Hom weak $\omega$-categories of a weak $\omega$-category

TL;DR

This paper constructs an underlying weak $oldsymbol{ extomega}$-category-enriched graph for Leinster's weak $oldsymbol{ extomega}$-categories, demonstrating functoriality with respect to weak $oldsymbol{ extomega}$-functors.

Contribution

It introduces a method to associate a hom weak $oldsymbol{ extomega}$-category to each weak $oldsymbol{ extomega}$-category, clarifying the structure of homs in these complex categories.

Findings

Constructs an underlying weak $ extomega$-category-enriched graph for each weak $ extomega$-category.

Shows the construction is functorial with respect to weak $ extomega$-functors.

Provides a new perspective on the internal structure of weak $ extomega$-categories.

Abstract

Classical definitions of weak higher-dimensional categories are given inductively; for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak $n$-categories for all natural numbers $n$, or of weak $\omega$-categories, take more sophisticated approaches, and the nature of the "hom" is often not immediate from the definitions. In this paper, we focus on Leinster's definition of weak $\omega$-category based on an earlier definition by Batanin, and construct for each weak $\omega$-category $\mathcal{A}$, an underlying (weak $\omega$-category)-enriched graph consisting of the same objects and for each pair of objects $x$ and $y$, a hom weak $\omega$-category $\mathcal{A}(x,y)$. We also show that our construction is functorial with respect to weak $\omega$-functors introduced by Garner.