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This course is part of the MAGIC core.


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.


Spring 2012 (Monday, January 16 to Friday, March 23)


  • Mon 10:05 - 10:55


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.


The topics covered are:
  1. Categories: definitions, examples, special kinds of arrows and objects, duality
  2. Functors: definitions, examples, full and faithful functors, subcategories, Hom-functors, contravariant functors
  3. Universal properties: examples including vector space bases, fields of fractions, tensor products, quotients, products, and coproducts
  4. Natural transformations: definitions and examples, functor categories, equivalence of categories, horizontal composition
  5. Limits: examples, general definition, computing limits in Set, complete categories
  6. Colimits: definition, examples, computing colimits in Set
  7. Adjunctions: vector space bases, formal definition, examples, unit and counit
  8. Limit preservation: right adjoints preserve limits
  9. Limit creation: general adjoint functor theorem, examples
  10. The category of Sets


Photo of Bana Al Subaiei
Bana Al Subaiei
Photo of Abdulsatar Al-Juburie
Abdulsatar Al-Juburie
Photo of Kalyan Banerjee
Kalyan Banerjee
Photo of Yumi Boote
Yumi Boote
Photo of Tom Brookfield
Tom Brookfield
Photo of Christopher Cave
Christopher Cave
Photo of Yu-Yen Chien
Yu-Yen Chien
Photo of Anthony Chiu
Anthony Chiu
Photo of Michael Dymond
Michael Dymond
Photo of Michal Ferov
Michal Ferov
Photo of Adam Firkin
Adam Firkin
Photo of Matthew Gadsden
Matthew Gadsden
Photo of Belema Gancarz
Belema Gancarz
Photo of Kostas Georgiadis
Kostas Georgiadis
Photo of Tom Harris
Tom Harris
Photo of Andrew Jones
Andrew Jones
Photo of Daniel Jones
Daniel Jones
Photo of Fiachra Knox
Fiachra Knox
Photo of Benjamin Lang
Benjamin Lang
Photo of John Lapinskas
John Lapinskas
Photo of Nicholas Loughlin
Nicholas Loughlin
Photo of Umberto  Lupo
Umberto Lupo
Photo of Keith McCabe
Keith McCabe
Photo of Gregory McKay
Gregory McKay
(East Anglia)
Photo of David OSullivan
David OSullivan
Photo of Laura Phillips
Laura Phillips
Photo of Alan READING
Photo of Andrew Steele
Andrew Steele
Photo of Simon StJohn-Green
Simon StJohn-Green
Photo of Dragomir Tsonev
Dragomir Tsonev
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Bryan Williams
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Dan Dan Yang
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Rida-e Zenab


Categories for the working mathematicianMac Lane
Handbook of Categorical Algebra: Basic category theoryBorceux
Category TheoryAwodey


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.)


The course will be assessed by a single exam. The paper is available under the Assignments tab, and the deadline is 13th April 2012.

Exam for MAGIC009: Category Theory Spring 2012

Files:Exam paper
Deadline: Friday 13 April 2012 (1323.7 days ago)