Advanced Topics in Category Theory

Teaching

The teaching style will be lecture-based, but supported by a practical component where students will learn to use a proof assistant for higher category theory, and build a small portfolio of proofs. Towards the end of the course we will explore some of the exciting computer science research literature on monoidal and higher categories, and students will choose a paper and present it to the class.

Aims

The module will introduce advanced topics in category theory. The aim is to train students to engage and start modern research on the mathematical foundations of higher categories, the graphical calculus, monoids and representations, type theories, and their applications in theoretical computer science, both classical and quantum.

Objectives

On completion of this module, students should:

• Be familiar with the techniques of compositional category theory.
• Have a strong understanding of basic categorical semantic models.
• Begun exploring current research in monoidal categories and higher structures.

Syllabus

Part 1, lecture course:
The first part of the course introduces concepts from monoidal categories and higher categories, and explores their application in computer science.
‐ Monoidal categories and the graphical calculus
‐ The proof assistant homotopy.io
‐ Coherence theorems and higher category theory
‐ Linearity, superposition, duality, quantum entanglement
‐ Monoids, Frobenius algebras and bialgebras
‐ Type theory for higher category theory

Part 2, exploring the research frontier:
In the second part of the course, students choose a research paper to study, and give a presentation to the class.
There is a nice varied literature related to the topics of the course, and the lecturer will supply a list of suggested papers. 

Classes

There will be four exercise sheets for homework, with accompanying classes by a teaching assistant to go over them.

Assessment

• Problem sheets (50%)
• Class presentation (20%)
• Practical portfolio (30%)

Reading List

Chris Heunen and Jamie Vicary, “Category for Quantum Theory: An Introduction”, Oxford University Press