Category Theory

A.Y. 2020/2021
Overall hours
MAT/01 MAT/02 MAT/03 MAT/04
Learning objectives
The aim of this course is to provide an introduction to Category Theory, with particular attention to its unifying significance and to its use in algebra, logic and topology.
Expected learning outcomes
Acquisition and mastery of the fundamental notions of Category Theory, being able to use them in various fields of application.
Course syllabus and organization

Single session

Lesson period
First semester
Lessons, as well as weekly meetings for extra explanations, will be online online on Microsoft Teams, Team: Teoria delle categorie A.A. 20/21, Canale: Lezioni
Course syllabus
Categories, functors, natural transformations. Universal properties, limits and colimits. Adjunctions, equivalences. Representable functors and Yoneda Lemma. Monads and algebras for a monad, monadic functors. Monoidal categories and closed monoidal categories. Monoids in a monoidal category. Regular and Barr-exact categories. Additive categories. Abelian categories. Grothendieck and elementary Toposes.
Prerequisites for admission
No specific prerequisites are requested.
Teaching methods
Online lessons on Microsoft Teams.
Teaching Resources
S. Mac Lane: Categories for the working mathematician, Springer, 1997, 2nd edition

S. Awodey: Category theory, Oxford University Press, 2006

J. Adamek, H. Herrlich, G. Strecker: Abstract and concrete categories, Wiley
Interscience Publ. 1990.

T. Leinster, Basic Category Theory, Cambridge University Press, 24/lug/2014

F. Borceux: Handbook of categorical algebra, 1-2-3, Cambridge University
Press, 1994

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.
Assessment methods and Criteria
The final examination consists of two parts a written exam and an oral exam.

- During the written exam, the student must solve some exercises in the format of open answer questions, with the aim of assessing the student's ability to solve problems in Category Theory. The duration of the written is usually about 2 hours.
The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding Category Theory in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
The complete final examination is passed if all two parts (written, oral ) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/01 - MATHEMATICAL LOGIC - University credits: 0
MAT/02 - ALGEBRA - University credits: 0
MAT/03 - GEOMETRY - University credits: 0
Lessons: 42 hours
Professor: Mantovani Sandra
Educational website(s)
Thursday 12.45-14.15, by appointment
Studio 1019, I Floor, Dipartimento di Matematica, Via Saldini, 50