programming and monads. At Google, >> however, we use Java, C++ and Python (collectively "JCP") for programs >> that run on our servers and JavaScript for programs that run in our >> webpages. So there's not a lot of call for learning a functional >> programming language either. > > It might be worth noting that JavaScript is a functional language. > (It has a lambda operator ("function"), closures, and can pass > functions as parameters and return values.) However, because it has > eager evaluation, the whole monad business does not apply, at least not > in the way it applies to Haskell. > > In fact, JavaScript is probably the most widely used functional language > on the planet. I think you're confusing the existence of first-class functions with functional programming. Functional programming avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state. It's certainly possible to write functional programs in any of these languages, but it takes a lot of conscious effort--in fact, I'd say it's harder to write a functional program in JavaScript because of the myriad of strange ways state changes occur. I'm not sure what you mean by "the whole monad business does not apply". There are lots of monads, each doing something different. There are several monadic parsers I know of in JavaScript, for instance. Here's a monad for making JavaScript be lazily evaluated instead of eager: function e(x) { return function() { return x; } } function m(x, y) { return function () { return x()(y()); } } > But there are two strange phenomena: > > - Functional-programming experts keep on overlooking JavaScript (probably > because it is so ugly from a theorists point of view) Probably because it's not functional. > - Most professional JavaScript programmers fail to see the enormous > functional potential of JavaScript. > > It is a very strange situation: the whole world uses a functional language > and almost nobody is aware of it. > > Anyway, even though I am very category-prone, I must admit that category > theory might be a very tough sell for the JavaScript crowd :) Yes--most JavaScript development is done by amateurs who cut and paste someone else's code and try to tweak it to do what they want. They are not mathematicians. > Best, > Carsten > -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-24 16:25 ` additions Mike Stay @ 2009-12-26 0:03 ` Toby Bartels 0 siblings, 0 replies; 47+ messages in thread From: Toby Bartels @ 2009-12-26 0:03 UTC (permalink / raw) To: categories Mike Stay wrote in part: >Yes--most JavaScript development is done by amateurs who cut and paste >someone else's code and try to tweak it to do what they want. They >are not mathematicians. Actually, some of us *are* mathematicians. But we are not programmers. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* Re: additions [not found] ` <7f854b310912240825s39f195b2x2db16cc8f3a5cde7@mail.gmail.com> @ 2009-12-25 8:18 ` Carsten Führmann 0 siblings, 0 replies; 47+ messages in thread From: Carsten Führmann @ 2009-12-25 8:18 UTC (permalink / raw) To: Mike Stay Dear Mike, >> It might be worth noting that JavaScript is a functional language. >> (It has a lambda operator ("function"), closures, and can pass >> functions as parameters and return values.) However, because it has >> eager evaluation, the whole monad business does not apply, at least >> not >> in the way it applies to Haskell. >> >> In fact, JavaScript is probably the most widely used functional >> language >> on the planet. > > I think you're confusing the existence of first-class functions with > functional programming. Functional programming avoids state and > mutable data. It emphasizes the application of functions, in contrast > to the imperative programming style, which emphasizes changes in > state. > > It's certainly possible to write functional programs in any of these > languages, but it takes a lot of conscious effort--in fact, I'd say > it's harder to write a functional program in JavaScript because of the > myriad of strange ways state changes occur. I used the term "functional [programming] language" on purpose (as opposed to "functional programming style"), because of your statement >> So there's not a lot of call for learning a functional programming language either. which I feel might be wrong. I meant that JavaScript is a functional programming language in the same way in which ML/OCaml/F#, Lisp, and Scheme are (just uglier, slower, and running in a sandbox called "browser"). These are considered functional languages by many, and their categorical semantics has been studied. (Well, the semantics of idealized versions.) JavaScript is just riddled with some syntactic and semantic ugliness that makes it unattractive for formal study, but that doesn't make it un-functional in principle. > I'm not sure what you mean by "the whole monad business does not > apply". There are lots of monads, each doing something different. > There are several monadic parsers I know of in JavaScript, for > instance. Here's a monad for making JavaScript be lazily evaluated >instead of eager: > function e(x) { return function() { return x; } } > function m(x, y) { return