Magnitude homology and homotopy type of metric fibrations

TL;DR

This paper demonstrates that metric fibrations sharing a base and fiber have identical magnitude homology and homotopy types, generalizing Leinster's product formula for finite metric spaces and connecting different definitions of magnitude homotopy type.

Contribution

It establishes the invariance of magnitude homology and homotopy type for metric fibrations and links various definitions of magnitude homotopy type.

Findings

Metric fibrations with common base and fiber have isomorphic magnitude homology.

The magnitude homotopy type is invariant under certain fibrations.

Different definitions of magnitude homotopy type are shown to be equivalent.

Abstract

In this article, we show that each two metric fibrations with a common base and a common fiber have isomorphic magnitude homology, and even more, the same magnitude homotopy type. That can be considered as a generalization of a fact proved by T. Leinster that the magnitude of a metric fibration with finitely many points is a product of those of the base and the fiber. We also show that the definition of the magnitude homotopy type due to the second and the third authors is equivalent to the geometric realization of Hepworth and Willerton's pointed simplicial set.