Unfolding of symmetric monoidal $(\infty,n)$-categories

Abstract

We prove Lurie's proposed description of left rigid symmetric monoidal $(\infty,n)$-categories in terms of purely $\infty$-categorical data. More precisely, unfolding gives an equivalence between left rigid symmetric monoidal $(\infty,n)$-categories and complete categorical chain complexes of rigid symmetric monoidal $\infty$-categories. More generally, unfolding identifies skeletal sequences with categorical chain complexes without the completeness condition. We extend these equivalences to $(\infty,\infty)$-categories and, stably, obtain an equivalence between complete unbounded categorical chain complexes and left rigid categorical spectra. We also analyze their groupoid completion and express the homotopy groups of the associated spaces and spectra in terms of categorical cycles and boundaries. As examples, we construct the categorical chain complexes associated to geometric and algebraic bordism categories, including their equivariant variants, and to iterated span categories. Their localizations recover classical bordism and $L$-groups, while the span examples are related to algebraic $K$-theory.