A well kept secret?

["Ellis D. Cooper" <xtalv1@netropolis.net>]

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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