Coming Fall 2018

MA 490-01 - Category Theory | Tuesday & Thursday, 1:00 - 2:00 P.M.

(Note: Skip this paragraph if you're not a math geek; skip to the last paragraph if you're in a hurry) Is there a set (denote it I) such that, for any other set (say, S), there exists a unique (total) function f : I → S? (Yes! One!) Is there a set (denote it T) such that, for any other set S, there exists a unique function f : S →! T? (Infinitely many!) What if, instead of sets and functions, the previous questions were about groups and homomorphisms? Would the answers to the first two questions change? (Spoiler: Yep, there's one group (and only one (up to isomorphism)) that satisfies these properties, and it satisfies both!)

Category Theory is the study of math from a bird's-eye view; that is, instead of focusing on the details of specific structures (sets/functions, sets with algebraic structures/homomorphisms, sets with a notion of nearness/continuous functions, any given group, etc.), in Category Theory one is interested in these concepts more generally, focusing on the common algebraic structure and patterns that arise in these structures, the mappings between these structures, the mappings between those mappings, and so on.

However, despite Category Theory having its origins in pure math (and being about as abstract as one can get), today many researchers outside of pure math are interested in the subject, as it has found applications in computer science, biology, physics, systems science, and more. In this course, we will focus on the math side of things (full disclosure: students will write proofs and construct examples/counterexamples) and take some excursions into the computer science applications (functional programming, automata, and maybe databases if time allows). More specifically, in this course we will cover universal properties, categories, categorical constructs, abstract structures, duality, functors, natural transformations, adjoints, and monads, with the applications sprinkled in and additional topics as time allows. If that doesn't look like a fun set of topics, then I don't know what does.

TL;DR Version: Have you ever thought, "I wish math was more abstract"? Then this is the class for you! If you've never thought that, but are fine with abstract things, then this might be the class for you, but you'll never know unless you take it. If, instead, you've thought, "This math instructor keeps doing proofs and teaching theory! What a jerk!" then... I'm surprised you're still reading this, but I applaud your dedication to reading, and will stop typing words now so that you can get back to whatever it was you were doing.

Prerequisites: MA 244 and one of MA 330 or MA 442