The universal Lipschitz path space of the Heisenberg group $\mathbb{H}^1$

The goal of this paper is to define and inspect a metric version of the universal path space and study its application to purely 2-unrectifiable spaces, in particular the Heisenberg group $\mathbb{H}^1$. The construction of the universal Lipschitz path space, as the metric version is called, echoes the construction of the universal cover for path-connected, locally path-connected, and semilocally simply connected spaces. We prove that the universal Lipschitz path space of a purely 2-unrectifiable space, much like the universal cover, satisfies a unique lifting property, a universal property, and is Lipschitz simply connected. The existence of such a universal Lipschitz path space of $\mathbb{H}^1$ will be used to prove that $\pi_{1}^{\text{Lip}}(\mathbb{H}^1)$ is torsion-free in a subsequent paper.