MATH 7400 - Homotopical Algebra

Yuri Berest, fall 2015.

Homotopical algebra is a non-abelian (or nonlinear) generalization of homological algebra that includes as special cases the classical homotopy theory of topological spaces as well as classical (abelian) homological algebra.

Created in the late 60s in seminal works of Quillen, Kan, Dold-Puppe and others, homotopical algebra originally served the needs of algebraic topology.

In recent years, however, its influence has extended to other areas of mathematics, especially algebraic geometry, representation theory, differential geometry and even parts of theoretical physics. This course will be an introduction to Quillen's theory of model categories, which is a foundation of modern homotopy theory.

References:

1. D. Quillen, "Homotopical Algebra", Lecture Notes in Math. 43, Springer-Verlag, 1967.

2. W. G. Dwyer and J. Spalinski, "Homotopy theories and model categories", Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126.

3. M. Hovey, "Model Categories", Mathematical Surveys and Monographs 63, American Mathematical Society, Providence, RI, 1999.

4. J. Lurie, "Higher Topos Theory" Annals of Mathematics Studies, 170. Princeton University Press, Princeton, NJ, 2009.

5. B. Toen, "Simplicial presheaves and derived algebraic geometry", in "Simplicial Methods for Operads and Algebraic Geometry", Birkhauser, 2010.