MAGIC009: Category Theory

Course details

A core MAGIC course


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


Live lecture hours
Recorded lecture hours
Total advised study hours


09:05 - 09:55


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


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.


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:
  • 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.


  • JK

    Dr Jonathan Kirby

    University of East Anglia


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Assessment information will be available nearer the time.


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