Introduction to category theory: a second-abstraction

When we first started talking about abstraction, we did so by observing the analogies between mathematical objects, such as integers and polynomials, the unit circle and remainders, etc. We spent the rest of the Abstract Algebra I series figuring out why these seemingly unrelated objects had similar behaviour -- what the fundamental properties were that resulted in this behaviour, and making these properties the "axioms" of various abstract algebraic structures.

But you may have later observed that even these various algebraic structures have analogies. For starters, every algebraic structure has the notion of homomorphisms -- things that commute with "structure". Then you have analogous object constructions, like trivial objects, product objects and quotient objects. And then you have the really neat stuff -- stuff like "normal subgroups are the kernels of group homomorphisms" vs. "ideals are the kernels of ring homomorphisms", etc. And perhaps most usefully of all, we've seen that some analogies themselves, such as between features of Lie groups and features of Lie algebras, might have a simpler and more abstract basis than the specific constructions of the theory.

You would be justified to believe that these analogies spring fro;m some shared principles -- and you would be justified to believe that these shared principles ought to be abstracted.

That much like how mathematical objects were found to have similar properties, and we'd categorise them as groups and rings and vector spaces and whatever -- these categories of objects too could have similar properties.

So this will be our approach: without stating the axioms beforehand and only general notions of what a category is and what a homomorphism (or "morphism" in category theory) is, we will try to prove theorems we know about specific categories like groups, for general categories -- and see what axioms we'll need.

(This, by the way, is called reverse mathematics. We've done this often here whenever dealing with something we must be rigorous about, e.g. in Topology.)

And the real idea we should have at the back of our heads is that we should stop thinking of groups, etc. as "sets with additional structure". They're really generalisations of sets, and homomorphisms are generalisations of functions. (I won't go here into exactly what I mean, but a good article to get your head wrapped around this is Sigma fields are Venn diagrams, for an illustration of how measurable functions, the morphisms in the measurable spaces category, are a "generalisation of functions".) So we won't try to force our objects to be sets and give them elements, or force our morphisms to be functions -- they will just be dots and arrows satisfying some axioms. This will require a bit of thinking, e.g. defining kernels without talking about identity elements.

Let's start.