Re: A well kept secret

Re: A well kept secret

[Andree Ehresmann]

Here are some comments on the discussion about category theory:

1. I testify that categories can be fruitfully introduced to  
undergraduates. In the late sixties, I followed 3 cohorts of students  
from their entrance at the university (just after the Baccalaureat) up  
to their graduation. I introduced category theory in the mid of their  
first year and thoroughly used it later on in my courses on Algebra,  
Topology, Differential Calculus and functional spaces, and Algebraic  
Topology. (All these courses have been multigraphed in Amiens.) Since  
the university in Amiens was only beginning to develop, there were not  
many students, but most of them seemed to enjoy categories and about  
10% of those who completed the cycle went on to do research (generally  
using categories) and obtained university positions.
However I had to stop my experiment because several of my colleagues  
did not appreciate categories:(t was a very bad time for them in  
France in the early 70's.

2. Applications of categories begin to be welcomed in the most varied  
scientific domains. An example is our general model "Memory Evolutive  
Systems" for 'natural' complex autonomous systems, such as biological  
or social systems; when we first introduced it in the 90's, people  
were somewhat skeptical, but now more people accept it, in particular  
cognitive scientists are taking a real interest in its application to  
cognitive systems (our model MENS).

3. Thus, though one of the older living "veterans" of the 60's "war",  
I am not pessimistic (as John Baez seems to imply). I think my  
generation has lost a battle, partially because of too much  
precipitation and not always enough diplomacy, but I feel that the  
youngsters are winning the war and I'ld try to help them as much as  
possible for I have kept all my enthusiasm.

So best wishes to the categories and to all their friends for 2010
Andree




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Re: A well kept secret

[Andrew Stacey]

I would just like to say thank you to all those who responded to my request
for some ideas for a story to tell on category theory.  I intended to reply to
each one but got swamped by grading exams and didn't get round to it before
the holiday (I still intend to reply but it will have to wait a week now) so
this is a "holding email" just to say that I'm grateful for the ideas and to
apologise for not responding (yet) individually.

God jul og godt nyttår!

Andrew


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Re: A well kept secret

[Reinhard Boerger]

Hello,


I still like to add some remarks. Category theory is one part of
mathematics, and it should be treated not better, but not worse than others.
It looks more important to me that categorical thinking becomes popular in
other areas in mathematics. Some years ago, a functional analyst needed half
an our to prove that homeomorphic Banach spaces have homemorphic duals, a
simple consequence of the fact that all functors preserve isomorphisms. 

Another example from my own experience: People, who worked about
orthomodular lattices noticed that they have no tensor product. So they
tried to weaken the notions and ended up with effect algebras, but
unfortunately they did not admit a tensor product either. Su people looked
for other notions. But they had already shown that a tensor product of
effect algebras exists if one admits 0=1; i.e. the tensor product my
collapse. But because they did not admit this, they had to formulate their
result more complicated.

Later I saw that tensor products of orthomodular posets exist if one admits
0=1; the easy proof uses the Adjoint Functor Theorem and does not give much
insight into the structure. It also seems to work for orthomodular lattices.

My preference for orthomodular posets rather than lattices is also inspired
by categorical thinking. The idempotents of an arbitrary ring with 1 form an
orthomodular poset, and this construction yields a functor. This is the
non-commutative analogue to the Boolean algebra of idempotents of an
arbitrary ring. But most people were inspired by quantum dynamics and were
looking for an abstraction for the set of projections of a Hilbert space.
Here joins and meets exist (somehow accidentially) because projections
correspond to closed subspaces. But they are not continuous and have no
physical meaning in general. I think it is often better to look for
functorial notions than to use ad-hoc-abtractions.


Greetings
Reinhard




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Re: A well kept secret

[F William Lawvere]

Dear  Andrew
The conceptual definition of lamba ring given by Andre',
(for which a presentation by useful but complicated identities 
is  theorem rather than a definition) solves an important one of
a  generic class of problems that I proposed in the article that is 
bundled with my thesis in TAC Reprints.

A simple pedagogically convincing example is given by the 
following dialogue :
 I have an example of a general category C and a functor U from it to finite sets;
moreover I have a particular object X in C : what information about X
can I find using U ? Well, you could count the points of U(X).Yes but
that is by no means all. The functor U has a group G of all natural
automorphisms, and so U can be lifted across the category of G-sets,
 thus  that number is actually a sum of more refined invariants
indexed by the subgroups of G.

The generic problem (for the doctrine of algebraic theories rather than
for the subdoctrine of permutation representations) considers a 
specific assignment of an algebraic theory to any morphism of algebraic 
theories (from Andre's example it should be clear which assignment) and asks 
for specific calculation (eg a presentation in terms of given presentations,
or indeed any information). The construction involves the natural structure
of a functor that is not representable in general but hope comes from
the  fact that this functor preserves filtered colimits and reflexive coequalizers
and that some examples are representable or otherwise computable.

