**Cours fondamental 1 ** (TA)

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

** Notes de cours :** https://www.math.univ-paris13.fr/~vallette/download/Homotopy%20Theories.pdf

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.

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)

- Homotopy theory of topological spaces
- Simplicial homotopy theory

- 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