So I have been wanting to understand Category Theory (see , , ) mainly because I thought it would help me understand advanced functional programming in Haskell and because of the book “Physics, topology, logic and computation: a Rosetta Stone” by Baez and Stay (2011). A number of the constructs in Haskell were derived from Category Theory so that was enough motivation for me to learn it. But also, monoidal categories are useful in several fields: physics, computer science, pure mathematics, and statistics. The biggest application for me is the idea of generalizing probabilistic graphical models through either Ologs or monoidal categories / tensor networks (see Brendan Fong’s thesis “Causal Theories: A Categorical Perspective on Bayesian Networks“).
To start my investigation of Category Theory, I began with the $20 thin book “Basic Category Theory for Computer Scientists” by Benjamin Pierce (see also the free online version “A taste of category theory for computer scientists” (1988)).
I really enjoyed the book. Dr. Pierce’s style is a little informal compared to pure math books like Mac Lane’s “Categories for the Working Mathematician” (pdf / Amazon), but I enjoy that more relaxed style of writing when I am first learning a field.
The biggest obstacle for learning category theory is the fact that category theory generalizes a lot of areas of pure mathematics like topology, abstract algebra, and geometry. It’s hard to generalize before you have examples to generalize, but the examples being generalized in category theory are mostly from higher level mathematics found in senior level undergraduate and graduate level courses. Pierce ameliorates this problem by introducing some of the most basic categories first: sets, ordered sets, partially ordered sets, groups, monoids, vector spaces, measure spaces, topological spaces, proofs, and a simple functional computer language. He takes the time to explicitly define most of these ideas, so, in theory, you could read this book without a background in theoretical mathematics, but it would be hard.
After defining categories and introducing the most basic categories, Pierce describes and defines the most basic ideas in category theory: subcategories, commutative diagrams, monomorphisms, epimorphisms, isomorphisms, initial/terminal objects, products, coproducts, universal constructions, equalizers, pullbacks, pushouts, limits, cones, colimits, cocones, exponentiation, and closed Cartesian categories. These ideas are spelled out over the thirty pages of chapter one including illuminating homework exercises. The homework exercises varied significantly in difficulty. Many of the exercises were trivial and there are two or three that I am still working on despite investing several hours of thought. Generally, I found the exercises to be a bit harder than those in Mac Lane’s book, but Pierce’s book required less of a background in mathematics. A couple of the exercises were incorrectly stated or impossible.
Chapter two introduced functors, natural transformations, adjoints, and F-algebras. After reading this chapter, I was finally able to understand the definition of monads which are an important part of the computer language Haskell! Pierce provides many examples of each of these ideas and enjoyable homework exercises to increase understanding. Pierce’s definition of adjoints is much easier to understand than the standard definitions using counit adjunction or Hom Sets.
The last major chapter concerns applications of category theory to computer science–specifically lambda-calculus and programming language design.
The first two chapters of the book give a reasonable condensed introduction to category theory for students that have taken a course in abstract algebra. A course in topology or linear algebra would be another useful prerequisite. I carried around the light 100 page book for a few months so that I could learn something whenever I had some extra time. I had hoped that when I had proven that several functors where monads, I would then really understand monads, but a full understanding still eludes me. Similarly, I have proven that several functor pairs are adjoint, but I still don’t feel as though I understand adjoint functors. I guess more work and learning are needed.
Have a great year. – Cheers, Hein