Category Theory
A.Y. 2025/2026
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.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
6 CFU:
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. Elementary toposes.
+3 CFU:
Some topics among the following:
Conditions for monadicity
Algebraic categories and their characterization
Enriched categories
Grothendieck topos
Internal properties of elementary toposes
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. Elementary toposes.
+3 CFU:
Some topics among the following:
Conditions for monadicity
Algebraic categories and their characterization
Enriched categories
Grothendieck topos
Internal properties of elementary toposes
Prerequisites for admission
No specific prerequisites are requested.
Teaching methods
Frontal lessons.
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. http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf
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.
S. Awodey: Category theory, Oxford University Press, 2006
J. Adamek, H. Herrlich, G. Strecker: Abstract and concrete categories, Wiley
Interscience Publ. 1990. http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf
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 exam consists of a written test and an oral test.
In the written test, some open-ended exercises will be assigned to verify the ability to solve problems in Category Theory. The duration of the written test is normally 2 hours.
Only students who have passed the written test can access the oral test. During the oral test, students will be asked to explain some results from the course program and solve some Category Theory problems, in order to assess their knowledge and understanding of the topics covered, as well as their ability to apply them.
The exam is considered passed if both the written and oral tests are passed. The grade is expressed on a scale of thirty and will be communicated immediately at the end of the oral test.
In the written test, some open-ended exercises will be assigned to verify the ability to solve problems in Category Theory. The duration of the written test is normally 2 hours.
Only students who have passed the written test can access the oral test. During the oral test, students will be asked to explain some results from the course program and solve some Category Theory problems, in order to assess their knowledge and understanding of the topics covered, as well as their ability to apply them.
The exam is considered passed if both the written and oral tests are passed. The grade is expressed on a scale of thirty and will be communicated immediately at the end of the oral test.
MAT/02 - ALGEBRA - University credits: 9
Practicals: 24 hours
Lessons: 49 hours
Lessons: 49 hours
Professors:
Mantovani Sandra, Montoli Andrea
Professor(s)
Reception:
Thursday 12.45-14.15, by appointment
Studio 1019, I Floor, Dipartimento di Matematica, Via Saldini, 50
Reception:
by appointment via e-mail
office 1014, Via Saldini 50