Towards a globular path object for weak $\infty$-groupoids

Abstract

The goal of this paper is to address the problem of building a path object for the category of Grothendieck (weak) $\infty$-groupoids. This is the missing piece for a proof of Grothendieck's homotopy hypothesis. We show how to endow the putative underlying globular set with a system of composition, a system of identities and a system of inverses, together with an approximation of the interpretation of any map for a theory of $\infty$-categories. Finally, we introduce a coglobular $\infty$-groupoid representing modifications of $\infty$-groupoids, and prove some basic properties it satisfies, that will be exploited to interpret all $2$-dimensional categorical operations on cells of the path object $\mathbb{P}X$ of a given $\infty$-groupoid $X$.