Every student who learned the basics of operator algebras knows the Gelfand-Naimark representation theorem, usually stated non- categorically as "every commutative unital C*-algebra is isomorphic to the algebra of continuous functions on a compact Hausdorff space". Asking such students to check that this is part of a dual equivalence of categories is probably a good idea, and based on this one can do exercises about particular algebras they know - for instance to compute presentations by generators and relations of C(T^n), the algebra of continuous functions on the n-dimensional torus, which by the duality is instantly reduced to finding a presentation of C(S^1), etc. Also, some students might be willing to work out how the existence of presentations of C*-algebras by generators and relations relates to the fact that the category of C*-algebras is algebraic over Sets. On Mar 4, 2008, at 3:20 PM, Andrej Bauer wrote:
This semester I am teaching rudimentary category theory at graduate level. It is somewhat scary that I should be doing this, but other faculty members do not seem to do much general category theory.
I have only few students (and they are very bright) but their areas of research are quite diverse: discrete math/computer science, algebra, algebraic topology, and functional analysis.
I can plenty motivate categories for discrete math and computer science, with things like "initial algebras are inductive datatypes, final coalgebras are coinductive (lazy) datatypes".
I also know enough general algebra to motivate algebraists with tquestions like "What is an additive category with a single object?". And we will study algebraic theories as well.
Algebraic topologists are self-motivated. Nevertheless, we'll do some sheaves towards the end of the course.
But how do I show the fun in categories to a student of functional analysis? I would like to give him a class project that he will find close to his interests. The course is covering (roughly) the following material: basic category theory (limits, colimits, adjoints, we mentioned additive and enriched categories), Lawvere's algebraic categories, monads (up to stating Beck's theorem and working out some examples), basics of presheaves and sheaves with a slant toward topology. There must be some functional analysis in there.
I would very much appreciate some suggestions.
Best regards,
Andrej