Basic Category Theory

Hi, I'm the author. Thanks for your interest in my book!

I agree, the words "relatively little background" are too vague. What I had in mind was that the book requires little background relative to many other introductions to category theory (such as the grand-daddy of them all, Mac Lane's Categories for the Working Mathematician). But I should have been more specific. If update the arXiv submission, I'll fix that.

As cokernel points out, the level of knowledge assumed is roughly what you'd get from an undergraduate mathematics degree at an ordinary university in Britain (and probably many other countries too). I know this because I used it several times to teach a master's course at the University of Glasgow. Probably the most famous master's-level category theory course is the one that Cambridge runs in its Part III (master's) programme, which I've also taught. But this book covers much less than the Cambridge course, and assumes less background too.

If you've taken either (i) enough algebra that you're comfortable with rings, groups and vector spaces, or (ii) any kind of topology course, then you should be able to understand enough of the examples that you can get a good grip on the general concepts. If you haven't, then it might not be the right book for you. As others have pointed out, Lawvere and Schanuel's book Conceptual Mathematics assumes much less background than mine, and there are also texts oriented towards readers with a computer science background.

The short answer is category theoretic monads. Once upon a time, Eugenio Moggi realized that monads could be used as the formal basis for assembling sequential actions. For example, when you write

    print("hello ");
    print("world")

you expect the output to be "hello world" and not "worldhello " or something weirder. Moggi realized that the semicolon there was a monadic operation that arranged for the two outputs to be printed sequentially.

That might seem like a small thing, but it isn't. In theory, so far as I understand, specifically in denotational semantics, the next best alternative is continuation passing, which adds an extra parameter and return value to the actions and is generally awkward.

The big advantage, though, is practically, in lazy programming languages like Haskell. In that case, the alternative was world passing, where the entire (metaphorical) universe is passed to the action to allow normal parameter passing conventions to sequentially associate the actions. (In contrast, monadic IO does the same thing, but hides the world in the definition of the monad.) The result is sequential code that we all know and love.

Then it was discovered that other bases than the (metaphorical) IO world made for meaningful, useful monads: the maybe monad aborts a computation on the first error, for example. And other data structures like lists...well, go look for a paper with a title like "turn your failures into a list of successes" and we were off and running.

[Edit]

And I'm thinking of the wrong Wadler paper. Try some of the ones from here [1], starting at the bottom.

[1] http://homepages.inf.ed.ac.uk/wadler/topics/monads.html

> Category theory is not widely used by those who program computers and thus produce computation in them.

Computation is bigger than just programming in industry. "Computing" has always referred to the overall discipline of computer science.

> you point out how many people productively write software without knowing anything about category theory (or even abstract algebra).

That wasn't my point at all. Read again. They can have even more productivity if they used category theoretic abstractions.

> Saying that people use category theory to write software without knowing it is like saying they use Maxwell's Laws to write software

Good thing that's not what I said either. You're making a habit of this.

People can understand a structure, in that they can grasp its semantics and its application, without knowing it's canonical name.

I assume the functional language you used was a Lisp or something not statically typed?

you might enjoy looking at category theory for the sciences by david spivak. there's a hardcover edition, but it's available online as well.

[html version] http://category-theory.mitpress.mit.edu

[old pdf version] http://math.mit.edu/~dspivak/CT4S.pdf