[Twisol]

> However, it turns out that category theory is equivalent to type theory (ie, what computers use).

Just to elaborate on this a little, a program of type B in context of variables of types A1, A2, ..., An, can be modeled as an arrow (A1 * A2 * ... * An) -> B. (More elaborate type theories get, well, more elaborate.) The various ways you might put programs together (e.g. if-expressions, loops, ...) become various ways you can assemble arrows in your category. Sequential composition in a category is the most fundamental, since it describes dataflow from the output side of one program to the input site of another; but the other ways of composing programs together appear as additional structure on the category.

Categories take "semicolon" as the most primitive notion, and that's all a category really is. In fact, if you've heard monads called "the programmable semicolon", it's because any monad induces a related category whose effectful programs `a ~> b` are really pure programs `a -> m b`, where `m` is the monad.