Categorically Speaking

In one of those moments of moral weakness that seem to be becoming more common, I bought a copy of Saunders MacLane's Categories for the Working Mathematician. My excuse was that it was cheap, and that they seemed to be fresh out of Categories for the Loafing Physicist. I had heard that Category Theory was largely about arrows and diagrams, so this sort of metageometric stuff seemed attractive to me.

I also knew that category theory had its origins in homological algebra, and that I wouldn't recognize a homological algebra if it hit me on the head, and this should have been a warning to me. MacLane, of course, is a very clear writer, and he presents his material in a lucid and well organized fashion.

The problem, it seems, is that he is actually writing for an audience of mathematicians, so that the examples presented are often (usually) unfamiliar to me, and the arguments often a bit too abstract for the very limited space in my mathematical abstractions registers.

Fortunately, it turns out that there really is a Category Theory for the Barely Sentient book, and I even found an unread copy on my bookshelves - it's actually called Conceptual Mathematics: A First Introduction to Categories by F. William Lawvere and Stephen Hoel Schanuel. I turned to it when MacLane got a bit heavy, and it's really quite nice.

The authors claim to have used the material with classes ranging from high school to graduate seminars, and, oddly enough, I find that believable.

I was taught that the most primitive notion in mathematics was the set. Of course sets aren't too interesting until we start thinking about functions from a set to another (or to itself). Category theory, to the extremely limited extent I apprehend it, turns that around a bit. Functions, or mappings, come to the fore. They can be considered not just to be mappings between sets, but between more general objects, provided that the mappings obey some simple rules. Sets, and functions between them are one category, but there are many others: the objects may be topological spaces, groups, and so on.

My first reaction was - hey, those are just sets too, right - and so they often are, but sets with structure. If my intuition about Categories is correct, the idea is to abstract into Category theory some general types of structure that don't depend on the specifics of the objects and their mappings.

Looks interesting so far.

The arrows are functions, by the way, AKA mappings, morphisms, and functionals.