A semi-model structure for Grothendieck weak 3-groupoids

TL;DR

This paper develops a semi-model structure for Grothendieck weak 3-groupoids using a new endofunctor that acts like a path object, advancing the understanding of higher-dimensional algebraic structures.

Contribution

It introduces a semi-model structure for Grothendieck 3-groupoids and a recognition principle for globular theories modeling Grothendieck n-groupoids.

Findings

Constructed an endofunctor  that behaves like a path object.

Proved a recognition principle for globular theories of Grothendieck n-groupoids.

Identified obstructions in constructing path objects in higher dimensions.

Abstract

In this paper we apply some tools developed in our previous work on Grothendieck $\infty$-groupoids to the finite-dimensional case of weak 3-groupoids. We obtain a semi-model structure on the category of Grothendieck 3-groupoids of suitable type, thanks to the construction of an endofunctor $\mathbb{P}$ that has enough structure to behave like a path object. This makes use of a recognition principle we prove here that characterizes globular theories whose models can be viewed as Grothendieck $n$-groupoids (for $0\leq n \leq \infty$). Finally, we prove that the obstruction in arbitrary dimension (possibly infinite) only resides in the construction of (slightly less than) a path object on a suitable category of Grothendieck (weak) $n$-categories with weak inverses. This also gives a sufficient condition for endowing an $n$-groupoid \`a la Batanin with the structure of a Grothendieck $n$-groupoid.