Talking to Undergraduates about Category Theory

Talking to Undergraduates about Category Theory

[Micah Blake McCurdy]

Hallo!

I have over the last several years repeatedly given to delegates of the
Canadian Undergraduate Mathematics Conference a talk about Category Theory,
all of which were very well received. I should mention at the outset that I
had the (debatable) advantage of _being_ an undergraduate for all three
talks. Some elements which went over especially well:

1) Historical considerations, namely, the role of category theory in the
development of algebraic topology. The use of category theory as a language
for making rigorous certain intuitions, as well as facilitating
calculations. This point of view resonates very well with undergraduates, to
whom _all_ of mathematics is a more or less hazy mass of intuitions and
proofs; who long for clarity and order.

2) Freeing constructions from set by diagrammatic descriptions. For
instance, defining the notion of a group object in an arbitrary category C,
and then noting that such gadgets are already studied for various C. This
appeals for two reasons: it gives an elegant explanation for _why_
similar-seeming things are similar, and, more importantly, it _suggests new
questions_, namely, for a new category of study, "what are the internal
wombats in this category" for various choices of wombat. Especially for
older undergraduates, who are thinking to themselves "Subject X is really
very fascinating, but what will I ever do with it?", this is a very
appealing notion.

3) Diagrammatic methods in proofs. The device of commuting diagrams to form
and illustrate proofs is generally both novel and wonderful to
undergraduates. This has many sub-parts, among them:
       i) One augments a symbolic intuition with a geometric intuition.
Thus, proving that a large diagram commutes becomes a sort of   tangram
puzzle.
       ii) Proofs become both easier to construct and, more importantly,
easier to communicate. This is especially near to the hearts of undergrads
who have difficulty constructing proofs, more difficulty understanding the
proofs of others, and yet more in having their proofs understood by others.

If you were so inclined, you might well introduce string diagrams. The third
point, considered strictly, is not really a part of category theory, but I
think it is cut from the same cloth.

On a perfectly peripheral note, I often place two bottles of (preferably
obscure) beer on the desk in front of me before I begin speaking; promising
one to the best question after the talk and the other to the best heckling
during the talk. I strongly encourage heckling, and I doubt that
undergraduates enjoy this any more than other mathematicians. If all goes
wrong, you can drink the beer yourself.

In any event, good luck.

Micah