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Iterated magnitude homology

Emily Roff

TL;DR

This work develops iterated magnitude homology to capture higher-order enrichment structures in categories, linking magnitude homology to bicategorical classifying spaces via the double magnitude nerve. It proves an Eilenberg–Zilber-type theorem and Künneth formulas for magnitude chain complexes, enabling computations for product and metric-space settings. For strict 2-categories, the iterated theory recovers the classical homology of the classifying space, and the framework is applied to enriched groups (strict 2-groups, partially ordered groups, and normed groups) to extract both topological and geometric invariants. The results illuminate how higher-order enrichment governs homological behavior and extend naturally to strict n-categories, offering a unified tool for probing algebraic and geometric structure in enriched categorical contexts.

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 $n > 2$.

Iterated magnitude homology

TL;DR

This work develops iterated magnitude homology to capture higher-order enrichment structures in categories, linking magnitude homology to bicategorical classifying spaces via the double magnitude nerve. It proves an Eilenberg–Zilber-type theorem and Künneth formulas for magnitude chain complexes, enabling computations for product and metric-space settings. For strict 2-categories, the iterated theory recovers the classical homology of the classifying space, and the framework is applied to enriched groups (strict 2-groups, partially ordered groups, and normed groups) to extract both topological and geometric invariants. The results illuminate how higher-order enrichment governs homological behavior and extend naturally to strict n-categories, offering a unified tool for probing algebraic and geometric structure in enriched categorical contexts.

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 -categories for .
Paper Structure (11 sections, 13 theorems, 39 equations)

This paper contains 11 sections, 13 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. The structure of the paper
  3. Acknowledgements
  4. Declaration of interest
  5. Enriched categories
  6. Conventions
  7. Magnitude homology
  8. The Künneth theorem
  9. Iterated magnitude homology
  10. Enriched groups
  11. Magnitude homology of enriched groups

Key Result

Lemma 3.4

Suppose $\mathbf{X}$ is a $\mathcal{V}$-category such that $\mathbf{X}(x,x) \cong I$ for every $x \in \mathbf{X}$. Then the normalized magnitude complex of $\mathbf{X}$ is given in degree $n$ by where the sum is over tuples $x_0, \ldots x_n$ such that $x_i \neq x_{i+1}$ for all $i$.

Theorems & Definitions (36)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Lemma 3.4
  • proof
  • Example 3.5
  • Theorem 4.1: Bisimplicial Eilenberg--Zilber theorem
  • ...and 26 more