Homotopical Algebra
A.Y. 2019/2020
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
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Homotopy & homology. Weak equivalences & quasi isomorphisms. Fibrations & cofibrations. Simplicial structures & geometric realisation. Model categories & homotopy categories. Quillen functors & derived functors. A1-homotopy & motivic homotopy theories.
Prerequisites for admission
We assume known the basic notions from algebraic topology & homological algebra and some fundamentals of the theory of schemes for the last part on motivic homotopy. The students should be familiar with fundamental group, singular homology/cellular homology, CW-complexes and related basic computations. For example, see A. Hatcher Algebraic topology Cambridge Univ. Press, 2002 In a few lectures we will treat some topology using J. P. May A concise course in algebraic topology Chicago Lectures in Mathematics 1999. For homological algebra we just assume known the basics on chain complexes as explained in the first chapter of C. Weibel's book An introduction to Homological Algebra Cambridge Univ. Press, 1994. For schemes just read Marc Levine Elementary Algebraic Geometry or any other introductory book on the subject.
Teaching methods
Lectures.
Teaching Resources
J.F. Jardine: Lectures on Homotopy Theory, University of Western Ontario, Canada.
M. Hovey: Model Categories Math Surveys & Monographs Vol. 63 AMS 1999
M. Hovey: Model Categories Math Surveys & Monographs Vol. 63 AMS 1999
Assessment methods and Criteria
The examination consists of a written homework and an oral exam.
MAT/02 - ALGEBRA - University credits: 6
Lessons: 42 hours
Professor:
Barbieri Viale Luca
Shifts:
-
Professor:
Barbieri Viale LucaProfessor(s)
Reception:
Email contact (usually for Tuesday h. 2-4 p.m.)
Office - Math Department