Bill



On Tue 12/15/09  3:14 PM , Joyal, André joyal.andre@uqam.ca sent: 
> Dear Andrew,
> 
> You wrote
> 
> >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).
> A colloquium is a good place for expressing wild ideas.
> But they must be related to something everyone can understand and
> touch.I suggest you talk about "The field with one element" if you
> think the subject can fit your audience.
> 
> http://en.wikipedia.org/wiki/Field_with_one_element
> Many things in this subject are very speculative
> but there are also a few concrete developpements. 
> One is the algebraic geometry "under SpecZ" of Toen and
> Vaquié.Another due to Borger is using lambda-rings.
> What is a lambda-ring?
> In their book "Riemann-Roch-Algebra" Fulton and Lang define a
> lambda-ringto be a pre-lambda-ring satisfying two complicated identities [(1.4) and
> (1.5)][Beware that F&L are using an old terminology: they call a lambda-ring
> a "special lambda-ring"and they call a pre-lambda-ring a "lambda-ring"]
> The notion of lambda-ring (ie of "special lambda-ring" in the
> terminology of F&L)can be defined in a natural way if we use category theory. 
> Let Z[]:CMon ---> CRing be the functor which associates to a commutative
> monoid M the ring Z[M] freely generated by M (it is the left adjoint to the forgetful
> functor in theopposite direction). If we compose the functot Z[] with the forgetful
> functor U:CRing --->Setwe obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings
> can be defined to be the theory of natural operations on the functor V.
> The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism
> Z[M]--->1+tZ[M][[t]] which takes an element x\in M to the power series 1+tx.
> 
> 
> Best,
> André
> 
> 



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categorical "varieties of algebras" (fwd)

[Michael Barr]

I am forwarding this to the categories list, where I am sure there will be
many answers, but I have never myself delved into these interesting
questions.  Please be sure to copy your answers to him (although he should
probably subscribe to the list).

--M

---------- Forwarded message ----------
Date: Sun, 13 Dec 2009 23:15:11 +0200
From: Alexandru Chirvasitu <chirvasitua@gmail.com>
To: barr@math.mcgill.ca
Subject: categorical "varieties of algebras"

Dear Prof. Barr,


My name is Alexandru Chirvasitu, and I am a first-year mathematics graduate
student at UC Berkeley. I apologize for bothering you wih this, especially
since you don't know me, but I was kind of at a loss: I don't really know
any people working in the areas I am interested in personally, so I thought
I'd give this a go :). I'm quite sure you'll be able to clear this out
straight away.

Before coming to Berkeley, I was interested in applying category-theoretic
methods to study coalgebras, Hopf algebras, and other such creatures:

http://arxiv.org/abs/0907.2881

It became clear later on that to get some further insight into the universal
constructions useful for these problems (Hopf envelopes of co (or bi)
algebras, free Hopf algebra with bijective antipode on a Hopf algebra,
etc.), it would be useful to apply some Tannaka reconstruction techniques
and move the free constructions "up the categorical ladder": free monoidal
category on a category, (left) rigid envelope of a monoidal category, etc.
Unfortunately, I couldn't find any results stating clearly (clearly for
someone who is perhaps not *too* familiar with the higher categorical
machinery) that such free categories always exist. Of course, the few
constructions I needed can easily be done by hand, but what I had in mind
was some kind of higher categorical analogue of the fact that the forgetful
functor from a variety of algebras to another variety of algebras with
"fewer operations" has a left adjoint.

There's also an issue of how strict things should be. For what I was doing,
the following setting is typical: consider the category whose objects are
(not necessarily strict, but that's not very important here) monoidal
categories with a specified left dual and specified (co)evaluation maps for
every object, and whose morphisms are the functors which preserve all of
this structure *strictly*. Then I wanted to conclude that the forgetful
functor from this to *Cat* has a left adjoint, which should be easy enough.

To state my question properly, I'm thinking about a category whose objects
are categories *C* endowed with certain "operations", consisting of functors
from *C^n x (C opposite)^m* to *C* (example: the specified left dual in the
above example is a contravariant functor), appropriately natural
transformations between such functors (example: the evaluation maps in a
rigid tensor category as above form, together, a dinatural transformation),
and equations involving these natural transformations; the morphisms are
functors which preserve all the structure *strictly*. Consider the forgetful
functor from this to a similar category, but with "fewer operations" (this
could easily be made precise). Then, is there a  result stating that such a
functor always has a left adjoint?

I expect it should be easy enough to employ a form of the Adjoint Functor
Theorem (working with universes say, to have everything set-theoretically
sound) to prove something like this, and I was thinking about writing it up
for further reference. My problem was that I can't seem to be able to tell
exactly what is well-known and has been written up, what is folklore and
trivial, etc. Also, even though a statement as outlined above and to which I
could refer would be completely satisfactory, I realize that after
destrictification things become much more interesting, and you probably get
some neat bicategorical results.

