Categories – What’s the Point?

Categories are relatively uninteresting without functors. Functors are more or less mappings or transformations between categories.

Category theory by some accounts initiated in the foundations of algebraic topology. In that subject, one wishes to codify some of the aspects of a high-dimensional shape or geometric object in algebraic terms. The hope is that the algebraic image of the shape is easier to understand than the shape itself. The mapping of shapes into their algebraic "shadows" is a functor.

For instance, you can take a torus and look at the collection of all closed loops drawn on the surface of the torus. Now identify two loops as the same if one can be smoothly deformed into the other by moving it along the surface of the torus. Long story short, this gives the set of such loops a structure known as a group. There is a very extensive theory developed around groups, so you might surmise that you could partially classify shapes by looking at this associated group (called the fundamental group or first homotopy group).

This mapping from geometric objects to groups of loops drawn on them is a functor from the category of topological spaces to the category of groups. This is the beginning of homotopy theory.

As for your comment on sets, there is the category Set which consists of all sets. That is, the objects of Set are sets and, given two sets X and Y, the morphisms between X and Y in Set are simply the functions f : X -> Y. An interesting technical point here (which is related to your quote) is that Set itself cannot be a set, as there is no set of all sets within Zermelo-Fraenkel set theory (the most common axiomatic formulation of set theory). Rather, there is a notion of class, and one says that the collection of all sets forms a class rather than a set.

That list is a list of examples of universal properties being used to define things. As such, it's a testament to how mindbogglingly convenient universal properties are.

There are a lot of areas of math that have no obvious connection to abstract algebra or real analysis (for instance combinatorics, logic, number theory...) yet the vast majority of mathematicians will know what you mean when you ask whether they are an algebraist or an analyst, and will know how they should be classified. In fact if I see a mathematician eating corn on the cob, I can generally tell which they are without asking. (Analysts eat in spirals, analysts in rows. My best guess for why is that analysts pay attention to the cue from how their teeth eat corn, while algebraists follow the visual cue from the lines on the corn.)

This classification is a matter of taste. Any professional mathematician is adept at both areas of mathematics, but will have a clear preference.

Category theory serves as an excellent litmus test. It seems to have a magnetic attraction for algebraists, and a similar repulsion for analysts. At its heart, category theory is an abstraction of common patterns of algebraic manipulation where the details of what you are manipulating become entirely irrelevant. (Indeed you can often replace one type of thing with an apparently unrelated type of thing. Such as a geometric problem about toruses with an algebraic question about pairs of integers.)

Speaking personally, I learned enough category theory to pass all of the courses I had to take, but words like "category", "monad" "cohomology" and "functor" are red flags warning me that I'm about to be uninterested in the rest of the discussion. But YMMV.

After a few years of graduate school, with lots of colleagues working on the same material as me, it has gradually become much easier. The best text I've seen so far is Paolo Aluffi's "Algebra: Chapter 0". It's an algebra textbook meant for first year graduate students in pure mathematics, but it was the major factor in clarifying category theory for me. Of course, the book also contains pretty much all topics in abstract algebra from basic groups to abelian categories. So if you're not pegging to be a research mathematician, chances are the book is not worth your time.

Part of my goal with this series is to demystify the subject as it was demystified for me, and to incorporate as many concrete applications to programming concepts as I can.

I hope you find some value in what I produce :)

I think that's the cleanest method.

    fizzbuzzers = [(3,"fizz"), (5,"buzz"), (7, "baz")]
    fizzbuzz k = 
      let str = concat [s | (n,s) <- fizzbuzzers, k `mod` n == 0]
      if str == "" then show k else str
    putStrLn $ intercalate "\n" $ map fizzbuzz [1..110]