**Duality and categorical logic** **M2 [LMFI](https://master.math.u-paris.fr/annee/m2-lmfi/), 2ème periode, 2024-2025** **[Université Paris Cité](https://www.u-paris.fr)** Practical information ========================== * **Lectures**: Thursday, 16h15-18h15, Olympe de Gouges, room 130 * **Lecture period**: 6 January - 27 March 2025 * **Exam period**: 7 - 11 April 2025 * **Lecturer**: [Sam van Gool](https://samvangool.net) * **Exercises**: There are five optional exercise sessions on Tuesday, 16:30-18:30, taught by [Joshua Wrigley](https://jlwrigley.github.io), see below. * **Evaluation**: Homework (50%) and final exam (50%). * **References**: + [(GG)](./dualitybook.html), textbook on topological duality for distributive lattices. Course content ========================== ### 1. 9 January 2025 * Introduction, order and lattices, duality for finite distributive lattices (GG, Chapter 1). ### 2. 16 January * Duality for morphisms finite posets - finite distributive lattices, adjunctions between posets, examples (GG, Chapter 1). ### TD1. 21 January (Sophie Germain 1014) * [Exercise sheet 1](./dcl-td1.pdf) ### 3. 23 January * Homework 1 available ### 4. 30 January * Homework 1 deadline at start of class (16:15) ### TD2. 4 February (Sophie Germain 1014) * Exercise sheet 2 ### 5. 6 February ### 6. 13 February ### TD3. 18 February (Halle aux Farines 280F) * Exercise sheet 3 ### 7. 20 February ### 8. 27 February ### TD4. 4 March (Halle aux Farines 280F) * Exercise sheet 4 ### 9. 6 March ### 10. 13 March ### TD5. 18 March (Halle aux Farines 280F) * Exercise sheet 5 ### 11. 20 March ### 12. 27 March Syllabus ========================== This is a course on Stone-Priestley duality theory and categorical logic. The main goal is to provide students with the necessary background to be able to start independently reading current research in this field. We will start from bounded distributive lattices, which are fundamental structures in logic, capturing an extremely basic language that contains as its only primitives "or", "and", "true", and "false". Stone showed that distributive lattices are in a duality with a class of topological spaces with non-trivial specialization order. Priestley re-framed this duality as one between distributive lattices and certain partially ordered topological spaces. Duality theory has since then found applications in a number of areas within logic and the foundations of computer science. The first part of the course will introduce the mathematical foundations of the theory, also introducing along the way the necessary order theory, topology, and category theory. In the second part of the course, we will discuss applications of the theory to logic, first to intuitionistic propositional and modal logics, and then to higher order logics. This last part will naturally lead to discussing concepts and methods from categorical logic and possibly also topos theory. The precise topics treated in this part will also depend on student interest. Some basic knowledge of category theory and topology will be helpful, although not strictly required.