A proof of the existence of Batanin's initial operad

Abstract

In this paper we prove the existence of the n-globular operad used in Batanin's definition of weak n-category. This operad is initial in the category of n-globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n-globular operad used in his definition of weak n-category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n-globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions.