On k-invariants for $(\infty, n)$-categories

TL;DR

This paper extends the concept of k-invariants to $( , n)$-categories, showing that their homotopy towers are classified similarly to spaces, and provides a framework for constructing these invariants in advanced categorical contexts.

Contribution

It introduces a classification of the stages of the homotopy tower of $( , n)$-categories using k-invariants, generalizing classical Postnikov towers.

Findings

Successive stages of the homotopy tower are classified by k-invariants.

Provides a construction of k-invariants for $ $-operad algebras.

Establishes an abstract framework for Postnikov-type towers in higher categories.

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.