We 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 (Séminaire Bourbaki
1959/1960 (1966) exposé 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
“-Čech
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”
of a metric
space
, which
is a group-like object in metric spaces, such that the conjugation classes of homomorphisms
correspond to the isomorphism classes of “principal
-fibrations”
over
.
In other words, the latter are classified like topological covering spaces.
