Lecturer: Emanuele Dotto

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 one hour lectures

Assessment: 85% examination by 3 hour written exam, 15% coursework

Formal registration prerequisites: None

Assumed knowledge:

• MA249 Algebra II
• MA3F1 Introduction to Topology

Useful background: The module will explore examples coming from various areas of mathematics. Some level of comfort with basic definitions from the following modules will be helpful to better appreciate the module content:

• MA3G6 Commutative Algebra
• MA377 Rings and Modules
• MA3E1 Groups and Represenations
• MA3H6 Algebraic Topology

Synergies: There are set theoretic subtleties at the foundations of category theory, as well as beautiful connections with logic, that we will not dive into. MA4J7 Cohomology and Poincaré Duality.

Leads to: This module provides essential background for postgraduate studies in modern algebraic topology and algebraic geometry.

Content: Mathematical structures come equipped with a notion of "map", or "morphism", 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. Thus category theory provides a framework to study systematically those properties and constructions which can be formulated purely in terms of maps. Perhaps surprisingly, many common mathematical constructions arise in this manner, for example products, 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.

The module will cover roughly the first 4 chapters of" Category theory in context", by Emily Riehl. This is a provisional list of contents:

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

In the last section we will 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.

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

Books:

Emily Riehl, Category Theory in Context , 2016

Brown, Topology and Groupoids, 2006

Additional Resources