A Short Introduction to Stone Duality

(under construction … work in progress)

I write this with an eye on later generalisations to category theory. No category theory is needed to understand the notes, but I will hint at category theoretic generalisations to help the reader who is interested in studying category theory later.

To not burden the text, I delegate some standard definitions to footnotes.

These notes do not contain proofs. They are intended for students who, with some guidance, want to find the proofs themselves.

Part 1: Finite Duality

• Introduction and first duality theorem
• Algebraic duality

Part 2: Perspectives on Duality (Ongoing)

• Category Theory
• Universal Algebra
• Logic

Part 3: Stone Duality for Boolean Algebras (Ongoing)

• Topological duality

Part 4: Generalisations (Planned)

• Universal Algebra
• Category Theory
• Enriched Category Theory
• Order Theory

Acknowledgements

Thanks to Sri Pranav Kunda for working through these exercises and helping to shape this material.

Further Thoughts

These notes originated from the desire to teach my own area (logic, category theory, theoretical computer science) to beginning mathematics students. I now think that Stone duality can be a good topic for a general introductory mathematics course. At the end the student will have not only learned many basic techniques^[1] and concepts^[2] but also been in touch with many areas of mathematics (algebra^[3], topology^[4], logic^[5], category theory^[6], combinatorics, rewriting theory^[7], algorithms, …). Most importantly students will have seen how all these different areas, often taught in isolation, come together in the study of a single topic.

Part 1 and 2 are a good starting point as one can get quite far starting from a basic understanding of sets and some ability in algebraic manipulation. From their different directions can be taken.

1. The topological generalisation of the finite duality.
2. An algebraic and logical investigation of propositional logic.
3. The order theoretic generalisation of the finite duality.

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1. Writing proofs, definitions of mathematical objects as equivalence classes, algebraic calculations using adjoints, … ↩︎

2. Structure preserving maps, free constructions, definitions by generators and relations, … ↩︎

3. Boolean algebra, distributive lattices, universal algebra, lattice theory, order theory, … ↩︎

4. Compactness, Cantor space, … ↩︎

5. Classical propositional logic, syntax and semantics, object-language and meta-language, soundness and completeness, … ↩︎

6. Adjunctions, equivalence of categories, monads, … ↩︎

7. Rewriting to normal form, … ↩︎