I must once again apologize for intruding on your time like this, for the
potential silliness of a newbie's question, and for the ramble factor and
length of this message :). I hope you do get to reply.


Thank you,


Alexandru


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Re: A well kept secret

[Dusko Pavlovic]

i am wondering why is the public image of category theory so important for
us.

i mean, if category theory is a powerful and useful tool, as it is, then
it should be able to take care for itself. bread does not need
advertising.

i have been with categories for many years. i think in categories, and i
used them in each and every one of my research projects, in every piece of
software that i designed, in every paper that i wrote. but sometimes it is
easier to get to the point without spelling out all definitions in full
generality. and without tackling the opprobium.

e.g., i worked on networks, and have papers about trust networks, and
reputation networks, and recommender systems. a network is a weighted
graph, and it composes to some extent, because a friend of a friend is
almost like a friend, but a friend of a friend of a friend etc, six hops
removed --- is probably not a friend. but you can do with networks a lot
of what you can do with categories: make arrow networks, adjoin
colimits... anyway, i defined all that, and mentioned categories, but did
not really advertise them. was that a mistake? maybe it wasn't such a good
work, and i would have done a disservice to category theory by advertising
it in a bad paper.

i have been using categories in my little crypto modules, and in serious
reduction proofs, for more than 5 years (cryptography is a theory of
functions after all!), but i first gave a crypto talk using categories
last week. and this was not a talk to hard-core cryptographers.

i think category theory sometimes suffers from our advertising. even in a
good paper, advertising is advertising. there are places for that, and
there are places where it is better not to do.

i do understand that we need to take care for the public image of our
work. funding depends on that, hiring depends on that. but maybe we should
clearly state that this is a matter of advocacy and of influence, and not
mix it up with Promoting the Truth. i somehow think that the truth can
take care for itself.

but as always, maybe i am wrong.

-- dusko



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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

[Joyal, André]

Dear Andrew,

You wrote

>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).

A colloquium is a good place for expressing wild ideas.
But they must be related to something everyone can understand and touch.
I suggest you talk about "The field with one element" if you think 
the subject can fit your audience.

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

Many things in this subject are very speculative
but there are also a few concrete developpements. 
One is the algebraic geometry "under SpecZ" of Toen and Vaquié.
Another due to Borger is using lambda-rings.
What is a lambda-ring?
In their book "Riemann-Roch-Algebra" Fulton and Lang define a lambda-ring
to be a pre-lambda-ring satisfying two complicated identities [(1.4) and (1.5)]
[Beware that F&L are using an old terminology: they call a lambda-ring a "special lambda-ring"
and they call a pre-lambda-ring a "lambda-ring"]
The notion of lambda-ring (ie of "special lambda-ring" in the terminology of F&L)
can be defined in a natural way if we use category theory. 
Let Z[]:CMon ---> CRing be the functor which associates to a commutative monoid M the 
ring Z[M] freely generated by M (it is the left adjoint to the forgetful functor in the
opposite direction). If we compose the functot Z[] with the forgetful functor U:CRing --->Set
we obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings 
can be defined to be the theory of natural operations on the functor V.
The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism Z[M]--->1+tZ[M][[t]] 
which takes an element x\in M to the power series 1+tx.


Best,
André



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Dangerous knowledge

[Joyal, André]

Dear all,

I wonder if you have seen the BBC "documentary" called "Dangerous knowledge"? 
It is divided in ten parts:

1) 
http://www.youtube.com/watch?v=Cw-zNRNcF90&feature=related

2)
http://www.youtube.com/watch?v=wpWXT9yMBnw&feature=related

3)
http://www.youtube.com/watch?v=1AAvWb5wYNk&feature=related

4)
http://www.youtube.com/watch?v=qUL-x8Gm1h4&feature=related

5)
http://www.youtube.com/watch?v=So9RAbBy1ps&feature=related

6)
http://www.youtube.com/watch?v=fqKQ0-T3swY&feature=related

7)
http://www.youtube.com/watch?v=oldUAw2Aux0&feature=related

8)
http://www.youtube.com/watch?v=0ZcErXdR_eQ&feature=related

9)
http://www.youtube.com/watch?v=BkezCyb7Lkw&feature=related

10)
http://www.youtube.com/watch?v=_8dczB1rY-Q&feature=related

What do you think?

André



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Re: Dangerous knowledge

[John Baez]

The BBC wrote:

 In this one-off documentary, David Malone looks at four brilliant
>> mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan
>> Turing - whose genius has profoundly affected us, but which tragically drove
>> them insane and eventually led to them all committing suicide.
>>
>
Jim Stasheff wrote:

At least the Turing implication is very misleading - see below.
>

It's also not true that George Cantor committed suicide!

