Introduction to Homotopy Theory (2019−2020)

Institution
Université Paris Cité.
Cursus
M2 Fundamental Mathematics (S2).
Responsibilities
Lectures: 24h.
Notes
 Notes.
Exam
 Exam.
Solved exam
 Solved exam.

Introduction

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.

Prerequisites

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

Plan

  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.

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

and 18
, 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

and 17
, in the room 1009 of the Sophie Germain building. It covered chapters 1 and 2. The solution can be found here.

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

Bibliography

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: