Beijing Institute of Mathematical Sciences and Applications

Simplicial homotopy thoery has always been one of simplices and their incidence relations, along with methods for constructing maps and homotopies of maps within these constrains. As such, the methods and ideas are algebraic and combinatorial and, despite the deep connection with the homotopy theory of topogical spaces., exist complitely outside any topological context. This point of view was effectively introduced by Kan, and later encoded by Quillen in the notions of a closed model category. Simplicial homotopy theory, and more generally the homotopy theories associated to closed model categories, can then be interpreted as a purely algebraic enterprise, which has had substantial applications throughout homological algebra, algebraic geometry, number theory and algebraic K-theory.