Abstract Homotopy Theory and the Thomason Model structure

There is a closed model structure on the category of small categories, called Thomason model structure, that is Quillen equivalent to the standard model structure on the category of topological spaces. We will give an introduction to the concepts necessary to understand the definition, as well as the purpose of the Thomason model structure. These concepts include category theory, classical homotopy theory on topological spaces, simplicial homotopy theory on simplicial sets and abstract homotopy theory via the use of model categories. We will show, that there is a model structure on the category of small acyclic categories, that is Quillen equivalent to the Thomason model structure. Both of these model structures share the same cofibrant objects, and we will show that these include finite semilattices, countable trees, finite zigzags and posets with five or less elements.