Dear Andrej, Conventional functional analysis is largely concerned with Banach spaces, and there's certainly alot that can be said about the category of Banach spaces and linear contractions (i.e., continuous linear transformations with norm less-than-or-equal-to 1). For example, it is symmetric monoidal closed (with internal hom, the space of _all_ continuous linear transformations!) and locally countably presentable. In fact, it is a countable-ary quasi-variety, and the full subcategory of its finite-dimensional objects is a good example of a *-autonomous category that is not compact closed. [Although it is not hard to prove the latter directly, it can also be seen as an interesting application of Robin Houston's theorem that products and coproducts can not differ in a compact closed category.] In fact, I think Ban is a fine example which can teach any student of category theory a number of salutary lessons: 1. in category theory, the meaning of isomorphism is fixed---so if you have a pre-existing class of isomorphisms in mind (in this case, the isometric (norm-preserving) isomorphisms), then you must take care in choosing an appropriate class of morphisms; 2a. there's more to defining internal homs than just slapping an extra structure on the external homs; 2b. forgetful functors don't have to be "the obvious thing"; 3. you can't always have your cake and eat it too!---the whole category can not hope to be self-dual, precisely because it is locally presentable (and not a poset). Of course there are also (unital) C*-algebras, and I can make an interesting point about them too---sometimes one needs to consider maps between C*-algebras which are not *-homomorphisms: for example, there are "completely positive maps" and "completely bounded maps". Now, as important as the b.o./f.f. factorisation may be in general, it seems fishy to speak of a category whose objects are C*-algebras but whose morphisms preserve only part of the C*-algebraic structure; and so it was that analysts were led to develop the notions of "operator space" and "operator system" which provide the correct level of structure to define c.b. maps and c.p. maps, respectively. In fact, these are quite interesting categories in their own right: operator spaces are said to model "non-commutative functional analysis" ---but I only have a tenuous grasp of what that is supposed to mean! I meant to discuss quantale theory and Banach sheaves too, but I've run out of time---perhaps someone else will pick up the thread. Cheers, Jeff. --- Andrej Bauer <Andrej.Bauer@fmf.uni-lj.si> 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
Looking for the perfect gift? Give the gift of Flickr! http://www.flickr.com/gift/