Category Theory: The Math Behind Mathematics

In the late 1800s, mathematics was undergoing a radical shift. Led by David Hilbert (among others), a new group of mathematicians became more interested in abstracting ideas rather than solving problems. Initially criticized for lacking practicality, this difference in philosophy has led to a vast amount of interesting results, one of them being Category Theory.

Category Theory was initially developed in the 1940’s by Samuel Eilenberg and Saunders Mac Lane as an attempt to create a general language that can be applied to any field of mathematics. Instead of focusing on exact definitions, Category Theory emphasizes the relationships between objects. This can lead to fascinating connections between a variety of seemingly unrelated concepts. The two main aspects of Category Theory are categories and functors. Let’s see what both of these are.

A category is defined as a collection of objects that have morphisms between them. This is super broad, some examples will help!

The category Set is the collection of all sets and the morphisms are functions that go between these sets.

The category Group is the collection of all groups and the morphisms are group homomorphisms between these groups.

The category Vector(K) is the collection of vector spaces (over the field K) and the morphisms are linear transformations between these vector spaces.

Hopefully these three examples gives you an idea of just how broad Category Theory is, we’ve just seen examples in three major fields of math!

I’ve left out a few details in this definition. First, there must always be an identity morphism, which just takes every object to itself. Second is the property of composition. This means that the morphisms in a category must be composible. We can easily think about this concept in the Set category by function composition. For example, if we have two functions:

Then we can compose these functions to give

The compsitionality function also holds for the homomorphisms of the Groupcategory and the linear transformations of the Vector(K) category.

The second major aspect of Category Theory are functors. These are maps between categories, so a functor must tell us both how to map both objects and morphisms of one category to another. A functor must respect morphism composition and map the identity morphism of the input category to the identity morphism of output category.

Hopefully you can see how powerful functors are. There are a variety of standard functors that can give insight into different aspects of math. I’ll give a basic one here, but more can be found in the suggested reading at the end.

The Forgetful Functor will “forget” some properties of the original object. When this functor goes from Group to Set, it maps every group to a set containing the elements of that group. Each group homomorphism is mapped to a function that brings elements from the domain group to elements of the image group. An example of the Forgetful Functor is shown below.

To make things even more abstract, we can look at the category of functors and examine the functors between functors. This can be done repeatedly, and this “layering” of categories is called Higher Category Theory. Working with the category Functor has led to fascinating results, such as the Yoneda Lemma. Studies such as this led mathematician Norman Steenrod to call it “abstract nonsense.”

Given its breadth, Category Theory has found itself into a wide array of fields such as theoretical physics, computational theory, and resource theory. It is still a young field, with so much more to offer.

If you want to learn more about Category Theory, here are some links to get you started.

Seven Sketches in Compositionality: An Invitation to Applied Category Theoryis a fantastic, approachable textbook that gives an introduction into Category Theory through its applications. One one occasion while reading, I found myself very confused by a passage and emailed David Spivak on a whim. He responded the next day with a fantastic explanation! My only complaint is that is can get lost in definitions without always having a clear big picture. I highly recommend this book.

What is Category Theory Anyway? is a series of blog posts that gives a very clear walkthrough the basics on Category Theory. While the previous textbook can get lost in the details, this series is the opposite! The greater picture is always clear in the writing, while sometimes lacking formality. I found the two works complemented each other nicely.

Categories for the Working Mathematician is commonly referred to as the “definitive” book on Category Theory. Written by one of the founders, it has a vast array of advancements that has been made since Category Theory was created. I would use this book for reference.

Category Theory in Context is fantastic, recent book written by one the world’s leading mathematicians in Category Theory: Emily Riehl. I have not personally read this book, but have heard great things about it.

Thanks for reading! Leave a comment if you have any thoughts or questions about this article.