MAGIC009: Category Theory

Course details

A core MAGIC course


Autumn 2023
Monday, October 2nd to Friday, December 8th


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55 (UK)

Course forum

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In mathematics, it is often useful to study mathematical objects (such as groups, vector spaces, and topological spaces) not only in isolation, but also in relation to each other by considering the appropriate kind of morphism (such as group homomorphisms, linear maps and continuous functions, respectively). This apparently simple idea led to the discovery of one of the most important concepts of 21st century mathematics, that of a category, and the development of the corresponding theory, Category Theory.
Category Theory allows us to make precise some informal analogies between different parts of mathematics and to discover unexpected connections between them, leading to deep applications in Algebra, Algebraic Geometry, Algebraic Topology and Mathematical Logic. As such, Category Theory should be of interest to a wide range of PhD students in Pure Mathematics.
In this course, you will learn about the fundamental notions and results of Category Theory, illustrated in a variety of examples. The main theme will be the `calculus' of functors and natural transformations. 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 algebra. 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. 


1. Categories
2. Functors and natural transformations
3. Equivalence of categories
4. Adjunctions
5. Equivalent characterisation of adjunctions
6. Limits
7. Duality and colimits
8. Preservation of limits
9. Presheaves 
10. The Yoneda lemma


  • NG

    N. Gambino

    University of Manchester


No bibliography has been specified for this course.


The assessment for this course will be released on Monday 8th January 2024 at 00:00 and is due in before Friday 19th January 2024 at 11:00.

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.


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