Yasuhiko Asao - Classification of metric fibration

Abstract

The notion of metric fibration is a metric space analogue of topological fiber bundle, which is  introduced by Leinster. In this talk, we classify metric fibrations in a way parallel to topological case. We give two equivalent definitions of metric fibration, which are by the lifting property ’ and  by transition functions’. The latter helps us to consider principal fibrations’ and classify them by the metric fundamental group’. Here the notion of group’ is not mere a usual group but is a group object in the category of metric spaces. Moreover, we can define the 1-Cech cohomology of metric spaces with the coefficient in metric groups’ that also classifies metric fibration.

Metric fibration

We first define the metric fibration.

Leinster showed that the magnitude of a metric fibration is the product of the base and fibre space. This allows us to generate spaces with the same magnitude.

Grothendieck fibration

A Fiset fibration or semi-normed cat is a small category with a norm on every morphism which are measure subadditvely.

examples:

1. Any small cat is a fiset-cat with $|f|=0$ for every morphism
2. Any metric space is -”- with the norm given by the metric

From this point of view a grothendieck fibration can be generalized to fiset-cat whose restriction to met is exactly a metric fibration.

Lax functor: A Lax functor from a metric space $X$ to the metric category is defined. We then construct a category of lax functors.

To classify the fibration we mimic the classification of the principal fibration. The classification conntects the categorical core if the Fibration category with an automorphism group and