Universität Regensburg   IMPRESSUM   DATENSCHUTZ
Fakultät für Mathematik Universität Regensburg
Derived functors and cohomology through higher categories
Semester
WiSe 2021 / 22

Lecturer
Denis-Charles Cisinski

Type of course (Veranstaltungsart)
Vorlesung

German title
Abgeleiteten Funktoren und Kohomologie durch höheren Kategorien

Contents
The aim of the course is at first to relate classical constructions of homotopy theory and homological algebra to the ones from infinity-category theory. In particular, we will see how to enhance the axioms of Eilenberg and Steenrod for singular homology theory, or to interpret sheaf cohomology. More generaly, we will see how to interpret any constructions from the theory of Quillen's theory of model categories. Many of the constructions/computations will be an illustration of the theory of Kan extensions in the context of higher categories. In particular, we will see how the concept of Kan extension is fundamental for the organization of mathematical discourse as well as for implementing basic computational tools. This is a natural continuation of the lectures on Higher Categories, but, for students willing to accept basic facts, this is also an oportunity to learn higher categories from examples, through what they do, and we will try our best to make the lecture accessible to an audience as wide as possible.

Leitfaden:

    1. Reminder of higher category theory
    1. Yoneda Lemma, limits and colimits
    2. Kan extensions
    3. Pullbacks and pushouts
    2. Homotopical Algebra
    1. Localization
    2. Calculus of fractions
    3. Derived functors
    4. Approximation Theorems
    5. Universal property of spaces
    6. Computation of mapping spaces
    3. Homological Algebra
    1. Stable infinity-categories
    2. Derived categories as localizations
    3. Spectra
    4. Additive and exact infinity-categories
    5. Derived categories as completions

A perspective of this course will be to understand derived Morita theory (studying functors between derived categories of quasi-coherent sheaves on schemes) which will be the subject a course in the Summer Semester.

Literature
  • D.-C. Cisinski, Higher Categories and Homotopical Algebra, Cambridge studies in advanced mathematics, vol. 180, Cambridge University Press, 2019
  • J. Lurie, Higher Topos Theory, Annals of mathematics studies, vol. 170, Princeton University Press, 2009
  • J. Lurie, Higher Algebra, https://www.math.ias.edu/~lurie/papers/HA.pdf
  • J. Lurie, Kerodon, https://kerodon.net/


    • Recommended previous knowledge
    Basics of Algebraic Topology, of Homological Algebra, and of Category Theory

    Time/Date
    Mo 16-18, Fri 16-18

    Location
    M 103 or online

    Course homepage
    http://www.mathematik.uni-regensburg.de/cisinski/lehre.html
    (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

    Registration
    • Preliminary registration for the organisation of exercise classes: at the end of the previous
      semester via EXA or LSF (see announcement by the department)
    • Registration for the exercise classes: via GRIPS in the first week of the lecture period
    • Registration for course work/examination/ECTS: FlexNow
    Course work (Studienleistungen)
    • Oral examination (without grade): Duration: , Date:
    Examination (Prüfungsleistungen)
    • Oral exam: Duration: 30 minutes, Date: individual, re-exam: Date: individual
    Modules
    BV, MV, MArGeo, MGAGeo, LA-GyGeo

    ECTS
    9