MMath Mathematics

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In Mathematics we study mathematical objects (such as groups and topological spaces) not only in isolation, but also in relation to each other, by considering the appropriate kind of morphisms between them (such as group homomorphisms and continuous functions, respectively). This idea leads naturally to Category Theory, one of the most important areas of modern Mathematic, with widespread applications.

Category Theory allows us to establish precise analogies between different parts of mathematics and to discover unexpected connections between them. Because of this, the unit should be of interest to a wide range of students of Pure Mathematics.

In the unit, you will first learn about the basic notions and results of Category Theory, namely categories, functors, natural transformations, equivalences, adjunctions, and limits. You will then see an application of Category Theory by exploring how fundamental notions of Logic can be phrased and studied using category-theoretic tools. Throughout the unit, definitions and theorems will be illustrated with concrete examples from across Mathematics.

The unit will assume working familiarity with rigorous treatment of the basic mathematical language (sets, functions, relations) and ability to understand, construct and write simple mathematical proofs. 
 

Pre-requisites: MATH21120 Groups and Geometry and MATH21112 Rings and Fields

The unit aims to introduce the fundamental concepts, techniques, and results of Category Theory, and to illustrate how they can be applied in Logic.

1. Define the notion of a category and solve simple exercises about it.
2. Define the notions of a functor and of a natural transformation and solve simple exercises about them.
3. State the equivalent characterisations of adjunctions and solve simple exercises about them.
4. Define the fundamental kinds of limits and colimits and solve simple exercises about them.
5. State and apply the fundamental results on preservation of limits by adjoint functors.
6. State the Stone duality theorem and explain the connection between Boolean algebras and topological spaces.
7. Explain the connection between Boolean algebras and propositional logic and solve simple exercises about them.

Basic notions (6 lectures, 3 weeks). Categories (2 lectures). Diagrammatic reasoning (1 lecture). Universal properties (2 lectures). Duality (1 lecture).

Functors and natural transformations (4 lectures, 2 weeks). Functors (1 lecture). Natural transformations (1 lecture). Equivalence of categories (1 lecture). Necessary and sufficient conditions for a functor to be an equivalence (1 lecture)

Adjunctions and limits (6 lectures, 3 weeks). Adjunctions (1 lecture). Characterisations of adjunctions (3 lectures). Limits (1 lecture). Preservation of limits by adjoint functors (1 lecture).

Categorical Logic (6 lectures, 3 weeks). Boolean algebras (1 lecture). Stone duality (3 lectures). Syntax and semantics of propositional logic (1 lecture). Soundness and completeness of propositional logic via Stone duality (1 lecture).

Review (2 lectures, 1 week). Review (2 lectures).

Feedback will be given on the weekly problem sheet assignments. Tutorials will provide an opportunity for students' work to be discussed and for feedback on their understanding to be given. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour. 

Other: Weekly formative homework

Final Exam - 100% - Generic feedback made available after exam period

Weekly formative homework - 0% - Individual feedback given

S. Awodey, Category Theory, Oxford University Press (2nd edition), 2010.

T. Leinster, Basic Category Theory, Cambridge University Press, 2014.  

E. Riehl, Category Theory in Context, Dover Publications, 2016.