Master II Lecture

Homotopy Theories

(November-December 2025)

Announcement

Dominik Schrimpel and Quentin Schroeder (PhD students at Paris XIII) will organise a reading seminar on infinity-categories starting mid-Januray 2026.
It should be made up of 12 talks. All the details, includling the email addresses to register are available here:

Abstract

The goal of this lecture will be to present various "concrete" homotopy theories. We will start with the classical homotopy theory of topological spaces (higher homotopy groups, cofibration and fibrations, cellular complexes, Whitehead and Hurewicz theorems). Then we will move to the homotopy theory of simplicial sets (definitions, simplex category, adjunction and cosimplicial objects, examples, fibrations, Kan complexes, and simplicial homotopy).

This course will directly follow the one of Bernhard Keller "Homologie, cohomologie et faisceaux" (September-October 2025) and it will be pursued further by the course of Grégory Ginot "Homotopie II" (January-February, 2026).

Organisation

The courses will take place

Monday 1.45pm-4.45pm (Jussieu 15-25 102): November 3, 10, 17, 24, December 1, 8.
Wednesday 3pm-6.15pm (Halle aux Farines 406 B): November 5, 12, 19, 26, December 3, 10.
Wednesday 3.30pm-6.45pm (Olympe de Gouge 255): November 12, 19.
Wednesday 3pm-6.15pm (Halle aux Farines 471 E): November 26, December 3, 10.

The exam will take place December 17 at 4pm in rooms Sophie Germain 0011 and 0013. You will received official exam notices from the administration.

Lecture Notes

The notes of the course are available here:

(version of November 27, 2025)
The notes of the course of 2024 including the chapter on simplicial sets in French are available here:

Layout

Homotopy theory of topological spaces
Simplicial homotopy theory

Worksheet

Worksheet 1
Worksheet 2
Worksheet 3
Worksheet 4
Worksheet 5
Worksheet 6

YouTube links (2020 course)

Course 1 [4 november]: https://youtu.be/tFEjsTGzVfk (1/2) & https://youtu.be/NV_Q8StHP6Y (2/2)
Course 2 [5 november]: https://youtu.be/W_Ze-h04fHo (1/2) & https://youtu.be/zChw5rzs1IY (2/2)
Course 3 [11 november]: https://youtu.be/O3rx2rHJZPM (1/2) & https://youtu.be/jR8yXHouaFk (2/2)

Course 4 [12 november]: https://youtu.be/7s8WEuuPrw0 (1/2) & https://youtu.be/KG3VEW5P09w (2/2)
Course 5 [18 november]: https://youtu.be/zrfybhj5xqk
Course 6 [19 november]: https://youtu.be/O6mTvYjDczY (1/2) & https://youtu.be/YWSsAR30IaY (2/2)

Course 7 [25 november]: https://youtu.be/5jTArhCNn4E

Course 8 [26 november]: https://youtu.be/tAdSwbQbqI8 (1/2) & https://youtu.be/ZMvw8b0SY78 (2/2)

Course 9 [2 december]: https://youtu.be/Z6l6vp-r8tk

Course 10 [3 december]: https://youtu.be/JjcYRKKT85Y

Course 11 [9 décember]: https://youtu.be/zFleVAuHq3E

Course 12 [10 december]: https://youtu.be/xz-nJ7NF59o (1/2) & https://youtu.be/T57GHF0Yy2M (2/2)

Course 13 [16 december]: https://youtu.be/66mFKD_vfa8

Exam (2024)

English version
Version française
Corrigé

Exam (2021)

English version
Version française
Corrigé

Exam (2020)

English version
Version française
Corrigé

Exam (2018)

English version
Version française
Corrigé

Beyond the lectures

Analysis situs: great website dedicated to the works of Poincaré on the algebraic topology of manifolds.
Voyages aux pays des maths: episode of this great series (watch the other ones!) dedicated to the Poincaré conjecture.

References

Algebraic Topology, Tammo tom Dieck, EMS Textbooks in Mathematics, 2008.
A concise course in Algebraic Topology, J. Peter May, Chicago Lectures in Mathematics, 1999.
Algebraic Topology, Allen Hatcher, Cambridge University Press, 2001.

An elementary illustrated introduction to simplicial sets, Greg Friedman, arXiv:0809.4221, 2008.
Simplicial homotopy theory, Paul G. Goerss and John F. Jardine, Progress in Mathematics, 2009.
Simplicial homotopy theory, Edward B. Curtis, Advances in Mathematics 6, 107-209, 1971.
Simplicial Objects in Algebraic Topology, J. Peter May, Chicago Lectures in Mathematics, 1992.

Prerequisistes

From Bernhard Keller's course: category, functor, adjunction, (co)limits, topological space, homeomorphism.

Follow-up

Homotopie II by Grégory Ginot (Master 2 Mathematics, Fundamental Course II, January-February 2024).

Seminars

Seminar of the Algebraic Topology team (Université Sorbonne Paris Nord).

Professor

Bruno Vallette (lectures/exercise sessions)

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Last updates: November 27, 2025.