The course aims to provide a broad introduction to category theory, an area of great relevance to many areas of contemporary mathematics, and to present the foundations of topos theory and categorical logic. By the end of the course, the student is expected to have become thoroughly familiar with categorical techniques and their applications in a wide variety of mathematical situations.
Course Prerequisites
No previous knowledge of category theory and the related subjects is required. Familiarity with the contents of the first two years of a Bachelor's degree in Mathematics is desirable, but not strictly necessary.
Teaching Methods
Frontal lectures and exercise sessions.
Assessment Methods
The student will be able to choose between two alternative exam modes: - Solutions (prepared at home) to exercises assigned by the lecturer and seminar on a suitable topic extending the contents of the course (chosen in agreement with the lecturer). The evaluation of this seminar will constitute the two thirds of the final grade, while the remaining third will be determined by the solutions to the exercises. - Taking two partial written exams, one administered half way through the course and the other immediately after the end of it.
Contents
The course will begin by presenting the basic notions and results of category theory, and will continue by explaining the fundamentals of topos theory and categorical logic. More specifically, the topics that will be addressed are the following: - Categories, functors and natural transformations - Representability - Universal properties, limits and colimits - Adjunctions - Categories of sheaves on a site and their properties - Introduction to first-order logic and its interpretation in categories
