Category Theory Course, MT2008

Monday and Wednesday, 11am, SR1

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, the most important use of categories. There are almost no formal prerequisites, so anyone with some "mathematicial maturity" is welcome, but the course will be aimed at graduate students.

Topics will include:

• Categories, Functors, and Natural Transformations
• Universal properties
• Representable Functors: Yoneda lemma, Yoneda embedding
• Limits and Colimits: Limits in Set, limit preservation, reflection and creation, Representable functors preserve limits
• Adjunctions: Unit and counit, triangle identities, definition via initial objects in comma categories, reflective subcategories
• Limits and adjunctions: Limits (and colimits) as adjoint functors, right adjoints preserve limits
• Adjoint Functor Theorems: applications

Books

Exercises

• Sheet 1: Categories and Functors
• Sheet 2: Universal Properties
• Sheet 3: Natural Transformations
• Sheet 4: Representable Functors
• Sheet 5: Limits and Colimits
• Sheet 6: Adjunctions
• Sheet 7: Miscellaneous Questions

Other resources

• Notes by Tom Leinster about the Yoneda Lemma (in ps format)
• The n-category cafe describes itself as "a group blog on math, physics and philosophy", and is mainly written by category theorists.
• Set theory for category theory is a survey paper by Michael Shulman on set-theoretic foundations.
• Some more links due to Jacob Biamonte