Simplicial sets

Abstract

This chapter introduces the basic elements of the homotopy theory of simplicial sets. Technically, the purpose is twofold: to prove that the category of simplicial sets has a homotopical structure in the sense that it admits the structure of a closed model category (Theorem 11.3), and to show that the resulting homotopy theory is equivalent in a strong sense to the ordinary homotopy theory of topological spaces (Theorem 11.4). Insofar as simplicial sets are algebraically defined, and the corresponding closed model structure is combinatorial in nature, we obtain an algebraic, combinatorial model for standard homotopy theory.