And I would not call Ludwig Boltzmann a mathematician.  I'd call him a
physicist.

But the documentary seems a bit more accurate than this summary.
And it could be good to have documentaries that sensationalize mathematics
and make it seem "edgy" and "dangerous".  We oldsters can tut-tut about the
inaccuracies and lack of serious content, but as a kid I would have enjoyed
it - and if it makes one youngster pursue a career in mathematics instead of
crime, that may justify its existence.

Best,
jb


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Re: Dangerous knowledge

[Vaughan Pratt]

Aren't artistic types alleged to be more prone to mental illness than
say sales clerks, real estate agents, auto mechanics, farmers,
lumberjacks, etc?   More generally, creative people?  On that basis
would one expect a higher prevalence of mental disorders among
theoretical physicists (Boltzmann) than experimental ones (Rutherford),
or among mathematicians than engineers, or among top chefs than short
order cooks?

Everyone seems to be a mental health expert today, just as everyone is
an expert on evolution and global warming (but not quantum mechanics or
ecology or anesthesiology, funny how that works).  I'd be uncomfortable
with any innuendos of this kind about theoreticians vs. practitioners,
or creatives vs servants, or prima donnas vs. choristers, without some
solid independent evaluation of this question by professionals with a
substantial track record in mental health.  Has any such evaluation been
made?

Vaughan Pratt


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A well kept secret

[Joyal, André]

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

[Steve Vickers]

Dear Andre,

I think category theorists have done an excellent job at publicizing the
secret. I am very much struck at category meetings what a variety of
backgrounds the participants come from, lots from computer science of
course, and now increasingly many physicists. It seems to me this is
exactly because category theory has the qualities you describe. It
enables the pure category theorists, the computer scientists, the
physicists to meet and talk together with a high degree of mutual
understanding.

I don't think of myself as a pure category theorist, but I can't imagine
trying to do what I do without it.

All the best,

Steve.

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

[Charles Wells]

To add to Steve Vickers' remarks:  Category theory is definitely out
of the closet.  A substantial number of the questions on MathOverflow
involve categorical concepts and many of them are questions about
category theory itself, not merely applications.  The n-category café
blog and its ancillary n-labs has lots of category stuff, both
applications to physics and about category theory itself.

Charles Wells



-- 
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A well kept secret

[Paul Taylor]

I'm not too sure what the context was, but Andre' Joyal said on 7
December,

 > Category theory is a powerful mathematical language.  It is extremely
good
 > for organising, unifying and suggesting new directions of research.

I completely agree.

 > It is probably the most important mathematical developpement of
 > the 20th century.

It is too early to tell.

[Comment attributed to Zhou Enlai (Chinese Communist leader 1949-76)
when asked his opinion of the French Revolution.]

 > But we cant say that publically.

I think we should be wary of slapping ourselves on the back too much.

The fact is that category theory alienated the rest of the mathematical
world.   Since the damage had been done in the 1970s, well before my
time,
I have never managed to work out how this happenned, or who was
responsible.

Probably it was the result of haughty claims about being the "most
important mathematical development",  and about being the foundations
of mathematics before any serious technical work was done to justify
this.
Of course the ignorance and arrogance of mathematicians outside our
subject
had a lot to do with it too.

Indeed, I believe that there is nothing wrong with pre-1980 category
theory that cannot be attributed to the fact that it was done by pure
mathematicians,  and nor is there anything wrong with the post-1980
subject that is not the result of its having been done by computer
scientists.

However,  discussion on that is not going to get us very far.  What is
more relevant and able to be fixed is the point in Andre's title, that
category theory is a
       WELL KEPT SECRET.

Secrecy, like charity, begins at home.   For example, the notion of
       ARITHMETIC UNIVERSE
was one of the most insightful developments of 1970s categorical logic.

It captures exactly what is taught as "discrete mathematics" to
computer science students (and is relevant to combinatorial
mathematics),
namely products, equalisers, stable disjoint sums, stable effective
quotients of equivalence relations and FINITE powersets.  It is the
least structure that is capable of constructing the free internal gadget
of the same kind,  so the original idea was to prove Godel's
incompleteness
theorem categorically.

Recently I was looking though the archives of the "Foundations of
Mathematics" (FOM) mailing list at   cs.nyu.edu/pipermail/fom/
and, amongst all of the personal abuse directed at Colin McLarty and
Steve Awodey, came across an interesting argument against category
theory, namely that the notion of elementary topos was merely an
aping of the axioms of set theory.   Arithmetic universes answer that
objection extremely well.

The work on arithmetic universes was done THIRTY SIX YEARS AGO, and
many people since then have been nagging the author to write it up,
indeed I myself have been doing so for half of that time now.

