Category theory in foundations of mathematics and computer science 1000-2M20TKJ
Category theory is based on the observation that many internal properties of mathematical objects can be externally described in terms of morphisms between objects. The advantage of this perspective is that it gives a direct way to transfer definitions, notions and ideas among different branches of mathematics and computer science. This often leads to the emergence of new fields of science. Examples include pointless topology, non-commutative geometry, Grothendieck motives, nominal automata, monadic second-order logic, and many more.
The aim of this course is to provide an introduction to category theory in foundations of mathematics and computer science. Purely theoretical topics are interspersed with practical ones including automata theory, languages and computations in various categories (such as: definable sets, vector spaces, topological spaces, etc.).
Programme:
1. Categories. Basic concepts and definitions.
2. Functors and natural transformations.
3. Yoneda lemma.
4. Adjunctions.
5. Automata in categories. Recognizable languages.
6. Fibrations.
7. Logic in categories.
8. Coherent categories and sets definable in positive-existential first-order theories.
9. Pretoposes and elimination of imaginaries.
10. Classifying toposes.
11. Computations in classifying toposes and pretoposes.
Type of course
Learning outcomes
Knowledge:
The student
1. knows basic notions and results in category theory;
2. knows basic notions and results in automata theory, languages theory and computability theory in categories;
3. understands the way category theory can be applied in foundations of mathematics and computer science.
Abilities:
The student
1. is able to prove basic theorems of category theory;
2. is able to prove basic theorems of automata theory, language theory and computability theory in categories;
3. is able to study effectiveness of computations in pretoposes;
4. is able to transfer basic definitions, notions and ides between different branches of mathematics and computer science.
Social competences:
The student
1. knows limitations of own knowledge and understands the need of further education, in particular in acquiring the knowledge out of the current field;
2. knows how to precisely formulate questions that serve to deepen own understanding of a given subject or to find missing elements of reasoning.
Assessment criteria
Written take-home exam.
Bibliography
• Adamek, Jiri, and Vera Trnková. Automata and algebras in categories. Vol. 37. Springer Science & Business Media, 1990.
• Awodey, Steve. Category theory. Oxford university press, 2010.
• Borceux, Francis. Handbook of Categorical Algebra: Volume 1, 2, 3. Cambridge University Press, 1994.
• Hodges, Wilfrid, and Hodges Wilfrid. Model theory. Cambridge University Press, 1993.
• Jacobs, Bart. Categorical logic and type theory. Elsevier, 1999.
• Johnstone, Peter T. Sketches of an Elephant: A Topos Theory Compendium: 2 Volume Set, Oxford University Press, 2002.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Computer Science
- Master's degree, second cycle programme, Computer Science
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: