Re: Help!

[Ross Street <street@ics.mq.edu.au>]

Dear Mike

Some categories that are easily described (even to talented high
school students) are:

the category fun of functions (where objects are natural numbers and
morphisms are functions);
the category mat of matrices (again the objects are natural numbers);
the category brd of braids; and,
the category tang of tangles.

There are enough functors amongst these to be interesting. They are
all monoidal categories.

One can try to discuss other structure the categories have in common
so that strong monoidal
functors (although I probably wouldn't introduce too much such
terminology) preserve it.
For example, duals in mat and tang; trace and braid closure; etc.

One could try to show how the specialized, seemingly ad hoc
Reidemeister moves translate
naturally into the braided monoidal setting.

A hint about how the "new" (mid 1980s)  polynomial link invariants
come from a functor
tang --> mat might be of interest.

Best wishes,
Ross

On 05/10/2007, at 10:52 PM, Michael Barr wrote:

> What would you say to an undergraduate math club about categories?
> I have
> been thinking about it, but I am not sure what to say.  Talk about
> cohomology, which is what motivated E-M?  I don't think so.  Talk
> about
> dual spaces of finite-dimensional vector spaces?  Maybe, but then
> what?