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, 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
Emmanuel
Wagner "Théorie de l'homologie" (September-October 2021); it will
open the doors to the one of Najib Idrissi Homotopie II"
(January-February, 2022) and to the one of Muriel Livernet
"Catégories supérieures" (March-April 2022).
Lecture Notes
The notes of the course will be
available here: 
(Final version, 1/1/21)
Bruno Vallette (lectures/exercise
sessions)