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 $S4$.

Lecturer
Shehtman Valentin Borisovich

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