This talk is meant to introduce the Quillen model categories. In short, model categories are used to give an effective construction of the localization of categories, where, similarly to localization of rings, the problem is to convert a class of morphisms, called weak-equivalences, into isomorphisms.
In the case of a model category, the localized category is identified with a homotopy category, a category whose morphism sets consist of equivalence classes of morphisms under a certain homotopy relation which is determined by the model structure.
We demonstrate this by constructing in Gap/CAP the bounded derived category of some categories with finite global projective dimension, for example the category of finite left presentations over a polynomial ring or the category of representations of an acyclic quiver.