Introduction to Category Theory

Welcome to the learning project Introduction to Category Theory.

Learning Project Summary

• Project code:
• Suggested Prerequisites:
  • None
• Time investment:
• Assessment suggestions:
• Portal:Mathematics
• School:Mathematics
• Department:
• Stream
• Level:Undergraduate

What is Category Theory?

Abstract nonsense is a popular term used by mathematicians to describe certain kinds of arguments and concepts in category theory. This term is believed to have been coined by the mathematician Norman Steenrod, one of the developers of the categorical point of view. This term is used by practitioners as an indication of mathematical sophistication or coolness rather than as a derogatory designation.

Certain ideas and constructions in mathematics display a uniformity throughout many domains. The unifying theme is category theory. Rather than enter an elaborate discussion on particulars of arguments, mathematicians will use the expression such and such is true by abstract nonsense. Typical instances are arguments involving diagram chasing, application of the definition of universal property, definition of natural transformations between functors, use of the Yoneda lemma and so on.

This course is an introduction to abstract nonsense.

Scope of the course

Category theory is usually considered a hard subject, and isn't typically part of an undergraduate curriculum. This doesn't need to be so. This course is primarily intended for undergraduate students in pure mathematics. It might be useful to computer theorists, linguists, physicists, and some others. We may sometimes sacrifice a little mathematical rigor in order to make the text more accessible and stay more focused on the right kind of abstract nonsense. In particular we try to avoid set theoretic nonsense.

It is still unclear how much material can be covered in a undergraduate course.

Contents

Lessons

The full list

• Lesson 1: Sets and Functions
• Lesson 2: Products and Coproducts of Sets
• Lesson 3: Monoids
• Lesson 4: Categories
• Lesson 4.5: Equivalence of categories
• Lesson 5: Products and Coproducts
• Lesson 6: Products are Limits
• Lesson 7: Equalizers
• Lesson 8: Pullbacks
• Lesson 9: Limits
• Lesson 10: Exponentials
• Ideas for a new lesson:
  • generalized elements?
  • Hom(C,-) functors?
  • Yoneda embedding?
  • F-algebras (and initial F-algebras)
  • catamorphisms

Related resources

• Category Theory
• Wikipedia article:
  • Category Theory
  • List of category theory topics
  • Category:Category theory
• Wikibooks textbook:
  • Discrete mathematics
  • Set Theory
  • Abstract Algebra
  • Category theory

How can you help?

• If you're a mathematician and know some category theory, you can write any lessons that haven't been written yet.
• If you're a mathematician or a wannabe-mathematician, but know little about category theory, you can fill in the gaps or fix the errors in lessons that have been written. We also need exercises (anything from trivial computations to miniprojects (say boolean algebras)).
• If you're not a mathematician, but you think you understand what is written, you can rewrite things in better, more readable English.
• If you don't quite understand anything, you can write comments/questions to talk pages.

Active participants

Active participants in this Learning Group

Link to any subpages this page might have