Homology of higher categories

Abstract

The classical Dold--Kan correspondence identifies simplicial abelian groups with connective chain complexes. On the level of homotopy theory, these provide models for strict grouplike $\mathbb{E}_\infty$-spaces and connective $H\mathbb{Z}$-module spectra. We establish a higher-categorical Dold--Kan correspondence identifying stratified simplicial commutative monoids with strict symmetric monoidal $\omega$-categories and with connective categorical $H\mathbb{N}$-module spectra. This provides a combinatorial, algebraic and stable homotopy-theoretic model for higher-categorical homological algebra. As a consequence, the free categorical $H\mathbb{N}$-module spectrum provides a natural notion of homology for higher categories. Through the categorical Dold--Kan correspondence, it is modeled by the free stratified simplicial commutative monoid on the Street nerve, the higher-categorical analogue of singular chains. We obtain a categorical Dold--Thom theorem, which leads to explicit computations of the categorical homology of the globes and shows that categorical homology detects genuinely higher-categorical information invisible to classical homology.