function () { return x()(y()); } } Doesn't very fact that JavaScript allows you to write down the delaying monad give away its functional-language nature? And doesn't the existence of monadic parsers in JavaScript underpin that it might be beneficial for real-life programmers to learn some functional programming? By "monad business" I meant using monads to introduce side effects to lazy languages like Haskell, I could have been clearer there. Categorically, your monad is of a different kind, as I shall now sketch. (Just in case anyone is interested.) First, we need to observe that it is not straightforwardly a monad in the categorical sense. The reason is that the naturality square of the "unit" e does not commute. Considering that underlying functor T of the monad-in-spe sends a morphism f to T f = lambda g.lambda().f(g()) the naturality square would be e \circ f == (lambda g.lambda (). f(g())) \circ e which fails iff the f has a side effect (in the widest sense, which includes going into an infinite loop): that effect would get invoked on the equation's left side but not on the right. However, your code *does* represent a monad on the subcategory of (denotations of) effect-free (and terminating) programs. Categorically, (T, m, e) corresponds to an attempt to define a strong monad on an *unspecified* subcategory of the symmetric premonoidal category (not CCC!) that models your eager language (long story...). Fortunately, such a categories exist: e.g. the maximum one is given by all morphisms w.r.t. which your unit-in-spe is natural, but again that's a long story. At any rate, from a categorical and conceptional point of view the delaying "monad" on an eager language differs from Haskell-style monads. Happy holidays, Carsten http://www.cs.bath.ac.uk/~cf/ [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
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* Re: additions [not found] ` <4B347567.9070603@bath.ac.uk> @ 2009-12-29 23:17 ` Mike Stay 2009-12-30 21:00 ` additions Greg Meredith 0 siblings, 1 reply; 47+ messages in thread From: Mike Stay @ 2009-12-29 23:17 UTC (permalink / raw) To: categories 2009/12/25 Carsten Führmann <c.fuhrmann@bath.ac.uk>: > I used the term "functional [programming] language" on purpose (as > opposed to "functional programming style"), because of your statement > >> So there's not a lot of call for learning a functional programming >> language either. > > which I feel might be wrong. OK, I worded that badly. I think there are lots of reasons to learn functional programming, and once you're doing functional programming, then you need to learn category theory to do it well. Most of the code we've got is not functional, and the languages we work with make it hard to use higher-order functions and closures. So there's some resistance to overcome in convincing people to use functional style. > I meant that JavaScript is a functional > programming language in the same way in which ML/OCaml/F#, Lisp, and > Scheme are (just uglier, slower, and running in a sandbox called > "browser"). These are considered functional languages by many, and > their categorical semantics has been studied. (Well, the semantics of > idealized versions.) JavaScript is just riddled with some syntactic > and semantic ugliness that makes it unattractive for formal study, but > that doesn't make it un-functional in principle. The syntax of those languages certainly encourages functional composition over imperative programming, and they make it easy to construct closures and higher-order functions. However, none of them are purely functional like Haskell. I suppose I don't see the point of making the distinction between functional and imperative unless you really can't cause side-effects. >> I'm not sure what you mean by "the whole monad business does not >> apply". There are lots of monads, each doing something different. >> There are several monadic parsers I know of in JavaScript, for >> instance. Here's a monad for making JavaScript be lazily evaluated >>instead of eager: >> function e(x) { return function() { return x; } } >> function m(x, y) { return function () { return x()(y()); } } > > Doesn't very fact that JavaScript allows you to write down the > delaying monad give away its functional-language nature? And doesn't I could write down the delaying monad in Java, too, but it would be much larger. If the only feature you require of a functional language is that the syntax makes it *possible* to create closures, then nearly any programming language will fit the bill. If it has to be easy, then Java and C/C++ are not functional, while Scheme, ML, JavaScript and Perl are. On the other hand, if you say that it should be hard to use the imperative style in a functional language, then Scheme and ML are functional, while Perl and JavaScript are not. > the existence of monadic parsers in JavaScript underpin that it might > be beneficial for real-life programmers to learn some functional > programming? Sure. See above. > By "monad business" I meant using monads to introduce side effects to > lazy languages like Haskell, I could have been clearer there. > > Categorically, your monad is of a different kind, as I shall now > sketch. (Just in case anyone is interested.) Thanks, that _was_ interesting! I suppose what I'm really looking for is cool algorithms like the one described in Backhouse's paper "Fusion on Languages" (thanks, Neel!) where they either wouldn't have been discovered without category theory, or where category theory is the only decent way to understand the algorithm. > Happy holidays, > Carsten Thanks! To you, too. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
