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

TL;DR

This paper constructs a path object for weak $ abla$-groupoids, advancing the proof of Grothendieck's homotopy hypothesis by developing composition, identities, and inverses within a globular set framework.

Contribution

It introduces a globular path object for weak $ abla$-groupoids, including systems of composition, identities, and inverses, and defines a coglobular $ abla$-groupoid for modifications.

Findings

Developed a globular set with composition, identities, and inverses.

Constructed a coglobular $ abla$-groupoid representing modifications.

Proved properties enabling interpretation of 2-dimensional categorical operations.

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 $X$.