
Now that I’ve hopefully got you interested in exploring abstraction a bit further (if you already haven’t read my first blog post about abstraction and word games, check it out here), allow me to introduce you to the particular brand of abstraction that I and dozens of mathematicians around the world are dedicating years of our lives to studying: categories.
But what is a “category” to a mathematician”? At its most basic level, a category is just a collection of things called “objects” which are connected by a bunch of arrows, sometimes called “morphisms”, that adhere to a few simple rules.
To illustrate these rules, imagine that you have a map of Canada laid out in front of you. Then for every (not necessarily distinct) pair of cities on the map, if there is a way of driving from one to the other, draw an arrow in the direction of travel. Notice that the “not necessarily distinct” condition means that you would end up drawing “do nothing” arrows that start and end in the same city, since you can certainly drive to a city that you’re already in! The most important thing to notice about the resulting arrows is that we are able to “compose” them, that is, join them head-to-tail. For example, if I can drive from Toronto to Ottawa, and from Ottawa to Montreal, then I can drive from Toronto to Montreal. Finally, suppose that we want to compose three arrows (say, Toronto to Ottawa to Montreal to Quebec City). Strictly speaking, there are two different ways of doing this, depending on which pair of arrows we decide to combine first. The thing to notice, though, is that it doesn’t actually matter which way we combine them, as we end up with the same arrow in the end regardless. The formal term for this property of being able to combine things in any order is “associativity”, and it is quite common among algebraic operations. For example, addition and multiplication are both associative; compare the two ways of adding up, say, 5+3+2.
These three properties – the ability to compose arrows, the existence of “do nothing”, or “identity”, arrows, and the associativity of arrow composition – are precisely what turn a bunch of objects and morphisms into a category.
The defining properties of a category may seem arbitrary, but they strike a remarkable balance. On the one hand, they are general enough to be applicable, not only across most of mathematics, but far beyond. Concepts from category theory are being applied to a multitude of fields, including computer science with languages like Haskell and Julia, chemistry, and even contract law. On the other hand, these few properties still manage to be rich and meaningful enough to have held the interest of dedicated researchers the world over for nearly a century, with no end in sight to the new things we can learn about them.
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