Connections on decorated path space bundles

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.