Classification of metric fibrations

Classification of metric fibrations YASUHIKO ASAOWe study a notion of "fibration for metric spaces", called metric fibration, that was originally introduced by Leinster (Doc.Math.18 (2013) 857-905) in the study of magnitude.He showed that the magnitude of a metric fibration splits into the product of those of the fiber and the base, which is analogous to the case for the Euler characteristic and topological fiber bundles.His idea and our approach are based on Lawvere's suggestion of viewing a metric space as an enriched category (Rend.Sem.Mat.Fis.Milano 43 (1973) 135-166).Actually, the metric fibrations are the restriction of the enriched Grothendieck fibrations (Sminaire Bourbaki 1959/1960(1966) expos 190) to metric spaces (arXiv 2303.05677).We give a complete classification of metric fibrations by several means, which are parallel to those used for topological fiber bundles.That is, the classification of metric fibrations is reduced to that of "principal fibrations", which is done by the "1-ech cohomology" in an appropriate sense.Here we introduce the notion of torsors in the category of metric spaces, and the discussions are analogous to those in sheaf theory.Further, we can define the "fundamental group" m 1 .X / of a metric space X , which is a group-like object in metric spaces, such that the conjugation classes of homomorphisms Hom.m 1 .X /; G/ correspond to the isomorphism classes of "principal G-fibrations" over X .In other words, the latter are classified like topological covering spaces.