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 2020); it will
open the doors to the one of Najib Idrissi Homotopie II"
(January-February, 2020) and to the one of Muriel Livernet
"Catégories supérieures" (March-April 2020)
The notes of the course will be
(Final version, 1/1/21)