Abelian Groups in omega-categories

TL;DR

This paper explores abelian group objects within omega-categories, extending the Dold-Kan correspondence to presheaves of omega-categories and characterizing descent conditions akin to higher stacks.

Contribution

It provides a comprehensive review of abelian groups in omega-categories and introduces a parameterized Dold-Kan correspondence for presheaves, filling gaps in existing literature.

Findings

Describes descent conditions in terms of glueing for higher stacks.

Extends Dold-Kan correspondence to presheaves of omega-categories.

Clarifies the structure of abelian group objects in omega-categories.

Abstract

We study abelian group objects in $\omega$-categories and discuss the well-known Dold-Kan correspondence from the perspective of $\omega$-categories as a model for strict $\infty$-categories. The first part of the paper is intended to compile results from the existing literature and to fill some gaps therein. We go on to consider a parameterized Dold-Kan correspondence, i.e. a Dold-Kan correspondence for presheaves of $\omega$-categories. The main result is to describe the descent or sheaf condition in terms of a glueing condition that is familiar for 1 and 2-stacks.