# Introduction to Category Theory

Welcome to the learning project Introduction to Category Theory.

## 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

• Lesson 9: Limits (Nov 5, 2007)
• Ideas for a new lesson:
• generalized elements?
• Hom(C,-) functors?
• Yoneda embedding?
• F-algebras (and initial F-algebras)
• catamorphisms