# Homotopy II (2021−2022).

## 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.

An introductory video for the course is available here.

### 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 will start the week of January 10^{th} and end on the week of February 18^{th}.
They will be located in the Grands Moulins campus in the 13^{th} arrondissement of Paris.
The schedule will be:

- Wednesdays from 11:15 to 13:15 in room 136 of the Olympe de Gouges building.
- Fridays from 16:15 to 18:15 in room 2016 of the Sophie Germain building.

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

The exam took place on Friday, March 4th, between 2PM and 5PM, in room 153 of the Olympes de Gouges building. The French version of the exam is here, while the English version is here. A partial solution is available here, you can find full solutions them in the following references:

- Exercise A:
**Hirschhorn, P.:**Section 7.6.12.*Model Categories and their Localizations*. Mathematical Surveys and Monographs 99. American Mathematical Society, Providence, RI. - Exercise B:
**Rezk, C.:**Section 2.*Fibrations and homotopy colimits of simplicial sheaves.*arXiv:math/9811038. - Exercise C:
**Larusson, F.:***The homotopy theory of equivalence relations.*arXiv:math/0611344v2.

There were a couple of typos in the exam (sorry!):

- A1: The category considered is not (co)complete in general. Either axiom (MC1) needs to be dropped, or one needs to consider the subcategory of triples $(X,f,g)$ such that $g \circ f = \phi$, where $\phi$ is some morphism $A \to Y$ fixed at the beginning.
- C2: In the French version, it was erroneously written that fibrations are defined object-wise, when it should have been cofibrations.
- C11: In the target, $A$ and $B$ were reversed; it should have been $[A,X] \times_{[A,Y]} [B,Y]$.

## Lectures

- Jan. 12 (⚠️ Lecture held online, watch): 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. 14 (⚠️ Lecture held online, watch): Definition of model categories and first examples (Section 1.3).
- Jan. 19: 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).
- Jan. 21: End of Section 1.4: explicit description of the homotopy category of a model category, Whitehead theorem.
- Jan. 24: Beginning of Section 1.5: projective model structures on chain complexes.
- Jan. 28: End of Section 1.5: projective model structure on chain complexes, cofibrantly generated model categories. Beginning of section 1.6: Quillen adjunctions.
- Feb. 02: End of Section 1.6: Quillen equivalences. Section 1.7: Homotopy limits and colimits.
- Feb. 04: 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. 09: Continuation of the proof of the existence of the Quillen model structure on simplicial sets, anodynes extensions, simplicial mapping space.
- Feb. 11: End of the proof of the Quillen equivalence between simplicial sets and topological spaces.
- Feb. 16: Left Bousfield localization and rational homotopy theory. Transferred model structure on CDGAs.
- Feb. 18: Overview of the equivalence between rational homotopy theory of spaces and CDGAs.

## Bibliography

### Course material

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, together with solutions for them.

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*.