On the categorification of homology

TL;DR

This work develops a rigorous program that categorifies classical homology theories by replacing spaces with $\infty$-categories and homotopy-theoretic constructions with higher-categorical analogues. Central to the approach is the notion of a categorical homology theory: a reduced, oplax excisive endofunctor on $\infty\mathrm{Cat}_*$ that preserves filtered colimits, represented by categorical spectra via an analogue of Brown representability. The paper proves foundational results including a categorified Dold–Thom theorem, a categorified Hurewicz theorem, and a categorified Whitehead theorem, and shows that categorical $R$-homology lifts to $(R,R)$-bimodule categorical spectra. A key technical framework is built around Gray-categories, the Gray tensor product, and the Street nerve, enabling a robust stability theory with spectrum objects, spectrification, and a tensorial higher-algebra structure on stable presentable Gray-categories. The constructions culminate in a versatile homological toolkit for higher categories, with applications to categorified cobordism and the study of categorical co/limits, while establishing foundational representability and density results via the Street nerve and related higher-categorical nerves.

Abstract

We categorify the concept of homology theories, which we term categorical homology theories. More precisely, we extend the notion of homology theory from homotopy theory to the realm of $(\infty,\infty)$-categories and show that several desirable features remain: we prove that categorical homology theories are homological in a precise categorified sense, satisfy a categorified Whitehead theorem and are classified by a higher categorical analogue of spectra. To study categorical homology theories we categorify stable homotopy theory and the concept of stable $(\infty,1)$-category. As guiding example of a categorical homology theory we study the categorification of homology, the categorical homology theory whose coefficients are the commutative monoid of natural numbers, which we term categorical homology. We prove that categorical homology admits a description analogous to singular homology that replaces the singular complex of a space by the nerve of an $(\infty,\infty)$-category. We show a categorified version of the Dold-Thom theorem and Hurewicz theorem, compute categorical homology of the globes, the walking higher cells, and prove that categorical $R$-homology with coeffients in a rig $R$ multiplicatively lifts to the higher category of $(R,R)$-bimodules.

Theorems & Definitions (496)

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