Notes on $n$-Transformations by Theories ($n\in {\mathbb{N}^*}$)

TL;DR

This paper extends the theory of weak omega categories by defining weak omega functors and transformations as models of homogeneous colored theories, building on Berger's and Weber's work on n-transformations.

Contribution

It introduces a new framework for modeling weak omega functors and transformations using homogeneous colored theories, generalizing previous results on n-transformations.

Findings

Weak omega functors are modeled as homogeneous colored theories.

Weak omega natural transformations are also modeled within this framework.

The approach unifies the theory of n-transformations with broader categorical structures.

Abstract

Clemens Berger showed that Weak Omega Categories of Michael Batanin can be defined as model of a certain kind of theories that he called "homogeneous theories". By using the work of Mark Weber on the Abstract Nerves for the specific case of the $n$-Transformations ($n\in {\mathbb{N}^*}$), we show that we can also define Weak Omega Functors, Weak Omega Natural Transformations, and so on, as models of certain kind of colored theories which are homogeneous as well.