2023 Faculty Courses School of Computing Department of Mathematical and Computing Science Graduate major in Mathematical and Computing Science
Topics on Mathematical and Computing Science OA
- Academic unit or major
- Graduate major in Mathematical and Computing Science
- Instructor(s)
- Kazuyuki Asada
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MCS.T424
- Number of credits
- 200
- Course offered
- 2023
- Offered quarter
- 1-2Q
- Syllabus updated
- Jul 8, 2025
- Language
- Japanese
Syllabus
Course overview and goals
basic category theory, and its relation to theoretical computer science and logic
Course description and aims
categorical universality, categorical treatment of algebra, and their application to theoretical computer science and logic
Keywords
category, functor, natural transformation, adjunction, representability, universality, Yoneda Lemma, limit, monad
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
whiteboard (and/or slide)
Schedule: the 4th and 5th periods on June 5 and the 3rd, 4th, and 5th periods from June 6 to June 9 (rooms will be notified to registered students by e-mail).
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | category, functor, natural transformation | their definitions and examples |
| Class 2 | (co)product, category equivalence, opposite category | a first step of category theory |
| Class 3 | examples of (co)limits | a first step of universality |
| Class 4 | limit and universal arrow | the basics of universality |
| Class 5 | whiskering, horizontal composition, adjunction | understanding adjunction |
| Class 6 | adjunction and limit | understanding their relation |
| Class 7 | limits in a functor category | basic properties of (co)limits in a functor category |
| Class 8 | presheaf, Yoneda Lemma, representability | category and set |
| Class 9 | universal algebra and monad | categorical generalization of algebra |
| Class 10 | Lawvere theory | categorical algebra and categorical semantics |
| Class 11 | cartesian closed category | categorical understanding of higher-order functions |
| Class 12 | another aspect of monad | an application of monad to computer science |
| Class 13 | initial algebra, final coalgebra, and (co)induction | categorical understanding of (co)induction |
| Class 14 | fibration | logical relation |
Study advice (preparation and review)
Textbook(s)
Basically, one can consult the following online free textbook, but some themes of the lecture are not treated in the book.
Basic Category Theory, Tom Leinster (Available freely at https://www.maths.ed.ac.uk/~tl/bct/ ）（There is also a version of this book translated into Japanese.）
Other books are also introduced in the lecture.
Reference books, course materials, etc.
A Resume is handed out at the lecture, by which one can prepare or review for the lecture with the textbook.
Evaluation methods and criteria
By a report.
Related courses
- MCS.T231 ： Algebra
- MCS.T221 ： Set and Topology II
Prerequisites
You should have elementary knowledge on set theory (function, product set, quotient set, poset, monotonic function, etc.), be able to write smoothly the definitions of any one of the notions of monoid, group, ring or field, and of homomorphism between them, and know at least three concrete examples of them.