Introductory description

Mathematical structures come equipped with a notion of "maps", or "morphisms", which are used to compare objects with such structure. For example, we may not want to distinguish between the sets {1,2,3} and {a,b,c} by noticing that we can map the elements of the first set to the second bijectively.
This principle extends to virtually all mathematical objects. To name a few: one studies groups via group homomorphisms, vector spaces via linear maps, spaces via continuous maps, manifolds via smooth maps, probability spaces via measurable functions, paths via homotopies.

A category consists of a collection of objects, a collection of morphisms, and a composition rule. Many common mathematical constructions only use the structure of a category, for example coproducts, direct sums, kernels, quotients, coinvariants, compactifications. The theme of the module will be to investigate how vaguely analogous constructions in various areas of mathematics are in fact instances of the same construction carried out at the level of a category.

Our primary reference will be "Category theory in context" by Emily Riehl.

The module will cover the notions of categories, functors, natural transformations, limits and colimits, adjunctions, and Freyd's adjoint functor theorem. We will then apply the category theoretical framework to formulate the Seifert Van-Kampen theorem for the fundamental groupoid, and use it to calculate the fundamental group of the circle.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

-Categories, functors and natural transformations
-Representable functors and the Yoneda Lemma
-Limits and colimits
-Adjunctions and Freyd's Theorem
-The fundamental groupoid

Learning outcomes

By the end of the module, students should be able to:

• Explain the definitions and properties of the basic notions of category theory
• Understand adjunctions and the importance of Freyd's adjoint functor theorem
• Use the framework of category theory to prove results by universal property
• Recognise common constructions as instances of categorical constructions
• Describe the fundamental groupoid of a gluing of spaces from that of its pieces

Interdisciplinary

Category theory is being increasingly applied in computer science, for example in semantics and functional programming . This module would provide a solid foundation for venturing into this area of modern computer science.

Subject specific skills

Ability to apply categorical principles to various areas of mathematics
Ability to work in the abstract formalism of category theory

Transferable skills

Ability to translate scientific ideas into mathematical language
Ability to communicate complex ideas and mathematical results clearly
Ability to analyse and solve abstract mathematical problems