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19223811 Forschungsmodul: Topologie "Homotopy Theories and Model Categories"

  • FU-Students should register via Campus Management.
  • Non-FU-students should register via MyCampus/Whiteboard.
  • The course will be held remotely using WebEx.

Summer Term 2021

Dozenten: Prof. Dr. Holger Reich, Prof. Dr. Elmar Vogt

  • Time and place:  Tuesday,  4pm -- 6pm, online

  • Leistungsnachweis/criteria for proof of performance:
    Grade and credit points will be awarded based on a presentation and written summary.

Prerequisites: We assume basic knowledge of topology as taught in Topology I and II. In fact, very little material from topology II is required: some familiarity with CW-complexes, chain complexes and some basic concepts of category theory.
This seminar might also be of interest for students with an algebraic leaning, since many categories, which are interesting for algebraists, admit model structures.

Content: The seminar will cover advanced topics from topology and homotopy theory.

Model Categories were introduced by D. Quillen in his Springer Lecture Notes volume "Homotopical Algebra" to extend the methods and philosophy of homotopy theory from the usual one in topology to a broad variety of subjects including several fields of algebra.

A model category is simply an ordinary category with three specified classes of morphisms called fibrations, cofibrations and weak equivalences which satisfy a few simple axioms reminiscent of the corresponding concepts in homotopy theory. This allows to set up a substantial portion of homotopy theory in many other fields of mathematics like homological algebra and algebraic geometry. But it also is still put to good use in parts of algebraic topology,for example in stable and equivariant stable homotopy theory, in an organized approach to homotopy limits and colimits, and giving a satisfactory answer to the concept of formally inverting weak equivalences in a category which avoids the usual set theoretic problems.

We will use the excellent survey article [DS] by W. G. Dwyer and J. Spalinsky of the same title as the seminar's as our primary source for the talks. Depending on the interest of participants we might include at the end some additional material.


We will decide the subjects of the last four talks later. There we want to include wishes of the participants. We should deal with homotopy limits and colimits in at least one of these talks.

13.04. Talk 1: Organization and overview Holger Reich/
Elmar Vogt
20.04. Talk 2: Review of some concepts of category theory Elmar Vogt
27.04. Talk 3: Model categories Karl Volkenandt
04.05. Talk 4: Cylinder objects and left homotopy Fabian Gringel
11.05. Talk 5: Path objects and right homotopy. Relating right and left homotopy Elmar Vogt
18.05. Talk 6: The homotopy category of model categories and localizations of categories. Kristoffer Rank Rasmussen
25.05. Talk 7: Chain complexes I Nikola Sadovek
01.06. Talk 8: Chain complexes II Evgeniya Lagoda
08.06. Talk 9: Topological Spaces Jessica Gonzalez Hurtado
15.06. Talk 10: Derived Functors Nicolas Gabriel Beck
22.06. Talk 11: Homotopy Limits and Colimits Ferry Saavedra
29.06. Talk 12: Homotopy colimits for topological spaces Ferry Saavedra
06.07. Talk 13: Homotopy colimits II Elmar Vogt
13.07. Talk 14: TBA N.N.


  • W. G. Dwyer, J. Spalinsky: Homotopy theories and model categories. In: Handbook of Algebraic Topology,North Holland 1995, pp. 73-126.
  • Mark Hovey: Model categories. Amer. Math. Soc. 1999