  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 $X$.
