# Introduction to Homotopy Theory (2019–2020)

M2 Fundamental Mathematics (S2) • Lectures • 24h • Last updated on

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 $$\infty$$-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 treatement 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.

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

Lecture notes are available here.

## Plan

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

## Organization

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), due on February 4th, is available here. You can find the solution there.

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.

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 ( unusual time slot: 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 bried introduction to $$\infty$$-categories.

## Bibliography

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.

Works on homotopy theory:

Reminders on algebraic topology and homological algebra:

(The password is homotopie.)