On the categorification of homology

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.