Homology of higher categories

TL;DR

This paper develops a homology theory for higher categories using a categorical analogue of the Eilenberg-Steenrod axioms, establishing a Dold-Kan correspondence and computing homology of globes.

Contribution

It introduces a categorical homology framework, proves a Dold-Kan correspondence, and computes homology for specific higher categorical structures.

Findings

Established a categorical Dold-Kan correspondence.

Proved a categorical Dold-Thom theorem with multiplicative structure.

Computed categorical homology of globes.

Abstract

Homology is characterized by the Eilenberg-Steenrod axioms. We define homology of higher categories via a categorical analogue of the Eilenberg-Steenrod axioms. We prove a categorical Dold-Kan correspondence, providing a combinatorial presentation of categorical homology in which the Street nerve plays the role of the singular complex. This implies a categorical Dold-Thom theorem that endows categorical homology with a multiplicative structure and leads to computations of categorical homology of the globes.