**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), M. Gehrke and S. v. Gool, [Topological duality for distributive lattices: Theory and Applications](./dualitybook.html) (2024). + (MM) S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory (1994). + (S) T. Streicher, [Introduction to Category Theory and Categorical Logic](https://www2.mathematik.tu-darmstadt.de/~streicher/CTCL.pdf) (2004). 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 * Some remarks on topology (GG, Chapter 2); Prime filter theorem, Stone representation of distributive lattices (GG, Chapter 3). * [Homework 1](./dcl-hw1.pdf) available. Due Thursday 30 January, 16h15. ### 4. 30 January * Priestley duality (GG, Chapter 3). * Homework 1 due at start of class, 16h15. ### TD2. 4 February (Sophie Germain 1014) * [Exercise sheet 2](./dcl-td2.pdf) ### 5. 6 February * Duality methods: duals of free distributive lattices, Cantor space, modal algebras, compatible relations (GG, Chapter 4). ### 6. 13 February * Compatible relations continued, intuitionistic logic, Heyting algebras, Esakia duality (GG, Chapter 4). ### TD3. 18 February (Halle aux Farines 280F) * [Exercise sheet 3](./dcl-td3.pdf) ### 7. 20 February * Class cancelled. * [Homework 2](./dcl-hw2.pdf) available. Due Thursday 6 March, 16h15. ### 8. 27 February * Categories of presheaves, Yoneda, finite limits (terminal object + pullbacks), subobjects, definition of elementary topos (parts of MM, Chapters 0 and 1). ### TD4. 4 March (Halle aux Farines 280F) * [Exercise sheet 4](./dcl-td4.pdf) ### 9. 6 March * Review of definition of subobjects and power object * Pullback of mono is mono * Subobject posets have finite meets * Unpacking the definition of elementary topos (MM Sec. IV.1) * Subobject classifiers (MM Sec I.3) * Examples of subobject classifiers: in Set, in the presheaves on the poset 2, in the presheaves on the poset N (MM Sec I.4) ### 10. 13 March * Cartesian closed categories and their relation to elementary toposes * Subobjects in a presheaf category * Exponentials in a presheaf category * Sheaves on a topological space * Roughly corresponding to these [notes](./toposnotes.pdf), excluding Section 5. ### TD5. 18 March (Halle aux Farines 280F) * [Exercise sheet 5](./dcl-td5.pdf) ### 11. 20 March * The internal language of a topos (MM, Sec. VI.5) * Logical connectives internally: conjunction and implication (S, Sec. 13) * [Homework 3](./dcl-hw3.pdf) available. Due Thursday 3 April, 16h15. ### 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.