The goal of this lecture is to present two “concrete” homotopy theories. We will start with the classical homotopy theory of topological spaces (homotopy groups, cellular complexes, Whitehead and Hurewicz theorems, fibrations). 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). The notion of a simplicial set will be introduced with a view toward a definition of infinity-categories.

Note

Exercise sessions are taught by Sylvain Douteau.

Content

Homotopy theory of topological spaces
Simplicial homotopy theory

I recommend reading the lecture notes written by Bruno Vallette for this course, as well as taking a look at his website for past exams.

Prerequisites

Introductory course “Cohomologie et faisceaux”.

Bibliographie

Tammo tom Dieck. Algebraic Topology. EMS Textbooks in Mathematics, 2008 https://www.ems-ph.org/books/book.php?proj_nr=86
Peter May. A concise course in Algebraic Topology. Chicago Lectures in Mathematics, 1999 https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf
Allen Hatcher. Algebraic Topology. Cambridge University Press, 2001 http://pi.math.cornell.edu/~hatcher/AT/ATpage.html
Greg Friedman. An elementary illustrated introduction to simplicial sets. An elementary illustrated introduction to simplicial sets. Rocky Mountain J. Math. 42 (2012), no. 2, 353–423 https://www.irif.fr/~mellies//mpri/mpri-ens/articles/friedman-an-elementary-illustrated-introduction-to-simplicial-sets.pdf
Paul G. Goerss and John F. Jardine. Simplicial homotopy theory. Progress in Mathematics, 2009 https://www.springer.com/gp/book/9783034601887 Edward B. Curtis. Simplicial homotopy theory. Advances in Mathematics 6, 107-209, 1971 https://www.sciencedirect.com/science/article/pii/0001870871900156