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Model Categories

General data

Course ID: 1000-1M23KMO
Erasmus code / ISCED: 11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title: Model Categories
Name in Polish: Kategorie modelowe
Organizational unit: Faculty of Mathematics, Informatics, and Mechanics
Course groups: Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores): 6.00 Basic information on ECTS credits allocation principles:
  • the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS;
  • the student’s weekly hourly workload is 45 h;
  • 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes;
  • weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS;
  • work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load.
view allocation of credits
Language: English
Type of course:

elective monographs
Requirements:

Topology II 1000-134TP2
Prerequisites:

Algebraic topology 1000-135TA
Elements of Category Theory 1000-1M07ET
Prerequisites (description):

Familiarity with basics of topology within the scope of the Topology I course. Understanding of the fundamental notions of homotopy theory (homotopy equivalence, fundamental group, chain complexes and singular homology) as discussed in Topology II. It will be helpful to have experience with category theory as in the course Elements of Category Theory. Understanding of concepts of the Algebraic Topology course such as singular cohomology, CW-complexes and homotopy groups is also recommended.
Short description:

This course is an introduction to abstract homotopy theory within the framework of model categories. The central concepts are weak factorization systems, model categories, Quillen functors and Quillen equivalences. The purpose of the lecture is to develop the theory of homotopy invariance inside model categories and methods of comparison between various homotopy theories and to discuss applications in algebraic topology and homological algebra.
Full description:

  1. Weak factorization systems, the small object argument, model categories, cofibrantly generated model categories.
  2. Homotopies, homotopy equivalences, the homotopy category, Quillen functors and Quillen equivalences.
  3. The model structure on the category of topological spaces.
  4. The Kan–Quillen model structure on the category of simplicial sets.
  5. Model structures on the category of chain complexes.
  6. Projective and injective model structures on categories of diagrams, homotopy limits and colimits.
  7. Reedy model structures, mapping spaces in model categories.
Bibliography:

  1. Mark Hovey Model Categories 1999
  2. Philip Hirschhorn Model Categories and Their Localizations 2002
  3. William Dwyer, Jan Spaliński Homotopy theories and model categories (Handbook of Algebraic Topology 1995)
Learning outcomes:

  1. Familiarity with basic concepts of abstract homotopy theory in the framework of model categories: homotopies, homotopy equivalences, fibrant and cofibrant replacements.
  2. Ability to recognize homotopy non-invariant constructions and to approximate them by homotopy invariant ones using Quillen functors.
  3. Understanding of the classical homotopy theory of topological spaces as a special case of abstract homotopy theory in model categories.
  4. Familiarity with the current developments in abstract homotopy theory sufficient for taking up independent research.
Assessment methods and assessment criteria:

Participation in classes, written homework assignments and oral exam.

Classes in period "Summer semester 2023/24" (in progress)

Time span: 2024-02-19 - 2024-06-16
Selected timetable range:
Navigate to timetable
Type of class:
Classes, 30 hours more information
Lecture, 30 hours more information
Coordinators: Karol Szumiło
Group instructors: Karol Szumiło
Students list: (inaccessible to you)
Examination: Examination
Course descriptions are protected by copyright.
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