General
This course is part of the MAGIC core.
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.
Semester
Spring 2012 (Monday, January 16 to Friday, March 23)
Timetable- Mon 10:05 - 10:55
Students
|
Bana Al Subaiei
(Southampton) |
Abdulsatar Al-Juburie
(Newcastle) |
Kalyan Banerjee
(Liverpool) |
Yumi Boote
(Manchester) |
|
Tom Brookfield
(Birmingham) |
Thomas Bullock
(York) |
Christopher Cave
(Southampton) |
Yu-Yen Chien
(Southampton) |
|
Anthony Chiu
(Manchester) |
Michael Dymond
(Birmingham) |
Michal Ferov
(Southampton) |
Adam Firkin
(Birmingham) |
|
Matthew Gadsden
(Sheffield) |
Belema Gancarz
(Durham) |
Kostas Georgiadis
(Loughborough) |
Oliver Goodbourn
(York) |
|
Thomas Harris
(Southampton) |
Andrew Jones
(Sheffield) |
Daniel Jones
(Durham) |
Fiachra Knox
(Birmingham) |
|
Benjamin Lang
(York) |
John Lapinskas
(Birmingham) |
Nicholas Loughlin
(Newcastle) |
Umberto Lupo
(York) |
|
Keith McCabe
(Durham) |
Gregory McKay
(East Anglia) |
David OSullivan
(Sheffield) |
Laura Phillips
(Manchester) |
|
Alan READING
(Birmingham) |
Felix Rehren
(Birmingham) |
Andrew Steele
(Nottingham) |
Simon StJohn-Green
(Southampton) |
|
Dragomir Tsonev
(Loughborough) |
Bryan Williams
(Liverpool) |
Dan Dan Yang
(York) |
Rida-e Zenab
(York) |
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: right adjoints preserve limits
- Limit creation: general adjoint functor theorem, examples
- The category of Sets
Bibliography
| Categories for the working mathematician | Mac Lane |
| Handbook of Categorical Algebra: Basic category theory | Borceux |
| Category Theory | Awodey |
Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will 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
The course will be assessed by a single exam. The paper is available under the Assignments tab, and the deadline is 13th April 2012.
Assignments
Exam for MAGIC009: Category Theory Spring 2012
| Files: | Exam paper | ||
| Deadline: | Friday 13 April 2012 (431.4 days ago) | ||
Files
Files marked L are intended to be displayed on the main screen during lectures.