Iterated magnitude homology

TL;DR

This paper introduces an iterated magnitude homology theory for higher-enriched categories, unifying concepts from categorical and classifying space homologies, with applications to groups with additional structures.

Contribution

It develops a new homology theory for higher-enriched categories, bridging magnitude homology and classifying space homology, and explores its properties for various structured categories.

Findings

Recovers homology of classifying spaces for strict 2-categories

Analyzes behavior for partially ordered and normed groups

Extends to strict n-categories for n > 2

Abstract

Magnitude homology is an invariant of enriched categories which generalizes ordinary categorical homology -- the homology of the classifying space of a small category. The classifying space can also be generalized in a different direction: it extends from categories to bicategories as the geometric realization of the geometric nerve. This paper introduces a hybrid of the two ideas: an iterated magnitude homology theory for categories with a second- or higher-order enrichment. This encompasses, for example, groups equipped with extra structure such as a partial ordering or a bi-invariant metric. In the case of a strict 2-category, iterated magnitude homology recovers the homology of the classifying space; we investigate its content and behaviour when interpreted for partially ordered groups, normed groups, and strict $n$-categories for $n > 2$.