leichter@yale-com.UUCP (Jerry Leichter) (06/02/83)
Category theory is described in many texts. Unfortunately, it's a rather dry and difficult subject to get into unless you have a fairly extensive math background - not so much because of inherent difficulty, but because it's all based on common generalizations of ideas in different areas of math. If you aren't familiar with the ideas being generalized, most of the generalization will seem pretty pointless. (The inventors of category theory refer to it as "abstract nonsense".) My own math work goes back 5-6 years; I'm sure many more recent texts exist. However, I'd recommend you try the introductions to any good, recent algebra or algebraic topology text - Lang's "Algebra" is an example. Often, their first chapter is a good introduction. A couple of other books worth trying are Rotman's "Notes on Homological Algebra" (Van Nostrand Mathematical Study #26, 1970 - I don't know if it's still available) and Hilton and Stammbach's "A Course in Homological Algebra" (Springer-Verlag Graduate Text in Math #4, 1971). I've heard that MacLane's (?) "Category Theory For the Working Mathe- matician" is a good text, although I've never read it; I think it's in the same Springer-Verlag series as H&S. Beyond this...you'll have to explore. There are - or were a couple of years ago - three main streams of category theory: CT for algebraic topology (fairly bounded, driven by topological questions; described in topology texts); CT for homological algebra (much more "CT for CT's sake"; driven by algebraic questions of a very general sort); CT for (non-standard) logics; terms here include "Toposes" and "Universal Algebras" (again, much "CT for CT's sake", but driven by more general kinds of questions). As to commutativity of diagrams: Very simple. Category theory considers functions (aka maps). Bunches of related maps are represented by drawings of directed graphs. Thus: f A ---> B means "f is a map from A to B". Consider the following diagram, with three "objects" - things like A and B above - and three maps: f A ---> B | | g\ //h \ / v v C i.e.: f is a map from A to B; g from A to C; h from B to C. To say this diagram is commutative is just to say that any two paths along directions of the arrows joining a given pair of points yield the same map; i.e. for this simple example, h o f = g where "o" is functional composition. For this simple diagram, it would have been just as easy to say "h o f = g"; however, the term "commutative" applies just as well to large, complex diagrams, in which many different paths are possible. Commutativity in that case again means the equivalence along EVERY pair of paths. Thus, the commutativity of the square: f A ---> B | | g| |h | | v v C ---> D i means i o g = h o f. Adding an extra arrow from C to B would add two new triangles, whose commutativity would provide two new equations. -- Jerry decvax!yale-comix!leichter leichter@yale