Re: A well kept secret?

Re: A well kept secret?

[peasthope]

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.



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Re: A well kept secret?

[John Baez]

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

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A well kept secret?

[Ellis D. Cooper]

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



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Re: A well kept secret?

[Ronnie Brown]

 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
>

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Re: A well kept secret?

[John Baez]

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


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Re: A well kept secret?

[Greg Meredith]

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
>

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Re: A well kept secret?

[Zinovy Diskin]

>>  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.


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Re: A well kept secret?

[zoran skoda]

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


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Re: A well kept secret?

[Tom Leinster]

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


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Re: A well kept secret?

[Colin McLarty]

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


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CatLab

[Joyal, André]

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


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Re: A well kept secret?

[jim stasheff]

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



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Re: A well kept secret?

[Zinovy Diskin]

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.


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Re: A well kept secret?

[Ronnie Brown]

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.
 >
...

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Re: A well kept secret?

[Zinovy Diskin]

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.



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Re: A well kept secret?

[Joyal, André]

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



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additions

[Joyal, André]

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


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Re: additions

[Bob Coecke]

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.




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Re: additions

[Joyal, André]

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

 


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Re: additions

[Thorsten Altenkirch]

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



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Re: additions

[Dusko Pavlovic]

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



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Re: additions

[Peter Selinger]

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


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quantum information and foundation

[Joyal, André]

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






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Re: quantum information and foundation

[Urs Schreiber]

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


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Re: quantum information and foundation

[zoran skoda]

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


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Re: additions

[Mike Stay]

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


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Re: additions

[Steve Vickers]

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


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Re: additions

[Mike Stay]

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


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Re: additions

[Carsten Führmann]

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


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Re: additions

[Dusko Pavlovic]

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


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Re: additions

[Thorsten Altenkirch]

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.
>

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RE : categories: additions

[Joyal, André]

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


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Re: additions

[Michael Barr]

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
>

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Re: quantum information and foundation

[Dusko Pavlovic]

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.



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Re: quantum information and foundation

[Bob Coecke]

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.



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Re: quantum information and foundation

[Bob Coecke]

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.



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Re: A well kept secret?

[KCHM]

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/
=====================================


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Re: A well kept secret?

[Vaughan Pratt]

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


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Re: additions

[Mark Weber]

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


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Re: additions

[Scott Morrison]

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
>

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Re: additions

[Mark Weber]

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

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Re: CatLab

[Urs Schreiber]

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


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Re: additions

[Mike Stay]

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


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Re: additions

[Toby Bartels]

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


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Re: additions

[Carsten Führmann]

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/


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Re: additions

[Mike Stay]

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


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Re: additions

[Greg Meredith]

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.
>

...


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RE: A well kept secret?

[Larry Harper]

Dear All,



As one of MacLane's working mathematicians who follows the catlist, I would
like to add some thoughts about perceptions of category theory (CT). I
earned a Bachelor's in Physics at Berkeley in 1960 and went to grad school
in Mathematics at the University of Oregon the following fall. In Frank
Anderson's graduate algebra course I was first exposed to CT and hated it.
My background and ability in algebra were marginal anyway and to have my
first definition of tensor product be in terms of commuting diagrams was
disastrous. Fortunately, I got a summer job at the Jet Propulsion Lab and
one of my coworkers, Gus Solomon, gave me the classical constructive
definition of tensor products. Sammy Eilenberg came by Eugene and gave a
lecture on CT which did nothing to change my opinion of it. When I heard of
Serge Langs's characterization of CT as "abstract nonsense" it reinforced
what I already thought (See however,

                           http://en.wikipedia.org/wiki/Abstract_nonsense

which does not mention Serge Lang in the body of the article).



