Homotopy Theory and Triangulated Categories

Chapter 1 is dedicated to (pre)cofibration, (pre)fibration, and (pre)model categories. In section 1.1 I give some preliminar definitions and properties; in section 1.2 I am comparing six types of precofibration, prefibration, and premodel categories; in section 1.3 I give the definition of ABC cofibration, ABC fibration, ABC model categories, and I am comparing the several definitions of model category with respect to Quillen model categories. In the last section I give many examples. Chapter 2 starts with the definition of homotopy category, and ends with the proof that the homotopy category is an additive category. In the middle I am analysing briefly the stability in section 2.2, and functors in homotopy theory in section 2.3. The last chapter is completely devoted to triangulated categories. I begin with the definition of triangulated category and the first fundamental properties. The second section contains some useful criteria to establish when a triangulated category is algebraic. In the last two sections I come back to triangulated categories, showing many results, in particolar I prove that the homotopy category of a stable cofibration category is triangulated. All this justifies the title of my thesis: “homotopy theory and triangulated categories”.