C2.7 Category Theory (2026-27)
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- Lecturer: Profile: Andre Henriques
Course information
General prerequisites:
There are no essential prerequisites, but familiarity with the basic theory of groups, rings, vector spaces, modules, and topological spaces would be very useful. Other topics such as Algebraic Geometry, Algebraic Topology, Homological Algebra and Representation Theory are relevant. Category Theory also has links with Logic and Set Theory, but this course will not stress these.
Course term: Michaelmas
Course lecture information: 16 lectures.
Course weight: 1
Course level: M
Assessment type: Written Examination
Course overview:
Category theory brings together many areas of pure mathematics (and also has close links to logic and to computer science). It is based on the observation that many mathematical topics can be unified and simplified by using descriptions in terms of diagrams of arrows; the arrows represent functions of suitable types. Moreover, many constructions in pure mathematics can be described in terms of 'universal properties' of such diagrams.
The aim of this course is to provide an introduction to category theory using a host of familiar examples, to explain how these examples fit into a categorical framework, and to use categorical ideas to make new constructions.
The aim of this course is to provide an introduction to category theory using a host of familiar examples, to explain how these examples fit into a categorical framework, and to use categorical ideas to make new constructions.
Learning outcomes:
Students will have developed a thorough understanding of the basic concepts and methods of category theory. They will be able to work with commutative diagrams, naturality and universality properties and adjoint functors, and to apply categorical ideas and methods in a wide range of areas of mathematics.
Course synopsis:
Introduction: universal properties in linear and multilinear algebra.
Categories, functors, natural transformations. Examples including categories of sets, groups, rings, vector spaces and modules, topological spaces. Groups, monoids and partially ordered sets as categories. Opposite categories and the principle of duality. Covariant, contravariant, faithful and full functors. Equivalences of categories.
Adjoints: definition and examples including free and forgetful functors and abelianisations of groups. Adjunctions via units and counits, adjunctions via initial objects.
Representables: definitions and examples including tensor products. The Yoneda lemma and applications.
Limits and colimits, including products, equalizers, pullbacks and pushouts. Monics and epics. Interaction between functors and limits.
Monads and comonads, algebras over a monad, Barr-Beck monadicity theorem (proof not examinable). The category of affine schemes as the opposite of the category of commutative rings.
Rudiments of higher category theory (non-examinable).
Categories, functors, natural transformations. Examples including categories of sets, groups, rings, vector spaces and modules, topological spaces. Groups, monoids and partially ordered sets as categories. Opposite categories and the principle of duality. Covariant, contravariant, faithful and full functors. Equivalences of categories.
Adjoints: definition and examples including free and forgetful functors and abelianisations of groups. Adjunctions via units and counits, adjunctions via initial objects.
Representables: definitions and examples including tensor products. The Yoneda lemma and applications.
Limits and colimits, including products, equalizers, pullbacks and pushouts. Monics and epics. Interaction between functors and limits.
Monads and comonads, algebras over a monad, Barr-Beck monadicity theorem (proof not examinable). The category of affine schemes as the opposite of the category of commutative rings.
Rudiments of higher category theory (non-examinable).
Section outline
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You are welcome to attempt these questions before the course starts as an introduction to some of the ideas involved, but your solutions will not be marked. For more details see the introduction to the book 'Basic Category Theory' by T. Leinster which is available online on arXiv.org at arXiv:1612.09375.
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The relevant section in the lecture notes is Section 1. It covers the most basic definitions of a category, functors, natural transformations, equivalences.
The problem sheet is divided in three section, called A, B, and C.
• Section A consists of Q1 and Q2.
• Section B consists of Q3 and Q4, and should be turned in for marking.
• Section C consists of Q5 and Q6. -
The relevant material for this problem sheet is in Chapter 2 of the lecture notes. It covers topics such as adjoint functors and Yoneda's embedding, representable functors. Do not turn in, this sheet will not be marked; solutions will be provided later.
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This problem sheet covers limits and colimits. The relevant part of the lecture notes is Chapter 3.
The problem sheet is divided in three section, called A, B, and C.
• Section A consists of Q1 and Q2.
• Section B consists of Q3 and Q4, and should be turned in for marking.
• Section C consists of Q5. -
This problem sheet covers functors and limits/colimits, monads, and the Barr-Beck Theorem. The relevant material is in Sections 4 and 5 of the lecture notes. Do not turn in, solutions will be provided later.
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Registration start: Monday, 5 October 2026, 12:00 PMRegistration end: Friday, 6 November 2026, 12:00 PM
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Class Tutor's Comments Assignment
Class tutors will use this activity to provide overall feedback to students at the end of the course.
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