Introduction to Homotopy Theory (2019−2020)

Université Paris Cité
M2 Fundamental Mathematics (S2)
Lectures: 24h.
Solved exam


The goal of this course is to give an introduction to modern homotopy theory, its tools, and its applications, as well as to introduce the notion of ∞-category. We will essentially follow two examples: the founding example of topological spaces and the example of chain complexes (in the sense of homological algebra and algebraic topology). We will present the modern axiomatic treatment of homotopy theory – Quillen’s model categories – and we will explain the equivalence between topological spaces and simplicial sets. We will illustrate these methods through the example of rational homotopy theory, showing how the multiplicative structure of cochains – singular or de Rham – encode topological spaces up to rational homotopy.


It is advised to have already taken a class on algebraic topology as well as an introduction to homological algebra.


  1. Model Categories.
  2. Quillen functors and derived functors.
  3. Comparison between simplicial sets and topological spaces.
  4. Rational homotopy.
  5. Notion of ∞-category theory.


Lectures will start on January 6th, 2020 and end on February 13th, 2020.

The first two weeks, they will happen on:

The last four weeks, they will happen on:

The session of February 6th is moved to Monday, February 3rd, between 16:15 and 18:15, in the room 137 of the Olympe de Gouges building.

The optional homework (to hand in if you would like some feedback) was due on February 4th.

The updated solution is available on the page of the 2020–2021 lecture.

The exam was on Tuesday, February 18th, between 14:00 and 17:00, in the room 1009 of the Sophie Germain building. It covered chapters 1 and 2. The solution can be found here.

Date Content
Monday January 6th Section 1.1: Motivation, parallels between topological spaces and chain complexes (homotopy equivalences, weak equivalences, Whitehead theorem(s), models).
Tuesday January 7th Section 1.2: Fibrations, cofibrations, lifting properties, long exact sequences.
Monday January 13th Section 1.3: Categorical reminders. Beginning of Section 1.4: Definition of model categories.
Tuesday January 14th Section 1.4: some examples of model categories, a few properties. Section 1.5: Localization in the general case, definition of left homotopies.
Tuesday January 21th Section 1.5: end of the description of the homotopy category as a quotient of the category of fibrant-cofibrant objects.
Thursday January 23rd Section 1.6: Cofibrantly generated model categories, small object argument.
Tuesday January 28th Section 1.7: Quillen adjunctions and equivalences. Section 1.8: Homotopy (co)limits.
Thursday January 30th Sections 2.1–2.3: Introduction to simplicial sets.
Monday February 3rd (⚠️ 16:15–18:15) Sections 2.4–2.5: Model structure on simplicial sets, beginning of the equivalence with topological spaces.
Tuesday February 4th Sections 2.5–2.6: End of the equivalence with topological spaces. Dold–Kan correspondence. Section 3.1: Localization with respect to rational equivalences.
Thursday February 6th Moved to February 3rd.
Tuesday February 11th (⚠️ room 2017) Sections 3.2–3.3: Model structure on CDGAs, Sullivan theory, comparison with simplicial sets up to rational equivalence.
Thursday February 13th Chapter 3: Some applications of rational homotopy theory. Chapter 4: Very brief introduction to ∞-categories.


Course materials

I have written lecture notes which are available here.

Grégory Ginot gave a course in 2017–2019 on the same topic. On his webpage, you can find his lecture notes, as well as past exercise sheets and past exams.

Homotopy theory

Algebraic topology and homological algebra

In case you need reminders about the above topics: