# MAGIC009: Category Theory

## Course details

A core MAGIC course

### Semester

Autumn 2010
Monday, October 4th to Friday, December 17th

### Hours

Live lecture hours
10
Recorded lecture hours
0
0

Fridays
09:05 - 09:55

### Announcements

The second lecture, on Friday 15th October, will be given by Dr Vanessa Miemietz (UEA).

## Description

Category theory is the language of much of modern mathematics. It starts from the observation that the collection of all mathematical structures of a certain kind may itself be viewed as a mathematical object - a category. This is an introductory course in category theory. The main theme will be universal properties in their various manifestations, one of the most important uses of categories in mathematics.

### Prerequisites

There are few formal prerequisites to the material. However, I will be giving examples from mathematics to motivate the ideas and demonstrate how they are used, so an undergraduate degree in mathematics (rather than for example computer science or philosophy) would be an advantage. In particular, I will assume some knowledge of algebra such as vector spaces and their bases, and groups, but undergraduate level knowledge of these subjects is sufficient.

### Syllabus

The topics covered are:
• Categories: definitions, examples, special kinds of arrows and objects, duality
• Functors: definitions, examples, full and faithful functors, subcategories, Hom-functors, contravariant functors
• Universal properties: examples including vector space bases, fields of fractions, tensor products, quotients, products, and coproducts
• Natural transformations: definitions and examples, functor categories, equivalence of categories, horizontal composition
• Limits: examples, general definition, computing limits in Set, complete categories
• Colimits: definition, examples, computing colimits in Set
• Adjunctions: vector space bases, formal definition, examples, unit and counit
• Limit preservation and creation: right adjoints preserve limits, general adjoint functor theorem, examples
If there is time I will also cover the Yoneda Lemma and the Yoneda Embedding.

## Lecturer

• JK

### Dr Jonathan Kirby

University
University of East Anglia

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

## Assessment

### Attention needed

Assessment information will be available nearer the time.