A particularly fascinating, and downright tough to grasp area of Mathematics is Category Theory. Basically, a Category is like a Set, but with more going on.
For a cool primer, click here.
Some of you might not know the precise definition of a category; let me state it just for completeness. A category consists of a collection of “objects” and a collection of “morphisms”. Every morphism f has a “source” object and a “target” object. If the source of f is X and its target is Y, we write f: X → Y. In addition, we have:
1) Given a morphism f: X → Y and a morphism g: Y → Z, there is a morphism fg: X → Z, which we call the “composite” of f and g.
2) Composition is associative: (fg)h = f(gh).
3) For each object X there is a morphism 1X: X → X, called the “identity” of X. For any f: X → Y we have 1X f = f 1Y = f.
See, not so bad! … well …
The cool bit, outside of Mathematics, is it’s connection to philosophy in the broader sense.
This must seem very boring to the people who understand it and very mystifying to those who don’t. I’ll need to explain it more later. For now, let me just say a bit about what’s going on. Sets are “zero-dimensional” in that they only consist of objects, or “dots”. There is no way to “go from one dot to another” within a set. Nonetheless, we can go from one set to another using a function. So the category of all sets is “one-dimensional”: it has both objects or “dots” and morphisms or “arrows between dots”. In general, categories are “one-dimensional” in this sense. But this in turn makes the collection of all categories into a “two-dimensional” structure, a 2-category having objects, morphisms between objects, and 2-morphisms between morphisms.
This process never stops. The collection of all n-categories is an (n+1)-category, a thing with objects, morphisms, 2-morphisms, and so on all the way up to n-morphisms. To study sets carefully we need categories, to study categories well we need 2-categories, to study 2-categories well we need 3-categories, and so on… so “higher- dimensional algebra”, as this subject is called, is automatically generated in a recursive process starting with a careful study of set theory.
Recursion is WAY cool. For more, read this at the n-category Cafe
To my mind, category theory and its higher-dimensional variety have been devised to deal with deep problems at the core of mathematical activity.