Hom $\omega$-categories of a computad are free

TL;DR

This paper introduces a new description of the hom functor for weak ω-categories, demonstrating it preserves freeness on computads, and explores the properties of opposites and their interaction with hom functors.

Contribution

It provides a novel characterization of the hom functor in weak ω-categories and shows it preserves freeness on computads, unlike in strict ω-categories.

Findings

Hom functor admits a left adjoint called the suspension functor.

Hom functor preserves freeness on computads in weak ω-categories.

Opposite constructions commute with hom functors.

Abstract

We provide a new description of the hom functor on weak $\omega$-categories, and we show that it admits a left adjoint that we call the suspension functor. We then show that the hom functor preserves the property of being free on a computad, in contrast to the hom functor for strict $\omega$-categories. Using the same technique, we define the opposite of an $\omega$-category with respect to a set of dimensions, and we show that this construction also preserves the property of being free on a computad. Finally, we show that the constructions of opposites and homs commute.