My fascination with, and love of, CT was ignited in 1966 when I was a
postdoc with Gian-Carlo Rota at the Rockefeller University. Ron Graham and I
were collaborating on a conjecture of Rota; that the lattice of partitions
of an n-set has the same property that Erwin Sperner had demonstrated for
the lattice of subsets of an n-set (the largest antichain is the largest
rank). We had some partial results on Rota's conjecture and in the course of
writing them up I realized that they implicitly involved a notion of
morphism for the Ford-Fulkerson maxflow problem. I thought this was a
promising insight and incorporated it with the other material. I gave the
finished paper to Ron for approval but when it came back to me it had been
rewritten and all mention of flowmorphisms eliminated. I took this as a
challenge to show that flowmorphisms could lead to further insight into
Sperner problems. The result, which I called The Product Theorem, was
natural conditions under which the product of Sperner posets must also be
Sperner. The key was to realize that the concept of normalized flow
introduced in Graham-Harper (which is stronger than the Sperner property) is
equivalent to a flowmorphism from the given poset to a chain (total order).
If two posets, P,Q, have normalized flows then product, being a bifunctor,
will induce a flowmorphism from their product to the product of their
chains. All I had to do then was to find natural conditions under which the
product of (weighted) chains has a normalized flow. The Product Theorem
generalized known theorems of  Sperner, deBruijn et al & Erdos. and has
since been applied to prove at least 3 new conjectures.



Having such a success with flowmorphisms motivated me to dig more deeply.  I
showed that flowmorphisms have pushouts and was asking about pullbacks
(though I did not use those terms because I did not know them) when (about
1971) my office mate at JPL, Dennis Johnson, introduced me to Saunders
MacLane's classic, Categories for the Working Mathematician. This was, of
course, a revelation and changed my mathematical universe.



I joined the faculty of the University of California at Riverside in the
fall of 1970. In alternate years I taught a 2-quarter graduate course on
combinatorics. Over the next 36 years it evolved into two independent
courses having a common thesis. One was on maximum flows in networks and
Sperner problems, the other on minimum paths in networks and combinatorial
isoperimetric problems. I believe it is no accident that maximum flow and
minimum path (aka dynamic programming) problems are central to algorithmic
analysis and that they both have nice notions of morphism. The common thesis
of the two courses is that morphisms can be effective in solving hard
problems. In 2004 the notes for one course were published under the title
Global Methods for Combinatorial Isoperimetric Problems. If I live long
enough its companion volume on Sperner problems will appear. It will show
how several steps in the eventual resolution of the Rota Conjecture were
illuminated by  flowmorphisms.



It has been a personal goal, since the early 1970s, to demonstrate the
existence and usefulness of morphisms for combinatorial problems. This often
comes down to questions of

            1) How to use symmetry to systematically simplify the problem?

            2) How to pass to a continuous limit?

I like to call this endeavor the relativity theory of combinatorics. Albert
Einstein asked "What are the symmetries of the universe and what do they
tell us about it?" To show the depth and subtlety of such questions,
consider that two of the leading mathematicians of his age, Henri Poincare
and Hendrick Lorentz, studied Lorentz transformations five years before
Einstein. However they both missed the epoch-making relation E = mc^2 that
is easily deduced from Lorentz transformations. In studying a problem
through its morphisms we need all the help we can get. CT is invaluable as
the road map to morphism country!



Regards,



Larry Harper

Professor Emeritus of Mathematics

University of California, Riverside



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RE: A well kept secret?

[Colin McLarty]

2009/12/19 Larry Harper <harper@math.ucr.edu>:

mentions

> When I heard of
> Serge Langs's characterization of CT as "abstract nonsense" it reinforced
> what I already thought (See however,
>
>                           http://en.wikipedia.org/wiki/Abstract_nonsense
>
> which does not mention Serge Lang in the body of the article).

Lang's Algebra uses "abstract nonsense" to describe his favorite kind
of one line proof that some given construction (such as tensor
product)  defines its results uniquely up to isomorphism.  It is
discussed on the discussion page of the wikipedia article.

best, Colin


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RE: A well kept secret?

[jim stasheff]

This seems like an excellent advertisement of `thinking categorically'
and not necessarily writing in that dialect.

jim
Larry Harper wrote:
> Dear All,
>
>
>
> As one of MacLane's working mathematicians who follows the catlist, I would
> like to add some thoughts about perceptions of category theory (CT). I
> earned a Bachelor's in Physics at Berkeley in 1960 and went to grad school
> in Mathematics at the University of Oregon the following fall. In Frank
> Anderson's graduate algebra course I was first exposed to CT and hated it.
> My background and ability in algebra were marginal anyway and to have my
> first definition of tensor product be in terms of commuting diagrams was
> disastrous. Fortunately, I got a summer job at the Jet Propulsion Lab and
> one of my coworkers, Gus Solomon, gave me the classical constructive
> definition of tensor products. Sammy Eilenberg came by Eugene and gave a
> lecture on CT which did nothing to change my opinion of it. When I heard of
> Serge Langs's characterization of CT as "abstract nonsense" it reinforced
> what I already thought (See however,
>
>                            http://en.wikipedia.org/wiki/Abstract_nonsense
>
> which does not mention Serge Lang in the body of the article).
>
>
>
...

