MAGIC009: Category Theory

Course details

A core MAGIC course


Autumn 2013
Monday, October 7th to Saturday, December 14th


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55


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.


Category theory is an abstract algebraic point of view of mathematics. Some familiarity with an algebraic way of thinking is important. It is therefore an advantage to have studied an undergraduate course in group theory or ring theory, or some other abstract algebra course. I will assume some knowledge of algebra such as vector spaces and their bases, and groups, but a basic 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


  • JK

    Dr Jonathan Kirby

    University of East Anglia


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The assessment for this course will be via a single take-home paper in January and with 2 weeks to complete and submit online. The intention is that a student who has studied throughout the term will be able to pass by spending 2 hours on the exam.

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