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Homotopy II (2021–2022)

Université de Paris. M2 Fundamental Mathematics (S2). lectures. 24 h. Notes


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.


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. The time slots will be:

 The first two lecture (January 12th and January 14th) will be held online on Zoom as I was tested positive for COVID on January 10th. The Zoom link will be made available to you by email and the lecture will be broadcast live in the room. On Wednesday 12, the room will be changed to room 0013 of the Sophie Germain building for technical reasons. You will have received a Zoom link by email. On Friday 14, the room will be room 2016 of the SG building as usual and the lecture will also be broadcast simultaneously.

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


Lecture 1

1/12/2022 11:15-13:15
 Lecture held online.

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)


Lecture 2

1/14/2022 16:15-18:15
 Lecture held online.

Definition of model categories and first examples (Section 1.3).


Lecture 3

1/19/2022 11:15-13:15
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).

Lecture 4

1/21/2022 16:15-18:15

Lecture 5

1/24/2022 11:15-13:15

Lecture 6

1/28/2022 16:15-18:15

Lecture 7

2/2/2022 11:15-13:15

Lecture 8

2/4/2022 16:15-18:15

Lecture 9

2/9/2022 11:15-13:15

Lecture 10

2/11/2022 16:15-18:15

Lecture 11

2/16/2022 11:15-13:15

Lecture 12

2/18/2022 16:15-18:15


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. 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.

Works on homotopy theory:

  • William G. Dwyer et Jan Spaliński. “Homotopy theories and model categories”. In: Handbook of algebraic topology. Amsterdam: North-Holland, 1995, pp. 73–126. DOI:10.1016/B978-044481779-2/50003-1. MR1361887. Zbl:0869.55018. (Introduction to model categories.)
  • Yves Félix, Stephen Halperin et Jean-Claude Thomas. Rational Homotopy Theory. Graduate Texts in Mathematics 205. New York : Springer, 2001, p. xxxiv+535. ISBN: 0-387-95068-0. DOI: 10.1007/978-1-4613-0105-9. (Reference book on rational homotopy theory.)
  • Paul G. Goerss et John F. Jardine. Simplicial homotopy theory. Progress in Mathematics 174. Basel: Birkhäuser , 1999, pp. xvi+510. ISBN: 3-7643-6064-X. DOI:10.1007/978-3-0348-8707-6. MR1711612. Zbl:0949.55001. (Book on simplicial sets and their homotopical properties.)
  • Phillip Griffiths et John Morgan. Rational homotopy theory and differential forms. 2nd ed. Progress in Mathematics 16. New York: Springer, 2013. 224 p. ISBN: 978-1-4614-8467-7. DOI: 10.1007/978-1-4614-8468-4. (Notes on rational homotopy theory)
  • Kathryn Hess. “Rational homotopy theory: a brief introduction”. In: Interactions between homotopy theory and algebra. Contemp. Math. 436. Providence, RI: Amer. Math. Soc., 2007, pp. 175–202. DOI:10.1090/conm/436/08409. arXiv:math/0604626. MR2355774. Zbl:1128.55010. (Introduction to rational homotopy theory)
  • Mark Hovey. Model categories. Mathematical Surveys and Monographs 63. Providence, RI: American Mathematical Society, 1999, pp. xii+209. ISBN: 0-8218-1359-5. MR1650134. Zbl:0909.55001. (Book on model categories.)
  • Jacob Lurie. Higher topos theory. Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press, 2009, pp. xviii+925. ISBN: 978-0-691-14049-0. MR2522659. Zbl:1175.18001 (Very complete book on ∞-categories.)

Reminders on algebraic topology and homological algebra:

  • Glen E. Bredon. Topology and geometry. Graduate Texts in Mathematics 139. New York: Springer, 1993, pp. xiv+557. ISBN: 0-387-97926-3. DOI:10.1007/978-1-4757-6848-0. MR1224675. Zbl:0791.55001. ((Algebraic) topology and (differential) geometry textbook.)
  • Allen Hatcher. Algebraic topology. Cambridge: Cambridge University Press, 2002, pp. xii+544. ISBN: 0-521-79160-X. MR1867354. Zbl:1044.55001 (Algebraic topology textbook.)
  • Henri Paul de Saint-Gervais. Analysis Situs. (Lecture notes on algebraic topology, in French.)
  • Pierre Schapira. Categories and homological algebra. (Lecture notes on derived categories.)
  • Edwin Spanier. Algebraic topology. Berlin: Springer, 1995, pp. xiv+528. ISBN: 978-1-4684-9322-1. DOI:10.1007/978-1-4684-9322-1. MR210112. Zbl:0145.43303. (Algebraic topology textbook.)
  • Charles A. Weibel. An Introduction to homological algebra. Cambridge Studies in Advanced Mathematics 38. Cambridge: Cambridge University Press, 1994, pp. xiv+450. ISBN: 0-521-43500-5. MR1269324. Zbl:0797.18001. (Homological algebra textbook.)