On the algebraic K-theory of model categories

Waldhausen defined the algebraic K-theory of categories with cofibrations and weak equivalences. In many examples, these categories arise as full subcategories of model categories in the sense of Quillen. We use parts of the additional structure coming from model categories to compare the algebraic K-theory of different categories. The main tool for this is a generalization of Waldhausenʹs Approximation Theorem.