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A well kept secret?

[Ronnie Brown]

In reply to André :


What seems reasonable to do is analysis, namely what is behind the 
success of category theory and how is this success  related to the 
progress of mathematics.
Which implies asking questions of mathematics, some of which have been 
aired in this discussion list. In this way, it should be possible to 
avoid seeming partisan, but to ask serious questions, which should help 
to steer directions, or suggest new ones. Of course lots of great maths 
does not arise in this way, but by following one's nose, but that does 
not mean that such analysis of direction is unhelpful.

I know some argue that this excursion into what might be called the 
theory of knowledge, or into methodology,  seems unnecessary to some. In 
reply I sometimes point to remarks of Einstein on my web site
www.bangor.ac.uk/r.brown/einst.html
or more mundanely retort that normal activities normally require some 
meta discussion: if you want to go on a holiday, you do some planning, 
you don't just rush to the station and buy some tickets. I develop this 
theme in relation to the teaching of mathematics in an article
What should be the output of mathematical education?
on my popularisation and teaching page.

I gave a talk to school children on `How mathematics gets into knots' in 
the 1980s, and a teacher came up to me afterwards and said: `That is the 
first time in my mathematical career that anyone has used the word 
`analogy' in relation to mathematics.' Yet abstraction is about analogy, 
and very powerful it is too. This was part of the motivation behind the 
article
146. (with T. Porter) `Category Theory: an abstract setting for analogy 
and comparison', In: What is Category Theory? Advanced Studies in 
Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274. pdf

There is also interest in the question of how category theory comes to 
be successful, and more successful than, say,  the theory of monoids. 
This seems connected with the underlying geometric structure being a 
directed graph, i.e. allowing a `geography of interaction'. A category 
is also a partial algebraic structure, with domain of definition of the 
operation defined by a geometric condition. Is this enough to explain 
the success?

It is worth noting that the article
Atiyah, Michael, Mathematics in the 20th century, Bull. London Math. 
Soc., {34},  {2002}, 1--15,
suggests that important trends in the 20th century were:
                              higher dimensions, commutative to non 
commutative, local-to-global, and the unification of mathematics,
but does not include the words `category' or `groupoid', let alone 
`higher dimensional algebra'!

This kind of analysis needs to be presented to other scientists, and to 
the public, not only to mathematicians. There is a hunger for knowing 
what mathematics is really up to, in common language as far as possible, 
what new concepts, ideas, etc., and not just `we have solved Fermat's 
last theorem'.

If your analysis of what category theory should do suggests some gaps, 
then that is an opportunity for work!

Good luck

Ronnie Brown


Joyal wrote:

Category theory is a powerful mathematical language.
It is extremely good for organising, unifying and suggesting new directions of research.
It is probably the most important mathematical developpement of the 20th century.

But we cant say that publically.

André Joyal

  



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Re: A well kept secret?

[Andrew Stacey]

This discussion has been very interesting.  I have a couple of comments and
a request, but first a little background.  I've only recently truly
encountered category theory - I describe myself as a differential topologist
and as yet see no reason to change that description, but I've increasingly
needed to use at least the language of category theory to express some of the
things that I come across in algebraic and differential topology and this has
led to me learning some category theory at last.

However, I sometimes feel as though I've stumbled into a party by mistake and
can't find the way out.  I'm quite enjoying being at the party, I ought to
say, but every now and then I sit down in a corner and wonder how I got in,
and also suspect that I missed the Big Announcement at the beginning that said
what the party was for.

All this discussion about a "well kept secret" has gone a bit over my head.
I'm not sure what the secret is!  My forays into the categorical landscape
have been two-fold: understanding operations in cohomology theories and
understanding smooth spaces.  The first, paradoxically, relates to trying to
un-categorify something ("decategorify" now has a mathematical meaning and
I don't intend that); namely, the previous description of what we wanted to
understand was extremely categorical and we wanted a much more "hands on"
description, but that actually just led us from one categorical description to
another (our own journey was quite tortuous, I should say).  The second foray
wouldn't have happened if those I'd been talking to hadn't already been
speaking in categories - I had to learn the language just to join the
conversation.

