Algèbre catégorique

The first part of the course consists of an introduction to category theory.
After a brief introduction to set theory, we will introduce the notion of a category. This notion allows us to speak about various mathematical theories, such as ring theory, topogy and group theory, all at the same time. Indeed, a category consists of (for example) "all" groups, where a group is treated as a non-specified "object" and the morphisms play a more important role. Next, we introduice the "morphisms of categories", that we will call functors, and the "morphisms of functors", that we will call natural transformations. We will provide many examples to illustrate that these concepts appear everywhere in mathematics. Category theory really begins when we start doing constructions inside a category and study objects and functors with special properties. More specifically, we will study limits and colimits (such as (co)products and (co)equalizers) and adjoint functors.

In the second part of the course, we will study an "application" of category theory in algebra or geometry. Depending of the interest of the students, one the following subjects could be treated:
. The theory of monads and monadicity of functors. This theory can be applied to (Galois) descent theory.
. Topological quantum field theories: these theories were motivated by Quantum Physics and allow to compute invariants of topological manifolds.
. Abelian and Grothendieck categories: these type of categories are of importance in algebraic geometry and homological algebra.

Show the links between different mathematical theories by climbing to a higher level of abstraction. Introduce and study the basic notions of category theory and illustratte these with numerous examples from algebra, geometry, topology, etc.

At the end of the course, the students are capable to recognize different categorical structures within mathematical models. They can perform calculations with complex constructions of categorical algebra.

Required and Corequired knowledge and skills

Basic notions in algebra and geometry from the Bachelor courses.

oral course

References, bibliography, and recommended reading

Lecture notes will be available on UV
Mac Lane, Saunders Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998. xii+314 pp. ISBN: 0-387-98403-8

Borceux, Francis

Handbook of categorical algebra. 1-3. (English summary)

Basic category theory. Encyclopedia of Mathematics and its Applications, 50-51.

Course notes

• Syllabus
• Université virtuelle

Contacts

Joost Vercruysse, email: jvercruy@ulb.ac.be, office: 2.O8.104 (campus Plaine)

Campus

Plaine

Method(s) of evaluation

• Oral examination

Oral examination

oral exam with written preparation.

Mark calculation method (including weighting of intermediary marks)

A note on 20 for the final exam.

Language(s) of evaluation

• french
• (if applicable english )