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Homotopical Algebra

A.Y. 2025/2026
6
Max ECTS
42
Overall hours
SSD
MAT/02
Language
English
Learning objectives
The main task of this course is to give an introduction to the methods of homotopical algebra.
Expected learning outcomes
Knowledge of the fundamentals of the abstract homotopy theory and applications.

Lesson period: Second semester

Lessons timetable

Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi

Exams calendar

Single course

This course can be attended as a single course.

Take a single course
Course syllabus and organization

Single session

Responsible
Barbieri Viale Luca
Lesson period
Second semester
Course syllabus
Homotopy & homology. Weak equivalences & quasi isomorphisms. Fibrations & cofibrations. Model categories & homotopy categories. Quillen functors, derived functors & equivalences. Simplicial homotopy & geometric realisation. Nerve of a category & ∞-categories. Universal homotopy & motivic homotopy.
Prerequisites for admission
We assume known the basic notions from category theory, algebraic topology & homological algebra.
Teaching methods
Lectures.
Teaching Resources
Notes: Jardine's Lectures on Homotopy Theory available online and Kerodon https://kerodon.net/ an online resource for homotopy-coherent mathematics
Book: M. Hovey - Model Categories, Math Surveys & Monographs Vol. 63 AMS, 1999.
Survey: W.G. Dwyer & J. Spalinski - Homotopy theories and model categories in Handbook of Algebraic Topology I.M. James (ed.) North-Holland, 1995.
Assessment methods and Criteria
Some written homework will be assigned during class. These homework, which count as a written exam, must be completed and returned before the oral exam. For the oral exam, a seminar will be assigned on a preferred subject, based on the topics discussed in class. During the seminar presentation, questions will be asked on course topics relevant to the seminar. The final grade will take into account the written exam (insufficient, sufficient, good, or excellent) and the oral exam, i.e., the quality of the presentation and answers to the questions.
MAT/02 - ALGEBRA - University credits: 6
Lessons: 42 hours
Professors: Barbieri Viale Luca, Oestvaer Paul Arne
Professor(s)
Barbieri Viale Luca
Web site
Reception:
Email contact (usually for Tuesday h. 2-4 p.m.)
Office 2092 - Math Department