What follows scarcely seems like mathematics at all. Real mathematicians frequently recoil at the extreme level of abstraction involved in this subject, but I maintain it is one that can be enjoyed by those will very little math training. In a sense, it is on the interface between math and philosophy. (I say this advisedly, as I hope to make clear).

So, You have all put up with my long-winded and boring dissertations on things like Groups, Vector Spaces, Topological Spaces, Manifolds etc. Actually, it comes as some surprise to learn that many of these constructions are relatively modern - one might say that the history of 19th and 20th century has been one of increasingly finer dissection of the subject into sub-disciplines.

Then, in 1945 a guy named Saunders Mac Lane asked the seemingly simple question: Is it possible to come up with asinglelanguage that will describe all of of these disciplines with equal accuracy? He decided the answer is "Yes", and that this language should be calledcategory theory.

So we had better start with some definitions, which I motivate as follows. When we do set theory, we handle "objects" called sets, with "arrows" pointing form one object to another called set functions. When we do group theory, we handle objects called groups, with arrows between them, called group homomorphisms, vector spaces have linear transformations as arrows etc etc..

So, my definitions:

AcategoryCconsists of a collection of mathematical objects X, Y, Z for which the following is true;

for any pair of objects X, Y, there is at least one "mapping" f: X → Y called amorphism, or, by me, an arrow;

for each object X ∈Cthere is a privileged arrow called the identity, id<sub>X</sub>: X → X;

if f: X → Y, and g: Y → Z, then g ⋅ f: X → Z; this composition is associative;

f ⋅ id<sub>X</sub> = f and f = id<sub>Y</sub> ⋅ f.

That's it!

Oh - notice that I compose arrows right-to-left, this is standard.

So, I will give a couple of examples, make an important comment, then leave you in peace.

In the categorySet, the objects are sets, the arrows are set functions. In the categoryGrpthe objects are groups, the arrows are group homomorphisms. In the categoryK-Vecthe objects are vector spaces over the field K, the arrows are linear transformations. Simple really.

Now note this well; category theory is not really interested in the objects themselves, rather the arrows are the main interest - I will show you why later, with an example.

If objects take back seat, the elements in the objects - set elements, group elements, vectors, in our examples above - don't even make it through the door!