Categorical Dualities in Logic and Algebra
A.Y. 2024/2025
Learning objectives
This course is an introduction to categorical duality theories between algebra and topology in various mathematical contexts, including Stone Duality between distributive lattice and spectral spaces and between Boolean algebras and Stone spaces, the Stone-Gelfand-Yosida theory on dualising the category of compact Hausdorff spaces, and Pontyagin Duality for Abelian groups. The course provides categorical and algebraic tools and methods of sufficient generality for the study of the mentioned duality theorems in a uniform conceptual framework.
Expected learning outcomes
Understanding of the theory of dual concrete adjunctions and its conceptual significance; development of specific detailed expertise on several classical duality theories.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
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.
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.
Prerequisites for admission
No specific requirements. While some familiarity with the language of category theory will be of help, all necessary concepts are presented during the course.
Teaching methods
Blackboard, slides, handouts.
Teaching Resources
For background in category theory, the relevant parts of:
1. S. MacLane, "Categories for the Working Mathematician", Springer, 1978
2. E. Riehl. "Category Theory in Context", Dover, 2014
For the main topics of the course:
3. P. Johnstone, "Stone Spaces", Cambridge University Press, 1986.
Further references on specific topics will be provided during the course.
1. S. MacLane, "Categories for the Working Mathematician", Springer, 1978
2. E. Riehl. "Category Theory in Context", Dover, 2014
For the main topics of the course:
3. P. Johnstone, "Stone Spaces", Cambridge University Press, 1986.
Further references on specific topics will be provided during the course.
Assessment methods and Criteria
Written test and subsequent interview on the topics of the course. Specific projects agreed upon with the instructor by way of partial or complete replacement of the final interview.
MAT/01 - MATHEMATICAL LOGIC - University credits: 6
Lessons: 42 hours
Professor:
Marra Vincenzo
Shifts:
Turno
Professor:
Marra VincenzoEducational website(s)
Professor(s)
Reception:
By appointment
Dipartimento di Matematica "Federigo Enriques", via Cesare Saldini 50, room 2048