# Introduction to Homotopy Theory (2019–2020)

## 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 $\infty$-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.

### Plan

- Model Categories.
- Quillen functors and derived functors.
- Comparison between simplicial sets and topological spaces.
- Rational homotopy.
- 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:

- Mondays 15:45–17:45, Sophie Germain building, room 1012.
- Tuesdays 14:00–16:00, Sophie Germain building, room 2016.

The last four weeks, they will happen on:

- Tuesdays 14:00–16:00, Sophie Germain building, room 2016 (
*except on February 11th: room 2017*). - Thursdays 9:00–11:00, Halles aux Farines, room 280F (
*except February 6th: moved to the 3rd*).

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. |

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:

**William G. Dwyer et 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 et 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 et 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*.**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.**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 $\infty$-categories.)*Higher topos theory*.

Reminders on algebraic topology and homological algebra:

**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*.