Operations in Leinster's Weak $\omega$-Category Operad

TL;DR

This paper explores Leinster's refinement of Batanin's weak ω-category operad, explicitly constructing higher category structures, composites, and coherence laws to deepen understanding of weak ω-categories.

Contribution

It provides an explicit construction of higher category structures from Leinster's operad with contraction, clarifying the coherence laws and braiding.

Findings

Constructed composites and associativity laws

Established coherence laws for weak ω-categories

Demonstrated Eckmann-Hilton braiding in this context

Abstract

Batanin defines a weak $\omega$-category as an algebra for a certain operad. Leinster refines this idea and defines the weak $\omega$-category operad as the initial object of a category of "operads with contraction". We demonstrate how a higher category structure arises from this definition by explicitly constructing various composites, associativity and coherence laws, and an Eckmann-Hilton braiding.