Magnitude and magnitude homology of filtered set enriched categories

TL;DR

This paper develops a unified framework for magnitude theory in filtered set enriched categories, extending concepts from metric spaces and finite categories to broader contexts, and explores their homotopy invariance and fibrations.

Contribution

It introduces a general magnitude theory for filtered set enriched categories, unifying Euler characteristic and magnitude concepts across geometry, topology, and combinatorics.

Findings

Extended magnitude to broader classes of metric spaces including infinite ones

Connected magnitude homology with homotopy invariance properties

Analyzed fibrations of graphs with identical magnitude but non-isomorphic structures

Abstract

In this article, we give a framework for studying the Euler characteristic and its categorification of objects across several areas of geometry, topology and combinatorics. That is, the magnitude theory of filtered sets enriched categories. It is a unification of the Euler characteristic of finite categories and it the magnitude of metric spaces, both of which are introduced by Leinster. Our definitions cover a class of metric spaces which is broader than the original ones, so that magnitude (co)weighting of infinite metric spaces can be considered. We give examples of the magnitude from various research areas containing the Poincar\'{e} polynomial of ranked posets and the growth function of finitely generated groups. In particular, the magnitude homology gives categorifications of them. We also discuss homotopy invariance of the magnitude homology and its variants. Such a homotopy includes digraph homotopy and r-closeness of Lipschitz maps. As a benefit of our categorical view point, we generalize the notion of Grothendieck fibrations of small categories to our enriched categories, whose restriction to metric spaces is a notion called metric fibration that is initially introduced by Leinster. It is remarkable that the magnitude of such a fibration is a product of those of the fiber and the base. We especially study fibrations of graphs, and give examples of graphs with the same magnitude but are not isomorphic.