Cours fondamental 1 (TA)

Homotopie I

Bruno Vallette

Contact : vallette à vallette@math.univ-paris13.fr

Page web du cours : https://www.math.univ-paris13.fr/~vallette/Course-MasterII-2024.html

Présentation

The goal of this lecture will be 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 an infinity-category.

Contenu

• Homotopy theory of topological spaces
• Simplicial homotopy theory

Prérequis

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