Two Models for the Homotopy Theory of Cocomplete Homotopy Theories

Abstract

There two major approaches to the problem of formalizing the notion of a "homotopy theory", they are homotopical algebra and higher category theory. The main result of this thesis is a precise comparison between these approaches, i.e. the theorem saying that the homotopy theory of cofibration categories (representing homotopical algebra) is equivalent to the homotopy theory of finitely cocomplete quasicategories (representing higher category theory).