34756 - Foundations of mathematics

Learning outcomes

At the end of the course, the student is able to understand and apply methods from logic and category theory in the context of pure mathematics as well as other disciplines such as computer science.

Course contents

This course provides an introduction to the foundations of mathematics as developed in the context of logic and category theory.

Particularly, the course will cover the following topics:

1. Category Theory
2. Homological Algebra
3. Sheaves and topoi
4. Simplicial methods and homotopy theory

Readings/Bibliography

As references, the students can consult the course notes by the lecturer as well as the following sources:

• Awodey, Steve. Category theory. Second edition. Oxford Logic Guides, 52. Oxford University Press, Oxford, 2010. xvi+311 pp. ISBN: 978-0-19-923718-0
• Mac Lane, Saunders. Categories for the working mathematician, Second Edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998. xii+314 pp. ISBN: 978-0-387-98403-2
  
• Mac Lane, Saunders ; Moerdijk, Ieke. Sheaves in geometry and logic. A first introduction to topos theory. Corrected reprint of the 1992 edition. Universitext. Springer-Verlag, New York, 1994. xii+629 pp. ISBN: 0-387-97710-4
• Goerss, Paul G.; Jardine, John F. Simplicial Homotopy Theory. Birkhäuser, Basel, 2010. ISBN: 978-3-0346-0188-7

Teaching methods

The course will be delivered through 48 hours of frontal lectures, dedicated to the discussion of the material as well as problems. Furthermore, practice problems will be given to students to verify their understanding of the course, and to practice solving problems.

Assessment methods

The assessment for the course consists in an oral exam.

This exam aims at assessing the comprehension of the topics covered in the course, and the ability of the student to apply them in the solution of problems.

Teaching tools

On the online platform Virtuale, the students will find:

• the course notes, to be consulted in addition to the textbook and the notes taken during frontal lectures by the student;
• practice problems list, to be solved independently for practice.

Office hours will be by appointment, either in person or online.

Office hours

See the website of Martino Lupini