Categorical Dualities in Logic and Algebra

Note: Only part of the syllabus is covered in class. Usually all topics up to and including item 7 are covered. The remaining items are only partially addressed, if time allows, depending on the audience's interests.

1. Short historical and conceptual introduction. Motivating examples of dualities.
2. Category theory background.
3. General theory of concrete dual adjunctions. Dualising objects. Descent to the dually equivalent full subcategories.
4. Background on lattices and Boolean algebras.
5. Background on spectral, Priestley, and Stone spaces.
6. Stone duality, I: Distributive lattices and Spectral spaces.
7. Stone Duality, II: Boolean algebras and Stone spaces. Hints to further topics (the duality between probability and measure).
8. Background on Riesz spaces (vector lattices).
9. Stone-Gelfand-Yosida Duality: Vector lattices and Compact Hausdorff spaces. Hints to further topica (commutative C*-algebras; states; Riesz Representation Theorem.)
10. Background on topological groups and the Peter-Weyl Theorem.
11. Pontryagin Duality: Abelian groups and compact Abelian groups. Hints to further topics (Haar measure; Fourier transform).
12. Hints to recent research developments in duality theory.