Thursday, July 25, 2013 - from 16:00 to 16:25
Approaches to abstract homotopy theory fall into roughly two types: classical and higher categorical. Classical models of homotopy theories are some structured categories equipped with weak equivalences, for example model categories or (co)fibration categories in the sense of Brown. From the perspective of higher category theory homotopy theories are the same as $(\infty, 1)$-categories and these have plenty of models, including quasicategories, complete Segal spaces and Segal categories. The higher categorical approach sheds new light on homotopy theory, namely, it allows us to consider the homotopy theory of homotopy theories. Thus we can use homotopy theoretic methods to compare various notions of homotopy theory. Most of the known notions of $(\infty, 1)$-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.