Lecturers: Fabian Hebestreit
Office: 3.023
eMail: f.hebestreit at math.uni-bonn.de
Christoph Winges
Office: 4.014
eMail: winges at math dot uni-bonn dot de
Slot 1: tuesdays, noon - 2pm
Location: Hörsaal XVI (Nussallee 17)
Slot 2: thursdays, noon - 2pm
Location: Zeichensaal (Wegeler Straße 10)

Introduction to higher categories


To all those interested in the follow-up course Higher categories and homotopical Algebra: Its webpage is now online!


The second batch will take place via Zoom. Those eligable should have been contacted via eMail. If you think yourself eligable and have not been, send us an eMail.


In the lecture we will go through the very basics of higher category theory. As preparation we will (briefly) review relevant bits of ordinary category theory, as well as (more leasurely) classical simplicial homotopy theory. In particular, we will cover:

  • Classical category theory and simplicial sets
  • Quasi-categories and the coherent nerve
  • Anodyne extensions and Kan fibrations
  • Objectwise criteria for invertibility of functors and transformations
  • Grothendieck's homotopy hypothesis
  • Notes

    see here

    Exercise sheets

    Sheet 1, Solutions
    Sheet 2, Solutions
    Sheet 3, Solutions
    Sheet 4, Solutions and the not-very-efficiently-drawn diagram for the extension of Exercise 1
    Sheet 5, Solutions
    Sheet 6, Solutions
    Sheet 7, Solutions
    Sheet 8, Solutions
    Sheet 9, Solutions
    Sheet 10, Solutions
    Sheet 11, Solutions
    Sheet 12, Solutions
    Sheet 13, Solutions
    Sheet 14, Solutions
    Sheet 15, Solutions

    Recommended Literature

  • Cisinski: Higher categories and homotopical algebra (available here)
  • Haugseng: Introduction to ∞-categories (lecture notes, available here)
  • Heuts, Moerdijk: Trees in Topology and Algebra (available here)
  • Joyal: Notes on Quasicategories (available here)
  • Joyal: The theory of quasi-categories and its applications (available here)
  • Land: Introduction to infinity-categories (lecture notes, available here)
  • Lurie: Higher Topos theory (available here)
  • Lurie: Kerodon (Wiki, available here)
  • Mac Lane: Categories for the working mathematician
  • Quillen: Homotopical Algebra
  • Riehl: A leisurely introduction to simplicial sets (available here)
  • Riehl: Categorical Homotopy Theory (available here)