The Homotopy Theory of (∞,1)-Categories

Abstract

The homotopy category of a stable (∞,1)-category can be endowed with a triangulated structure. The main objective of this thesis is to give a proof of this fact. First it will be discussed some ideas of higher category theory, before (∞,1)-categories and models of (∞,1)-categories will be studied. In particular, topological categories and simplicial categories will be mentioned, but the main focus will be on quasi-categories, which all are models for (∞,1)-categories. The theory of (∞,1)-categories, which is required in order to define stable (∞,1)-categories, is then discussed, in particular functors, subcategories, join constructions, undercategories, overcategories, initial objects, terminal objects, limits and colimits are formally discussed for quasi-categories. Finally, the definition of a stable (∞,1)-category will be discussed. Then the main theorem will be proved, after the required properties of stable (∞,1)-categories are discussed. Background theory from ordinary categories and simplicial sets are collected in the appendices.