Higher path groupoids and the holonomy of formal power series connections

TL;DR

This paper develops a framework for constructing higher holonomy functors from formal power series connections, enabling the study of path groupoids and their higher analogs, with applications to configuration spaces.

Contribution

It introduces a novel method to build higher holonomy functors from formal power series connections, extending previous ideas to higher groupoids.

Findings

Constructed functors from path, path 2-, and 3-groupoids of manifolds.

Developed a Gray functor for the path 3-groupoid of configuration spaces.

Extended the holonomy framework to higher categorical structures.

Abstract

Building on ideas of Kohno, we develop a framework for the construction of higher holonomy functors via the transport of formal power series connections. Using these techniques, we obtain functors from the path groupoid, the path 2-groupoid, and the path 3-groupoid of a manifold. As an application, we construct a Gray functor from the path 3-groupoid of the configuration space of $m$ points in $\mathbb{R}^n$ for $n\geqslant 4$.