Categories in Algebra and Topology

Recommended Prerequisites

Linear Algebra, Abstract Algebra and Topology.

Teaching Methods

The more relevant theoretical concepts and results are presented with detail and accuracy, followed by illustrative examples, motivating applications and historical notes.

Homework assignments, consisting of several exercises, will be proposed.

Active participation of the students in class discussions should be strongly encouraged.

Learning Outcomes

This course aims to be an introduction to the ideas and methods of category theory, which permeate so much of mathematics today,  and some of its applications. Since it became an indispensable tool for anyone doing research in abstract algebra, topology, logic or theoretical computer science, to name but a few, to become familiar with basic technics and ways of thinking of category theory is our main goal.  The main competences to be developed are : ability for generalization and abstraction; ability to formulate and solve problems; to design, analyse and use the categorical methods and technics; critical thinking.

Work Placement(s)

No

Syllabus

Part I: Introduction to Category Theory:

Categories, functors and natural transformations. Isomorphism and equivalence of categories. Construction of new categories: subcategories, product of categories and dual category. Categorical duality principle. Limits and colimits. Functor categories. Representable functors. Yoneda Lemma and Yoneda embedding. Adjoints and limits. Existence of adjoints (Freyd's Theorem).

Part II: It includes topics from the list below, chosen according to the interests of the students:

Monads and categories of Eilenberg-Moore algebras.

Cartesian closed categories. Toposes.

Locales.

Exact and regular categories. Additive, abelian, semi-abelian categories and homological categories.

Assessment Methods

Assessment
Attendance: 15.0%
Resolution Problems: 25.0%
Exam: 60.0%

Bibliography

S. Mac Lane, Categories for the Working Mathematician, 2nd edition, Springer-Verlag, New York, 1998.

F. Borceux, Handbook of Categorical Algebra, Vols 1-3, Cambridge University Press, Cambridge, 1994.