Abstract Nonsense 101

1 What is a Category?

A category is just a bunch of dots with arrows going between them. The tricky part is interpreting what those dots and arrows mean.

In general, there are three conditions that these dots and arrows need to satisfy. Every dot needs a special arrow that goes from the dot to itself. We call this special arrow the identity.

Also, if we have an arrow from dot $A$ to dot $B$ and an arrow from dot $B$ to dot $C$, then we have to be able to combine these arrows to get an arrow from dot $A$ to dot $C$.

Finally, combining an arrow with the identity arrow shouldn't change it.

And that's all a category is. Just a collection of dots and arrows satisfying these three rules.

2 What Things Are Secretly Categories?

One simple category is Set, the category of sets. The dots in Set are, of course, sets. The arrows are functions between sets. Every set has an identity map to itself, functions between sets can be composed, and composing a function with the identity map does nothing. So the arrows in Set follow our rules.

A lot of categories follow this model, where the dots are some sort of collection and the arrows are functions between them. For example, the dots in the category Grp are groups and the arrows are group homomorphisms. Every group has an identity homomorphism to itself, the composition of two homomorphisms is a homomorphism, and composing a homomorphism with the identity does nothing. So Grp is also a category.

Along the same lines, we have Ring, the category of rings and homomorphism between them, and Top, the category of topological spaces and continuous maps between them. The list goes on.

But not every category is of this form. As a more exotic example, you can look at a group as a one-dot category. Each element of the group corresponds to an arrow in the category. The identity element of the group is the identity arrow. And combining two arrows corresponds to the group operation. This object is still a category, but in this case it doesn't make sense to view the dot as a set and the arrows as structure-preserving transformations.

3 Why Is This A Useful Notion?

To give some motivation, let's just consider Grp for the moment. The categorical viewpoint is that instead of studying how the individual elements of a group fit together, we should study structure-preserving transformations between groups. Often, when we look at a group, we want to know about its subgroups or quotient groups. But since every subgroup is the image of a homomorphism into the group, we can just study subgroups by studying homomorphisms into the group. And by the first isomorphism theorem, every quotient of a group is the image of a homomorphism out of the group. So we can really study subgroups and quotient groups just by studying group homomorphisms.

The idea of studying structured objects by studying maps that preserve that structure pops up all over in mathematics. Category theory just takes this idea to the extreme by studying only these maps and forgetting everything else about the objects.

The first image is from here