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Course by V. B. Shehtman "Algebraic logic and categories"
(September 14–December 14, 2023, Steklov Mathematical Institute, Room 104 (8 Gubkina))

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  1. Definition and examples of categories. Dual category. Isomorphic objects.
  2. Definition and examples of functors. Isomorphism and equivalence of categories. Concretazibility of small categories.
  3. Limits in categories. Complete categories.
  4. Adjoint functors. Freyd's theorem.
  5. The category of $\Omega$-algebras. Term algebras.
  6. The freedom functor as a left adjoint to the forgetting functor.
  7. Equational theories and varieties of algebras. Completeness of varieties.
  8. The freedom functor for a variety. Lindenbaum-Tarski algebras.
  9. Birkhoff's theorem on varieties.
  10. Finite, locally finite, and finitely approximable varieties. Harrop's theorem on decidability.
  11. Semilattices, lattices, distributive lattices.
  12. The spectrum of a distributive lattice. Birkhoff-Stone theorem on representability of distributive lattices.
  13. Boolean algebras. Stone representation theorem for Boolean algebras. Finite Boolean algebras.
  14. Modular and projective lattices.
  15. Heyting lattices and Heyting algebras.
  16. Superintuitionistic propositional logics. Soundness and completeness theorems with respect to Heyting algebras.
  17. Intuitionistic Kripke frames and their Heyting algebras.
  18. The canonical Kripke frame of a Heyting algebra. Completeness of intuitionistic logic wrt Kripke frames.
  19. Normal modal algebras. Modal algebras of Kripke frames.
  20. Tarski-Jonsson representation theorem. The canonical Kripke frame of a modal algebra.
  21. McKinsey-Tarski theorem on translation of intuitionistic logic into $S4$.

Program

Lecturer
Shehtman Valentin Borisovich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no.  075-15-2022-265).



Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center


Course by V. B. Shehtman "Algebraic logic and categories", September 14–December 14, 2023

December 14, 2023 (Thu)
1. Lecture 14. Algebraic logic and categories
V. B. Shehtman
December 14, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

December 7, 2023 (Thu)
2. Lecture 13. Algebraic logic and categories
V. B. Shehtman
December 7, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

November 30, 2023 (Thu)
3. Lecture 12. Algebraic logic and categories
V. B. Shehtman
November 30, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

November 23, 2023 (Thu)
4. Lecture 11. Algebraic logic and categories
V. B. Shehtman
November 23, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

November 16, 2023 (Thu)
5. Lecture 10. Algebraic logic and categories
V. B. Shehtman
November 16, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

November 9, 2023 (Thu)
6. Lecture 9. Algebraic logic and categories
V. B. Shehtman
November 9, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

November 2, 2023 (Thu)
7. Lecture 8. Algebraic logic and categories
V. B. Shehtman
November 2, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

October 26, 2023 (Thu)
8. Lecture 7. Algebraic logic and categories
V. B. Shehtman
October 26, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

October 19, 2023 (Thu)
9. Lecture 6. Algebraic logic and categories
V. B. Shehtman
October 19, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

October 12, 2023 (Thu)
10. Lecture 5. Algebraic logic and categories
V. B. Shehtman
October 12, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

October 5, 2023 (Thu)
11. Лекция 4. Алгебраическая логика и категории
V. B. Shehtman
October 5, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

September 28, 2023 (Thu)
12. Лекция 3. Алгебраическая логика и категории
V. B. Shehtman
September 28, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

September 21, 2023 (Thu)
13. Лекция 2. Алгебраическая логика и категории
V. B. Shehtman
September 21, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  

September 14, 2023 (Thu)
14. Лекция 1. Алгебраическая логика и категории
V. B. Shehtman
September 14, 2023 18:00, Steklov Mathematical Institute, Room 104 (8 Gubkina)
V. B. Shehtman
  
 
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