Monoidal Categories
INTRODUCTION
Ever since I updated my website (be sure to take a look!), I’ve been getting a lot of emails about what exactly my work with the MIT Mathematics Department is. Category theory has long been hailed as one of those extremely abstract and theoretical fields of math, but recently it’s been making its move into more applied disciplines. We’ve found that we can use it in understanding information flow, making category theory relevant for computer science and even quantum engineering. But first, it’s important to try to understand what categories are and more specifically, what monoidal categories (the focus of my research) are.
CATEGORY
So, what even is a category? Well, we have some intuitive working definition that we use in every day life. But in the case of math, categories are specific collections of objects. More accurately, a category is a structure that has objects, linked by arrows. A key feature of these arrows is that we must be able to compose them associatively and there must exist an identity arrow for each object.
What does it mean to compose arrows? Well, suppose we had one arrow going from A to B. We should be able to replace that arrow with a composition of functions, the result of which is representative of what the original arrow was doing. When we replace the arrow with a composition of two or more arrows, say A to X to Y to B, we should be able to do this associatively. If you think of arrows as functions (which is a reasonably accurate approximation), then the composition of these three functions must be associative.
Pretty simple, right? There is a lot of hidden structure within categories that makes studying them very exciting, but for now let's look at an example.
CATEGORY OF SETS
If we take the arrows to be functions, we can call the objects sets in a particular category. In the category of sets, we have essentially what we understand as entire functions. We have the domain mapping to a codomain through a function, and the domain and codomain can be represented as sets. That's fairly intuitive, because most graphs, such as y=2x+4, can be rewritten as a category of sets. Do you know what the sets would be? What would the arrows be?
MONOIDAL CATEGORIES
Brace yourselves, because this isn't always the easiest to understand. A monoidal category is a type of category. Formally, it must have a bifunctor (known as the monoidal product), an identity object, and three natural isomorphisms. These isomorphisms essentially boil down to the fact that the bifunctor must be associative and has the identity object as a left and right identity (that is, if we do I x A, it is the same as A x I, and both are A).
EXAMPLES OF MONOIDAL CATEGORIES
Okay, so you may have lost me there. But we can consider a few simple examples of monoidal categories. The category of sets with the Cartesian product and with the one-element set as the identity object is one that we have all surely run into. Can you think of others? If you're familiar with linear algebra, maybe you can think of one involving vector spaces?
WRAPPING UP
I hope that helps you learn a little bit more about category theory. Categories aren't really so intimidating -- we interact with them all the time in our daily lives. The way we transform and understand categories can get complex though. In a later post, I'll discuss the notion of quotient-free (which was completely made up on the spot) and why we care about quotient-free monoidal categories.
