Nested homotopy models of finite metric spaces and their spectral homology

TL;DR

This paper introduces the concept of $r$-homotopy equivalence in quasimetric spaces, constructs minimal models, and develops spectral homology invariants that unify various existing homology theories for directed graphs and metric spaces.

Contribution

It defines $r$-homotopy equivalence and minimal models in quasimetric spaces, and introduces spectral homology as a unifying invariant for multiple homology theories.

Findings

Existence and uniqueness of $r$-minimal models for finite quasimetric spaces.

Decomposition of the magnitude-path spectral sequence into simpler components.

Spectral homology generalizes path, magnitude, blurred magnitude, and reachability homologies.

Abstract

For a real $r\geq 0,$ we consider the notion of $r$-homotopy equivalence in the category quasimetric spaces, which includes metric spaces and directed graphs. We show that for a finite quasimetric space $X$ there is a unique (up to isometry) $r$-homotopy equivalent quasimetric space of the minimal possible cardinality. It is called the $r$-minimal model of $X$. We use this to construct a decomposition of the magnitude-path spectral sequence of a digraph into a direct sum of spectral sequences with certain properties. We also construct an $r$-homotopy invariant ${\rm SH}^r_{n,I}(X)$ of a quasimetric space $X,$ called spectral homology, that generalizes many other invariants: the pages of the magnitude-path spectral sequence, including path homology, magnitude homology, blurred magnitude homology and reachability homology.