* Re: additions 2009-12-29 23:17 ` additions Mike Stay @ 2009-12-30 21:00 ` Greg Meredith 0 siblings, 0 replies; 47+ messages in thread From: Greg Meredith @ 2009-12-30 21:00 UTC (permalink / raw) To: Mike Stay Dear Mike, I suppose what I'm really looking for is cool algorithms like the one described in Backhouse's paper "Fusion on Languages" (thanks, Neel!) where they either wouldn't have been discovered without category theory, or where category theory is the only decent way to understand the algorithm. While not quite what you are looking for Rydeheard and Burstall<http://www.cs.manchester.ac.uk/~david/categories/book/book.pdf>might provide a good jumping off point. Best wishes, --greg On Tue, Dec 29, 2009 at 3:17 PM, Mike Stay <metaweta@gmail.com> wrote: > 2009/12/25 Carsten Führmann <c.fuhrmann@bath.ac.uk>: > > I used the term "functional [programming] language" on purpose (as > > opposed to "functional programming style"), because of your statement > > > >> So there's not a lot of call for learning a functional programming > >> language either. > > > > which I feel might be wrong. > > OK, I worded that badly. I think there are lots of reasons to learn > functional programming, and once you're doing functional programming, > then you need to learn category theory to do it well. > > Most of the code we've got is not functional, and the languages we > work with make it hard to use higher-order functions and closures. So > there's some resistance to overcome in convincing people to use > functional style. > ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 47+ messages in thread
end of thread, other threads:[~2009-12-30 21:00 UTC | newest] Thread overview: 47+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2009-12-17 23:30 A well kept secret? peasthope 2009-12-18 4:09 ` John Baez 2009-12-18 22:25 ` Ellis D. Cooper 2009-12-19 17:45 ` Ronnie Brown 2009-12-19 22:16 ` John Baez 2009-12-20 22:52 ` Greg Meredith 2009-12-21 15:46 ` Zinovy Diskin 2009-12-22 16:59 ` zoran skoda 2009-12-23 1:53 ` Tom Leinster 2009-12-23 14:15 ` Colin McLarty 2009-12-23 19:10 ` CatLab Joyal, André 2009-12-20 21:50 ` A well kept secret? jim stasheff [not found] ` <d4da910b0912220859q3858b68am4e58749f21ce839d@mail.gmail.com> 2009-12-23 4:31 ` Zinovy Diskin 2009-12-23 14:35 ` Ronnie Brown [not found] ` <4B322ACA.50202@btinternet.com> 2009-12-25 20:06 ` Zinovy Diskin 2009-12-20 17:50 ` Joyal, André [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E2159B6AA@CAHIER.gst.uqam.ca> 2009-12-21 8:43 ` additions Joyal, André 2009-12-21 14:16 ` additions Bob Coecke 2009-12-22 2:24 ` additions Joyal, André 2009-12-23 20:51 ` additions Thorsten Altenkirch 2009-12-24 23:55 ` additions Dusko Pavlovic 2009-12-26 2:14 ` additions Peter Selinger [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E370F5626@CAHIER.gst.uqam.ca> [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E370F5636@CAHIER.gst.uqam.ca> [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E370F5638@CAHIER.gst.uqam.ca> 2009-12-28 17:54 ` quantum information and foundation Joyal, André 2009-12-29 12:13 ` Urs Schreiber 2009-12-29 15:55 ` zoran skoda 2009-12-22 0:39 ` additions Mike Stay 2009-12-23 11:19 ` additions Steve Vickers 2009-12-23 18:06 ` additions Mike Stay 2009-12-24 13:12 ` additions Carsten Führmann 2009-12-24 19:23 ` additions Dusko Pavlovic 2009-12-23 19:06 ` additions Thorsten Altenkirch [not found] ` <Pine.LNX.4.64.0912211413340.15997@msr03.math.mcgill.ca> [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E2159B6B3@CAHIER.gst.uqam.ca> 2009-12-23 17:08 ` RE : categories: additions Joyal, André 2009-12-21 19:20 ` additions Michael Barr 2009-12-27 23:14 ` quantum information and foundation Dusko Pavlovic [not found] ` <Pine.GSO.4.64.0912272037140.28761@merc3.comlab> 2009-12-28 16:38 ` Bob Coecke [not found] ` <Pine.GSO.4.64.0912281630040.29390@merc4.comlab> 2009-12-28 18:17 ` Bob Coecke 2009-12-18 10:48 ` A well kept secret? KCHM 2009-12-19 20:55 ` Vaughan Pratt 2009-12-22 12:21 ` additions Mark Weber 2009-12-23 0:05 ` additions Scott Morrison 2009-12-23 14:13 ` additions Mark Weber [not found] ` <B3C24EA955FF0C4EA14658997CD3E25E2159B6B8@CAHIER.gst.uqam.ca> 2009-12-23 21:04 ` CatLab Urs Schreiber [not found] ` <4B3368C1.3000800@bath.ac.uk> 2009-12-24 16:25 ` additions Mike Stay 2009-12-26 0:03 ` additions Toby Bartels [not found] ` <7f854b310912240825s39f195b2x2db16cc8f3a5cde7@mail.gmail.com> 2009-12-25 8:18 ` additions Carsten Führmann [not found] ` <4B347567.9070603@bath.ac.uk> 2009-12-29 23:17 ` additions Mike Stay 2009-12-30 21:00 ` additions Greg Meredith
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