MAGIC009: Category Theory

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. Limits 
4. Duality and colimits 
5. Preservation of limits and colimits
6. Equivalences of categories
7. Adjunctions8. Equivalent characterisation of adjunctions
9. Presheaves 
10. The Yoneda lemma

Recommended textbooks:

Leinster, Tom. Basic category theory. Cambridge Studies in Advanced Mathematics, 143. Cambridge University Press, Cambridge, 2014. Available as a preprint: https://arxiv.org/abs/1612.09375

Mac Lane, Saunders. Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998.

Adámek, Jiří; Herrlich, Horst; Strecker, George E. Abstract and concrete categories: the joy of cats. Reprint of the 1990 original [Wiley, New York; MR1051419]. Repr. Theory Appl. Categ. No. 17 (2006), 1--507. Available at: http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf

Awodey, Steve. Category theory. Second edition. Oxford Logic Guides, 52. Oxford University Press, Oxford, 2010. xvi+311 pp. ISBN: 978-0-19-923718-0