Abstract:

Abstract: Approaches to abstract homotopy theory fall roughly into two types: classical and higher categorical. Classical models of homotopy theories are some structured categories equipped with weak equivalences, e.g. model categories or (co)fibration categories. From the perspective of higher category theory homotopy theories are the same as (infinity,1)-categories, e.g. quasicategories or complete Segal spaces. The higher categorical point of view allows us to consider the homotopy theory of homotopy theories and to use homotopy theoretic methods to compare various notions of homotopy theory. Most of the known notions of (infinity,1)-categories are equivalent to each other. This raises a question: are the classical approaches equivalent to the higher categorical ones? I will provide a positive answer by constructing the homotopy theory of cofibration categories and explaining how it is equivalent to the homotopy theory of (finitely) cocomplete quasicategories. This is achieved by encoding both these homotopy theories as fibration categories and exhibiting an explicit equivalence between them.