Cofibration categories and quasicategories

**Abstract**

Classically, homotopy theories are described using homotopical
algebra, *e.g.* as model categories or (co)fibration categories. Nowadays, they
are often formalized as higher categories, *e.g.* as quasicategories or
complete Segal spaces. These two types of
approaches highlight different aspects of abstract homotopy theory and
are useful for different purposes.
Thus it is an interesting question whether homotopical algebra and higher
category theory are in some precise sense equivalent.
In this talk I will concentrate on cofibration categories and
quasicategories. I will discuss some basic features of
both notions building up
to a result that the homotopy theory of cofibration categories is
indeed equivalent to the homotopy theory of cocomplete quasicategories.