Derek Ross <math@antiquark.com> said,
The impression I got from reading about categories (on the internet of course) is that category theory is a sort of grand unified theory of mathematics.
If only, Derek, if only! I am taking the liberty of assuming, on the basis of your email address, that your background is in fundamental physics. Whilst we may not yet have THE "GRAND Unified Theory" in a technical sense in Physics, we do have some extremely good ones. But the principal difference between pure mathematics and physics is that physics has an absolute point of reference, namely Nature or experiment, whereas pure mathematics navigates in dense fog. Several centuries have passed since anyone thought that mathematics had a major goal such as Euclidean geometry or solving algebraic equations. Mathematics is far less of an "exact science" than sociology in this respect. So the reason why category theory cannot be "a sort of grand unified theory of mathematics" is that, if you ask two mathematicians what such a theory might be expected to DO, you'll get at least three incompatible answers (and that's assuming they're sober). Category theory has no faults that cannot be directly attributed to the fact that it has been done by pure mathematicians.
Categories has the ability to describe any of the fields of mathematics, and is even able to link together totally disconnected fields and show that they are fundamentally the same thing.
After substituting "many" for "any", yes, you are absolutely right there, and the best way to see that is to attend an international category theory conference. (Unfortunately, there hasn't been one since Como in 2000, though I gather that the recent Galois Theory meeting in Toronto was a good substitute.) There you will find people from physics, computer science, topology, algebra and many other mathematical disciplines, using categorical methods AND LEARNING FROM EACH OTHER. Computer science conferences (the ones I go to otherwise) and, so far as I can gather, other mathematics conferences, have none of this interaction, being (so they claim) "more focused". One of the sources of confusion about the subject is that the "proportion" (for example) category theory : category :: group theory : group is not valid. The subject, or rather the community, is defined by its ATTITUDE to doing mathematics, not by a particular set of axioms. It is a willingness to admit new examples from different disciplines into the canon of one's subject - and being prepared to rewrite that subject FROM SCRATCH to accommodate those examples. For example that rescheduling of a computational process is a homotopy - but paths have to be defined in a directed way rather than using the (two-way) Euclidean interval. This should have been done from the start for pure mathematical reasons. Unfortunately, 35 years after Ronnie Brown had made the (previous) step of defining homotopy theory with groupOIDs, textbooks are STILL being written about fundamental GROUPs. My own guiding principle is in recognising that a simple notion in category theory often matches something in some application. My first experience of this was matching stable coequalisers with WHILE programs (Section 6.5 of my book). I am currently interested in seeing how a certain monadic adjunction provides various notions of "completeness" in topology. There are examples of this situation in set theory (the Lindenbaum--Tarski--Pare' theorem) and order theory (leading to constructively completely distributive lattices). That there may also be such situations in algebraic geometry and quantum computation is pure speculation on my part. However, this speculation is a better bet than might otherwise be made by someone from outside a discipline, because of the fact that the methods of category theory have been distilled from decades of experience in algebraic geometry and elsewhere. Category theory is the vehicle that carries INTUITIONS from one area of mathematics to another. Another completely silly argument that can only be appreciated sociologically is the one over diagrams. As we all know, drawing graphs of real-valued functions is the quickest way of landing up with a major fallacy in analysis. Unfortunately, this leads to a paralysis of symbols, itself prone to numerous elementary errors, when a diagram would have put across the idea much more easily and accurately. Our ancestors have had language for maybe 30,000 years, but they have had eyes for 300 MILLION years. For example, I have often heard theoretical computer scientists say that they "don't understand diagrams", and so write their proofs as extended chains of equalities between lambda terms. It is very difficult to see which of fifty successive symbols has changed from one line to the next, let alone to check that the step was valid. And yet the same people insist on the importance of types. The vertices of a commutative diagram state the TYPES of the sub-terms of the terms that are defined by paths around the diagram. A chain of equations corresponds to "dragging" the path/term across the diagram. Each "detour" (following one side of a cell instead of the other) makes a change within the term, and is justified by an axiom of the system. So the cells also have names. More interestingly, there may be many ways of dragging the path across the diagram, whereas the equational proof presents only one of them. Moreover, the converse of the argument often uses the same diagram, giving slightly different logical status to the cells and ways of filling in edges (which we often indicate by making them dotted). The geometry of the lay-out of the diagram may even tell us something about the structure of the argument. For example, the final chapter of my book uses parallelograms to express substitution or pullback. For me, a diagram is a MAP of the argument. Just as in Euclid (go and look at it!) we begin with the diagram, to tell us the plan of the proof. I annoy referees by putting too much information into the diagram. On the other hand, there is an equally silly attitude from certain categorists, who would prefer to cover three pages with unlabelled diagrams and clumsy verbal descriptions of "exponential transposes" rather than write a lambda term. However, if you force me to choose between diagrams and symbols as a way of thinking, I prefer diagrams. Symbolic calculi are usually designed (and therefore better suited) to PARTICULAR disciplines or situations, besides being notorious for concentrating on the trees rather than the forest. Diagrams tend to be less biased to one way of thinking, especially when the central idea is to "reverse" some usual way of doing things. I think this is what Dusko Pavlovic had in mind when he was talking about coalgebras. If you want to read more of my ramblings, you might like to see: * my book: "Practical Foundations of Mathematics", Cambridge University Press, 1999, ISBN 0-521-63107-6; * my current research programme: "Abstract Stone Duality", about which there are three papers in "Theory and Applications of Categories"; * the first year computer science course that I taught when I had a job: "Introduction to Algorithms"; * not to mention my (La)TeX package for drawing commutative diagrams. All available via http://www.di.unito.it/~pt Paul Taylor