Homotopy II (2021−2022)

Université Paris Cité
M2 Fundamental Mathematics (S2)
Lectures: 24h.
Homework w/ solution
Exam FR
Exam EN
Partial solution


The goal of this course is to introduce modern homotopy theory, its tools and applications. We will be particularly interested in two examples: chain complexes (see the previous Homology course) and topological spaces. We will present Quillen’s model categories, and we will explain the equivalence between topological spaces and simplicial sets. We will illustrate these methods with rational homotopy theory by showing that multiplicative structures of cochains encodes rational homotopy types of topological spaces.

An introductory video for the course is available here.


It is recommended to have taken the courses Homology (by Emmanuel Wagner) and Homotopy I (by Bruno Vallette). It will be useful to have a certain familiarity with categorical language and with basic notions of algebraic topology and homological algebra.

Plan of the course

  1. Model categories.
  2. Chain complexes.
  3. Simplicial sets and topological spaces.
  4. Rational homotopy theory.

After this course


Lectures will start the week of January 10th and end on the week of February 18th. They will be located in the Grands Moulins campus in the 13th arrondissement of Paris. The schedule will be:

My own office is in the Sophie Germain building. I can be contacted at najib.idrissi-kaitouni@imj-prg.fr.

The exam took place on Friday, March 4th, between 2PM and 5PM, in room 153 of the Olympes de Gouges building. The French version of the exam is here, while the English version is here. A partial solution is available here, you can find full solutions them in the following references:

There were a couple of typos in the exam (sorry!):


  1. Jan. 12 (⚠️ Lecture held online, watch): Analogies between different homotopy theories: topological spaces, simplicial sets, chain complexes. Motivation for the definition of model categories. (Co)fibrations vs. injections/surjections. Definition of model categories. (Sections 1.1 and 1.2)
  2. Jan. 14 (⚠️ Lecture held online, watch): Definition of model categories and first examples (Section 1.3).
  3. Jan. 19: Definition and construction of the localization of a category with respect to a class of weak equivalence, definition of left homotopies, first properties (Section 1.4.1).
  4. Jan. 21: End of Section 1.4: explicit description of the homotopy category of a model category, Whitehead theorem.
  5. Jan. 24: Beginning of Section 1.5: projective model structures on chain complexes.
  6. Jan. 28: End of Section 1.5: projective model structure on chain complexes, cofibrantly generated model categories. Beginning of section 1.6: Quillen adjunctions.
  7. Feb. 02: End of Section 1.6: Quillen equivalences. Section 1.7: Homotopy limits and colimits.
  8. Feb. 04: Reminders on simplicial sets and beginning of the proof of the existence of the Quillen model structure on the category of simplicial sets (Sections 2.1, 2.2, 2.3, and beginning of 2.4).
  9. Feb. 09: Continuation of the proof of the existence of the Quillen model structure on simplicial sets, anodynes extensions, simplicial mapping space.
  10. Feb. 11: End of the proof of the Quillen equivalence between simplicial sets and topological spaces.
  11. Feb. 16: Left Bousfield localization and rational homotopy theory. Transferred model structure on CDGAs.
  12. Feb. 18: Overview of the equivalence between rational homotopy theory of spaces and CDGAs.


Course material

I have written lecture notes which are available here. They were written two years ago and the content of the lecture has changed since then; in particular, the fourth chapter will not be covered in my lecture this year.

I gave a similar course last year, and also the year before that. You can in particular find past exams on these pages, together with solutions for them.

Grégory Ginot gave a course in 2017–2018–2019 on the same subject. You can find on his page his lecture notes, as well as past exercise sheets and exams.

Homotopy theory

Algebraic topology and homological algebra

In case you need reminders about the above topics: