Orientals as free algebras

TL;DR

This paper offers an alternative construction of Street's orientals as free algebras in strict ω-categories, utilizing monadic structures and Steiner's chain complexes to establish their equivalence.

Contribution

It introduces a new approach to constructing orientals via monads on strict ω-categories, connecting algebraic structures with cosimplicial objects.

Findings

The cosimplicial object constructed matches Street's orientals.

Monadic approach provides a new perspective on orientals.

Comparison results for polygraphs facilitate the proof.

Abstract

The aim of this paper is to give an alternative construction of Street's cosimplicial object of orientals, based on an idea of Burroni that orientals are free algebras for some algebraic structure on strict $\omega$-categories. More precisely, following Burroni, we define the notion of an expansion on an $\omega$-category and we show that the forgetful functor from strict $\omega$-categories endowed with an expansion to strict $\omega$-categories is monadic. By iterating this monad starting from the empty $\omega$-category, we get a cosimplicial object in strict $\omega$-categories. Our main contribution is to show that this cosimplicial object is the cosimplicial objects of orientals. To do so, we prove, using Steiner's theory of augmented directed chain complexes, a general result for comparing polygraphs having same generators and same linearized sources and targets.