RE: A well kept secret?

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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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?

[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?

[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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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?

[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?

[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?

[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?

[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?

[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?

[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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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?

[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: 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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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?

[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?

[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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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?

[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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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?

[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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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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