Some Basic Examples of Categories

For me, the frustrating thing about learning category theory was that
it seemed to be full of definitions, but that I couldn't see why I should care.
What were these category things, and what could I really talk about using this
strange new mathematical language of categories?

To avoid that in my presentation, I'm going to show you a couple of examples up front of things we can talk about using the language of category theory: sets, partially ordered sets, and groups.

Sets as a Category

We can talk about sets using category theory. The objects in the category of sets are, obviously, sets. Arrows in the category of sets are total functions between sets.

Let's see how these satisfy our definition of categories:

• Given a function f from set A to set B, it's represented by an arrow f : A → B.
• º is function composition. It meets the properties of a categorical º:
  • Associativity: function composition over total functions is associative; we know that from set theory.
  • Identity: for any set S, 1_S is the identity function: (∀ i ∈ S) 1_S(i) = i. It should be pretty obvious that for any f : S → T, f º 1_S = f; and 1_T º f = f.

Partially Ordered Sets

Partially ordered sets (that is, sets that have a "<=" operator) can be described as a category, usually called PoSet. The objects are the partially ordered sets; the arrows are monotonic functions (a function f is monotonic if (∀ x,y &isin domain(x)) x <= y ⇒ f(x) <= f(y).). Like regular sets, º is function composition.

It's pretty easy to show the associativity and identity properties; it's basically the same as for sets, except that we need to show that º preserves the monotonicity property. And that's not very hard:

• Suppose we have arrows f : A → B, g : B → C. We know that f and g are monotonic
  functions.
• Now, for any pair of x and y in the domain of f, we know that if x <= y, then f(x) <= f(y).
• Likewise, for any pair s,t in the domain of g, we know that if s <= t, then g(s) <= g(t).
• Put those together: if x <= y, then f(x) <= f(y). f(x) and f(y) are in the domain of g, so if (f(x) <= f(y)) then we know g(f(x)) <= g(f(y)).

Groups as a Category

There is a category Grp where the objects are groups; group homomorphisms are arrows. Homomorphisms are structure-preserving functions between sets; so function composition of those structure-preserving functions is the composition operator º. The proof that function composition preserves structure is pretty much the same as the proof we just ran through for partially ordered sets.

Once you have groups as a category, then you can do something very cool. If groups are a category, then functors over groups are symmetric transformations. Walk it through, and you'll see that it fits. What took me a week of writing to be able to explain when I was talking about group theory can be stated in one sentence using the language of category theory. That's a perfect example of why cat theory is useful: it lets you say some very important, very complicated things in very simple ways.

Miscellaneous Comments

There've been a couple of questions about from category theory skeptics in the comments. Please don't think I'm ignoring you. This stuff is confusing enough for most people (me included) that I want to take it slowly, just a little bit at a time, to give readers an opportunity to digest each bit before going on to the next. I promise that I'll answer your questions eventually!