On k-invariants for $(\infty, n)$-categories
Yonatan Harpaz, Joost Nuiten, Matan Prasma
TL;DR
The paper develops a higher-categorical Postnikov theory for (\infty, n)-categories by constructing a tower of homotopy (m,n)-categories controlled by k-invariants. It introduces abstract Postnikov towers in SM and stable contexts, and shows how these towers extend to algebras over ∞-operads and, crucially, to enriched ∞-categories, via tangent bundles and square-zero extensions. The central output is that each step ho_{(n+a,n)}(C)→ho_{(n+a-1,n)}(C) is classified by a cohomology class k_a in $\mathrm{H}^{a+1}(ho_{(n+a-1,n)}(C), \pi_a(C))$, with $\pi_a(C)$ a local system of abelian groups and with the corresponding Eilenberg–MacLane parametrized spectra $\mathrm{H}\pi_a(C)$ appearing in the heart of a t-structure on the tangent category. By inductively applying multiplicative abstract Postnikov towers to $\mathcal{V}$ and to enriched $\infty$-categories, the authors obtain a coherent framework for k-invariants across $(\infty,n)$-categories, enriched $\infty$-categories, and algebras over $\infty$-operads. This provides a structurally robust obstruction-theoretic mechanism for classifying and reconstructing higher-categorical data from abelian-local-system coefficients, with potential impact on orbifold-type invariants and higher-categorical algebra. The construction emphasizes the central role of local systems and parametrized spectra in encoding higher k-invariants and their multiplicative compatibilities.
Abstract
Every $(\infty, n)$-category can be approximated by its tower of homotopy $(m, n)$-categories. In this paper, we prove that the successive stages of this tower are classified by k-invariants, analogously to the classical Postnikov tower for spaces. Our proof relies on an abstract analysis of Postnikov-type towers equipped with k-invariants, and also yields a construction of k-invariants for algebras over $\infty$-operads and enriched $\infty$-categories.