I don't want anybody to read this as a personal attack -- it is
simply an example of a general phenomenon, albeit an important example
because of the importance of the material.   Anybody in my generation
or younger can cite lots of examples of "well known"  "folklore"
results that were supposedly discovered in the 1970s but have never
been written up.   The worst thing is that any younger person who
is so impertinent as to write out a proof of one of these results
has their paper rejected.

To give another example, the theory of continuous lattices is crucial
as background for my work on Abstract Stone Duality.   I asked exactly
the people who should have written it whether there was an introduction
to continuous lattices suitable for analysts.   There isn't, so
I had to write my own.   In this, I stated without proof that the
evaluation map   Sigma^X x X --> Sigma   is continuous (when the
topology Sigma^X is itself given the Scott topology)   iff  X is locally
compact,  and in this case Sigma^X is itself locally compact and
obeys the adjunction   Yx(-) -| Sigma^(-).    The referee quite
reasonably asked for a reference to a proof, but, so far as I can
gather, no such proof exists in the literature.

Two more examples: when is some Australian going to write
"2-categories for the working categorist"?
Where is the textbook on universal algebra based on monads?

So, to answer Andre's question about why category theory is such a well
kept secret -- it is because category theorists KEEP it as a secret.

Each of us can help to leak this secret by doing two things:

PUBLISH (= make freely available on the Web) all of the papers that
you PRIVATISED by handing them over to commercial journals.

WRITE textbook or encyclopedia accounts of your work for resources
like the "n-cat lab",   ncatlab.org/nlab/show/HomePage

Paul Taylor



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re: A well kept secret

[Michael Barr]

There are several comments I could make to this posting, but I will
confine myself to two.  First, I was around all through the 70s (and most
of the 60s) and I have no idea what categorists did to earn the opprobrium
described below.  A colleague of mine commented one day maybe 25 years
ago, that it seemed in the 60s that category theory would be important,
but it hasn't turned out that way.  I am not sure what didn't turn out
that way, but that seemed to have been the general opinion.

Second, a couple of papers by Linton and Manes in the Zurich Triples Book
(LNM #80) makes very explicit the connection between triples and universal
algebraic theories.  Although doubtless out of print, a dozen of us put a
lot of effort into retyping it in tex and republishing it as a TAC
reprint.  If someone wants to go ahead and replace every instance of
"triple" by "monad" go ahead.  Also Beck's tripleableness theorem is in
Beck's thesis, another TAC reprint also retyped by volunteers.

Incidentally (although Paul is well aware of this) every paper of mine
later than 1985 and every earlier paper of which I had an electronic
trace, is available on my personal ftp site.  Also incidentally my wife
and I retyped Grothendieck's Tohoku paper and are waiting only for
proof-reading by the Van Osdols to post it (hint, hint, since I know Don
reads this group).

Michael

On Thu, 10 Dec 2009, Paul Taylor wrote:

> I'm not too sure what the context was, but Andre' Joyal said on 7
> December,
>
>>  Category theory is a powerful mathematical language.  It is extremely
> good
>>  for organising, unifying and suggesting new directions of research.
>
> I completely agree.
>
>>  It is probably the most important mathematical developpement of
>>  the 20th century.
>
> It is too early to tell.
>
> [Comment attributed to Zhou Enlai (Chinese Communist leader 1949-76)
> when asked his opinion of the French Revolution.]
>
>>  But we cant say that publically.
>
> I think we should be wary of slapping ourselves on the back too much.
>
> The fact is that category theory alienated the rest of the mathematical
> world.   Since the damage had been done in the 1970s, well before my
> time,
> I have never managed to work out how this happenned, or who was
> responsible.
>
> Probably it was the result of haughty claims about being the "most
> important mathematical development",  and about being the foundations
> of mathematics before any serious technical work was done to justify
> this.
> Of course the ignorance and arrogance of mathematicians outside our
> subject
> had a lot to do with it too.

...

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re: A well kept secret

[jim stasheff]

Michael Barr wrote:
> There are several comments I could make to this posting, but I will
> confine myself to two.  First, I was around all through the 70s (and most
> of the 60s) and I have no idea what categorists did to earn the
> opprobrium
> described below.

I have my suspicions as to what categorists did to earn the opprobrium
described below.

The high density of new vocabulary in many research papers.

