Algebraic and Categorical Logic
A.Y. 2025/2026
Learning objectives
The course is an introduction to the algebraic and categorical aspects of logic, including the Lindenbaum-Tarski construction, algebraic representation theory, the internal logic of a doctrine/category, and syntactic categories. (Conceptual) completeness theorems for various logics will be studied from an algebraic and categorical perspective. The course presents sufficiently general methods and tools for the study of these concepts in a uniform framework.
Expected learning outcomes
Understanding of the fundamental tools of algebraic and categorical logic; development of specific skills on propositional and first-order logics, both classical and non-classical.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
Note: Only part of the syllabus indicated below is covered in class. Up to point 8 the programme is usually covered in its entirety. Subsequent topics, time permitting, are also partially covered depending on the interests expressed by the students.
1. Brief historical and conceptual introduction.
2. Background on posets and adjunctions.
3. Background on distributive lattices, Heyting algebras, and Boolean algebras.
4. Lindenbaum-Tarski construction and category theory background.
5. Representation theorems, and completeness for propositional logics (classical, coherent, intuitionistic). Interpolation and definability.
6. Doctrines and internal language.
7. Syntactic categories.
8. Conceptual completeness for first-order logics (classical, coherent).
9. Grothendieck toposes and classifying toposes.
10. Localic toposes and Deligne's theorem.
11. Hints to recent research developments.
1. Brief historical and conceptual introduction.
2. Background on posets and adjunctions.
3. Background on distributive lattices, Heyting algebras, and Boolean algebras.
4. Lindenbaum-Tarski construction and category theory background.
5. Representation theorems, and completeness for propositional logics (classical, coherent, intuitionistic). Interpolation and definability.
6. Doctrines and internal language.
7. Syntactic categories.
8. Conceptual completeness for first-order logics (classical, coherent).
9. Grothendieck toposes and classifying toposes.
10. Localic toposes and Deligne's theorem.
11. Hints to recent research developments.
Prerequisites for admission
Mathematical Logic 1. Familiarity with the language of category theory will be helpful, but all necessary notions will be presented in class.
Teaching methods
Blackboard and lecture notes.
Teaching Resources
For background in category theory, the relevant parts of:
T. Leinster, "Basic Category Theory", Cambridge University Press, 2014 (available online: https://arxiv.org/abs/1612.09375)
T. Leinster, "Basic Category Theory", Cambridge University Press, 2014 (available online: https://arxiv.org/abs/1612.09375)
Assessment methods and Criteria
Oral examination.
MAT/01 - MATHEMATICAL LOGIC - University credits: 6
Lessons: 42 hours
Professors:
Pasquali Fabio, Reggio Luca
Professor(s)