OVERVIEW

Category theory is a young branch of mathematics. Born officially in 1945 from a seminal paper by Samuel Eilenberg and Saunders Mac Lane, it soon found important applications in algebra, geometry and logic thanks to ideas of Alexander Grothendieck and William Lawvere. Category theory is currently expanding its areas of application, including physics and computer science, and it is imposing itself as a language to allow these disciplines, and the people working in them, to efficiently communicate with each other. This is achieved by isolating the common aspects of definitions and proofs and allowing, for example, to give a precise meaning to notions like 'natural' or 'canonical'.

AIMS AND CONTENT

LEARNING OUTCOMES

The course introduces the basic notions in category theory: categories, functors, natural transformations, adjunctions, limits, colimits. It also presents some fundamental results, such as Yoneda Lemma and the Special Adjoint Functor Theorem.

AIMS AND LEARNING OUTCOMES

The aim of the course is to provide a solid knowledge of basic concepts in category theory, and their main applications in mathematical practice. After attending the course, the student will be able to:
- rephrase concepts and results, already known or learned afterwards, using the language of category theory;
- define mathematical objects by means of their universal property, and use it to derive specific properties of such objects;
- exploit the Yoneda lemma and representability of functors to derive results on categories from results on sets;
- prove closure properties for classes of objects by means of adjoint functor theorems and monadicity theorems.

TEACHING METHODS

Lectures and exercise classes.

SYLLABUS/CONTENT

1. Fundamental concepts: categories, functors, natural transformations.
2. Classes of morphisms: epic, monic, iso; full and faithful functors.
3. Limits and colimits, universal objects: initial and terminal objects, (co)products, (co)equalisers, pullback and pushout squares.
4. Yoneda lemma, properties of representable functors.
5. Adjunctions and their properties, adjoint functor theorems, cartesian closed categories.
6. Possible additional topics:
 - Monads, Eilenberg--Moore and Kleisli constructions, algebras and monadicity theorem.
 - Monoidal categories and closed monoidal categories: definitions, examples, applications.

RECOMMENDED READING/BIBLIOGRAPHY

E. Riehl. Category theory in context. Dover, 2015
S. Awodey. Category theory, 2nd edition. Oxford Logic Guides 52, 2010
S. Mac Lane. Categories for the working matematician, 2nd edition. Springer, 1998

TEACHERS AND EXAM BOARD

LESSONS

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The final exam consists of an oral examination. In order to apply for adjustments to the rules, DSA students should folow the instructions available on AulaWeb at <https://2023.aulaweb.unige.it/course/view.php?id=12490#section-3>.

ASSESSMENT METHODS

The oral examination will verify that the student has learned the basic notions from category theory, the results treated during the course, and their proofs and applications. The final evaluation will take into account the correcteness of the exposition, its clarity, and the appropriateness of reasoning.