Not enough published at the level of Saunders Cats for teh Working
mathematician

Too many papers doing category theory for its own sake

apologies ahead of time to anyone whose ox is being gored

jim



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RE: A well kept secret

[Wojtowicz, Ralph]

Jim Stasheff wrote:
> we need to build that bridge

The paper linked below is an example of a "success story" that I have recently described to sponsors who want to know what use category theory has been to other parts of mathematics.  In my opinion, the fact that category theory provided not only new insights into semantics of full first-order S4 modal logic but also semantics of higher-order S4 helps build the bridge mentioned above.
http://www.andrew.cmu.edu/user/awodey/preprints/FoS4.phil.pdf

One sponsor had his own example (see the link below which he brought to my attention) which has generated a lot of interest among my colleagues at work since "network analysis" is one of our primary business areas.
http://comptop.stanford.edu/preprints/clust-functorial.pdf

I am still working through the following but think it also contributes to the bridge.
http://www.andrew.cmu.edu/user/awodey/preprints/homotopy.pdf

Ralph Wojtowicz
Metron, Inc.
1818 Library Street, Suite 600
Reston, VA  20190
www.metsci.com


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re: A well kept secret

[Vaughan Pratt]

Michael Barr wrote:
>  First, I was around all through the 70s (and most
> of the 60s) and I have no idea what categorists did to earn the opprobrium
> described below.

My experience with CT may give some insight here.

When I joined the MIT faculty in 1972 it was already 8 years since I'd
taken Max Kelly's category theory class, and the only bit of it that I
had retained by that time was that categories weren't required to have
an underlying set functor (and I couldn't in 1972 even phrase that much
of my recollection in the language of CT I used just now).

Now skip the next two paragraphs (which are only there to preserve
chronological order) unless you want all the boring details.

In between I spent a year taking mathematics honours (fourth year), in
which Max taught us topology (he had planned to teach us algebraic
topology but realized we weren't prepared for it) and we got many other
courses from John Mack (number theory), T.G. Room (axiomatic geometry),
Bruce Barnes (group theory), etc.  I then spent a year doing physics
honours (a second fourth year, I did maths honours first because I
wanted to be a theoretical physicist and had sensed that without maths
honours the physics honours year would be insufficient grounding for a
theoretical physicist).

But then the next year I noticed that computer science was an
as-yet-untapped gold mine of important yet easily solved problems,
whence my career move from physics into CS.  (A pity in some respects
since I've always been good at solving hard problems once they engage my
interest and the problems in physics had by then become quite hard and
therefore should have been right up my alley.)  So I became a CS grad
student at Sydney then Berkeley then Stanford, and then Don Knuth's
postdoc in 1971-72, and then spent a few months at IBM Yorktown Heights
in a visiting faculty position in 1972.

In 1972 the only person in the whole of 545 Technology Square (a 9-story
building on the "other side of the tracks" from the main body of MIT)
who talked about categories was Mitch Wand.  Mike Fischer was his
advisor.  I lived out Mike's way and commuted to work with him much of
the time, a half hour ride each way, so we got to discuss many things,
but I don't recall category theory ever coming up.  We mostly talked
about algorithms and program verification and programming language
design and group theory and other technical things, along with the
vegetable gardens we were growing in our back yards as a joint project
that also involved Albert Meyer.  We both were totally oblivious to
politics, which never came up.  And it never occurred to either of us
that we should discuss CT.

While still a grad student Mitch gave a graduate course on CT.  I didn't
attend any of it, being rather busy as a junior faculty and having no
occasion to, but I would occasionally hear feedback from those who did.
  The general feeling seemed to be that this was "mathematics made
difficult," a way of obfuscating the obvious.  I had no reason to defend
CT at the time and simply accepted these reports as putting CT in the
same ballpark that Rene Thom's chaos theory was later put by some of its
detractors.

In 1979, finding logic problems becoming more challenging, I
(re)discovered algebra by way of universal algebra.  I learned UA from
Rasiowa and Sikorski, which I found to my surprise I could speed-read
(must have been the excellent Sydney algebra courses), and successfully
applied it to the logic problem I'd previously been stuck on.

In 1983 I realized that category theory was the algebra of functions.  I
tried very very hard to understand Chapter 1 of CWM, which seemed far
more obscure than universal algebra.  Speed-reading that chapter was out
of the question for me.

Eventually I gave up and moved on to Chapter 2 and beyond, and after
that it was just as easy as universal algebra.

----------
So I would say that the opprobrium could well have originated from the
impression that CT was obfuscation, which Chapter 1 of CWM did nothing
to dispel.  Two four-syllable words beginning with the same letter, one
leading to the other.
----------

So what do I think today?  Well, I would rank three related concepts as
being of fundamental but not equal importance, in the following order,
most important first.

1.  2-categories

2.  Dense functors

3.  Natural transformations

The algebra of 2-categories is barely algebra, it is really the
associativity intrinsic to geometry.  If you cut a string, even one with
colored ink marks on it, in two places you can't tell after the fact in
which order the cuts were made.  If you cut a painting by Picasso
vertically then horizontally into four pieces, the same holds even
though the painting has depreciated.  These are respectively
associativity and middle-interchange.  I hardly recognize these as
algebra, they're geometry as far as I'm concerned, but they're the
algebra of 2-categories.  They suck, e pur si muove.

Dense functors are important because they expose what is "natural" about
natural transformations as an instance of 2-cells.  To see how this
works see http://boole.stanford.edu/pub/yon.pdf , "The Yoneda lemma as a
foundational tool for algebra."  I imagine Steve Lack et al have some
equivalent way of describing this viewpoint which I'm still waiting to
hear about (my 1962-1965 classmate Ross Street promised he'd get back to
me on this but that was a while back).  Meanwhile I've received
enthusiastic feedback about it from Ronnie Brown and also a response
from William Boshuk ("very enjoyable pamphlet"), though that's it so far.

I've felt for at least 15 years that the notion of natural
transformation as traditionally defined is a complicated concept.  This
I believe whether one thinks of them as a category theorist or (in their
manifestation as homomorphisms) as an algebraist.  Either way the idea
is subtle.  This subtlety of the concept is why I don't rank it higher.

My ranking makes it ironic that transformations that are called
"natural" should end up third.  But then that's just my ranking, YMMV as
they say.

Vaughan

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Re: A well kept secret

[Tom Leinster]

On Thu, 10 Dec 2009, Paul Taylor wrote:

> Two more examples: when is some Australian going to write
> "2-categories for the working categorist"?

This seems like a good candidate:

Stephen Lack, "A 2-categories companion"

http://arxiv.org/abs/math/0702535 (73 pages)

There's also Kelly and Street's excellent "Review of the elements of
2-categories" (1974), but doubtless you know about that, and I guess it's
not as comprehensive as what you're envisaging.

Best wishes,
Tom


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Re: A well kept secret

[Michael Fourman]

On 10 Dec 2009, at 14:49, Paul Taylor wrote:

>    In this, I stated without proof that the
> evaluation map   Sigma^X x X --> Sigma   is continuous (when the
> topology Sigma^X is itself given the Scott topology)   iff  X is locally
> compact,  and in this case Sigma^X is itself locally compact and
> obeys the adjunction   Yx(-) -| Sigma^(-).    The referee quite
> reasonably asked for a reference to a proof, but, so far as I can
> gather, no such proof exists in the literature.

Not in the compendium?

Professor Michael Fourman FBCS CITP
Informatics Forum
10 Crichton Street
Edinburgh
EH8 9AB 
http://homepages.inf.ed.ac.uk/mfourman/
For diary appointments contact :
mdunlop2(at)ed-dot-ac-dot-uk
+44 131 650 2690



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Re: A well kept secret

[Greg Meredith]

Dear Paul,

Two more examples: when is some Australian going to write

"2-categories for the working categorist"?

Where is the textbook on universal algebra based on monads?

Absolutely. The latter would so greatly simplify a number of
cross-disciplinary conversations.

Best wishes,

--greg

On Thu, Dec 10, 2009 at 6:49 AM, Paul Taylor <pt09@paultaylor.eu> wrote:

> I'm not too sure what the context was, but Andre' Joyal said on 7
> December,
>
>
> > Category theory is a powerful mathematical language.  It is extremely
> good
> > for organising, unifying and suggesting new directions of research.
>
> I completely agree.
>
>
> > It is probably the most important mathematical developpement of
> > the 20th century.
>
> It is too early to tell.
>
> [Comment attributed to Zhou Enlai (Chinese Communist leader 1949-76)
> when asked his opinion of the French Revolution.]
>
>

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Re: A well kept secret

[Zinovy Diskin]

On Thu, Dec 10, 2009 at 9:49 AM, Paul Taylor <pt09@paultaylor.eu> wrote:

> The fact is that category theory alienated the rest of the mathematical
> world.   Since the damage had been done in the 1970s, well before my
> time,
> I have never managed to work out how this happenned, or who was
> responsible.
>

Had it been really  *done*?  It may be just in "the nature of things"
when a community A provides abstract models for community B. Something
similar appears in relations between physicists and mathematicians, or
between physicists/computer scientists and engineers.

When a mathematician is building a math model for some physical
theory, his main driving force is a question "What do they *really*
do?"  As the work is progressing, the question develops into a thesis
"they don't actually understand what they do", and with this attitude,
the mathematician finally brings something structurally neat to
physicists. In a typical good case, the reaction would be like Manin
recently formulated in his interview "we always knew that but thank
you for attention". In a bad case, ... you know.

Isn't it similar to the math vs.category theory case?  For some people
structural clarity and elegance is a matter of life and death, for
others it's a dispensable luxury (it's a rephrasing of Edsger Dijkstra
if I remember right).

zd






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Re: A well kept secret

[Zinovy Diskin]

On Mon, Dec 7, 2009 at 9:13 AM, Joyal, André <joyal.andre@uqam.ca> wrote:
> Category theory is a powerful mathematical language.
> It is extremely good for organising, unifying and suggesting new directions of research.

one important point is missing: design and design patterns. Design
from scratch is for geniuses while ordinary people design by adapting
and developing preexisting patterns. Category theory created a
powerful system of design patterns for math and beyond (computer
science, physics, engineering). It seems it changed the very
nature of mathematical design.

