A categorical characterization of strong Steiner $\omega$-categories

TL;DR

This paper provides a new categorical characterization of strong Steiner $oldsymbol{ extomega}$-categories, connecting their algebraic chain complex models with intrinsic categorical properties.

Contribution

It introduces a loop-freeness condition on polygraphs that characterizes strong Steiner $oldsymbol{ extomega}$-categories without relying on chain complex formalism.

Findings

Characterization of strong Steiner $oldsymbol{ extomega}$-categories via loop-freeness.

Bridging algebraic models with categorical intuition.

Simplification of conditions defining these $oldsymbol{ extomega}$-categories.

Abstract

Strong Steiner $\omega$-categories are a class of $\omega$-categories that admit algebraic models in the form of chain complexes, whose formalism allows for several explicit computations. The conditions defining strong Steiner $\omega$-categories are traditionally expressed in terms of the associated chain complex, making them somewhat disconnected from the $\omega$-categorical intuition. The purpose of this paper is to characterize this class as the class of polygraphs that satisfy a loop-freeness condition that does not make explicit use of the associated chain complex and instead relies on the categorical features of $\omega$-categories.