# Introduction to Homotopy Theory (2019–2020)

M2 Fundamental Mathematics (S2) • Lectures • 24h • Last updated on

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 being in January 2020 and end in February 2020. More information to come.

## 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-Verlag, 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 Verlag, 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-Verlag, 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-Verlag, 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*.

(The password is `homotopie`

.)