Geometric Realization of Simplicial Sets

Abstract

In this thesis, I aim to proof that the geometric realization functor preserves simplicial homotopies between Kan complexes form the very basics of category theory. Therefore, the first chapter is concerned with basic definitions and theorems of category theory, such as the Yoneda Lemma. In the second chapter, I introduce simplicial sets and several examples, after which several properties of simplicial sets are discussed. The third chapter is concerned with the definition of the geometric realization and several alternate constructions, as well as some intuitive examples. In the fourth and fifth chapter, I work towards proving that the geometric realization functor preserves simplicial homotopies. In particular, in the fourth chapter, I give a proof that the geometric realization preserves finite limits, and in the fifth chapter, Kan complexes are introduced.