Z.


> 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

[David Spivak]

I think we should say it publicly.  Gays get to have gay pride, why
shouldn't categorists get to have category-theory pride?  Perhaps
we've just been in the closet too long.  I think it's the right thing
to do to explain to people that this stuff is interesting and
worthwhile to us.

Worst of all would be to let the fear of shame keep us from saying
what we hold as true.  If I consider category theory to be good,
beautiful, valid math, then I shouldn't be shy about saying as much.
If someone else doesn't consider it to be "real math," he or she can
challenge me -- I'm up for that discussion.  The worst they can do is
not give me a job, but this is not an issue because I don't belong at
a place that doesn't respect category theory.

Andre is right -- category theory is probably the most important
mathematical developpement of the 20th century.

David


On Mon, Dec 7, 2009 at 6:13 AM, Joyal, André <joyal.andre@uqam.ca> 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

[jim stasheff]

> this stuff is interesting and
> worthwhile to us.
>
> but that doesn't imply

> category theory is probably the most important
> mathematical developpement of the 20th century.
>   

we need to build that bridge

jim


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re: A well kept secret

[Robert Seely]

Well, we might not say that, but Voevodsky did.  Link on the triples
page: http://www.math.mcgill.ca/triples/  (or directly
http://claymath.msri.org/voevodsky2002.mov) where he says "Categories:
one of the most important ideas of 20th century mathematics".

BTW - the Farmelo book, The Strangest Man, is one I recommend to my
students - it's well worth looking at.  But one thing that struck me
was how *little* Farmelo plays the "strange man" theme - Dirac was
indeed strange, but that's not what makes him worth reading about, nor
was it what made him a great theoretician.  Farmelo doesn't (IMO) make
the same mistake so many documentary producers do ...

-= rags =-


On Mon, 7 Dec 2009, Joyal, André 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

[Mehrnoosh Sadrzadeh]

I have read that  Dirac had no empathy, not even for his family. I think the
story goes the same  for many other famous mathematicians/scientists. Why is
it being so promoted that being a good mathematician and a good human being
is impossible? Is it really true?

-Mehrnoosh

On Tue, Dec 8, 2009 at 5:23 AM, Robert Seely <rags@math.mcgill.ca> wrote:

> Well, we might not say that, but Voevodsky did.  Link on the triples
> page: http://www.math.mcgill.ca/triples/  (or directly
> http://claymath.msri.org/voevodsky2002.mov) where he says "Categories:
> one of the most important ideas of 20th century mathematics".
>
> BTW - the Farmelo book, The Strangest Man, is one I recommend to my
> students - it's well worth looking at.  But one thing that struck me
> was how *little* Farmelo plays the "strange man" theme - Dirac was
> indeed strange, but that's not what makes him worth reading about, nor
> was it what made him a great theoretician.  Farmelo doesn't (IMO) make
> the same mistake so many documentary producers do ...
>
> -= rags =-
>
>
>
> On Mon, 7 Dec 2009, Joyal, André 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
>>

-- 
Mehrnoosh Sadrzadeh
EPSRC Postdoctoral Research Fellow
Oxford University Computing Laboratory
Research Fellow of Wolfson College
http://web.comlab.ox.ac.uk/people/Mehrnoosh.Sadrzadeh/


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re: A well kept secret

[Robert Seely]

On Wed, 9 Dec 2009, Mehrnoosh Sadrzadeh wrote:

> I have read that  Dirac had no empathy, not even for his family. I think the
> story goes the same  for many other famous mathematicians/scientists. Why is
> it being so promoted that being a good mathematician and a good human being
> is impossible? Is it really true?

Certainly Farmelo's book gives that impression - Dirac wasn't exactly
warm and cuddly.  But I'd say that from my experience with
mathematicians, there's no reason to assert that one cannot be both a
good mathematician and a good human being - but maybe that's just
because most mathematicians I know are category theorists ...

... (there are exceptions, of course) ...

-= rags =-


> -Mehrnoosh
>
> On Tue, Dec 8, 2009 at 5:23 AM, Robert Seely <rags@math.mcgill.ca> wrote:
>
>> Well, we might not say that, but Voevodsky did.  Link on the triples
>> page: http://www.math.mcgill.ca/triples/  (or directly
>> http://claymath.msri.org/voevodsky2002.mov) where he says "Categories:
>> one of the most important ideas of 20th century mathematics".
>>
>> BTW - the Farmelo book, The Strangest Man, is one I recommend to my
>> students - it's well worth looking at.  But one thing that struck me
>> was how *little* Farmelo plays the "strange man" theme - Dirac was
>> indeed strange, but that's not what makes him worth reading about, nor
>> was it what made him a great theoretician.  Farmelo doesn't (IMO) make
>> the same mistake so many documentary producers do ...
>>
>> -= rags =-
>>

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