# Introduction to Homotopy Theory (2019–2020)

Université de Paris. M2 Fundamental Mathematics (S2). lectures. 24 h.

## 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.

This is the page for the 2019–2020 lecture. See here for the 2021–2022 page.

### 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 $\infty$-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) 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.

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 ( 16:15–18:15)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 bried introduction to $\infty$-categories.

## Bibliography#

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.

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. Rational Homotopy Theory. 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.)
• Paul G. Goerss et John F. Jardine. Simplicial homotopy theory. 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.)
• 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. Model categories. 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.)
• Jacob Lurie. Higher topos theory. 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.)

Reminders on algebraic topology and homological algebra:

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