Duality and categorical logic

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.

Bibliography

The course is mainly based on parts of the textbook [1]. For the categorical logic part, we may also draw from parts of [2].

[1] Gehrke, M. and Gool, S. v., Topological Duality for Distributive Lattices: Theory and Applications, Cambridge University Press (2023).

[2] Lambek, J. and Scott, P. J., Introduction to Higher Order Categorical Logic. Cambridge University Press (1986).

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• Bâtiment Sophie Germain
  8 place Aurélie Nemours
  75013 Paris