The algebra of the nerves of omega-categories

TL;DR

This paper introduces an algebraic framework called sets with complicial identities to describe the nerve of strict omega-categories, establishing an equivalence between these structures and strict omega-categories.

Contribution

It provides an algebraic description of the nerve of strict omega-categories and proves an equivalence between strict omega-categories and sets with complicial identities.

Findings

Algebraic description of the nerve of strict omega-categories

Construction of an equivalence between categories of strict omega-categories and sets with complicial identities

Introduction of sets with complicial identities as a new algebraic structure

Abstract

We show that the nerve of a strict omega-category can be described algebraically as a simplicial set with additional operations subject to certain identities. The resulting structures are called sets with complicial identities. We also construct an equivalence between the categories of strict omega-categories and of sets with complical identities.