Wojciech Chacholski: Mapping spaces and classification of fibrations

Classification questions are often about understanding components of a category. Objects of this category are the objects we would like to classify and the morphisms correspond to the relation up to which we would like to classify the objects. Such classification problems are then divided into two steps: first one needs to show that the components of the category form a set, and second enumerate this set. It is not unusual however that with such a category one can associate a unique homotopy type of a topological space, its NERVE, whose set of components coincide with the set of components of the category. Such a space carries much more information about our category that just the set of its components. In this talk I will illustrate such classification results by describing the homotopy type of the nerve of the category of fibrations with a given base and fiber. This will involve discussing continuity of mapping spaces and how it can be described combinatorially.