# Homotopy II (2020−2021).

## Introduction

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.

### Prerequisites

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

- Model categories.
- Chain complexes.
- Simplicial sets and topological spaces.
- Rational homotopy theory.

### After this course

- The course
*Higher categories*(by Muriel Livernet) is a natural continuation of this course and a conclusion to the “homotopy” sequence of the master’s degree (Homology – Homotopy I/II – Higher categories). - I gave in 2020 a Peccot lecture entitled
*Real homotopy of configuration spaces*at the Collège de France which gave concrete applications of the methods presented here.

## Organization

Lectures happened on Mondays 10:45–12:45 and Fridays 15:45–17:45.
They began on January 11^{th} and ended on February 19^{th}.
All lectures were online (via Zoom).
The sessions were recorded and they are now available on YouTube.

The exam was on Friday, March 5th, 14:00–17:00, in room 207C of the Halle aux Farines. It was an open-book exam: students had access to printed and handwritten lecture notes. Electronic devices were forbidden.

The retake exam (*rattrapage*) was be on June 15^{th}.
It consisted in oral exams lasting around 30 minutes total per student.
In-person sessions were in the morning between 9AM and 1PM, in the Sophie Germain building, room 1014.
Online sessions for students that cannot be physically present in Paris at that time were in the afternoon.

## Lectures

- Jan. 11 (watch, blackboard): 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). |
- Jan. 15 (watch, blackboard): Definition of model categories and first examples (Section 1.3). |
- Jan. 18 (watch, blackboard): Definition and construction of the localization of a category with respect to a class of weak equivalence, definition of left homotopies, first properties, dual case of right homotopies (Sections 1.4.1 and 1.4.2). |
- Jan. 22 (watch, blackboard): End of Section 1.4: explicit description of the homotopy category of a model category, Whitehead theorem. |
- Jan. 25 (watch, blackboard): Section 1.5.1: The proof of the existence of the projective model structure on the category of bounded-below chain complexes. |
- Jan. 29 (watch, blackboard): Existence theorem for cofibration generated model categories (Section 1.5.2), Quillen adjunctions (beginning of Section 1.6). |
- Feb. 1 (watch, blackboard): Definition of Quillen equivalences and characterization, homotopy limits and colimits. |
- Feb. 5 (watch, blackboard): 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). |
- Feb. 8 (watch, blackboard): Continuation of the proof of the existence of the Quillen model structure on simplicial sets, anodynes extensions, simplicial homotopy groups. |
- Feb. 12 ([watch](https://youtu.be/gckfIqqkO-, blackboard): End of the proof of the Quillen equivalence between simplicial sets and topological spaces. Left Bousfield localization and rational homotopy theory. |
- Feb. 15 (watch, blackboard): Transferred model structure on CDGAs and Sullivan algebras. |
- Feb. 19 (watch, blackboard): Equivalence between CDGAs and rational homotopy types through the PL forms. Applications of Sullivan’s theory: models of spaces, dichotomy theorem, etc. |

## Bibliography

### Course materials

I have written lecture notes which are available here. They were written last year 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 also gave a similar course last year; you can in particular find past exams on that webpage.

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

**William G. Dwyer and 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 and Jean-Claude Thomas.**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.)*Rational Homotopy Theory*.**Paul G. Goerss and John F. Jardine.**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.)*Simplicial homotopy theory*.**Phillip Griffiths and John Morgan.**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)*Rational homotopy theory and differential forms*.**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.**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.)*Model categories*.**Jacob Lurie.**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.)*Higher topos theory*.

### Algebraic topology and homological algebra

In case you need reminders about the above topics:

**Glen E. Bredon.**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.)*Topology and geometry*.**Allen Hatcher.**Cambridge: Cambridge University Press, 2002, pp. xii+544. ISBN: 0-521-79160-X. MR1867354. Zbl:1044.55001 (Algebraic topology textbook.)*Algebraic topology*.**Henri Paul de Saint-Gervais.**(Lecture notes on algebraic topology, in French.)*Analysis Situs*.**Pierre Schapira.**(Lecture notes on derived categories.)*Categories and homological algebra*.**Edwin Spanier.**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.)*Algebraic topology*.**Charles A. Weibel.**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.)*An Introduction to homological algebra*.