MAGIC009: Category Theory

Course details

A core MAGIC course


Autumn 2022
Monday, October 3rd to Friday, December 9th


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55 (UK)


Category theory begins with the observation that the collection of mathematical objects of a given kind (groups, topological spaces, graphs, etc...) together with the appropriate mappings between them (group homomorphism, continuous function, graph morphism) is an interesting mathematical structure in its own right: a category. 

Category theory, i.e. the study of categories, provides tools that can be applied uniformly to different kinds of mathematical structures and a convenient language to relate them precisely. This course will be an introduction to category theory. The main theme will be universal properties in their various manifestations, which is one of the most important uses of category theory in mathematics. Apart from introducing specific concepts and presenting the key results, one of the goals of the module is to teach you how to reason categorically.


It will be useful to have taken an undergraduate course in group theory or commutative algebra or some other abstract algebra course.

I will try to illustrate notions with examples from different areas, but you may find it useful to come up with examples from your preferred areas. 


  • Categories and functors. 
  • Universal arrows. 
  • Natural transformations and functor categories. 
  • Colimits. 
  • Duality and limits. 
  • Adjunctions. 
  • Preservation of limits. 
  • Presheaves and the Yoneda lemma.
  • Representables and limits
  • Kan extensions.


  • NG

    N. Gambino

    University of Manchester


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.


The assessment for this course will be released on Monday 9th January 2023 at 00:00 and is due in before Sunday 22nd January 2023 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.


Only current consortium members and subscribers have access to these files.

Please log in to view course materials.


Please log in to view lecture recordings.