So when you all talk of a "well kept secret" and something that "went wrong in
the 60s" (didn't everything?), please remember that some of us weren't even
born in the 60s, let alone thinking about mathematics, so haven't a clue
what's going on.  And, as I've tried to say above, I'm an outsider but one
with a favourable view of category theory so if it's hard for me to figure out
what the fuss is about, I'm not surprised that it's hard for anyone further
out.

Let me make these remarks a little more concrete with a request (or
a challenge if you prefer).  In my department, the colloquium is called
"Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; I've
been writing too many scripts lately!).  I'm giving this talk in January.  My
original plan was to say something nice and differential, with lots of fun
pictures of manifolds deforming or knots unknotting, or something like that.
However, the discussion here has set me to thinking about saying something
instead about category theory.  It is a pearl of mathematics, it does have
a certain beauty, there's certainly a lot that can be said, even to a fairly
applied audience as we tend to have here (it is the Norwegian university of
Science and Technology, after all), even without talking about programming
(about which I know nothing).

But for such a talk, I need a story.  I don't mean a historical one (I'm not
much of a mathematical historian anyway), I mean a mathematical one.  I want
some simple problem that category theory solves in an elegant fashion.  It
would be nice if there was one that used category theory in a surprising way;
beyond the idea that categories are places in which things happen (so perhaps
about small categories rather than large ones).

I'm not trying to get anyone to write my talk for me!  It's just that as
someone who only recently engaged with category theory then I'm aware
- painfully aware - that I often miss the point.  But to counter that, then as
  someone who only recently engaged with category theory then I can still
remember fairly vividly why I like it and what convinced me that it was worth
thinking about (and learning about), which will hopefully give the talk
a little more omph.

Thanks in advance for your suggestions,

Andrew Stacey

PS I just remembered something else I was going to mention.  Someone else
mentioned MathOverflow.  Well, there was a question about what was missing
from undergraduate mathematics.  I said "category theory".  It currently lies
5th in the list (out of 28, my other suggestion "how to write with chalk so it
doesn't squeak" is 12th).  More interesting than it's placing is the vast
number of comments that followed, mostly saying that too much "abstract
nonsense" would be off-putting to students.  You can read it at:

http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn



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Re: A well kept secret?

[John Baez]

Andrew Stacey wrote:

> All this discussion about a "well kept secret" has gone
> a bit over my head.  I'm not sure what the secret is!

We could tell you...

... but then it wouldn't be a secret, now, would it?

Seriously, I think the so-called "secret" is the power and glory of
category theory.  And I think some of the older category theorists on
this mailing list have a different attitude than youngsters like you
and me.  They fought to convince the world that category theory was
worthwhile. Some feel they lost that fight.  We came along later and
are a bit puzzled by that attitude: if you look around at the
landscape of mathematics today, categories are everywhere!  From
Grothendieck to Voevodsky to Lurie, etc., much of the most exciting
mathematics of our era would be inconceivable without categories.

I don't think I'll try to tell you the old war stories: others are
better qualified.  But I hope the veterans of those wars take heed of
your comments and realize many young mathematicians naturally find
categories interesting,  exciting, and/or useful.  Certainly there is
much about categories that these youngsters don't understand.  But
they can learn it if you explain it.`

Best,
jb


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Re: A well kept secret?

[Ross Street]

On 15/12/2009, at 5:41 AM, Andrew Stacey wrote:

>  In my department, the colloquium is called
> "Mathematical Pearls"
> I'm giving this talk in January.
>
> But for such a talk, I need a story.

Dear Andrew

Back in the early 90s Todd Trimble gave a beautiful colloquium talk to
our Mathematics Department at Macquarie. It was based on a question in
a book by Halmos which involved finding some group (topological I
think) doing something or other. It was not a categorical problem as
such.

Todd spoke about groups in a category with finite products. The only
categorical theorem he needed was that finite product preserving
functors take groups to groups. I believe he took the definition of
category as known but defined functor, product and internal group.

My vague memory is that he found a group solving the analogous problem
in some fairly combinatorial (presheaf?) category, then found a
product preserving functor to topological spaces to obtain the desired
group.

I hope Todd is reading this and I have jogged his memory enough to
write in more detail. It takes work and ingenuity to design such pearls.

Best wishes,
Ross

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