Connections on decorated path space bundles

TL;DR

This paper investigates decorated path space bundles over a principal bundle with a connection, constructing parallel transport and holonomy, and relating these to categorical geometry and connections.

Contribution

It introduces a framework for decorated path space bundles with parallel transport and holonomy, linking differential geometry with categorical structures.

Findings

Constructed parallel transport processes on decorated path space bundles

Analyzed holonomy bundles in the decorated setting

Connected categorical geometry with differential geometric concepts

Abstract

For a principal bundle $P\to M$ equipped with a connection ${\bar A}$, we study an infinite dimensional bundle ${\mathcal P}^{\rm dec}_{\bar A}P$ over the space of paths on $M$, with the points of ${\mathcal P}^{\rm dec}_{\bar A}P$ being horizontal paths on $P$ decorated with elements of a second structure group. We construct parallel transport processes on such bundles and study holonomy bundles in this setting. We explain the relationship with categorical geometry and explore the notion of categorical connections on categorical principal bundles in a concrete differential geometric way.