Re: Help!

[Michael Barr <barr@math.mcgill.ca>]

I want to thank all who replied and I will take all your comments
seriously.  I would love to talk about Stone duality and such but I don't
think many of our undergrads have ever heard of a topological space.  They
have heard of topology of course, but mostly they think it concerns things
like toruses and Klein bottles.  So they know nothing of the point set
underpinnings of algebraic topology.  My last term before retirement I
taught a course called topology and spent exactly 6 lectures on point-set
theory (taking a beeline to the Tychonoff theorem) before introducing pi_1
and covering spaces.  The students were last year undergrads and a couple
of grad students.

Do they know what a boolean algebra is?  Probably some do, some don't.
Groups and abelian groups they will know about, probably modules, etc.
Vector space duality is a familiar example, for finite dimension at least.

Hi-tech whiteboards and even video-taping are out.  I don't think we have
any of the former and the one case that I know of a lecture that was
video-taped (a fascinating lecture by Conway in the early '70s in which he
showed how the game of Life allowed the simulation of self-reproducing
Turing-power automata) seems to have disappeared without a trace.  I will
probably use a blackboard (or greenboard) and chalk, my favorite medium.

One suggestion that does appeal is to start with universal mapping
properties to explain products and sums.  One thing that always struck me
was Bill Lawvere's observation that the dual of the usual definition of
function as a subset of a product s.t.... namely as a quotient of a sum
s.t.... actually corresponds closely to the usual picture we draw when we
introduce functions for the first time.    I guess I could do worse than
build the whole lecture around universal mapping properties.  I could
mention the somewhat unmotivated definition of (infinite) product of
topological spaces as a perfect example of the universal viewpoint.
Especially as topologists had come up with that definition independent of
category theory.

Michael