Early in our mathematical education, we learn about a strong interplay between algebra and geometry—algebraic equations give rise to graphs and geometric figures, and geometric features can be encoded in algebraic expressions. It’s almost as if there’s a portal or bridge connecting these two realms in the grand landscape of mathematics: whatever occurs on one side of the bridge is mirrored on the other.
So although algebra and geometry are very different areas of mathematics, this connection suggests that they are intrinsically related. Incidentally, the `bridge’ that spans them is a but a dim foreshadow of much deeper connections that exist between other branches of mathematics that also may, a priori, seem unrelated: set theory, group theory, linear algebra, topology, graph theory, differential geometry, and more. And what’s amazing is that these relationships—these bridges—are more than just a neat observation. They are mathematics, and that mathematics has a name: category theory.
What is a category?
Martin Kuppe once created a wonderful map of the mathematical landscape (see facing page) in which category theory hovers high above the ground, providing a sweeping vista of the terrain. It enables us to see relationships between various fields that are otherwise imperceptible at ground level, attesting that seemingly unrelated areas of mathematics aren’t so different after all. This becomes extraordinarily useful when you want to solve a problem in one realm (topology, say) but don’t have the right tools at your disposal. By ferrying the problem to a different realm (such as group theory), you’re able to see the problem in a different light and perhaps discover new tools that may help the solution become much easier. In fact, this is precisely how category theory came to be. It was birthed in the 1940s in an attempt to answer a difficult topological question by recasting it in an easier, algebraic light.
Thinking back to the mathematical landscape, you’ll notice that each realm consists of some objects (set theory has sets, group theory has groups, topology has topological spaces…) that relate to each other (sets relate via functions, groups relate via homomorphisms, topological spaces relate via continuous functions…) in sensible ways (such as composition and associativity).
Martin Kuppe’s map of the mathematical landscape (used with permission)
This common thread weaves throughout the landscape and unites the various fields. The mathematics of category theory formalises this unification. More concretely, a category is a collection of objects with relationships between them (called morphisms) that behave nicely in terms of composition and associativity. This provides a template for mathematics, and depending on what you feed into that template, you’ll recover one of the mathematical realms: the category of sets consists of sets and relationships (ie functions) between them; the category of groups consists of groups and relationships (ie group homomorphisms) between them; the category of topological spaces consists of topological spaces and relationships (ie continuous functions) between them; and so on.
The analogy between a category and a template is due to Barry Mazur from his wonderfully written, non-technical introduction to category theory, When is one thing equal to some other thing? In it he writes, “This concept of category is an omni-purpose affair… There is hardly any species of mathematical object that doesn’t fit into this convenient, and often enlightening, template.” Indeed, as category theorist Eugenia Cheng so aptly put it in her treatise Higher-dimensional category theory, “category theory is the mathematics of mathematics”.
It’s all about relationships
One of the main features of category theory is that it strips away a lot of detail: it’s not really concerned with the individual elements in your set, or whether or not your group is solvable, or if your topological space has a countable basis. So you might be thinking, “Eh, category theory sounds so abstract. Can any good come from this?” Happily, the answer is yes! An advantage of ignoring details is that our attention is diverted away from the individual objects and turned towards the relationships—the morphisms—that exist between them. And as any category theorist will tell you, relationships are everything.
Indeed, one of the main maxims of category theory is that a mathematical object is completely determined by its relationships to all other objects. To put it another way, two objects are essentially indistinguishable if and only if they relate to every object in the category in the same way. This theme (which is a consequence of a famous result called the Yoneda lemma) isn’t too different from what we observe in life. You can learn a lot about people by looking at their relationships—their Facebook friends, who they follow on Twitter, who they hang out with on Friday nights, for instance. And if you ever meet two people who have the exact same set of friends, and whose interactions on social media are exactly the same, and who hang out with the exact same people on Friday nights, then you might jokingly say, “You can’t even tell them apart”. Category theory informs us that, all jokes aside, this is actually true mathematically!
So you might wonder “Hmm, if mathematical relationships are that important, then what about relationships between categories? Do they exist?” Great question. The answer is: absolutely! In fact, these particular relationships have a name—they are called functors. But why stop there? What about the relationships between those relationships? They, too, have a name: natural transformations.
In fact, we can keep on going: “What about the relationships between the relationships between the relationships between the…?” Doing so will land us in higher-dimensional category theory, which is where much of Cheng’s research lies.
As abstract as this may sound, these constructions—categories, functors, and natural transformations—comprise a treasure trove of theory that shows up almost everywhere, in mathematics and in other disciplines! Since its inception, category theory has found natural applications in computer science, quantum physics, systems biology, chemistry, dynamical systems, and natural language processing, just to name a few. (The website `Applied category theory’ contains a list of applications.) So even though category theory might sound a little abstract, it is highly applicable. And that’s no surprise. Category theory is all about relationships, and so is the world around us!
Conclusion
Categories are a little bit like anchovies: some folks love ’em, while for other folks they’re an acquired taste. So yes, it’s true that category theory may not help you find a delta for your epsilon, or determine if your group of order 520 is simple, or construct a solution to your PDE. For those endeavours, we do have to put our feet back on the ground. But thinking categorically can help serve as a beacon—it can strengthen your intuition and sharpen your insight—as you trek through the nooks and crannies of your favourite mathematical realms. And these days it’s especially hard to escape the pervasiveness of category theory throughout modern mathematics. So whatever your mathematical goals may be, learning a bit about categories will be well worth your time!
This article has been adapted from “What is Category Theory, Anyway?” first published on 17 January 2017 at math3ma.com
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