Towards a globular path object for weak ∞-groupoids

The goal of this paper is to address the problem of building a path object for the category of Grothendieck (weak) infinity-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 infinity-categories. Finally, we introduce a coglobular infinity-groupoid representing modifications of infinity-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 PX of a given infinity-groupoid X. (C) 2019 Elsevier B.V. All rights reserved.