Master II Lecture

Homotopy Theories

(November-December 2018)

WARNING: from Wednesday November 21, the course will begin at 9am and finish at noon on Wednesdays and Thursdays.


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, cellular complexes, Whitehead and Hurewicz theorems, Eilenberg—MacLane spaces, fibrations, and Postnikov towers). 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). Finally, we will study the rational homotopy theory via the homotopy theory of differential graded Lie or commutative (co)algebras (Sullivan approach : minimal model, Quillen approach : Whitehead Lie bracket, bar and cobar constructions, complete Lie algebra-Hopf algebras-groups).

This course will directly follow the one of Chrisitan Ausoni on Homology Theory (September-October 2018); it will open the doors to the one of Gregory Ginot on Abstract Homotopy Theory (January-February, 2018) and to the one of Yonatan Harpaz on Higher Algebra (March-April 2018)

Lecture Notes

The notes of the course are typed by Johan Leray: thank you very much!    (Version of 17/11/18: I still have to read them again).


  1. Homotopy theory of topological spaces
  2. Simplicial homotopy theory
  3. Rational homotopy theory




The lectures will take place every Wednesday 9am-12am (room 2013, Sophie Germain building) and every Thursday 9-12am (room 2016, Sophie Germain building) from November 7 to December 13 2018. Exercise sessions will be organised every Wednesday 9am-10.30am.


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


       Bruno Vallette (lectures/exercise sessions)

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Last updates : November 17th, 2018